Maria Alejandra

Erazo Diaz

Sorravich Kingsuwankul

Jona Krutaj

Maxime Le Bihan

Valentin Guigon

Alexandre Angelloznicoud

Sébastien Doustaly

Kamel Ismail

Jocelyn Maillard

Siwar Khelifa

Marie Aurélie Lapierre

Carla Morvan

Bruno Zingan

Marchais Kittika

Louis Magblenou

Brice Corgnet

Adam Zylbersztejn

All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

Keywords: Prospect theory, semi-parametric elicitation, risk attitudes, weighting function, loss aversion. JEL codes : D81, C91 Chapter 1. All at Once H prospect theory, risk attitudes elicitation, smoothing spline JEL codes: D81, C91 Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components Chapter 2. Smoothing Spline Method Subjective beliefs, ambiguity attitudes, sources of uncertainty, trust game, coordination game. JEL codes: D81, C91 Chapter

Finally, I would like to

This thesis would not have been possible without the help and support of many people. First of all, I would like to express my deepest gratitude to my supervisors, Brice Corgnet and Adam Zylbersztejn. They have shared with me their expertise on many of the questions raised in this thesis. Moreover, they have shared with me much more than that and helped me grow as a researcher.

I thank Roberto Hernán-González and Maria Alejandra Erazo Diaz for their contributions as co-authors. I thank Mohammed Abdellaoui, Olivier LŠHaridon, Dorothea Kübler, and Béatrice Rey-Fournier for their involvement in the evaluation of my thesis. I am especially thankful to Mohammed Abdellaoui and Olivier LŠHaridon for accepting to be reviewers of the thesis.

I also thank Richard Gonzales and George Wu for sharing the data from their own experiment and from the experiment of Amos Tversky and Daniel Kahneman.

I would like to thank Izabela Jelovac, Béatrice Rey-Fournier, and Corinne Autant-Bernard. To be more speciĄc, they opened the doors of this PhD to me Ąve years ago by accepting my applications to the Msc Program in Economics and Finance and the IDEXLYON scholarship at GATE. I thank Taï Dao, Yamina Mansouri, and Cindy Arman for facilitating the administrative procedures involved in the PhD program.

I

Résumé de la thèse

Au cours de leur vie, les gens font des choix en matière dŠinvestissement, dŠépargne, dŠalimentation, dŠéducation, de sport et de carrière professionnelle. Dans la plupart des cas, ces choix ressemblent à des loteries en ce sens quŠils produisent des résultats stochastiques.

Par exemple, un employeur ne sait pas, au moment où il embauche un salarié, si ce dernier accomplira avec succès les tâches pour lesquelles il est embauché.

Le choix de lŠemployeur peut donc être associé à deux conséquences possibles :

(i) lŠemployé accomplit les tâches avec succès ou (ii) lŠemployé nŠaccomplit pas les tâches.

De même, lors dŠune séance de tirs au but au football, le capitaine dŠune équipe doit choisir lŠun des deux côtés (pile ou face) dŠune pièce de monnaie équitable. Son choix permet, après que lŠarbitre a tiré à pile ou face, de savoir si le premier tir sera effectué par lŠéquipe du capitaine ou par lŠéquipe adverse.

Dans cet exemple, le choix du capitaine a deux conséquences : (i) le visage qui apparaît après le tirage au sort est le visage choisi par le capitaine et (ii) le visage qui apparaît après le tirage au sort est lŠopposé du visage choisi par le capitaine.

Dans de rares cas, les individus disposent dŠune mesure objective de la probabilité de chaque conséquence possible de leurs choix. CŠest le cas dans lŠexemple du capitaine. Il sait que le visage quŠil a choisi a 50% de chances i dŠapparaître et 50% de chances de ne pas apparaître à lŠissue du jeu de pile ou face.

Cependant, dans la plupart des cas, les individus ne disposent pas dŠune mesure objective de la probabilité de chaque conséquence possible de leurs choix.

CŠest par exemple le cas de lŠemployeur. Bien quŠil ne dispose pas dŠune mesure objective des probabilités associées aux conséquences, lŠemployeur peut se forger des probabilités subjectives (ou croyances) sur la base des informations (par exemple, le diplôme, lŠuniversité, lŠexpérience professionnelle) quŠil a recueillies au cours du processus de recrutement.

Suivant [START_REF] Knight | Risk, uncertainty and profit[END_REF], en économie, on parle de risque lorsque les individus ont des probabilités objectives des conséquences possibles de leurs choix. Les probabilités sont objectives dans le sens où elles sont indépendantes de la personne qui prend la décision. On parle de incertitude ou de ambiguïté, lorsque les individus ne disposent pas de probabilités objectives des conséquences possibles de leurs choix et que les individus doivent au contraire se forger des probabilités (ou croyances) subjectives. Les probabilités sont subjectives dans le sens où elles dépendent de la personne qui prend la décision.

Les probabilités associées aux conséquences, quŠelles soient objectives ou subjectives, inĆuencent le choix des individus entre plusieurs alternatives. Par exemple, le choix dŠun individu de frauder le Ąsc est fortement déterminé par son appréhension (objective ou subjective) de la probabilité dŠêtre pris par le Ąsc. De même, le choix dŠun individu dŠutiliser frauduleusement des services de transport payants est fortement déterminé par son appréhension (objective ou subjective) de la probabilité dŠêtre pris par les contrôleurs.

Mais au-delà des probabilités, il existe dŠautres caractéristiques propres aux individus qui déterminent leur choix entre plusieurs alternatives. Une première caractéristique est la valeur subjective (représentant le plaisir ou le bonheur) ii que les individus retirent de la réalisation de chaque conséquence. En économie, cette valeur subjective est appelée utilité.

La théorie de lŠutilité espérée et la théorie de lŠutilité subjective éspérée traitent respectivement de la prise de décision en situation de risque (les probabilités objectives des conséquences sont connues) et en situation dŠincertitude (les probabilités objectives des conséquences sont inconnues). Ces deux théories standard considèrent que les choix des individus entre plusieurs alternatives sont entièrement et uniquement déterminés par les probabilités et les utilités que les individus associent aux conséquences des différentes alternatives. Ces théories associent à chaque alternative la somme pondérée de lŠutilité des conséquences, les poids étant les probabilités (objectives ou subjectives). Ces théories postulent quŠun individu choisira lŠalternative qui maximise sa somme pondérée.

Cependant, les théories standard se heurtent à des difficultés empiriques qui suggèrent que les probabilités et lŠutilité ne sont pas les seuls déterminants des choix des individus. Une difficulté empirique importante des théories standard est le paradoxe de Ellsberg (1961). Selon ces théories, les sujets devraient être indifférents entre une urne connue contenant 50 boules rouges et 50 boules noires et une urne inconnue comprenant 100 boules rouges et noires dans une proportion inconnue, quelle que soit la couleur gagnante. Or, contrairement à cette indifférence, les sujets ont tendance à préférer lŠurne connue à lŠurne inconnue, quelle que soit la couleur gagnante. Dans le cadre de ces théories standard, le fait que les sujets tendent à préférer lŠurne connue à lŠurne inconnue (quelle que soit la couleur gagnante), implique que la somme des croyances dŠavoir une boule rouge et dŠavoir une boule noire dans lŠurne inconnue est inférieure à un.

Les difficultés empiriques des théories standard ont justiĄé le développement de théories de lŠutilité non attendue dans des conditions de risque et iii dŠincertitude qui intègrent de nouvelles caractéristiques individuelles en tant que déterminants des choix. LŠutilité dépendante du rang (RDU ; voir Quiggin, 1982;Schmeidler, 1989a) et la théorie cumulative des perspectives (CPT ; voir Tversky and Kahneman, 1992a, désormais TK92) sont apparues comme les deux principales alternatives aux théories traditionnelles. Elles expliquent les violations empiriques en introduisant des distorsions de probabilité (Bleichrodt and Pinto, 2000) et lŠaversion aux pertes (Wakker, 2010a).

Cette thèse contribue à la littérature dŠéconomie comportementale et expérimentale sur le risque et lŠincertitude. En particulier, cette thèse (i) propose de nouvelles méthodes pour mesurer la fonction dŠutilité, la fonction de pondération, lŠaversion aux pertes et les croyances et (ii) dérive de nouvelles idées théoriques dans le domaine de la déĄnition des incitations ainsi que le développement dŠun nouveau paradigme expérimental.

Le chapitre 1 établit une méthode semi-paramétrique complète qui satisfait aux quatre propriétés souhaitables des méthodes paramétriques : traçabilité, efficacité des données, robustesse des erreurs et facilité. Le chapitre 2 propose une version non paramétrique de la méthode semi-paramétrique du chapitre 1 avec une approximation de la fonction dŠutilité par une spline de lissage. Les applications des chapitres 1 et 2 à des données existantes concernant le risque conĄrme les écarts par rapport à la théorie de lŠutilité espérée en mettant en évidence la fonction de pondération des probabilités et lŠaversion pour les pertes.

Le chapitre 3 étend la méthode semi-paramétrique du chapitre 1 pour mesurer les croyances et les attitudes dŠambiguïté à lŠégard de sources discrètes dŠincertitude. Les sujets prennent des décisions dans ce type de situations incertaines au quotidien. Dans diverses applications, la méthode passe avec succès les tests de validité et fournit des résultats intéressants dans le cas des jeux de conĄance et de coordination. Ce chapitre constate des déviations par rapport iv à la théorie de lŠutilité attendue subjective grâce à des preuves dŠattitudes ambiguës. Il souligne que les gens sont plus insensibles à la probabilité en présence dŠévénements asymétriques quŠen présence dŠévénements symétriques, ce qui suggère que la formation des croyances est cognitivement exigeante. À sources dŠincertitude égales, les individus présentent l'aversion à la dépendance des gains et la recherche d'une variété de gains. Aversion à la dépendance des gains signiĄe que les gens nŠaiment pas que leurs propres gains dépendent des préférences des autres. Ce comportement est représenté par une fonction dŠutilité plus concave.

Recherche de la variété des gains signiĄe que les sujets préfèrent un plus grand nombre de gains possibles lorsque ces gains dépendent des préférences des autres.

Ce comportement se traduit par un plus grand optimisme.

Le chapitre 4 étudie lŠexistence du compromis risque-incitations (RIT) en fonction de lŠutilité dépendante du rang (RDU) et de la moyenne-variance-squewness (MVS). Les analyses théoriques montrent que le RIT est remarquablement robuste sous RDU mais pas sous MVS. Avec des données basées sur un nouveau modèle expérimental qui élimine les facteurs de confusion, le chapitre 4 fournit des preuves de lŠexistence de la RIT même dans le cas dŠagents qui recherchent le risque, ce qui est une prédiction distincte de la RDU. Les résultats conĄrment lŠexistence de lŠIRT et suggèrent quŠelle sŠapplique à un large éventail de situations, y compris les cas où les agents recherchent le risque (par exemple, la rémunération des dirigeants).

List of Figures

General introduction

Over their lifetime, people make choices about investments, savings, food, education, sports and professional careers. In most cases, these choices resemble lotteries in that they produce stochastic outcomes.

For example, an employer does not know at the time he hires an employee whether the employee will successfully perform the tasks for which he is hired.

The employerŠs choice can therefore be associated with two possible consequences:

(i) the employee successfully completes the tasks or (ii) the employee fails to complete the tasks.

Also, during a penalty shoot-out in football, a team captain must choose one of the two sides (heads or tails) of a fair coin. His choice allows, after the referee has tossed the coin, to know if the Ąrst shot will be taken by the captainŠs team or the opposing team. In this example, the captainŠs choice has two consequences: (i) the face that appears after the coin toss is the face chosen by the captain and (ii) the face that appears after the coin toss is the opposite of the face chosen by the captain.

In rare cases, individuals have an objective measure of the probability of each possible consequence of their choices. This is the case in the captainŠs example.

He knows that the face he has chosen has a 50% chance of appearing and a 50% chance of not appearing at the end of the toss of the fair coin.

1

General introduction

However, in most cases, individuals do not have an objective measure of the probability of each possible consequence of their choices. This is for example the case of the employer. Although he does not have an objective measure of the probabilities associated with the consequences, the employer can form subjective probabilities (or beliefs) based on the information (e.g. diploma, university, professional experience) that he has collected during the recruitment process.

Following [START_REF] Knight | Risk, uncertainty and profit[END_REF], in economics, we speak of risk when the individuals have objective probabilities of the possible consequences of their choices. The probabilities are objective in the sense that they are independent of the person who makes the decision. One speaks about uncertainty or ambiguity, when the individuals do not have objective probabilities of the possible consequences of their choices and that individuals must on the contrary form subjective probabilities (or beliefs). The probabilities are subjective in the sense that they depend on the person who makes the decision.

The probabilities associated with the consequences, whether objective or subjective, inĆuence the choice of individuals between several alternatives. For example, an individualŠs choice of whether to evade taxes is strongly determined by his or her apprehension (objective or subjective) of the probability of being caught by the tax department (Dhami, 2016). Similarly, an individualŠs choice of whether to use fare-paying transport services fraudulently is strongly determined by his or her apprehension (objective or subjective) of the probability of being caught by ticket inspectors.

But, beyond probabilities, there are other characteristics speciĄc to individuals that determine their choice between several alternatives. A Ąrst characteristic is the subjective value (representing pleasure or happiness) that individuals get from the realization of each consequence. In economics, this subjective value is called utility.

Expected utility theory and subjective expected utility theory deal respectively with decision-making under risk (objective probabilities of consequences are known) and under uncertainty (objective probabilities of consequences are unknown). These two standard theories consider that the choices of individuals between several alternatives are entirely and solely determined by the probabilities and utilities that individuals associate with the consequences of the different alternatives. These theories associate with each alternative the weighted sum of the utility of the consequences, the weights being the probabilities (objective or subjective). These theories posit that an individual will choose the alternative that maximizes his weighted sum.

However, standard theories face empirical difficulties that suggest that probabilities and utility are not the only determinants of individualsŠ choices. An important empirical difficulty of standard theories is the paradox of Ellsberg (1961). According to these theories, subjects should be indifferent between a known urn containing 50 red and 50 black balls and an unknown urn that comprises 100 red and black balls in an unknown proportion, irrespective of the winning color. However, contrary to this indifference, subjects tend to prefer the known urn to the unknown urn, irrespective of the winning color. Under these standard theories, the fact that subjects tend to prefer the known urn to the unknown urn (irrespective of the winning color), implies that the sum of the beliefs of having a red ball and having a black ball in the unknown urn is less than one.

Empirical difficulties of standard theories (see Starmer, 2000, for a review) jus-tiĄed the development of non-expected utility theories under risk and uncertainty that incorporate new individual characteristics as determinants of choices. Rank Dependent Utility (RDU; see Quiggin, 1982;Schmeidler, 1989) and Cumulative Prospect Theory (CPT; see Tversky and Kahneman, 1992, henceforth TK92) emerged as two main alternatives to the traditional theories. They explain empirical violations by introducing probability distortions (Bleichrodt and Pinto, 2000) and loss aversion (Wakker, 2010).

All chapters of the thesis are related to decision-making under risk and uncertainty. Three chapters provide measurement methods of decision models under risk and uncertainty. The Ąnal chapter derives new theoretical insights in the realm of incentive setting along with the development of a new experimental paradigm.

Measuring risk and uncertainty attitudes is of great value in many economic applications. For example, measures of attitudes toward risk and uncertainty can help explain behaviors in Ąnance (e.g. Baillon et al., 2018b), incentive setting (e.g. [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF], health choices (e.g. Attema et al., 2018), human behaviors in face of uncertainty generated by other humans or strategic uncertainty (e.g. Li et al., 2018), the behaviors of sport professionals (e.g. Bleichrodt et al., 2018), tax evasion (e.g. Dhami and Hajimoladarvish, 2020), election or vote (e.g. [START_REF] Kemel | An econometric estimation of prospect theory for natural ambiguity[END_REF].

I present in section 0.1 the prevalent theories that are widely used to model decisions under risk and uncertainty. Section 0.2 reviews measurement methods for risk attitudes. Finally, section 0.3 outlines the main goals of this thesis.

Decision theories under risk and uncertainty

Three prevailing theories are widely used to model decisions under risk and uncertainty: (subjective) expected utility theory, rank dependent utility and cumulative prospect theory. Before presenting them, we lay out the following notations.

Notations

Assume that consequences or outcomes are monetary, with R denoting the outcome set. Let denote under risk a lottery that gives outcome x i with known probability p i as follows:

L = (x 1 , p 1 ; x 2 , p 2 ; ..., x n , p n ) (1)
Denote by S a source of uncertainty, i.e. a group of events generated by a common mechanism of uncertainty. Denote by L a lottery that gives outcome x i if event E i ⊂ S occurs as follows:

L = (x 1 , E 1 ; x 2 , E 2 ; ..., x n , E n ) (2)
where the objective probabilities of E i (i = 1, 2, ..., n) are unknown. For notational convenience, we assume that outcomes are ordered: x 1 < x 2 < ... < x n .

Traditional models

The traditional theory to model decisions under risk is expected utility theory (Von Neumann and Morgenstern, 1947, EU henceforth). This theory postulates that decision maker values the lottery speciĄed in (1) in term of expected utility:

EU (L) = n i=1 p i u(x i ) (3)
with u(.) the utility function over outcomes.

For uncertainty, the traditional theory is the Subjective Expected Utility (SEU)

of [START_REF] Savage | The foundations of statistics[END_REF]. Because objective probabilities of events are not available, the decision maker forms subjective probabilities on events: P (E i ), i = 1, 2, ..., n.

The value of the prospect speciĄed in (2) under uncertainty is given by the sum of the utility of each of the outcomes weighted by its subjective probabilities:

SEU (L) = n i=1 P (E i )u(x i ) (4)
A distinctive feature of EU and SEU, is that objective and subjective probabilities are used linearly in the valuation of prospects. The paradox of Allais (1953) under risk and the paradox of Ellsberg (1961) under uncertainty challenged this feature. Rank Dependent Utility (RDU) and Cumulative Prospect Theory (CPT) emerged as two main alternatives to the traditional models.

0.1.3 Rank Dependent Utility (RDU) Quiggin (1982) proposes a rank dependent utility theory to evaluate the risky lottery speciĄed in (1). The value of the prospect is given by the sum of the utility of each of the outcomes weighted by a non-additive transformation of probabilities:

RDU (L) = n i=1 w n j=i p j -w n j=i+1 p j u(x i ) (5)
with w(.) the probability weighting function which is strictly increasing, transforms probability from [0, 1] to [0, 1] and satisĄes w(0) = 1w(1) = 0.

Schmeidler (1989) provides a rank dependent utility theory to evaluate under uncertainty the lottery speciĄed in (2) as follows:

RDU (L) = n i=1 W n j=i E j -W n j=i+1 E j u(x i ) (6)
with the event weighting function that satisĄes:

1. if E ⊂ F , then W (E) < W (F )
2. W (∅) = 1 -W (S) = 0 Abdellaoui et al. (2011a) show that event weighting function W (.) can be decomposed in terms of a source function w S (.) and belief P (.):

W (E) = w S (P (E)) (7)
The source function w S (.) is strictly increasing, transforms subjective probability (belief) from [0, 1] to [0, 1] and satisĄes w S (0) = 1w S (1) = 0.

Cumulative Prospect Theory (CPT)

Tversky and Kahneman (1992) considers that a person evaluate outcomes x i (i = 1, 2, ..., n) of the lotteries speciĄed in (1) and ( 2) in deviation to a reference point, say R:

x i -R (8)
The outcome x i is considered as gain (loss) if x i ≥ R (x i < R). Denote by m ≥ 0 (resp. t ≥ 0) the number of loss (resp. gain) outcomes: y -m < y -m+1 < ... < y -1

Losses < y 0 = 0 < y 1 < y 2 < ... < y t

Gains

The risky prospect speciĄed in (1) becomes under CPT: L = (y -m , p -m ; y -m+1 , p -m+1 ; ...; y -1 , p -1 ; y 0 , p 0 ; y 1 , p 1 ; ....; y t , p t ) (9)

with y -m = x 1 -R, y -m+1 = x 2 -R, ..., y t = x n -R; and p 1 = p -m , p 2 = p -m+1 , ..., p n = p t . The valuation of ( 9) is given by 7 CP T (L) = Similarly, the prospect (2) under uncertainty becomes:

L = (y -m , E -m ; y -m+1 , E -m+1 ; ...; y -1 , E -1 ; y 0 , p 0 ; y 1 , p 1 ; ....; y t , p t ) ( 11)

with

E 1 = E -m , E 2 = E -m+1 , ..., E n = E t
The valuation of ( 11) is given by

CP T (L) = 0 i=-m W - i j=-m E j -W - i-1 j=-m E j v(y i )+ t i=1 W + t j=i E j -W + t j=i+1 E j v(y i ) (12)
with v(.) the value function, W + (.) the event weighting function in the gain domain and W -(.) the event weighting function in the loss domain. The value function v() is continuous, strictly increasing and satisĄes v(0) = 0. The event weighting functions satisfy:

1. if E ⊂ F , then W + (E) < W + (F ) and W -(E) < W -(F ) 2. W + (∅) = 1 -W + (S) = 0 and W -(∅) = 1 -W -(S) = 0
Following the seminal study by TK92, as well as the subsequent developments in Köbberling and Wakker (2005) and Abdellaoui et al. (2008), the value function v(.) is composed of the loss aversion index λ > 0 which reĆects the exchange rate between gain and loss utility units, and the utility function u(.) that reĆects the intrinsic value of outcomes:

v(x) =          u(x) if x ≥ 0 λu(x) if x < 0 (13)
Similarly to RDU, the event weighting functions W + (.) and W -() can be decomposed in terms of source functions w + S (.) and w - S (.), and belief function P (.) (e.g. Attema et al., 2018).

Measurement methods

This section provides a literature review of the main methods to elicit utility functions, weighting functions, loss aversion and beliefs. The methods use binary lotteries of the form L = (x, y; p, 1p) that gives outcome x with probability p and outcome y with probability 1p. This notation corresponds to decision under risk. In decision under uncertainty, we just replace p and 1p by E and E c respectively. E denotes an event of state space S and E c denotes the complement of E in S. For notational convenience, we assume that all prospects L = (x, y; p, 1p) are rank-ordered. If a non-mixed prospect involves only gains (resp. losses), we assume that x ≥ y ≥ 0 (resp. x ≤ y ≤ 0). For mixed prospect we assume y ≤ 0 ≤ x.

Three types of approachŮparametric, semi-parametric and nonparametricŮ in decisions situation under risk and uncertainty are used.

Parametric approach

The parametric approach assumes parametric forms for utility and weighting functions. This approach is followed for example by Fehr-Duda et al. (2006) and Tversky and Kahneman (1992) and is not applicable for decisions under uncertainty.

In this approach, the authors start by collecting certainty equivalents ce i g for N non-mixed binary lotteries L g = (x g , y g ; p g , 1p g ) that involves different probabilities and outcomes (x g and y g ) and probabilities p g , with g = 1, 2, ..., N .

Under cumulative prospect theory (10), these certainty equivalents satisfy

ce i g = u -1 w i (p g )u(x g ) + (1 -w i (p))u(y g ) (14)
Second, the parametric forms of the utility and weighting functions are speciĄed.

For example, one can assumes a power utility function and the Prelec (1998) two-parameters weighting function:

u(x) =          x α if x ≥ 0 -λ(-x) β if x < 0 (15) w i (p) = exp -δ i (-ln(p)) γ i (16)
With these parametric forms, the equation (1.16) becomes in gain and loss domains ce + g = exp -δ + (-ln(p g )) γ + x α gy α g + y α g 1/α

(17)

ce - g = -exp -δ -(-ln(p g )) γ -(-x g ) β -(-y g ) β + (-y g ) β 1/β (18)
The parameters α, β, δ i and γ i can then be estimated in the equations ( 17) and ( 18) by nonlinear least squares or maximum likelihood. Also, Bayesian estimation can be used by making prior assumptions about parameters.

The link between the utility functions of the two domains (gain and loss) is required to elicit the loss aversion index (Abdellaoui et al., 2007b). Abdellaoui et al. (2008) makes this link with one indifference. The indifference consists in using a gain ce + g elicited in the Ąrst step and in determining the loss X g that satisĄes (0.5, ce + g ; X g ) ∼ 0. This implies in CPT framework that X g and the loss aversion index λ satisfy:

η + v(ce + g ) + η -λv(X g ) = u(0) = 0 (19)
For the power utility function, this relation becomes:

η + ce + g α -η -λ -X g β = u(0) = 0 (20)
or equivalently

λ = η + ce + g α η --X g β ( 21
)
Application of this method can be done under uncertainty [START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF][START_REF] Kemel | An econometric estimation of prospect theory for natural ambiguity[END_REF]Baillon et al., 2018a). In this context, subjective probabilities (beliefs) need to be estimated too.

This approach has four desirable properties: tractability, data-efficiency, easiness, error-robustness. We deĄne these properties below.

Tractable: the method allows for estimating risk attitudes with standard tools, like OLS or NLS (Abdellaoui et al., 2007a(Abdellaoui et al., , 2008)).

Error-robust: the method accounts for the fact that subjects make response errors when answering questions (Abdellaoui et al., 2008).

Easy (not cognitively demanding): the method relies on simple choices involving the lowest possible number of outcomes, that is 3 (Abdellaoui et al., 2008). A useful benchmark comes from TK92 who ask their subjects to make simple choices between a sure outcome and a binary lottery.

Data-efficient: the method requires few measurements (observations) to estimate the parameters of the utility function and the probability weights. As a rule of thumb, we take TK92 (with 28 measurements for 9 probability weights and one utility parameter per domain) as benchmark, and consider a method data-efficient if it requires no more than three measurements per estimate of a probability weight.1

In addition to these four desirable properties, the parametric methods are also Comprehensive in the sense that they allow for estimating utility function, loss aversion and weighting function under risk as well as under uncertainty.

Nevertheless, this approach has several drawbacks. First, empirical estimates are sensitive to the speciĄcation of the utility and probability weighting functions (Abdellaoui, 2000). Second, parametric methods only provide an overall measure of the goodness of Ąt of the model, rather than separate measures for each of its components Ű one for the utility function and one for the probability weighting function (see Gonzalez and Wu, 1999).

Semi-parametric approach

The semi-parametric approach assumes parametric forms for the utility function without imposing any parametric restriction on the probability weight. Methods that use this approach can be found Abdellaoui et al. (2008) and Abdellaoui et al. (2011b).

The semi-parametric method of Abdellaoui et al. (2008) entails three steps.

It resembles the previous parametric method that we had presented with two important differences. First, the probabilities p in the binary lotteries is kept Ąxed so that the decision weight w i (p) can be estimated directly as parameter alongside those of the utility function. Second, an additional step is introduced to estimate the loss aversion parameter λ. More precisely, the three steps are as follows.

First step: Utility and probability (or event) weight elicitation in the gain domain

The step starts by two choices from the researcher. First, the researcher has to choose a probability p that will be kept Ąxed throughout the elicitation of the utility function on the gain domain. Second, the authors have to choose a parametric form for the utility function.

To elicit the probability (or event) weight and parameters related that deĄne the selected utility function, the researcher collects certainty equivalents ce + g for N non-mixed binaries lotteries L g = (x g , y g ; p, 1p) that involves different outcomes (x g and y g ) and the Ąxed probability p, with g = 1, 2, ..., N . Under cumulative prospect theory, these certainty equivalents satisfy:

ce + g = u -1 η + u(x g ) + (1 -η + )u(y g ) (22)
with η + = w + (p). If for example, the authors choose p = 0.5 and the power utility function 15, the equation ( 22) becomes:

ce + g = η + x α g -y α g + y α g 1/α (23)
with η + = w + (0.5). The parameter α and η + = w + (0.5) in the relation 23 can be estimated by nonlinear least squares.

Second

Step: Utility and probability (or event) elicitation in the loss domain This step is similar to the previous one. The probability p in the series of lottery L g = (x g , y g ; p, 1p) indexes g = 1, 2, ..., N that will be kept Ąxed throughout the elicitation of the utility function on the loss domain is such that p = 1p. This condition is necessary in the third step that elicits the loss aversion paramater. Second, the authors have to choose a parametric form for the utility function. Under prospect theory functional, the certainty equivalents of L g = (x g , y g ; p, 1p) satisfy:

ce - g = u -1 η -u(x g ) + (1 -η -)u(y g ) (24)
If authors have selected p = 0.5 in the Ąrst step and power utility function (15), the equation ( 24) becomes

ce - g = -η --x g β --y g β + -y g β 1/β (25)
with η -= w -(0.5). The parameter β and η -= w -(0.5) in the equation ( 25) can be estimated by nonlinear least squares.

Third Step: Loss aversion elicitation

The elicitation of the loss aversion follows the same procedure as in the previous parametric methods.

This method has been applied by Attema et al. (2013) with an exponential utility function.

We can point out several limitations of this semi-parametric approach. First, the estimation resultsŮnamely the estimations of the loss aversion index and of the probability weightŮdepend on the choice of the utility function. Second, it allows us to know only one probability weight in each domain (gain and loss) so that the shape of the probability weighting function is not estimated. Abdellaoui et al. (2011b) extends the method to the elicit a broader range of points of the probability weighting function. Abdellaoui et al. (2011b) start by eliciting the utility function and one probability weight as in Abdellaoui et al. (2008). Having the estimation of the utility function, they elicit the certainty equivalents ce k for all desired probabilities p k , k = 1, 2, ..., K as follows:

ce k ∼ (x * , 0; p k , 1 -p k ) for k = 1, 2, ..., K (26)
with x * a Ąxed outcome that belongs to the elicited utility domain. The probability weights can then be estimated as:

w i (p k ) = u(ce k ) u(x * ) for k = 1, 2, ..., K (27)
The semi-parametric approach can be applied to uncertainty (e.g. Baillon et al., 2018a;Abdellaoui et al., 2011a). The semi-parametric method of Abdellaoui et al. ( 2008) keeps all appealing properties of parametric methods. Nevertheless, when extending it as in Abdellaoui et al. (2011b) to estimate probability weighting function, the method is no longer error-robust (e.g. Etchart-Vincent, 2004).

Non-parametric approach

The non-parametric approach does not make any parametric assumption about the utility and the weighting functions. Several methods are proposed in this approach.

The gamble-tradeoff method

The Ąrst nonparametric method that offers the possibility to elicit utility function in CPT frameworks is the tradeoff method proposed by Wakker and Deneffe (1996). The utility elicitation using the tradeoff method is applicable in uncertainty and risky contexts. Abdellaoui (2000), Bleichrodt and Pinto (2000) and Etchart-Vincent (2009a) have extended this tradeoff method to be able to estimate probability and/or event weights.

First step: The utility function

The method consists in eliciting equally-spaced outcomes in terms of utility. To do this, two outcomes R and r that have the same sign are Ąxed. In addition, a probability p ∈ (0, 1) and initial outcome x 0 are chosen. Once these values are chosen, the outcome x 1 is elicited from the following indifference:

(x 0 , R; p, 1 -p) ∼ (x 1 , r; p, 1 -p)
In the gain (resp. loss) domain the outcomes have to be chosen such that

0 ≤ r ≤ R ≤ x 0 ≤ x 1 (reps. 0 ≥ r ≥ R ≥ x 0 ≥ x 1 )
. This indifference implies:

w i (p) u(x 1 ) -u(x 0 ) = (1 -w i (p)) u(R) -u(r) (28)
After eliciting x 1 , the outcome x 2 is elicited from the following indifference:

(x 1 , R; p, 1 -p) ∼ (x 2 , r; p, 1 -p)
This indifference implies in the gain domain:

w i (p) u(x 2 ) -u(x 1 ) = (1 -w i (p)) u(R) -u(r) (29)
From equations 28 and 29, we have the following equality:

u(x 2 ) -u(x 1 ) = u(x 1 ) -u(x 0 ) (30)
From equation 30, x 1 is the midpoint of x 0 and x 2 in terms of utility2 . The previous procedure is repeated to have the sequence of outcomes x 0 , x 1 , ..., x n with the following indiferrences

(x j-1 , R; p, 1 -p) ∼ (x j , r; p, 1 -p) , j = 1, 2, ..., n
By setting u(x 0 ) = 0, u(x n ) = 1 in the gain domain, the valuations of elicited outcomes in this domain are given by

u(x j ) = j n , j = 1, 2, ..., n (31)
Proceeding similar thing in the loss domain by setting u(

x n ) = -1 leads to u(x j ) = - j n , j = 1, 2, ..., n (32)

Second step: the weighting function

The gamble-tradeoff method does not specify how to estimate the weighting function. Authors like Abdellaoui (2000), Bleichrodt and Pinto (2000) and Etchart-Vincent (2009a) have extended the tradeoff-method to be able to estimate the weighting function.

After determining the sequence x 0 , x 1 , ..., x n from the tradeoff method, Abdellaoui (2000) proposed to elicit the sequence of probabilities p 1 , p 2 , ..., p n-1 from the following indifferences

(x n , x 0 ; p j , 1 -p j ) ∼ x j
From these indifferences, the weighting function is given by

w i (p j ) = u(x j ) -u(x 0 ) u(x n ) -u(x 0 ) = j n , j = 1, 2, ..., n -1 (33)
Also, Bleichrodt and Pinto (2000) proposed another way to elicit the weighting function after determining the sequence x 0 , x 1 , ..., x n from the tradeoff method.

The authors proposed to determine probability weights with two types of questions. For any low probability p ∈ (0, 1), the probability weight w i (p) is elicited by asking for outcome z such that:

(x i , x j ; p, 1 -p) ∼ (x k , z; p, 1 -p) with x k ≥ x i ≥ x j in the gain domain, x k ≤ x i ≤
x j in the loss domain and

x k , x i and x j are elements of the sequence elicited in the tradeoff part. From the above indifference, the probability weight is given by

w i (p) = u(x j ) -u(z) u(x j ) -u(z) + u(x k ) -u(x i ) (34)
Similarly, for any higher probability p ∈ (0, 1), the probability weight w i (p) is elicited by asking for outcome z in the following indifference:

(x m , x n ; p, 1 -p) ∼ (x q , z; p, 1 -p)
with x m ≥ x n ≥ x q in the gain domain, x m ≤ x n ≤ x q in the loss domain and x m , x n and x q elements of the sequence elicited in the tradeoff part. The above indifference implies that the probability weight is given by

w i (p) = u(x n ) -u(x q ) u(z) -u(x m ) + u(x n ) -u(x q ) (35)
The elicited outcomes z and z do not necessary belong to the sequence of outcomes elicited in the tradeoff part. In this case their utilities are estimated from the utilities of elements of the sequence obtained in the tradeoff part. More precisely, a linear approximation is performed or parametric assumptions are made on utility function to obtain utilities of z and z when these outcomes do not belong to the sequence of outcomes. Doing so, this probability weighting function elictation is semi-parametric 3 as that of Etchart-Vincent (2009a).

The method of Etchart-Vincent (2009a) consists in determining CE j for any Ąxed probability p j with the following indifference

CE j ∼ (x k , x i ; p j , 1 -p j ) (36) with x i ≤ CE j ≤ x k in the gain domain, x k ≤ CE j ≤ x i in the loss domain and
x k , x i are elements of the standard sequence elicited in the tradeoff part. This indifference implies:

w i (p j ) = u(CE j ) -u(x i ) u(x k ) -u(x i ) (37) so that w -(p j ) = n u(CE j ) + i i -k (38) w + (p j ) = n u(CE j ) -i k -i (39)
3 The methods of Bleichrodt and Pinto (2000) and Etchart-Vincent (2009a) start by eliciting nonparametrically utility points using the tradeoff method. To elicit probability weights, they make use of parametric fits or linear interpolation to the nonparametric utility points obtained from the tradeoff method. The use of parametric fit (resp. linear interpolation) to find utility value to compute a probability weight make the probability weight dependent on the non linear utility function (resp. linear utility function over small interval) used to fit data. Doing so these methods are semi-parametric even though Bleichrodt and Pinto (2000) qualify their method to be parameter-free.

When choosing x k = x n and x i = x 0 as in Etchart-Vincent (2004), then w -(p j ) = -u(CE j ) and w + (p j ) = u(CE j ). As in the case of Bleichrodt and Pinto (2000), the CE j is unlikely to be an elements of the previously elicited sequence. Hence, u(CE j ) is estimated from parametric Ątting of utility points obtained from elements of the sequence (in the tradeoff part).

These methods are neither error-robust nor easy. Furthermore, it is worth to mention that the utility function in the gain domain (31) and that in the loss domain (32) are not linked since they are elicited separately. This is due to the fact that two values on the utility function (in addition to the utility of the reference point) Ů i.e u(x n ) = 1 in the gain domain and u(x n ) = -1 in the loss domainŮ are chosen. To link between the utility in the gain domain to the utility in the loss domain, only one value on the utility function has to be chosen and so that all the other values only depend on this choice. Consequently, these methods do not allow for the elicitation of loss aversion. Abdellaoui et al. (2007b) provided the Ąrst non-parametric method to elicit the utility functions in the gain and loss domains that are linked so that they allow for the estimation of loss aversion. This method is applicable only in decision under risk. The method elicits the probability that has one-half as weight and utility function in the full domain (loss and gain simultaneously) in four steps.

Method of mirror image applicable only in a risky context

The summary of the method is given in table 1.

First step: Elictation of p g and p ℓ with w + (p g ) = 0.5 and w -(p ℓ ) = 0.5

(x 0 , R; p, 1 -p) ∼ (x 1 , r; p, 1 -p) u(x 2 ) -u(x 1 ) = u(x 1 ) -u(x 0 ) p = 0.33 x 2 (x 1 , R; p, 1 -p) ∼ (x 2 , r; p, 1 -p) r = -100 p ℓ x 1 ∼ (x 2 , x 0 ; p ℓ , 1 -p ℓ ) w -(p ℓ ) = 0.5 R = -600 x 0 = -1000 Gain domain x 1 (x 0 , R; p, 1 -p) ∼ (x 1 , r; p, 1 -p) u(x 2 ) -u(x 1 ) = u(x 1 ) -u(x 0 ) p = 0.33 x 2 (x 1 , R; p, 1 -p) ∼ (x 2 , r; p, 1 -p) r = 100 pg x 1 ∼ (x 2 , x 0 ; p ℓ , 1 -p ℓ ) w + (pg) = 0.5 R = 600 x 0 = 1000 Elicitation of utility function Step 2 Lr ∈ [L 1 , 0] Lr ∼ (L A , L B ; p ℓ , 1 -p ℓ ) u(Lr) = 0.5u(L A ) + 0.5u(L B ) L 1 = -100, 000 u(L 1 ) = -1 Step 3 ℓ L 1 ∼ (ℓ, 0; 0.5, 0.5) u(ℓ)w -(0.5) = -1 g 0 ∼ (ℓ, g; 0.5, 0.5) u(g)w + (0.5) = 1 G 1 G 1 ∼ (g, 0; 0.5, 0.5) u(G 1 ) = u(g)w + (0.5) = 1 Step 4 Gr ∈ [0, G 1 ] Gr ∼ (G A , G B ; pg, 1 -pg) u(Gr) = 0.5u(G A ) + 0.5u(G B ) * All monetary amounts are in French francs (FF).
Each elicitation requires three indifferences. First, three outcomes x 0 , x 1 and x 2 equally spaced in terms of utility are elicited by using ( 28) and ( 29) as in the tradeoff method. These two indifferences entail u(x 1 ) = 0.5u(x 0 ) + 0.5u(x 2 ) (40)

In the case of the loss domain, where

x 2 ≤ x 1 ≤ x 0 , p ℓ is elicited with the following indifference (x 2 , x 0 ; p ℓ , 1 -p ℓ ) ∼ x 1 This indifference entails u(x 1 ) = w -(p ℓ )u(x 2 ) + (1 -w -(p ℓ ))u(x 0 ) (41)
Using ( 40) and ( 41), by identiĄcation, w -(p ℓ ) = 0.5 as wanted. We proceed similarly in the gain domain, where x 2 ≥ x 1 ≥ x 0 . The second indifference consists in eliciting a gain g that satisĄes (ℓ, g; 0.5, 0.5) ∼ 0. This indifference implies:

Second

u(g)w + (0.5) = 1 (44)
Finally the gain G 1 is elicited from the following indifference (g, 0; 0.5, 0.5) ∼ G 1 .

This indifference implies

u(G 1 ) = 1 (45)
G 1 has utility 1 and is the Şmirror imageŤ of L 1 in terms of utility.

Fourth step:

Once p g and G 1 are elicited from step 1 and step 3, the fourth and Ąnal step of the elicitation determines utility on the interval [0, G 1 ] in a similar way as it is done in the loss domain in step 2.

Even though this method elicits utility function simultaneously in the gain and loss domains and then offer the possibility to estimate loss aversion, this method has three drawbacks. First, this method is not applicable in uncertainty contexts. Recognizing this limitation, Abdellaoui et al. (2016) extend this method in uncertainty contexts. Second, the method only tells us how to elicit probabilities that have a weight of 0.5. We can point out two solutions to this drawback. The Ąrst solution is to follow Abdellaoui (2000) procedure to elicit other points in the weighting function. The second solution is to generalize how w + (p g ) = 0.5 and w -(p ℓ ) = 0.5 are elicited in the Ąrst step of this method as it is the case in the midweight method proposed by Van De Kuilen and Wakker (2011). The third drawback that this method shares with the previous (Abdellaoui, 2000;Bleichrodt and Pinto, 2000;Etchart-Vincent, 2009a) is due to the fact that before completely eliciting the weighting function, the utility function has to be elicited. This can be time-consuming for the experimenter if he is only interested in eliciting some features of the weighting function. It was the case for example of Etchart-Vincent (2009a,b). To avoid this, it is important to have a method that allows to elicit the weighting function without knowing utility function. The midweight method proposed by Van De Kuilen and Wakker (2011) was an important contribution in that direction.

Method of mirror image applicable in risky and uncertainty contexts

Since the mirror image method of Abdellaoui et al. (2007b) is only applicable in decision under risk, Abdellaoui et al. (2016) extended this method so that it could be applied in uncertainty and risky contexts. 4 As pointed out by Abdellaoui et al. (2016) and Bleichrodt et al. (2018), this method constitutes the Ąrst one that makes it possible to completely measure all the ingredients of prospect theory without making simplifying assumptions. The method is split in three steps.

First step: Connecting utility for gains and utility for losses (mirror image step)

This step determines the loss x - 1 and a gain x + 1 that have the same utility (in absolute value). The steps start by Ąxing a gain X and an event E (or probability p in a risky context). Then, the loss Y that satisĄes the following indifference is

elicited 0 ∼ (X, Y ; E, E c ) ( 46
)
This indifference means

w + (E)u(X) + w -(E c )u(Y ) = u(0) = 0 ( 47
)
The gain x + 1 is determined from the following indifference

x + 1 ∼ (X, 0; E, E c ) (48) so that u(x + 1 ) = w + (E)u(X) (49)
The loss x - 1 is determined from the following indifference

x - 1 ∼ (Y, 0; E c , E) (50) so that u(x - 1 ) = w -(E c )u(Y ) (51)
The relations (47), ( 49) and (51) imply:

u(x + 1 ) = -u(x - 1 ) (52)

Second step: Measurement of utility for gains

The second step serves to elicit the remainder standard sequence of gains, i.e

x + 2 , x + 3 , x + 1 , ..., x + n . The step starts by Ąxing a loss ℓ and eliciting the loss L < ℓ that satisĄes:

(x + 1 , L; E, E c ) ∼ (ℓ, 0; E c , E) (53) so that w + (E)u(x + 1 ) + w -(E c )u(L) = w -(E c )u(ℓ) (54)
or equivalently

u(x + 1 ) -u(0) =0 = w -(E c ) w + (E) u(ℓ) -u(L) (55)
Next, the gain x + 2 is elicited with the following indifference

(x + 2 , L; E, E c ) ∼ (x + 1 , ℓ; E, E c ) (56) so that w + (E)u(x + 2 ) + w -(E c )u(L) = w + (E)u(x + 1 ) + w -(E c )u(ℓ) (57)
or equivalently

u(x + 2 ) -u(x + 1 ) = w -(E c ) w + (E) u(ℓ) -u(L) (58)
Then, we continue the elicitation sequentially as follows:

(x + j , L; E, E c ) ∼ (x + j-1 , ℓ; E, E c ) , j = 2, 3, ..., n (59)
so as to get

u(x + j ) -u(x + j-1 ) = w -(E c ) w + (E) u(ℓ) -u(L) (60)
Relations ( 55) and (60) imply that 0,

x + 1 , x + 2 , x + 3 , x + 1 , ..., x + n is a standard sequence of gains, that is 0, x + 1 , x + 2 , x + 3 , x + 1 , ..., x + n are equally spaced in term of utility u(x + 1 ) -u(0) = u(x + j ) -u(x + j-1 ) , j = 2, 3, ..., n (61) u(0) = 0 and u(x + n ) = 1 lead to u(x + j ) = j n , for j = 1, 2, ..., n (62)

Third step: Measurement of utility for losses

Similar to the second step, the third steps serves to elicit the remainder standard sequence of losses, i.e x - 1 , x - 2 , x - 3 , ..., x - n . The step starts by Ąxing a gain g and eliciting the gain G > g that satisĄes

(G, x - 1 ; E, E c ) ∼ (g, 0; E, E c ) (63) so that w + (E)u(G) + w -(E c )u(x - 1 ) = w + (E)u(g) (64)
or equivalently

u(0) =0 -u(x - 1 ) = w + (E) w -(E c ) u(G) -u(g) (65)
Next, the loss x - 2 is elicited with the following indifference

(G, x - 2 ; E, E c ) ∼ (g, x - 1 ; E, E c ) (66)
so that

w + (E)u(G) + w -(E c )u(x - 2 ) = w + (E)u(g) + w -(E c )u(x - 1 ) (67)
or equivalently

u(x - 1 ) -u(x - 2 ) = w + (E) w -(E c ) u(G) -u(g) (68)
Then, we continue the elicitation sequentially as follows:

(G, x - j ; E, E c ) ∼ (g, x - j-1 ; E, E c ) , j = 2, 3, ..., n (69)
so as to get

u(x - j-1 ) -u(x - j ) = w + (E) w -(E c ) u(G) -u(g) (70)
Relations ( 68) and ( 70) imply that 0, x - 1 , x - 2 , x - 3 , x - 1 , ..., x - n is a standard sequence of losses, that is:

u(0) -u(x - 1 ) = u(x - j-1 ) -u(x - j ) , j = 2, 3, ..., n (71)
Because u(0) = 0 and u(x + 1 ) = 1 n , it follows from the Ąrst step that u(x - 1 ) = -1 n . So, it also follows from (71) that:

u(x - j ) = - j n , for j = 1, 2, ..., n (72)
Attema et al. ( 2018) and Bleichrodt et al. (2018) complete this method by one further step to elicit probability (or event) weights. This fourth step consists in determining the following indifference

x + E ∼ (x + n , 0; E, E c ) and x - E ∼ (x - n , 0; E, E c ) (73)
These indifferences means that

w + (E) = u(x + E ) and w -(E) = -u(x - E ) (74)
The values of u(x + E ) and u(x - E ) are approximated using the utility function elicited in steps 2 and 3. This way to elicit probability (or event) weight can also be found in Etchart-Vincent (2004, 2009a). Finally, it is worth to notice that this method is neither easy nor error-robust.

Midweight method

The midweight method proposed by Van De Kuilen and Wakker (2011) allows to elicit weighting function when the utility midpoint for at least one pair of outcomes is known. This method is applicable both in risky and uncertainty context.

In the risky context, this method can be viewed as a generalization of how the probability that has one half as weight is elicited in Abdellaoui et al. (2007b) and can be split in two steps.

First step:

The Ąrst step consists in determining the sequence of three outcomes x 0 , x 1 , x 2 equally spaced in terms of utility by the tradeoff method proposed by Wakker and Deneffe (1996) to obtain the equation 40. Recall this equation:

u(x 1 ) = u(x 2 ) + u(x 0 ) 2 (75)
with x 0 < x 1 < x 2 in gain domain and x 2 < x 1 < x 0 in the loss doamain.

Second step:

Once x 0 , x 1 and x 2 are determined in the Ąrst step, the method generalizes the relation (41) in Abdellaoui et al. (2007b). Formally, a lottery

L = (x 2 , x 1 , x 0 ; p 2 , p 1 , p 0
) is constructed and the quantity ϵ is determined by looking for the following indifference

(x 2 , x 1 , x 0 ; p 2 , p 1 , p 0 ) ∼ (x 2 , x 0 ; p 2 + ϵ, p 0 + p 1 -ϵ) (76)
This indifference implies

u(x 1 ) = w i (p 2 + ϵ) -w i (p 2 ) w i (p 2 + p 1 ) -w i (p 2 ) u(x 2 ) + 1 - w i (p 2 + ϵ) -w i (p 2 ) w i (p 2 + p 1 ) -w i (p 2 ) u(x 0 ) (77)
Using 75 and 77, by identiĄcation p 2 , p 2 + ϵ and p 2 + p 1 are equally spaced in terms of weight w i as follows

w i (p 2 + ϵ) = w i (p 2 ) + w i (p 2 + p 1 ) 2 (78)
To use this formula, the weight of p 2 and p 2 + p 1 have to be known before computing w i (p 2 + ϵ). Then, the experimenter has to start with the case (p 0 , p 1 , p 2 ) = (0, 1, 0) that corresponds to eliciting ϵ so that w i (ϵ) = 0.5. Then, use this ϵ to elicit other points on the weighting function and so on.

We can point out two drawbacks for this method. First, it only minimizes the need to elicit the utility function before eliciting the weighting function because the utility midpoint for at least one pair of outcomes is still required. Second this method does not allow direct measurement of the weight of any desired probability p in (0, 1). For example, if the experimenter is only interested to measure the weight w i (p) of p = 0.2, the midweight method will force him to elicit several points of the weighting function before reaching w i (0.2). This kind of criticism applies also to Abdellaoui (2000). Again, this can be particularly time-consuming for the experimenter.

Similarly to the decision under risk, one can elicit three outcomes x 0 , x 1 , x 2 equally spaced in terms of utility by the tradeoff method proposed by Wakker and Deneffe (1996) in a context of uncertainty. Once the outcomes x 0 , x 1 , x 2 are elicited, the events G ⊂ D and B ⊂ D with D = G ∪ B are determined as follows

(x 2 , x 1 , x 0 ; A, D, C) ∼ (x 2 , x 0 ; A ∪ G, C ∪ B) (79)
This indifference implies under CPT that:

w i (A ∪ G) = w i (A) + w i (D ∪ A) 2 (80)
Similar to the risky context, this formula can be used by starting with (A, D, C) = (∅, Ω, ∅) so as to exploit the fact that w i (∅) = 0 and w i (Ω) = 1.

A main drawback of the midweight method under uncertainty is that it is not always applicable. In fact, to be able to elicit G and B in the indifference 79, the event space Ω needs to be sufficiently rich such as a continuum (Van De Kuilen and Wakker, 2011). Thus, if for example the event space Ω contains only two events the midweight method will not be able to provide the probability (or event) weights for that events. Finally, it is worth to notice that this method is neither easy nor not error-robust.

Alternating least squares estimation method

This method is proposed by Gonzalez and Wu (1999). To implement this method, the experimenter Ąxes the probabilities p 1 , p 2 , ..., p K for which he wants to elicit probability weights. He also Ąxes n pairs of positive (or negative) outcomes (x 1 , y 1 ), (x 2 , y 2 ),...,(x n , y n ) with 0 ≤ y j < x j , j = 1, 2, .., n. The experimenter elicits the following K × n certainty equivalents

ce jk ∼ (x j , y j ; p k , 1 -p k ) for j = 1, 2, ..., n and k = 1, 2, ..., K
The method assumes the equation that characterizes the certainty equivalent with an additive and normally distributed error term at the scale of the utility u as follows:

u(ce jk ) = w + (p k )u(x j ) + (1 -w + (p k ))u(y j ) + ϵ u jk (81)
The values of w + (p k ), u(x j ) and u(y j ) are determined based on an alternating least squares estimation method. The idea of the estimation is the following.

First, the starting values of w + (p k ), u(x j ) and u(y j ) are chosen. Based on these starting values of u(x j ) and u(y j ), the values of u(ce jk ) are approximated through linear interpolation. Finally, the values of w + (p k ), u(x j ) and u(y j ) are determined iteratively.

The method is error-robust and easy. However, the method is not dataefficient and tractable (Van De Kuilen and Wakker, 2011). Furthermore, the method is applicable in the gain and loss domains separately so that the there is no link between utility function in the two domains. As such, loss aversion cannot be measured (Abdellaoui et al., 2007b). Most importantly, as pointed out by Van De Kuilen and Wakker (2011), this method is not data-efficient (i.e inefficient) and tractable. More importantly, deĄning the response error at the utility scale is problematic because it produces solutions that are characterized by unrealistic concavity of the utility and probability weighting functions. To

Aims and outline of the thesis

The thesis is at the intersection of behavioral economics and decision theory. It has two main objectives. The Ąrst is to develop new methods to estimate the different components of decision models under risk and under uncertainty. For risk, the idea is to develop new methods for estimating (i) utility function, (ii) probability weighting function and, (iii) loss aversion. In the case of uncertainty, a fourth component must also be measured: (iv) beliefs (i.e. subjective probabilities) that the decision maker forms about the different possible events. The second objective is to use these methods to understand in a concrete way the decision-making of individuals in various Ąelds of economics. For this second objective, the thesis focuses on applications related to incentive setting and strategic uncertainty in game theory.

To provide further details, Chapter 1, co-authored with Brice Corgnet and Adam Zylbersztejn, establishes a comprehensive semi-parametric method that satisĄes the four desirable properties of parametric methods: tractable, dataefficient, error-robust, easy. Doing so, we develop a method that increases the precision of parameter estimates of parametric methods while being easy to implement and estimate. The method can be seen as an extension of the semi-parametric method of Abdellaoui et al. (2008) with respect to three points.

First, our method uses a single step to estimate the probability weights and the utility function in the full domain, whereas Abdellaoui et al. (2008) propose a two-step procedure. This feature of our method allows for testing several important restrictions (partial reĆection, identical probability weighting functions across domains, and duality) as well as imposing these restrictions whenever necessary. Imposing partial reĆection helps avoid the problem of arbitrary measurement of loss aversion with power utility functions (see Wakker, 2010).

The method allows for testing and imposing identical probability weighting functions across domains (i.e., w + (p) = w -(p)) thus allowing us to test a key assumption of OPT. Also, the method allows for testing and imposing duality (i.e., w

+ (p) = 1 -w -(1 -p))
. 5 By allowing us to impose duality as well as identical probability weighting across domains, our method can be applied under RDU (Quiggin, 1982;Gilboa, 1987;Schmeidler, 1989) and Original Prospect

Theory (Kahneman and Tversky, 1979).

Second, unlike Abdellaoui et al. (2008), we can estimate multiple probability weights and thus elicit the shape of the probability weighting function. This is done without losing error-robustness.

Third, certainty equivalents for mixed prospects are obtained using a different procedure than the one proposed by ABL. In ABL, subjects are asked to provide a loss amount L for which they are indifferent between the status-quo (0) and a binary lottery (G, L; p g , 1p g ) where G is a Ąxed gain and L ∈ (-∞, 0] is a loss. In this elicitation procedure, the researcher does not know the lower bound of the loss interval. By contrast, our method keeps track of the upper and lower bounds of the loss interval because ce k belongs to the interval (y k , x k ). This is an appealing property of our method for two reasons. First, asking subjects to provide indifference values on unbounded intervals can be cognitively demanding (Wakker and Deneffe, 1996;Abdellaoui et al., 2007b). This may lead to errors that potentially inĆate the estimates of loss aversion, as reported by ABL (see Chapter 2, single-authored, establishes a nonparametric method for measuring utility function, weighting function and loss aversion under risk and uncertainty.

It satisĄes three desirable properties of parametric methods: tractable, errorrobust, and easy. The method builds upon the smoothing spline literature.

As such, the method is more data consuming than parametric methods (e.g. Ahamada and Flachaire, 2010;Green and Silverman, 1993). This method should thus be used in two main cases. First, behavioral studies often elicit few observations per subject. In such cases, the method can be used to derive estimates at the aggregate level by pooling subjects as in Tversky and Kahneman (1992). Second, the method can be performed at the individual level if the number of observations per subject is sufficiently large as in Gonzalez and Wu (1999).

The method includes smooth parameters. These parameters allow to control for the collinearity between the value and weighting functions (e.g. Zeisberger et al., 2012;lŠHaridon and Vieider, 2019) by penalizing deviation from the linear value function.

Chapter 3, co-authored with another PhD student Maria Alejandra Erazo Diaz, develops a new method to measure beliefs, utility function and weighting function towards events that are not necessarily equally likely and belong to a discrete set (i.e., discrete sources of uncertainty). It is a multi-stage method.

In the Ąrst stage, we only specify utility and estimate events weights nonparametrically. Based on event weights from the Ąrst stage, the method allows to estimate the parameters of any weighting function. Our method thus allows for more Ćexibility in the parametric choices of weighting function in comparison to existing methods (e.g. Baillon et al., 2018bBaillon et al., , 2021Baillon et al., , 2018a) that rely on the neo-additive weighting function of Chateauneuf et al. (2007). Using simulations, we show that the multi-stage approach is more robust to misspeciĄcation issues than the one-stage approach that speciĄes simultaneously the functional form for the utility and weighting functions (e.g. Baillon et al., 2018a). The method is based on simple choices that involve the lowest possible number of outcomes (i.e., three). As such, this method is easy for subjects, compared to methods that are based on exchangeable events or matching probabilities (e.g. Baillon et al., 2018b;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]Abdellaoui et al., 2020), in which each choice involves four outcomes [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Abdellaoui et al., 2008). Finally, contrary to previous methods (e.g. Baillon et al., 2018b,a), our method accounts for response errors that are pervasive in experimental data [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF].

We implement the method experimentally to both equal and different sources of uncertainty in two contexts: trust and coordination games. The method successfully passes validity tests and provides plausible results, showing the reliability of the results derived from it. People are more insensitive to likelihood in the presence of asymmetric events than symmetric events, suggesting that belief formation is cognitively demanding. For equal sources of uncertainty, people exhibit payoff dependence aversion and variety of payoffs seeking. Payoff dependence aversion means that people dislike that their own payoffs depend on the preferences of others. This behavior is captured by a more concave utility function. Variety of payoffs seeking means that subjects prefer a greater number of possible payoffs when such possible payoffs depend on the preferences of others.

This behavior is captured by more optimism.

Chapter 4, co-authored with Brice Corgnet, Roberto Hernán-González and Adam Zylbersztejn, provides new insights about the Risk Incentives Tradeoff (RIT) originally established under EUT. The RIT is a fundamental result of principal-agent theory (e.g. Holmström, 1979;[START_REF] Milgrom | Economics, Organization and Management[END_REF]).

Yet, empirical evidence has been elusive [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF][START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF]. This could be due to a lack of robustness of the theory outside of the standard expected utility framework (EUT) or to confounding factors in the empirical tests. We Ąrst study RIT under alternative theories:

Rank-Dependent Utility (RDU) and Mean-Variance-Skewness (MVS). We show that, under RDU, RIT is remarkably pervasive because it applies not only when agents are risk-averse but also when they are risk-neutral or risk-seeking.

For example, RIT is present when risk-neutral or risk-seeking agents who are moderately risk-seeking in the probability domain (i.e., they moderately overweight probabilities) are risk-averse in the utility domain (i.e., their utility is concave). This result suggests RIT might be more widespread than predicted by EUT. This observation contrasts with the limited empirical evidence for RIT. However, MVS provides a setup in which, in line with existing evidence, RIT is less pervasive. Although RIT applies whenever agents are risk-averse under EUT, this is not the case under MVS. Indeed, RIT might not hold for risk-averse agents who value positive skewness. Furthermore, under MVS the optimal variable pay (Ąxed pay) can increase (decrease) with the shock, which is what we refer to as reversed RIT. This occurs when the distribution of the shock is positively skewed and the aversion to variance is less pronounced than the value of skewness for a risk-averse MVS agent.

These theoretical results show that non-EUT models provide reasons for both hope and despair regarding the robustness of RIT. To test the predictions of the alternative theories, we develop a novel lab test of RIT to eliminate confounding factors. To do so, we directly elicit the minimum Ąxed pay agents are ready to accept given a value of the variable pay. In this setup, the focus is on agentsŠ decisions because principals do not make contractual decisions (see e.g., [START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF]. This BareBone design (BB design, henceforth) allows us to discard confounding factors related to the risk attitudes of principals. It also eliminates any asymmetry of information between the principal and the agent whose preferences are unknown. We also use monetary effort instead of a real-effort task (e.g., Anderhub et al., 2002;Keser and Willinger, 2007;[START_REF] Gächter | Design a contract: A simple principalagent problem as a classroom experiment[END_REF] to discard other confounding factors often present in experimental data such as social motives and reference points (see [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF]. More generally, our design is such that common confounding factors, such as organizational hierarchies, delegation, implicit incentives, tacit knowledge, uncertainty and market dynamics (Jensen and Meckling, 1995;[START_REF] Raith | SpeciĄc knowledge and performance measurement[END_REF][START_REF] Devaro | An empirical analysis of risk, incentives and the delegation of worker authority[END_REF][START_REF] Edmans | Dynamic ceo compensation[END_REF]He et al., 2014), are absent. We do not mean to underplay the importance of these confounding factors but implement a testbed for the basic mechanism underlying RIT. It follows that a lack of evidence supporting RIT in our setup would be a deĄnitive blow for the theory. To ensure that our BB design can be used to study RIT, we analyze a BareBone (BB) principal-agent model.

Our experiment shows that RIT is remarkably robust and more pervasive than predicted by EUT. In line with RDU but in contrast with EUT and MVS, RIT arises even when agents are risk-seeking. This Ąnding has direct implications for various applications of the theory in which agents are risk-seeking agents, as is the case of executive compensation [START_REF] Garen | Executive compensation and principal-agent theory[END_REF][START_REF] Edmans | The effect of risk on the ceo market[END_REF][START_REF] Edmans | Dynamic ceo compensation[END_REF][START_REF] Edmans | Executive compensation: A survey of theory and evidence[END_REF] and highly-paid work packages [START_REF] Ma | Portfolio manager compensation in the us mutual fund industry[END_REF].

Risk-seeking is likely to be pervasive in these applications because of selection arguments [START_REF] Maccrimmon | Characteristics of risk taking executives[END_REF][START_REF] Brenner | The risk preferences of us executives[END_REF]. Furthermore, executive packages are often positively skewed due to, for example, the use of stock options [START_REF] Edmans | Executive compensation: A survey of theory and evidence[END_REF]. As a result, an agent who appears to be risk-averse when rewarded according to linear contracts might appear to be risk-seeking when facing skewed compensation packages.

Introduction

There is now a large body of empirical evidence showing systematic violations of expected utility theory (EUT; see Starmer, 2000, for a review). The original version of prospect theory (OPT; see Kahneman and Tversky, 1979, henceforth KT79) and its subsequent reĄnements, most notably cumulative prospect theory (CPT; see Tversky and Kahneman, 1992, henceforth TK92), explain these empirical violations by introducing probability distortions (Bleichrodt and Pinto, 2000) and loss aversion (Wakker, 2010). Classical methods for eliciting risk attitudes (such as Holt and Laury, 2002) that are based on EUT may lead to biased estimates (Abdellaoui et al. (2011a) and Abdellaoui et al. (2008, Figure 1; henceforth ABL)) and produce incoherent results [START_REF] Bleichrodt | Making descriptive use of prospect theory to improve the prescriptive use of expected utility[END_REF][START_REF] Hershey | Probability versus certainty equivalence methods in utility measurement: Are they equivalent?[END_REF]. New methods have thus been proposed that build on CPT instead of EUT (ABL). These methods fall into three broad categories: parametric (with parametric form of utility and probability weighting functions), semi-parametric (with parametric form of the utility function and parameter-free probability weighting function) and non-parametric (no parametric form for either function).

While all parametric methods rely on a parametric speciĄcation of both utility and probability weighting, they may differ in terms of the required data structures and the employed estimation procedures. The data used by parametric methods usually involve certainty equivalents (e.g. Fehr-Duda et al., 2006;[START_REF] Bruhin | Risk and rationality: Uncovering heterogeneity in probability distortion[END_REF]lŠHaridon and Vieider, 2019, TK92), binary choices (e.g. Harrison andRutström, 2008, 2009;[START_REF] Harrison | Cumulative prospect theory in the laboratory: A reconsideration[END_REF], as well as indifference between non-degenerate lotteries from the so-called tradeoff method (Booij et al., 2010). Then, commonplace estimation procedures range from simple arithmetic calculations [START_REF] Tanaka | Risk and time preferences: Linking experimental and household survey data from vietnam[END_REF] [START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF]Harrison andRutström, 2008, 2009;[START_REF] Harrison | Cumulative prospect theory in the laboratory: A reconsideration[END_REF], nonlinear least squares (NLS)(e.g.TK92, Gonzalez and Wu, 1999, henceforth GW99), OLS [START_REF] Fox | Prospect theory and the brain[END_REF], and Bayesian methods (e.g. Nilsson et al., 2011;[START_REF] Toubia | Dynamic experiments for estimating preferences: An adaptive method of eliciting time and risk parameters[END_REF]Murphy and ten Brincke, 2018;Spiliopoulos and Hertwig, 2019;Baillon et al., 2020;Gao et al., 2020).

Parametric methods are more often used in applied research than nonparametric and semi-parametric methods (ABL) because they have four appealing properties: tractable, data-efficiency, easy and error-robust (see Section 4.2 for deĄnitions), which are associated with minimal data requirements and simple estimation procedures.

These virtues, however, come at a cost. First, empirical estimates are sensitive to the speciĄcation of the utility and probability weighting functions (Abdellaoui, 2000). Second, parametric methods only provide an overall measure of the goodness of Ąt of the model, rather than separate measures for each of its components Ű one for the utility function and one for the probability weighting function (see GW99).

The aim of this paper is to establish a comprehensive semi-parametric method that satisĄes the four desirable properties of parametric methods (see Section 4.2 for deĄnitions). Doing so, we develop a method that increases the precision of parameter estimates of parametric methods while being easy to implement and estimate. Our method also comes with an additional advantage because it can be implemented not only under CPT but also under OPT and alternative theories such as rank dependent utility theory (henceforth RDU; see Quiggin, 1982;Gilboa, 1987;Schmeidler, 1989) This paper proceeds as follows. Section 4.2 outlines the existing semiparametric and non-parametric methods. Section 4.3 presents our elicitation method under cumulative prospect theory. Section 1.4 focuses on identiĄcation

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components and misspeciĄcation. Section 4.4 provides two applications of the method.

Section 3.7 concludes.

Existing semi-parametric and non-parametric methods

In this section, we review the existing semi-and non-parametric methods of eliciting risk attitudes under CPT. Table 1.1 provides an overview of these methods along the four desirable properties of parametric methods (tractability, data-efficiency, easiness, error-robustness) and comprehensiveness. We deĄne these properties below.

Tractable: the method allows for estimating risk attitudes with standard tools, like OLS or NLS (Abdellaoui et al., 2007a, ABL).

Error-robust: the method accounts for the fact that subjects make response errors when answering questions (ABL).

Easy (not cognitively demanding): the method relies on simple choices involving the lowest possible number of outcomes, that is 3 (ABL). A useful benchmark comes from TK92 who ask their subjects to make simple choices between a sure outcome and binary lottery.

Data-efficient: the method requires few measurements (observations) to estimate the parameters of the utility function and the probability weights. As a rule of thumb, we take TK92 (with 28 measurements for 9 probability weights and one utility parameter per domain) as benchmark, and consider a method as well as under uncertainty (i.e. unknown probability).

The existing methods typically use the tradeoff approach of Wakker and Deneffe (1996) in three steps. 2 In the Ąrst step, the tradeoff method estimates the utility function by eliciting a sequence of outcomes (x 0 , x 1 , x 2 , ..., x n ) which are equally spaced in terms of utility: u(x i )-u(x i-1 ) = constant for i = 1, 2, ..., n. In the second step, the utility function obtained in the Ąrst step is used to estimate the weighting function. If the utility function in the gain and loss domains is elicited on the same scale (Abdellaoui et al., 2007b(Abdellaoui et al., , 2016)), the loss aversion can then be inferred in a third step. The tradeoff method is not error-robust because it assumes that the Ąrst-step values (x 1 , x 2 , ..., x n ) are elicited without errors. This assumption is particularly restrictive because the elicitation of these values is subject to error propagation: any error in the Ąrst-stage elicitation of a given value (x i ) affects the subsequent estimates of values (x i+1 , ..., x n ). 3 It 1 ABL use the term "efficient" instead. Since this property only relates to the physical resources (such as time and money) used for data collection, we coin the term data-efficiency to avoid confusion with the (unrelated) statistical notion of the efficiency of an estimator.

2 See, e.g. Abdellaoui (2000); Bleichrodt and Pinto (2000); Etchart-Vincent (2004); Abdellaoui et al. (2007b);Etchart-Vincent (2009a); Booij and Van de Kuilen (2009); Van De Kuilen and Wakker (2011); Abdellaoui et al. (2016); Attema et al. (2018); Bleichrodt et al. (2018); Blavatskyy (2021). The main reason of the popularity of the tradeoff method is that the elicitation of the utility function is robust to probability distortions.

3 Another issue with the standard tradeoff method is strategic responding (Harrison and Rutström, 2008;Abdellaoui et al., 2020). For the sake of illustration, suppose that the researcher is looking for outcomes x 1 and x 2 by eliciting a pair of chained indifference values x 1 and x 2 such that (x 1 , $1; 0.5, 0.5) ∼ ($10, $5; 0.5, 0.5) and (x 2 , $1; 0.5, 0.5) ∼ (x 1 , $5; 0.5, 0.5). One of these four lotteries is picked at random for payoff at the end of the experiment. Hence, the decision-making problem boils down to calibrating the following lottery: R(x 1 , x 2 ) = ($1, $5, $10, x 1 , x 2 ; 1/4, 1/4, 1/8, 1/4, 1/8). This, in turn, provides incentives to overstate the values of x 1 and x 2 , since x * 1 > x 1 and x * 2 > x 2 yields a lottery R(x The literature proposes three notable alternatives to the tradeoff method (GW99; ABL; Abdellaoui et al., 2011c).5 However, GW99 is not a comprehensive method because it does not elicit loss aversion. Also, Van De Kuilen and Wakker (2011) point to the lack of tractability and efficiency of this method. Furthermore, their method tends to produce an extremely concave (resp. convex) utility function in the gain (resp. loss) domain (see footnote 13).

The semi-parametric method of ABL, in turn, satisĄes the four appealing properties of parametric methods, and provides information on the goodness of Ąt of the functional form chosen for estimating the utility function. Yet, ABL is not a comprehensive method because it cannot estimate the weighting function.

Achieving comprehensiveness by including an additional step to estimate the weighting function, as in Abdellaoui et al. (2011c), comes at the cost of potentially multiplying response errors (Etchart-Vincent, 2004, pp. 221).

In addition, the approach in ABL and Abdellaoui et al. (2011c) function separately in the gain and loss domains makes it impossible to impose partial reĆection (i.e., identical utility curvature in both domains) which is often required to circumvent the arbitrary measurement of loss aversion (see Wakker, 2010, pp. 267-270).6 By allowing for a joint estimation of the utility function in both domains, our method allows for testing and imposing partial refection whenever needed. 7 The second problem, as pointed out by Wakker andDeneffe (1996, pp. 1148), comes from the fact that the elicitation of loss aversion in ABL and Abdellaoui et al. (2011c) is based on asking subjects to provide a loss amount L on an unbounded interval (-∞, 0]. This procedure is more cognitively demanding than stating L on a bounded interval (Abdellaoui et al., 2007b).

Then, not knowing the lowest possible value of the loss amount L could lead to large response errors that potentially inĆate the estimates of loss aversion, as reported in ABL (see Table 11,.

CP T (L) = v(x) -v(y) w i (p) + v(y) (1.1) CP T (L) = w + (p)v(x) + w -(1 -p)v(y) (1.2)
where w i and v are both continuous, strictly increasing and satisfying v(0) = 0, w i (0) = 0 and w i (1) = 1, and i =Ş+Ť (i =Ş-Ť) stands for the gain (loss) domain.9

Following the seminal study by TK92, as well as the subsequent developments in Köbberling and Wakker (2005) and ABL, we assume that the value function v(.) is composed of the loss aversion index λ > 0 which reĆects the exchange rate between gain and loss utility units, and the utility function u(.) that reĆects the intrinsic value of outcomes:

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

v(x) =          u(x) if x ≥ 0 λu(x) if x < 0 (1.3)
However, without further assumptions, loss aversion (λ) as deĄned in (1.3)

is not empirically identiĄable. Indeed, we can rescale the utility function u(.) in the gain and loss domains with two different linear transformations u * (x) = ρu(x) for x ≥ 0 and u * (x) = τ u(x) for x < 0, so that we have a linear transformation v * (x) = ρv(x) for the value function by deĄning λ * = λ ρ τ (Wakker, 2010, p. 248):

v * (x) =          u * (x) if x ≥ 0 λ * u * (x) if x < 0 with λ * = λ ρ τ
As a result, as long as ρ ̸ = τ , we have two different values for the loss aversion index (λ and λ * ) that represent the same underlying preferences. However, one can avoid this arbitrary measurement of loss aversion using utility functions that are differentiable at 0 with u ′ (0) ̸ = 0 (e.g. exponential utility in equation (3.17)). In the case of power utility functions (see equation (3.16)), the issue of identiĄcation of loss aversion is present unless partial reflection (i.e., identical utility curvature in the full domain) is imposed.

Elicitation

We start by considering two standard utility functions that have been previously

shown to provide a good Ąt to experimental data: the power utility function (see, e.g. GW99; [START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF] and the exponential utility function (see, e.g. Attema et al., 2013).10 Next, we detail the steps for estimating the probability weighting function, the curvature of the utility function, and the loss aversion index.

Utility function

As in Booij et al. (2010), we use the following notation for the power (3.16) and exponential (3.17) utility functions:

u(x) = (1 (x≥0) -1 (x<0) ) ♣x♣ αp1 (x≥0) +βp1 (x<0) (1.4) u(x) = (1 (x≥0) -1 (x<0) ) 1 -exp (β e 1 (x<0) -α e 1 (x≥0) )x α e 1 (x≥0) + β e 1 (x<0) (1.5)
where 1 (.) refers to the indicator function. The important properties of these functions are related to domain-speciĄc curvature, loss aversion, and partial reĆection, and can be summarized as follows. For gains (losses), the power function in (3.16) is concave if α p < 1 (β p > 1), linear if α p = 1 (β p = 1), and convex if α p > 1 (β p < 1). For gains (losses), the exponential function in (3.17) is concave if α e > 0 (β e < 0), linear if α e -→ 0 (β e -→ 0), and convex if α e < 0

(β e > 0). Furthermore, the two functions imply two different deĄnitions of loss aversion. For (3.16), the loss aversion index is λ = -v(-$1) v($1) which corresponds to the standard deĄnition in TK92. For (3.17), the loss aversion index is given by the deĄnition in Köbberling and Wakker (2005) In addition, we use the following method for curvature comparisons of different utility functions for a given interval, such as [0, x] in the gain domain or [x, 0] in the loss domain. We compute the following measure of utility curvature (see also Abdellaoui et al., 2016):

, that is λ = v ′ ↑ (0) v ′ ↓ (0) with v ′ ↑ (
α = 1 xu(x) x 0 u(t)dt if x ≥ 0 (1.6) β = 1 xu(x) 0 x u(t)dt if x < 0 (1.7)
For power (exponential) utility function, this yields α = 1 1+αp and β = -

1 1+βp ( α = 1 1-exp(-αex) -1 αex and β = 1 exp(βex)-1 -1 βex .
) Then, α > 0.5 / α = 0.5 / α < 0.5 correspond to concave / linear / convex utility functions in the gain domain. In the loss domain we have β > -0.5 / β = -0.5 / β < -0.5 that correspond to concave / linear / convex utility functions.

Estimating probability weighting functions and utility curvature

The Ąrst step of our estimation procedure consists of three parts. First, we select the set of probabilities ¶p k : k = 1, 2, ..., K♢ for which weights are estimated in the gain and loss domains, with p k < p k+1 . For any p k , its complement 1p k must also be included in the set of probabilities, so that p K-k+1 = 1p k for k = 1, 2, ..., K. Then, in a given domain, one elicits (at least) two certainty equivalents for each probability p k :

ce i j,k ∼ (x i j,k , y i j,k ; p k , 1 -p k ) , j = 1, 2, ..., N i k and N i k ≥ 2 (1.8)
where N i k stands for the number of certainty equivalents for p k in domain i ∈ ¶Ş+Ť;Ş-Ť ♢, and x i j,k and y i j,k are outcomes such that x + j,k > y + j,k ≥ 0 and x - j,k < y - j,k ≤ 0.12 Thus, in total one needs to elicit N + = K k=1 N + k ≥ 2×K certainty equivalents in the gain domain and N -= K k=1 N - k ≥ 2 × K certainty equivalents in the loss domain. For invertible u and using (2.1) and (1.3), these certainty equivalents satisfy the following condition:

ce i j,k = u -1 u(x i j,k ) -u(y i j,k ) w i (p k ) + u(y i j,k ) (1.9)
Let ce, x and y be column vectors containing all the realizations of ce i j,k , x i j,k and y i j,k , respectively. Any column vector z ∈ ¶ce, x, y♢ is constructed as follows:

      z + 1 z + 2 . . . z + K z - 1 z - 2 .
.

As in the literature (e.g. ABL; [START_REF] Hey | Noise and bias in eliciting preferences[END_REF]Bruhin et al., 2010, henceforth BFE10), we assume that certainty equivalents are observed with additive response error with mean 0. Thus, the empirical counterpart of (2.9) is given by: 13

13 An alternative approach would be to introduce the response error term at the utility level (GW99, eq. 7):

u(ce l ) = (u(x l ) -u(y l )) ×   K k=1 (δ + k D + l + δ - k D - l )I k l   + u(y l ) + e i l .
However, defining the response error at the utility level is problematic when using certainty equivalents data because it produces solutions that are characterized by unrealistic concavity of the utility and probability weighting functions. To illustrate this point, suppose that we are interested in eliciting utility only over strictly positive outcomes with a power utility function u(z) = z α . For an extremely concave utility function (i.e., α > 0 and α -→ 0) and an extremely concave weighting function (i.e., δ + k = 1 for k = 1, 2, .., K ) along with the PT assumptions w + (0) = 0 and w + (1) = 1, we have e + l = 0 for all l = 1, 2, ..., N + . For the non-parametric method of GW99 which aims at estimating u(z) for z ∈ A(z) ≡ ¶$25, $50, $75, $100, $150, $200, $400, $800♢ and the probability weights w(p) for p ∈ B(p) ≡ ¶0. 01, 0.05, 0.10, 0.25, 0.40, 0.50, 0.60, 0.75, 0.90, 0.95, 0.99♢, it follows that an extremely concave utility function (i.e., u(z) = constant > 0 for z ∈ A(z) and u(0) = 0) and an extremely concave weighting function (i.e., w(p) = 1 for p ∈ B(p) and w(0) = 1w(p) = 0) are solutions of the Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

ce l = u -1   (u(x l ) -u(y l )) ×   K k=1 (δ + k D + l + δ - k D - l )I k l   + u(y l )    + e i l (1.10)
where I k is a dummy variable set to 1 if the probability equals p k and 0 otherwise, D + (D -) is a dummy variable set to 1 for a positive (negative) certainty equivalent and 0 otherwise, e i the response error term and l is the l th line in ce, x, y, I k and e i . Probability weights correspond to: 14 15

w + (p k ) = δ + k and w -(p k ) = δ - k for k = 1, ..., K (1.11)
Note that u -1 in (1.10) should be written in the full domain. To do that, one can Ąrst write u in the full domain using indicator functions as in (3.16) and

(3.17), and u -1 can be derived using standard algebra. Finally, the parameters in (1.10) can be estimated as long as one provides a functional form for u (and thus for u -1 ), such as (3.16) or (3.17). 16 The estimation of equation (1.10) can be done with either NLS or MLE. Under standard assumptions of normally distributed error terms and homoscedasticity, optimization problem. A similar issue can be found in Section 4.3 of Green and Silverman (1993).

14 Take a probability p s for s ∈ 1, 2, ..., K. I s l equals 1 for any observation l that involves p s , and all the other probability dummy variables I c l for c ̸ = s are set to 0. In that case, we have

K k=1 (δ + k D + l + δ - k D - l )I k j = δ + s D + l + δ - s D - l .
In the gain domain, the dummy variable D + l equals 1 while D - l equals 0 so that we get

K k=1 (δ + k D + l + δ - k D - l )I k j = δ + s D + l + δ - s D - l = δ + s . Hence, w + (p k ) = δ + k in (1.10) for k = 1, 2, ..., K. Analogously, w -(p k ) = δ - k in (1.10) for k = 1, 2, ..., K.
15 Note that we do not require monotonicity of the weighting function as is commonly done in the literature (see e.g. GW99, p. 147). Requiring monotonicity in our method can be achieved by adding the following restriction on weights:

δ i k = δ i 1 + k j=2 exp(a i j ) for k ≥ 2.
In this case, the estimated parameters are δ i 1 , a i 2 , a i 3 , ..., a i K . 16 One can choose the functional form that best performs in terms of goodness of fit (e.g. [START_REF] Hey | Investigating generalizations of expected utility theory using experimental data[END_REF]Fehr-Duda et al., 2006;[START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF].

both methods provide identical point estimates for risk-attitude components, with MLE being a more efficient method (see, e.g. Wooldridge, 2010, p. 470).17

In addition, two sources of heteroscedasticity can be present at the level of individual data: the variance of the error term may vary (i) with respect to the range ♣x ly l ♣ of a lottery, but also (ii) across domains i = +, -(gains vs. losses). Both of them can be accounted for by MLE (lŠHaridon and Vieider, 2019, BFE10). Herein, we adopt a more general form of heteroscedasticity than these two studies by assuming that σ i,l = σ i ♣x ly l ♣ ψ with ψ ≥ 0 where ψ ̸ = 0 implies there is heteroscedasticity due to the range of outcomes. 18Our method allows us to account for these various sources of heteroscedasticity by applying MLE to equation (1.10). The log-likelihood function is:

logL(θ i , δ i , σ i , ψ) = -(N + + N -)log √ 2π - N + +N - l=1 log(σ i ♣x l -y l ♣ ψ ) - 1 2 N + +N - l=1   e i l σ i ♣x l -y l ♣ ψ   2 (1.12)
where θ i stands for the parameters associated with the utility function in domain i. Maximizing the log-likelihood with respect to all the parameters provides a simultaneous estimation of the utility function and probability weights for the gain and loss domains.

We use two criteria to assess the achieved goodness of Ąt: Akaike Information Criterion (AIC) and leave-one-out Cross Validation (CV). AIC is a standard measure in the literature (see, e.g. Fehr-Duda et al., 2006;[START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF][START_REF] Hey | Investigating generalizations of expected utility theory using experimental data[END_REF] and is given as AIC = 2n p -2logL, where logL is the log- The (absolute) difference between the predicted and actual certainty equivalents is the (absolute) prediction error for that lottery. This out-of-sample prediction procedure is repeated over N + + N -steps such that each lottery is left out of the sample once. Then, the value of the criterion is the mean absolute prediction error.

The certainty equivalent in equation (1.10) is given by equations (1.13) and

(1.14) for power and exponential utility functions, respectively

ce l =(D + l -D - l )   ♣x l ♣ αpD + l +βpD - l -♣y l ♣ αpD + l +βpD - l ×   K k=1 (δ + k D + l + δ - k D - l )I k l   + ♣y l ♣ αpD + l +βpD - l   1 αpD + l +βpD - l + e i l (
ce l =ln   exp((β e D - l -α e D + l )x l ) -exp((β e D - l -α e D + l )y l ) ×   K k=1 (δ + k D + l + δ - k D - l )I k l   + exp((β e D - l -α e D + l )y l )   × 1 β e D - l -α e D + l + e i l
(1.14)

Estimating loss aversion

As a second step, we measure the loss aversion index λ as deĄned in (1.3) based on the estimates of the utility function and the probability weights outlined in subsection 1.3.2.2. Following Abdellaoui et al. (2007b), the estimation of the loss aversion index can be done using a set of K indifference relationships that involve mixed binary prospects:

ce k ∼ (x k , y k ; p k , 1 -p k ) , k = 1, 2, ..., K (1.15)
where y k < 0 < x k . Under CPT, these indifferences imply that:

ce k = v -1 w + (p k )v(x k ) + w -(1 -p k )v(y k ) (1.16)
Because ce k belongs to the interval (y k , x k ), it could either be a gain or a loss. Also, note that for each k both w + (p k ) and w -(1p k ) are known since they have been elicited in the previous step. Echoing our previous notation, let ce, x and y contain the realizations of ce k , x k and y k . In addition, denote by δ + and δ -the column vectors such that δ

+ ′ ≡ (δ + 1 , δ + 2 , ..., δ + K ) and δ -′ ≡ (δ - 1 , δ - 2 , ..., δ - K ) = (δ - K , δ - K-1 , ...,
ce k = v -1 δ + k v(x k ) + δ - k v(y k ) + e k (1.17)
v -1 in (1.17) can be derived similarly to u -1 in (1.10). Then, the respective certainty equivalent equations for power and exponential utility functions become:

ce k = (D + k -D - k )   δ + k (x k ) αp -λδ - k (-y k ) βp D + k -λD + k   1 αpD + k +βpD - k + e k (1.18) ce k = ln 1 - αeD + k +βeD - k D + k -λD - k δ + k 1-exp(-αex k ) αe -λδ - k 1-exp(βey k ) βe β e D - k -α e D + k + e k (1.19)
Using the values of the probability weights and the parameters of u(.) from the Ąrst step, we can estimate (1.18) or (1.19) by NLS or MLE to obtain λ.

Key properties of our method

Comparison with ABL. We reĄne and extend the method previously proposed by ABL in several ways.

First, unlike ABL, we can estimate multiple probability weights and thus elicit the shape of the probability weighting function.

Second, our method uses a single step to estimate the probability weights and the utility function in the full domain, whereas ABL propose a two-step procedure. This feature of our method allows for testing several important restrictions (partial reĆection, identical probability weighting functions across domains, and duality) as well as imposing these restrictions whenever necessary.

Imposing partial reĆection helps avoid the problem of arbitrary measurement of loss aversion with power utility functions (see Wakker, 2010). Testing for identical probability weighting functions across domains (i.e., w

+ (p k ) = w -(p k )
for all k) allows us to test a key assumption of OPT. In addition, this assumption must also be made under CPT whenever loss aversion is present and preferences are homogeneous (Al-Nowaihi et al., 2008). 20 Our method allows for testing and imposing duality (i.e., w 21 By allowing for testing and imposing duality as well as identical probability weighting across domains, our method can be applied under RDU (Quiggin, 1982;Gilboa, 1987;Schmeidler, 1989) and OPT (KT79). This is not the case for existing parametric, semi-parametric, or non-parametric methods.

+ (p k ) = 1 -w -(1 -p k ) for all k).
Third, certainty equivalents for mixed prospects are obtained using a different procedure than the one proposed by ABL. In ABL, subjects are asked to provide a loss amount L for which they are indifferent between the status-quo (0) and a binary lottery (G, L; p g , 1p g ) where G is a Ąxed gain and L ∈ (-∞, 0] is a loss.

In this elicitation procedure, the researcher does not know the lower bound of the loss interval. By contrast, equation (1.15) keeps track of the upper and lower bounds of the loss interval because ce k belongs to the interval (y k , x k ). This is an appealing property of our method for two reasons. First, asking subjects to provide indifference values on unbounded intervals can be cognitively demanding (Wakker and Deneffe, 1996;Abdellaoui et al., 2007b). This may lead to errors 20 Homogeneity of preferences holds whenever multiplying all the payoffs of a non-mixed lottery by a positive constant c also leads the certainty equivalent of the lottery to be multiplied by c. 21 As pointed out by Abdellaoui (2000Abdellaoui ( , pp. 1509Abdellaoui ( -1510)), testing for duality with parametric methods and non-parametric methods based on the tradeoff approach requires using the specific probability weighting function of Goldstein and Einhorn (1987) and [START_REF] Lattimore | The inĆuence of probability on risky choice: A parametric examination[END_REF].

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components that potentially inĆate the estimates of loss aversion, as reported by ABL (see Table 11,. Second, eliciting indifference values on bounded intervals allows us to use a standard switching outcome procedure (Booij and Van de Kuilen, 2009).

Comparison to standard parametric methods. Our method retains all the appealing properties of parametric methods. First, it is as data-efficient as parametric methods because, for K probability weights to be elicited, the smallest number of certainty equivalents required to measure all the three components of risk attitudes in the full domain is 5K (i.e., 2K certainty equivalents in the gain domain, 2K in the loss domain and K for mixed lotteries). Second, we use simple choices (comparisons of certain outcomes and binary lotteries) so that the method is not cognitively demanding for subjects. Third, our method is tractable because we can measure risk attitudes using standard econometric tools. Fourth, our estimation method accounts for response errors.

In parametric methods, it is key to assess the validity of the functional forms used for the probability weighting and utility functions. However, parametric methods do not allow the researcher to separately assess the goodness of Ąt of each of these functions (GW99). In contrast, our semi-parametric method does not make any parametric assumption regarding the probability weighting function and allows the researcher to evaluate the goodness of Ąt of the utility function separately. Thus, one can select the utility function with the best Ąt, further improving the accuracy of the elicitation of risk attitudes of the semi-parametric method compared to the parametric one.

Applicability to unknown probabilities. Our method is also directly applicable to cases of uncertainty where probabilities are unknown. It does not require setting any speciĄc conditions on the event space, and hence can be Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components applied to real-life uncertainty situations (Baillon et al., 2018). 22 Extending our method to the case of uncertainty can thus be done by replacing probability dummy variables by event dummy variables in equations (1.10) and (1.17).

Robustness to monotonicity problem. We further note that the monotonicity problem raised by [START_REF] Apesteguia | Monotone stochastic choice models: The case of risk and time preferences[END_REF], which could lead to identiĄcation issues, does not apply to our case. First, our method is not based on binary choices (as in random utility models), but on eliciting certainty equivalents. Second, we deĄne the error term at the certainty equivalent level and not at the utility level. For power and exponential utility functions, the certainty equivalent is monotonic in the utility parameters and probability weights. This is also true for a broad range of utility functions once we use the Arrow-Pratt approximation for binary lotteries in the context of RDU and CPT. A related problem to [START_REF] Apesteguia | Monotone stochastic choice models: The case of risk and time preferences[END_REF] can arise with the certainty equivalent method when the error term is deĄned at the utility level, as in the non-parametric method of GW99 (see footnote 13 for a discussion of these issues).

Spline extension. Our semi-parametric method requires specifying a utility function which allows us to keep the data-efficiency property of parametric methods. However, in the case of an extensive dataset (as in GW99), our semi-parametric method can be extended to use spline approximation of the utility function. In appendix 1.C, we provide a linear spline extension for our semi-parametric method.

22 Non-parametric methods (e.g. Van De Kuilen and Wakker, 2011;Abdellaoui et al., 2020) require the event space to be rich (e.g. continuous). This means that in applied contexts, the universal event is an interval on a real scale (e.g. temperature in town). The semi-parametric method of Abdellaoui et al. (2011a) requires equally likely events either with (i) preset priors (like in Ellsberg's urn experiments), or (ii) a rich event space analogous to non-parametric methods.

Parameter recovery and misspecification

Following Gao et al. (2020), Nilsson et al. (2011) and Murphy and ten Brincke (2018), we report in this section two types of simulation exercises: parameter recovery and robustness to misspeciĄcation. In the parameter recovery exercise, we estimate a model assuming that we know the speciĄcation of the utility and weighting functions used to simulate the data. Our aim is to assess the extent to which an estimation method can identify the parameters underlying the simulated data. In the robustness to misspeciĄcation exercise, we estimate a model assuming an incorrect speciĄcation of the utility and weighting functions and check for the extent to which an estimation method identiĄes the underlying parameters from simulated data. For the sake of comparison, the simulations are made for both our semi-parametric method and parametric methods. Subsections 1.4.1 and 1.4.2 explain the simulation exercises and results are presented in subsection 1.4.3.

Parameter recovery

The calibration of lotteries follows TK92. We consider 9 probabilities Ű 0.01, 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, and 0.99 Ű along with possible outcomes (x, y) in each domain: (0, 100), (0, 400) and (50, 150) for the gain domain, and (-100, 0), (-400, 0) and (-150, -50) for the loss domain. Each of the 9 probabilities is combined with each pair of outcomes, resulting in a lottery (x, y; p, 1p).

This gives a total of 27 lotteries in each domain, and each probability occurs three times in the dataset.

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

Our method

Simulation 1: Power utility function. We simulate data for 1000 (s = 1, 2, ..., 1000) hypothetical subjects. For each subject s, we generate random parameters of the power utility functions in the gain and loss domains from a uniform distribution U (0.1, 2.1) (Spiliopoulos and Hertwig, 2019, ABL).

For the 9 probabilities involved in the lotteries, we generate probability weights using a uniform distribution U (0, 1), making sure that higher probabilities are assigned to higher weights.

Then, these simulated power utility parameters (one per domain) and probability weights (9 per domain) are plugged into CPT formulas to generate noiseless certainty equivalents. In the last step of the data generation process, we use two random variables from U (0, 0.025) to simulate standard deviations σ s,i in each domain i ∈ ¶+, -♢, and then draw 27 random values from N (0, σ s,i ♣xy♣)

which we add to the previously generated 27 noiseless certainty equivalents to obtain the noisy ones. 23 Finally, we use the noisy certainty equivalents as input data and compute MLE outcomes for our semi-parametric method.

Simulation 2: Exponential utility function. This simulation exercise is based on the same principles as Simulation 1, the sole difference being the utility function. This time, we draw exponential utility parameters for each domain from U (-0.01, 0.01).24

Parametric methods

We consider eight common parametric speciĄcations. These parametric speciĄcations arise from the combination of the two standard utility functions (power and exponential) (see [START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF] and four popular weighting functions (see TK92; Goldstein and Einhorn, 1987;[START_REF] Lattimore | The inĆuence of probability on risky choice: A parametric examination[END_REF]Prelec, 1998). These weighting functions are:

w i (p) = p (a1 (i=+) +c1 (i=-) ) p (a1 (i=+) +c1 (i=-) ) + (1 -p) (a1 (i=+) +c1 (i=-) ) 1 a1 (i=+) +c1 (i=-) (1.20)
where a, c ∈ (0, 1] is the two-parameter weighting function of Goldstein and Einhorn (1987) and [START_REF] Lattimore | The inĆuence of probability on risky choice: A parametric examination[END_REF]. The vast majority of parametric estimations in the literature rely on one of these four weighting functions in combination with a standard utility function [START_REF] Stott | Cumulative prospect theoryŠs functional menagerie[END_REF]. Henceforth, we refer to (1.20), (1.21),

w i (p) = exp --ln(p) b1 (i=+) +d1 (i=-) (1.21) where b > 0, d > 0 w i (p) = exp -b1 (i=+) + d1 (i=-) × -ln(p) a1 (i=+) +c1 (i=-) (1.22) where a > 0, b > 0, c > 0, d > 0 w i (p) = (b1 (i=+) + d1 (i=-) )p (a1 (i=+) +c1 (i=-) ) (b1 (i=+) + d1 (i=-) )p (a1 (i=+) +c1 (i=-) ) + (1 -p) (a1 (i=+) +c1 (i=-) ) (1.23) where a > 0, b > 0, c > 0, d > 0.
Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

(1.22) and (1.23) as TK92, P98-I, P98-II and GE87 respectively.25

Simulation 3 and 4. We then run two additional series of simulations (Simulations 3 and 4) that are based on the same principles as Simulations 1 and2 (respectively), the sole difference being the weighting function. Depending on the simulation, we specify one of the four parametric weighting functions. For the weighting function of TK92, we draw the values for a and c from U (0.2, 1) (e.g. Dhami, 2016, p. 122). For the remaining weighting functions, we draw a, b, c and d from U (0.1, 1.5).

Parameter recovery under model misspecification 1.4.2.1 Our method

Simulation 5: Power utility function. We consider the data generated in Simulation 4 under an exponential utility function in combination with each of the four weighting functions in turn. We apply our estimation procedure to each of these four simulated datasets, while misspecifying the utility function which is assumed to be power instead of exponential.

Simulation 6: Exponential utility function. We follow the same procedure as in Simulation 5, this time using the data generated in Simulation 3 with power utility function. For the sake of model misspeciĄcation, we assume utility to be exponential.

Parametric methods

Simulation 7 and 8. As before, we run two additional series of simulations for parametric methods. In Simulation 7 (8), we rely on data previously generated

Result of parameter recovery and misspecifcation

Table 1.2 shows the average absolute error in parameter estimates, which is deĄned as the absolute difference between the true parameter value and its estimate. Figure 1.1 plots the average absolute error in parameter estimates across all parameters and all simulations. When a point is above the 45 • line, this means that the average absolute error in the parametric speciĄcation is greater than the corresponding simulation in the semi-parametric speciĄcation.

For parameter recovery exercises (Figure 1.1, panel (a)), parametric speciĄcations tend to perform better than the semi-parametric method. However, for misspeciĄcation exercises, the semi-parametric method performs better than parametric methods (Figure 1.1, panel (b)). Taking both types of simulations into account (Ągure 1.1, panel (c)), we highlight that average absolute errors in parameter recovery exercises are substantially smaller than misspeciĄcation errors. Overall, we conclude that the semi-parametric method (i) is less sensitive to model misspeciĄcation and (ii) produces more reliable estimates when the model is misspeciĄed than standard parametric methods.

Applications

In this section we use existing experimental data to compare our semi-parametric method and parametric methods. Because our method relies on the elicitation of certainty equivalents, we compare it to parametric methods that also make use of certainty equivalents. Since our exercise requires the use of datasets that allow for the elicitation of certainty equivalents, we start this section by detailing our choice of datasets before comparing the methods in terms of goodness of Ąt and providing estimation results.

Data

To apply our method, we need to elicit certainty equivalents in the gain and loss domains for two-outcome lotteries. These lotteries should vary each of the outcomes as well as the corresponding probabilities, and at least two certainty equivalents should be elicited for each probability.

We made an extensive search of the literature to identify available datasets from which we could make individual estimates. We reviewed the datasets fully matching our selection criteria. However, this dataset does not include mixed lotteries, thus not allowing us to estimate loss aversion. In addition, TK92

provide median data that contain several certainty equivalents per probability in the gain and loss domains, also in line with our selection criteria. They also include mixed lotteries so that we can elicit loss aversion. We thus apply our method to the median data of TK92 and to the individual data of BFE10.

Goodness of fit across models

We start by evaluating the goodness of Ąt of our method relative to parametric alternatives. The corresponding values of AIC and CV are reported in Tables 1.3 and1.4.

For the data of TK92 (BFE10), power (exponential) utility function best

Ąts the data under our semi-parametric method according to both AIC and CV. The best parametric speciĄcation is the combination of a power utility function and the one-parameter weighting function of TK92 (an exponential utility function and the two-parameter weighting function of GE87) for the data of TK92 (BFE10).

For each dataset and each criterion, the best-Ątting speciĄcation under our semi-parametric method outperforms the best-Ątting parametric speciĄcation.

This implies that our semi-parametric method Ąts the data better than standard parametric methods. 1998)),P98-II (two-parameter weighting function of Prelec (1998), LG92 (two-paramete weighting function of [START_REF] Lattimore | The inĆuence of probability on risky choice: A parametric examination[END_REF])) ‡ The semi-parametric method with exponential utility function provides smaller AIC and CV

AIC * CV * AIC CV u(.) w(.) TK92 † P98-I † P98-II † LG92 † TK92 † P98-I † P98-II † LG92 † - -
AIC * CV * AIC CV u(.) w(.) TK92 † P98-I † P98-II † LG92 † TK92 † P98-I † P98-II † LG92 † - -

Results

In this section, we focus our analysis on the best-Ątting speciĄcations under parametric and semi-parametric methods, as highlighted in the previous section.

Curvature of the utility function for gains and losses

Table 1.5 summarizes the semi-parametric and parametric estimates of the main components of the CPT value function (curvature in each domain and loss aversion) for the data of TK92 and BFE10. 27 In the remainder of the results section, we use z-tests to assess whether a coefficient is equal to a speciĄc value and whether two coefficients are equal, and χ 2 -tests for joint hypotheses. Tests are two-sided, unless stated otherwise. Furthermore, we reject partial reĆection (H 0 : α + β = 0, pvalues < 0.0058).

Loss aversion

With our semi-parametric method, we replicate the standard Ąnding of loss aversion, with λ = 1.751. 28 Our estimate of the loss aversion index is close to the estimated value of 1.6 that was elicited in both Booij et al. (2010) who use 27 The utility curvature is computed based on equations (2.14) and (2.19). Detailed results for the semi-parametric method are reported in Appendix 1.A.2 for TK92 and in Appendix 1.B for BFE10.

28 In section 4.3, we propose to estimate loss aversion λ in a second step after estimating utility and weighting functions in a first step. This two-step procedure we propose can be applied regardless of whether elicitation of loss aversion is of interest (which requires both steps) or not (which only requires the first step of estimation). In the former case, one could alternatively apply a one-step procedure (e.g. l'Haridon and Vieider, 2019) that simultaneously estimates equations (1.10) and (1.17). Table 1.A.11 in appendix 1.A.4.1 summarizes the maximum likelihood estimates of utility curvature, probability weights and loss aversion obtained through a one-step procedure. In the one-step procedure, estimated loss aversion is 1.688 compared to the median value of 1.751 in the two-step procedure.

structural estimation techniques, and ABL for pooled data. It is also similar to the estimates reported by [START_REF] Tom | The neural basis of loss aversion in decision-making under risk[END_REF], Pennings and Smidts (2003) , andBooij andVan de Kuilen (2009): λ = 1.93, λ = 1.8, andλ = 1.87, respectively. Note that a large meta-analytical study by Brown et al. (2021) suggests that the mean loss aversion coefficient lies between 1.8 and 2.1. From that perspective, our estimates are on the conservative side and come close to the lower bound of that interval. and then shifting to underweighting for p ∈ (0.1, 1). We cannot reject H 0 :

w -(0.1) = 0.1 (p -value = 0.7042).
Over the 9 probabilities in the data of TK92, the hypothesis of identical probability weights across domains (w The hypothesis of duality (w In the gain domain, probabilistic risk neutrality w + (p) = p is rejected for all probabilities (all pvalues < 0.0001), except for 0.5 (pvalue = 0.9496).

+ (p k ) = w -(p k ))
+ (p k ) = 1 -w -(1 -p k ))
Overall, we reject the hypothesis of linearity of the probability weighting function over the whole range of probabilities in the gain domain. The resulting weighting function is once again inverse S-shaped with overweighting for p ∈ (0, 0.5] and underweighting for p ∈ (0.5, 1).

Similar patterns emerge in the loss domain with overweighting for p ∈ (0, 0.5],

and underweighting for p ∈ (0.5, 1). We cannot reject H 0 : w -(0.5) = 0.5 (pvalue = 0.8881).

Over the 7 probabilities in the data of BFE10, the hypothesis of identical probability weights across domains (w

+ (p k ) = w -(p k ))
is rejected for the probability p = 0.05 (p-values < 0.0204), but not for others (all p-values > 0.0553).

For tail probability (p = 0.05 in this dataset), overweighting is more pronounced for gains that for losses. Using a joint test, the hypothesis of identical probability weights across domains is rejected (pvalue < 0.0001).

Finally, the hypothesis of duality (w

+ (p k ) = 1 -w -(1 -p k )) is rejected
for p = 0.9 (pvalue < 0.0146) but not for the remaining probabilities (all pvalues > 0.1159). A joint test rejects the duality hypothesis (pvalue < 0.0467).

Discussion

Table 1.6 summarizes the discussion for both datasets. Echoing the seminal Ąndings in TK92,30 the parametric estimates reported in Table 1.5 imply concavity in the gain domain (α = 0.544, statistically different from 0.5 with pvalue < 0.0001) and convexity in the loss domain (β = -0.525, statistically different from -0.5 with p-value = 0.001). In addition, partial reĆection cannot be rejected at the 5% signiĄcance level (H 0 : α+β = 0, p-value = 0.0936). When considering the data of BFE10, the parametric estimates imply concavity in both gain (α = 0.589, statistically different from 0.5 with pvalue < 0.0001) and loss

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components (β = -0.428, statistically different from -0.5 with pvalue < 0.0001) domains.

In addition, partial reĆection is rejected (H 0 : α + β = 0, pvalue < 0.0001).

We note that parametric estimates lead to inconsistent results across datasets for utility curvature in the loss domain and partial reĆection. By contrast, semi-parametric estimates lead to different Ąndings: (i) concave utility in both domains and (ii) rejection of partial reĆection. Unlike parametric methods, these Ąndings are consistent across datasets. The absence of convexity of the utility function in the loss domain is consistent with a number of previous studies (ABL;

BFE10 Abdellaoui et al., 2011c;Attema et al., 2013Attema et al., , 2016;;[START_REF] Kemel | An econometric estimation of prospect theory for natural ambiguity[END_REF]. ABL, Attema et al. (2013Attema et al. ( , 2016)), Abdellaoui et al. (2011c) and [START_REF] Kemel | An econometric estimation of prospect theory for natural ambiguity[END_REF] use the semi-parametric method developed by ABL, whereas BFE1031 use a parametric method with the two-parameter probability weighting function of GE87. In contrast, studies based on the tradeoff method Ąnd support for the convexity of the utility function in the loss domain (Abdellaoui, 2000;Etchart-Vincent, 2004, 2009b;Abdellaoui et al., 2007bAbdellaoui et al., , 2013Abdellaoui et al., , 2016;;Booij and Van de Kuilen, 2009;Hajimoladarvish, 2017;Attema et al., 2018;Bleichrodt et al., 2018). From a theoretical standpoint, people who exhibit a concave utility in the loss domain can still be risk seeking [START_REF] Chateauneuf | Risk seeking with diminishing marginal utility in a non-expected utility model[END_REF]. Our

semi-parametric method thus allows for such a possibility.

The empirical evidence on partial reĆection in the literature is mixed. Our rejection of partial reĆection is consistent with some studies (Abdellaoui et al., 2013(Abdellaoui et al., , 2016;;Attema et al., 2013, 2016, ABL) but not others (e.g. Abdellaoui, 2000;[START_REF] Andersen | Choice behavior, asset integration and natural reference points[END_REF]Abdellaoui et al., 2007b;Booij and Van de Kuilen, 2009;Harrison and Rutström, 2009;Booij et al., 2010).

Parametric estimates on the data of TK92 lead to a rejection of equality of probability weighting function across domains (pvalue < 0.0001). As observed

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components by TK92, parametric estimates imply that both w + (0.5) and w -(0.5) are less than 0.5 so that the duality condition is rejected. When considering the data of BFE10, identical probability weighting cannot be rejected (pvalue < 0.0001).

Using data of BFE10, parametric estimates imply the equality of probability weighting functions across domains (pvalue = 0.9172) and the rejection of duality (pvalue < 0.0001). Again, parametric estimates lead to inconsistent results across datasets for the comparison of probability weighting functions across domains.

In contrast, our method provides consistent results for the comparison of probability weighting functions across domains. We reject duality (RDU, Quiggin, 1982;Gilboa, 1987;Schmeidler, 1989) and identical probability weights across domains across datasets(OPT, Kahneman and Tversky, 1979). Tests of duality and identical probability weightings that cover the whole range of probabilities are scarce in the literature. Our Ąndings echo Abdellaoui (2000) who reject both duality and identical probability weighting functions across domains under risk. However, under uncertainty, Abdellaoui et al. (2005) do not reject duality, although they reject identical weighting functions across domains. Importantly, our rejection of both duality and identical probability weights provides support for CPT.32

Our method also reveals an interesting pattern in the probability weighting function Ű more overweighting of tail probabilities in the gain domain than in the loss domain Ű which fully stands in line with CPT, but does not arise under the parametric approach.33 Hence, the level of optimism for very small probabilities of gains is more pronounced than the level of pessimism for very small probabilities of losses.

Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

Two important similarities with TK92 also emerge. First, our estimates of loss aversion (λ) are close to 2, in line with the estimate provided by TK92.

Second, echoing the central tenets of CPT, we Ąnd that the probability weighting function is domain-speciĄc. Furthermore, in both domains it is characterized by the overweighting of small probabilities and the underweighting of large ones.

Altogether, our results mesh well with CPT. In both domains, the inverse S-shaped probability weighting function affects risk preferences alongside the utility function.

Conclusion

ABL and Abdellaoui et al. (2011b) deploy a semi-parametric method to elicit the utility function and loss aversion. In this paper, we go one step further by developing a semi-parametric method that elicits all dimensions of risk attitudes, including the whole range of probability weights. Importantly, it retains the four appealing properties of the parametric methods that have been discussed at length in the literature.

Our method is also Ćexible because it can be applied to both risk and uncertainty. Furthermore, it can be used to extend the popular elicitation technique of Holt and Laury (2002) to the case in which probabilities are distorted, following the approach of Abdellaoui et al. (2011b). Finally, even though our method does not readily apply to the context envisioned by Kőszegi and Rabin (2007),34 one can speculate on a possible procedure combining Köszegi and RabinŠs approach and our semi-parametric method. This procedure could start by introducing probability weighting functions in Kőszegi and Rabin (2007) following the work of Baillon et al. (2020). We see this as a promising avenue for future research. We use all the median observations from non-mixed prospects (see their Table 3) as well as the Ąrst six median observations from mixed prospects (see their Table 6).

b) Procedure

The data are generated via the switching outcome procedure in which an indifference value is inferred through a list of equally spaced certain outcomes, ranging from the admissible maximum indifference value to the admissible minimum indifference value. Note that an alternative approach, the direct matching procedure in which subjects are directly asked to provide their indifference values, tends to produce more inconsistencies [START_REF] Bostic | The effect on the preference-reversal phenomenon of using choice indifferences[END_REF]Booij and Van de Kuilen, 2009). Internal consistency of the responses to each prospect is monitored by the computer software to reduce response errors.

c) Data for the Ąrst step

All outcomes are expressed in US dollars. In Table 3 of TK92, there are 28 median values of certainty equivalents for binary lotteries that involve 7 pairs of positive monetary outcomes (0, 50), (0, 100), (0, 200), (0, 400), (50, 100), (50, 150) and (100, 200), and 9 probabilities of getting the higher outcome: 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95 and 0.99.

Also, the same Table 3 in TK92 provides 28 median values of certainty equivalents for binary lotteries that involve 7 pairs of negative monetary outcomes (0, -50), (0, -100), (0, -200), (0, -400), (-50, -100), (-50, -150) and (-100, -200),

and the same list of 9 probabilities as in the gain domain. These probabilities are now associated to losing the higher outcome.

As required by our method, at least two certainty equivalents for each of the nine probabilities are available per domain so as to perform a simultaneous measurement of the utility function and the probability weighting function in the full domain.

c) Data for the second step

All outcomes are expressed in US dollars. In Table 6 of TK92, there are 6 cases of indifferences involving mixed prospects. The Ąrst four items consist in eliciting the values of gains x to make subjects indifferent between the mixed prospects (x, y; 0.5, 0.5) and 0. The values of y are -25, -50, -100 and -150.

The two others cases consist in eliciting gains x that make subjects indifferent between two mixed prospects (x, y; 0.5, 0.5) and (z, w; 0.5, 0.5). The triplets (y, z, w) take the values of either (-50,50,-20) or (-125,150,-50). Note that here the experimenter has no control over the maximum level of x which may hinder the use of the switching outcome procedure for Ąnding indifference value. For this reason, we make changes in the third step of the original method of ABL ). Also, our method is based on the comparisons of binary lotteries and sure outcomes. Hence, these data do not exactly Ąt our method. With the present data, we compute loss aversion for each of the six questions and take the median as estimated value to account for response error (following ABL). For the Ąrst four items, we compute loss aversion as follows:

λ = w + (0.5) w -(0.5) × x αp (-y) βp (1.24) λ = w + (0.5) w -(0.5) × 1 -exp(-α e x) 1 -exp(β e y) × β e α e (1.25)
Formulas (1.24) and (1.25) correspond to power and exponential speciĄcations, respectively.

For the last two questions, we compute loss aversion as follows:

λ = w + (0.5) w -(0.5) × z αp -x αp (-w) βp -(-y) βp (1.26) λ = w + (0.5) w -(0.5) × exp(-α e x) -exp(-α e z) exp(β e y) -exp(β e w) × β e α e (1.27)
Formulas (1.26) and (1.27) correspond to the power and exponential spec-iĄcations, respectively. Finally, following ABL, we compute the median loss aversion.

1.A.2 Our main semi-parametric measurements

Our semi-parametric measurements are presented in Tables 1.A.1 and 1.A.

1.A.4 Parametric measurements

We consider the following parametric speciĄcations (1.28) and (1.29)

ce l =(D + l -D - l )   ♣x l ♣ αpD + l +βpD - l -♣y l ♣ αpD + l +βpD - l × W l + ♣y l ♣ αpD + l +βpD - l   1 αpD + l +βpD - l + e i l
(1.28)

ce l =ln   exp((β e D - l -α e D + l )x l ) -exp((β e D - l -α e D + l )y l ) × W l + exp((β e D - l -α e D + l )y l )   1 β e D - l -α e D + l + e i l (1.29)
with W l can be one of the four speciĄcations of TK92, P98-I, P98-II and GE87:

W l = p (aD + l +cD - l ) l p (aD + l +cD - l ) l + (1 -p l ) (aD + l +cD - l ) 1 aD + l +cD - l W l = exp --ln(p l ) aD + l +cD - l W l = exp -bD + l + dD - l × -ln(p l ) aD + l +cD - l W l = (bD + l + dD - l ) × p (aD + l +cD - l ) l (bD + l + dD - l ) × p (aD + l +cD - l ) l + (1 -p l ) (aD + l +cD - l )
Equation (1.28) and (1.29) allow us to elicit the utility and probability weighting function parameters in the full domain.

Once the parameters of the utility function and the probability weighting function are obtained, we estimate loss aversion as described in equations (1.24)

-(1.27).

1.B.1 Individual results

This appendix provides results based on individual estimates. We focus on the exponential utility function which is found to provide a better Ąt to the data than the power utility function under both our semi-parametric method and the parametric one. Analyses presented below are based on median comparisons of coefficients using Sign Rank tests. All tests are two-sided, unless stated otherwise.

1.B.1.1 Curvature of the utility function

The For gains, the estimated median exponential utility parameter under the semiparametric method is 0.0068 and signiĄcantly greater than 0 (pvalue < 0.0001, one-sided Sign Rank test). In the loss domain, the median exponential utility parameter is estimated at -0.0065 and signiĄcantly below 0 (pvalue < 0.0001).

Furthermore, partial reĆection is rejected: comparing the median estimates of both coefficients yields pvalue < 0.0001. The parametric method leads to very similar results.35

1.B.2 Probability weighting function

1.C Spline extension

Our semi-parametric method requires specifying a utility function. Using spline instead may further serve to reduce or eliminate such parametric assumptions.

In this section, we provide a linear spline extension for our semi-parametric method.37

1.C.1 Estimating probability weighting functions and utility curvature

We consider utility function over a range q, 0 ∪ [0, q]. We divide this range in arbitrarily small intervals [q j , q j+1 ] over which utility is assumed to be linear, with q ≡ q -τ -< q -τ -+1 < ... < q 0 ≡ 0 < q 1 < ... < q τ + -1 < q τ + ≡ q and

j ∈ ¶-τ -, -τ -+ 1, ..., -1, 0, 1, ...., τ + -1, τ + ♢. Notations (ce l , x l , y l , I k l , D + l
and D - l ) are the same as in Sections 1.3.2.2 and 1.3.2.3. We assume that all the outcomes x l and y l in the binary lotteries are such that ¶x l , y l ♢ ∈ ¶q ≡ q -τ -, q -τ -+1 , ..., q 0 ≡ 0, q 1 , ..., q τ + -1 , q τ + ≡ q♢. A linear spline approximation of the utility of any certainty equivalent ce l ∈ [q j , q j+1 ] is thus given by:

u(ce l ) = u(q j ) + u(q j+1 ) -u(q j ) q j+1 -q j (ce l -q j ) (1.30)
where u(0) = 0. The subsequent identiĄcation of loss aversion as in Köbberling and Wakker (2005) requires the utility function to be differentiable at 0 with u ′ (0) = 1. This implies that u(q -1 ) = q -1 and u(q 1 ) = q 1 . We assume response errors at the level of the utility ratio and get the equation:

38 u(ce l ) -u(y l ) u(x l ) -u(y l ) = K k=1 (δ + k D + l + δ - k D - l )I k l + e i l (1.31)
with the scaling u(0) = 0, u(q -1 ) = q -1 and u(q 1 ) = q 1 . Equation (1.31) then allows us to estimate the probability weights and the utility evaluated at the knots: q -τ -, q -τ -+1 , ..., q -2 , q 2 , ..., q τ + -1 , q τ + .

38

We choose to define the error term at the level of the utility ratio rather than the utility itself so as to circumvent the problem of extreme utility curvature (see footnote 13 for further explanation).

1.C.2 Estimating loss aversion

With utility function and probability weights estimated in the previous steps, we can rewrite equation (1.16) as follows:

1 (ce k ≥0) + λ1 (ce k <0) u(ce k ) = w + (p k )u(x k ) + λw -(1 -p k )u(y k ) (1.32)
where ¶x k , y k ♢ ∈ ¶q ≡ q -τ -, q -τ -+1 , ..., q 0 ≡ 0, q 1 , ..., q τ + -1 , q τ + ≡ q♢ and 1 (.)

refers to the indicator function. Using the same notation as in Section 1.3.2.3

and assuming an additive error at the basic utility scale (e k ), the empirical counterpart of equation ( 2.21) then becomes:

D + k + λD - k û(ce k ) = δ+ k û(x k ) + λ δ- k û(y k ) + e k (1.33)
We can then estimate the loss aversion index of Köbberling and Wakker (2005) from equation (2.22) by minimizing the sum of squared errors with respect to λ.

1.C.3 Comparing our semi-parametric method with its non-parametric spline version

To conduct spline estimations, we need data with a high number of certainty equivalents per subject. Among the several existing datasets that we reviewed (e.g. GW99; BFE10; lŠHaridon and Vieider, 2019;Harrison and Rutström, 2009;[START_REF] Andersson | Robust inference in risk elicitation tasks[END_REF][START_REF] Pedroni | The risk elicitation puzzle[END_REF][START_REF] Eisenberg | Uncovering the structure of self-regulation through data-driven ontology discovery[END_REF], the one of GW99 is best suited for our analysis because it includes 165 certainty equivalents per subject.

The 165 values of certainty equivalents correspond to binary lotteries that involve 15 pairs of positive monetary outcomes (0, 25), (0, 50), (0, 75), (0, 100), (0, 150), (0, 200), (0, 400), (0, 800), (25, 50), (50, 75), (50, 100), (50, 150), (100, 150), (100, 200) and (150, 200) and 11 probabilities of obtaining the higher outcome: 0.01, 0.05, 0.1, 0.25, 0.4, 0.5, 0.6, 0.75, 0.9, 0.95 and 0.99.

The subjects are 10 graduate students in psychology. GW99 use the switching outcome procedure for eliciting certainty equivalents, resulting in a total of 1650 certainty equivalents.

1.C.4 Results

We perform both semi-parametric and linear spline estimations using individuallevel and median data. At the individual level, the utility function is also found to be predominantly concave in both methods. The typical inverse S-shaped weighting function is also pervasive across methods, with the exception of Subject 6 whose weighting function is weakly concave.

1.D Hierarchical Bayesian Parameter Estimation

The goal of this application is to illustrate how to deploy Bayesian techniques with our method. To that end, we use the data of lŠHaridon and Vieider (2019) who elicit risk parameters from individual decisions of 2,939 subjects across 30 countries.

We use Hierarchical Bayesian Parameter Estimation (HBPE) (Nilsson et al., 2011;Murphy and ten Brincke, 2018;Baillon et al., 2020;Gao et al., 2020) ) is the normal cumulative distribution function, c refers to the country of residence and e i s,c,l is a normally distributed error term N (0, σ 2 i ). As in lŠHaridon and Vieider (2019), we consider a country-speciĄc exponential utility function u c (.) characterized by α c e and β c e . Following [START_REF] Rouder | An introduction to bayesian hierarchical models with an application in the theory of signal detection[END_REF], Nilsson et al. (2011) and Gao et al. (2020), we assume the following prior and hyperprior distributions. As priors we take:

ce s,c,l = u -1 c   (u c (x l ) -u c (y l )) ×   K k=1 (δ c,+ k D + l + δ c,- k D - l )I k l   + u c (y l )    + e i s,c,l (1.34) with δ c,+ k = F (z c,+ k ), δ c,- k = F (z c,- k ), F (.
α c e ⇝ N (α e , σ α 2 e ), β c e ⇝ N (β e , σ β 2 e ), z c,i k ⇝ N (z i k , σ 2 z i k ), σ 2 + ⇝ IG(0.001, 0.001) and σ 2
-⇝ IG(0.001, 0.001) where IG(.) stands for inverse gamma distribution. As hyperpriors, we take: α e ⇝ N (0, 10), β e ⇝ N (0, 10), z i k ⇝ N (0, 1), σ 2 αe ⇝ IG(0.001, 0.001), σ 2 βe ⇝ IG(0.001, 0.001), σ 2 z i k ⇝ IG(0.001, 0.001). We estimate the posterior distributions of world-and country-speciĄc parameters by using Markov Chain Monte Carlo (MCMC) with blocked Gibbs sampling (Baillon et al., 2020;Gao et al., 2020). After discarding a burn-in of 10000 samples, we collect 40000 samples to approximate the posterior distributions of the parameters of interest. We conĄrm the convergence of the MCMC chain by visual inspection of the trace plots, the autocorrelation plots and kernel densities of parameters based on the Ąrst and second halves of the sample.

We then use mean estimates of utility curvature and probability weights as inputs to estimate loss aversion. Like Nilsson et al. (2011), Spiliopoulos and Hertwig (2019) and Gao et al. (2020), we assume that the country-speciĄc loss

Results

Utility and probability weighting functions

Introduction

Empirical violations of expected utility theory (EUT; see Starmer, 2000, for a review) explain the development of alternative theories of decision makings.

Cumulative Prospect Theory (CPT; see Tversky and Kahneman, 1992, henceforth TK92) emerged as the theory with more descriptive validity (e.g. Blavatskyy, 2021;Attema et al., 2013).

Under CPT, Risk attitudes result simultaneously from the value and weighting functions. As result, several combinations of the shapes of the value and weighting functions can lead to the same level of risk-attitudes. A main challenge when measuring CPT is how to deal with the collinearity between the value and weighting functions (e.g. Zeisberger et al., 2012;Abdellaoui et al., 2011a).

Measurement methods of CPT can be done under three approaches: parametric (with parametric speciĄcation of the utility and probability weighting functions)1 , semi-parametric (with parametric speciĄcation of the utility function and parameter-free probability weighting function) and non-parametric (no parametric speciĄcation for either function).

Parametric (e.g. TK92) and semi-parametric (Abdellaoui et al., 2008) methods are more often used because their implementations are quick and easy. Yet, collinearity between the value and the weighting functions makes estimation results sensitive to parametric speciĄcations (e.g. Abdellaoui, 2000;Abdellaoui et al., 2008). A pragmatic and limited way to get rid of this collinearity is to assume a linear value function (e.g. lŠHaridon and Vieider, 2019) in line with the dual theory of Yaari (1987).

The so-called tradeoff method of Wakker and Deneffe (1996) is an alternative.

This method is non-parametric and the collinearity between utility and weighting functions does not play a role in the elicitation procedure. This explains why most non-parametric methods (e.g. Abdellaoui, 2000;Abdellaoui et al., 2007bAbdellaoui et al., , 2016;;Blavatskyy, 2021) are built upon the tradeoff method (see [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] for detailed discussions on elicitation methods). The nonparametric method of Gonzalez and Wu (1999, henceforth GW99) is a notable exception which does not rely on the tradeoff method. 2 However, the collinearity between utility and weighting function takes a stronger form in this method. Irrespective to the data under consideration, the method can lead to unrealistic concave value function and probability weights close to 1 (see footnote 4).

This paper establishes an alternative nonparametric method to the tradeofftype methods. The proposed method approximates the value function with smoothing spline. The smooth parameter allows to control for the collinearity between the value and weighting functions by penalizing deviation from the linear value function. In contrast to the tradeoff-type methods, the proposed method accounts for response errors and rely on simple questions. The method remains applicable under uncertainty where probabilities of events are unknown.

The method is applied on the two datasets of TK92 and GW99. Results indicate that the probability weighting function is not sign-dependent. The value function is S-shaped with a loss aversion coefficient of 1.6.

The rest of the paper is organized as follows. Section 2.2 presents the spline value function. Section 2.3 develops the smoothing spline method for eliciting cumulative prospect theory components. Section 2.4 presents the key features of the method. We illustrate the method in Section 2.5 using the data of TK92 and GW99. Sections 2.6 and 2.7 provide discussion and conclusion.

Spline value function for CPT

CP T (L) = v(x) -v(y) w i (p) + v(y) (2.1) CP T (L) = w + (p)v(x) + w -(1 -p)v(y) (2.2)
where w i and v are both continuous, strictly increasing and satisfying v(0) = 0, w i (0) = 0 and w i (1) = 1, and i =Ş+Ť (i =Ş-Ť) stands for the gain (loss) domain.

CPT makes no explicit link between weighting functions w + (.) and w -(.) which makes it more general than the original version of prospect theory (OPT, Kahneman and Tversky, 1979) in which w + (p) = w -(p), or rank dependent utility theory (RDU, Quiggin, 1982;Gilboa, 1987;Schmeidler, 1989) that includes the duality condition w

+ (p) = 1 -w -(1 -p).
Following the seminal study by TK92 and the meta-analysis of Brown et al.

(2021), the value function v(.) is composed of the loss aversion index λ > 0, which reĆects the exchange rate between gain and loss utility units, and the (basic) utility function u(.) that reĆects the intrinsic value of outcomes:

v(x) =          u + (x) if x ≥ 0 -λu -(-x) if x < 0 and u(x) =          u + (x) if x ≥ 0 -u -(-x) if x < 0 (2.3) with u + : R + -→ R + , u -: R + -→ R + and u + (0) = u -(0) = 0.
LetŠs assume that the basic utility function u is twice differentiable over R - ¶0♢

and differentiable at 0. Assuming that the utility function is differentiable at 0 means that the loss aversion index in the relation 2.3 corresponds to the ratio of the left and right derivative of the value function as deĄned by Köbberling and Wakker (2005). Loss aversion (loss seeking) corresponds to λ > 1(λ < 1), whereas λ = 1 captures loss neutrality.

Spline value function

The spline approximation of the function u i (.) corresponds to (e.g., Ahamada and Flachaire, 2010;[START_REF] Ruppert | Selecting the number of knots for penalized splines[END_REF]Green and Silverman, 1993):

u i (z) = a i 0 + J i j=1 a i j z j + Q i t=1 b i t (z -q i t ) J i + for z ≥ 0 and i = +, - (2.4)
with J i ≥ 1 the order of the spline, Q i the number of internal knots,

q i 1 < q i 2 < ... < q i T and (z -q i t ) J i + =          (z -q i t ) J i if z ≥ q i t 0 otherwise
To accommodate with u(0) = 0 of Prospect Theory, the scaling a i 0 = 0 is used. Without loss of generality, the scaling a i 1 = 1 is used to allow identiĄcation of the loss aversion index à la Köbberling and Wakker (2005). This scaling means that the utility function is differentiable at 0, with u ′ (0) = 1.

A main advantage of (2.4) is that the shape of the utility function is very Ćexible. The following three quadratic splines (2.5) -( 2 In the gain domain, the method can be implemented by following three substeps.

u i (z) = z -0.47z 2 -0.01(z -0.25) 2 + -0.01(z -0.25) 2 + -0.01(z -0.25) 2 + (2.5) u i (z) = z + z 2 + 0.1(z -0.25) 2 + + 0.1(z -0.50) 2 + + 0.1(z -0.75) 2 + (2.6) u i (z) = z -1.75z 2 + 3(z -0.25) 2 + -2.5(z -0.25) 2 + + 2.5(z -0.25) 2 + (2.7)
First, a set of probabilities ¶p k : k = 1, 2, ..., K♢ are selected, with p k < p k+1 .

Second, at least two certainty equivalents for each probability p k are elicited:

ce j,k ∼ (x j,k , y j,k ; p k , 1 -p k ) , j = 1, 2, ..., N + k and N + k ≥ 2 (2.8)
where N + k stands for the number of certainty equivalents for positive outcomes x j,k and y j,k such that x j,k > y j,k ≥ 0. Thus, in total

N + = K k=1 N + k ≥
2 × K certainty equivalents are elicited. Using (2.1) and ( 2.3), these certainty equivalents satisfy the following condition:

u + (ce j,k ) = u + (x j,k ) -u + (y j,k ) w + (p k ) + u + (y j,k ) (2.9)
Let ce, x and y be the column vectors containing all the realizations of ce j,k , x j,k and y j,k , respectively. Let also I k be a dummy variable set to 1 if the probability equals p k and 0 otherwise. For k = 1, .., K, denote δ + k ≡ w + (p k ) ∈ (0, 1). Arranging the terms in equation ( 2.9) and taking the log leads to the following equation:

log u + (ce j,k ) -u + (y j,k ) = log u + (x j,k )) -u + (y j,k ) + log w + (p k ) (2.10)
Adding an error term e to the log-transformation (2.10) leads to the following empirical equation:

4 log u + (ce l ) -u + (y l ) = log u + (x l ) -u + (y l ) + log K k=1 δ + k I k l + e l (2.

11)

4 An alternative approach would be to introduce the response error term at the utility level (GW99, eq. 7): u(ce l ) = (u(x l )u(y l ))

K k=1 δ + k I k l + u(y l ) + e l .
However, defining the response error at the utility level is problematic when using certainty equivalents data because it produces solutions that are characterized by unrealistic concavity of the utility and probability weighting functions. To illustrate this point, suppose that we are interested in eliciting utility only over strictly positive outcomes with a power utility function u(z) = z α . For an extremely concave utility function (i.e., α > 0 and α -→ 0) and an extremely concave weighting function (i.e., δ + k = 1 for k = 1, 2, .., K ) along with the PT assumptions w + (0) = 0 and w + (1) = 1, we have e l = 0 for all l = 1, 2, ..., N + . For the non-parametric method of GW99 which aims at estimating u(z) for z ∈ A(z) ≡ ¶$25, $50, $75, $100, $150, $200, $400, $800♢ and the probability weights w(p) for p ∈ B(p) ≡ ¶0.01, 0.05, 0.10, 0.25, 0.40, 0.50, 0.60, 0.75, 0.90, 0.95, 0.99♢, it follows that an extremely concave utility function (i.e., u(z) = constant > 0 for z ∈ A(z) and u(0) = 0) and an extremely concave weighting function (i.e., w(p) = 1 for p ∈ B(p) and w(0) = 1w(p) = 0) are solutions of the optimization problem [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF].

where l is the l th line in ce, x, y, I k and e. Denote by θ + the vector of parameters ( ¶a

+ j ♢ J + j , ¶b + t ♢ Q + t=1
) associated with the smooth approximation of the utility function u + (.). Minimizing the following penalized sum of squared error allows to estimate the utility function and the probability weights as follows:

5 min θ + ,δ + 1 ,...,δ + K N + l=1 e 2 l + ρ + x 0 u ′′ + (z) 2 dz (2.12)
where ρ + ≥ 0 is the smooth parameter, u ′′ + (.) the concavity (second derivative) of the utility function and [0, x] the range over which the utility function is elicited.

Each level of the smooth parameter ρ + corresponds to a speciĄc combination of utility curvature and probability weighting function. For example, when subject exhibits risk-aversion, the case ρ + -→ +∞ corresponds to the linear utility function and the less elevated probability weighting function that will result from an estimation based on the dual theory of Yaari (1987). Then, the smooth parameter allows to chose the combination of shapes of the utility and weighting functions by penalizing deviation from the linear utility function.

Following the literature (e.g. Ahamada and Flachaire, 2010; Green and Silverman, 1993), the optimal values of the smooth parameter ρ + and the order of the spline Q + corresponds to the ones that minimize the leave-one crossvalidation (CV). Formally, the optimal ρ + and Q + provide the smallest value for:

CV = 1 N + N + l=1 ce -l -ce l (2.13)
where ce -l is the predicted value of ce l based on data without the l th observation. 6

The following measure of utility curvature over [0, x] in the gain domain (see [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Abdellaoui et al., 2016) is used to classify the utility function as concave or convex:

α = 1 xu + (x) x 0 u + (t)dt (2.14)
The utility function u + (.) is considered to be concave, linear and convex on the interval [0, x] when α > 0.5, α = 0.5 and α < 0.5 respectively.

Step 2: utility and weighting functions in the loss domain

This step is similar to the Ąrst step. At least two certainty equivalents for each probability p k are elicited:

ce j,k ∼ (x j,k , y j,k ; p k , 1 -p k ) , j = 1, 2, ..., N - k and N - k ≥ 2 (2.15)
where N - k stands for the number of certainty equivalents for negative outcomes x j,k and y j,k such that x j,k < y j,k ≤ 0. Thus, in total N -= K k=1 N - k ≥ 2 × K certainty equivalents are elicited.

6 Denote by u + (.) and δ + k (k = 1, 2, ..., K) the estimated utility function and decision weights. The predicted certainty equivalent ce l is the solution to the equation u

+ ( ce l ) = u + (x l ) -u + (y l ) K k=1 δ + k I k l + u + (y l ).
Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components

Consider the following log-transformation of prospect theory functional form with an additive error term:

log u -(-ce l ) -u -(-y l ) =log u -(-x l ) -u -(-y l ) + log K k=1 δ - k I k l + e l (2.16)
where l is the l th line in ce, x, y, I k and e. Denote by θ -the vector of parameters ( ¶a

- j ♢ J - j , ¶b - t ♢ Q - t=1
) associated with the smooth approximation of the utility function u -(.). Minimizing the following penalized sum of squared error provides the estimate of the utility function and the probability weights:

7 min θ -,δ - 1 ,...,δ - K N - l=1 e 2 l + ρ --x 0 u ′′ -(z) 2 dz (2.17)
where ρ -≥ 0 is a (Ąxed) smooth parameter, u ′′ -(.) the concavity (second derivative) of the utility function and [x, 0] the range on which the utility function is elicited. As in the gain domain, the smooth parameter controls the concavity of the utility function by penalizing deviation from linear utility function or the dual theory of Yaari (1987).

As in the gain domain, the optimal values of the smooth parameter ρ -and the order of the spline Q -correspond to the ones that minimize the leave-one cross-validation (CV):

CV = 1 N - N - l=1 ce -l -ce l (2.18)
where ce -l is the predicted value of ce l based on data without the l th observation. 8

As in the gain domain, the following measure of utility curvature over the loss interval [x, 0] is used to classify the utility function as concave or convex (see [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Abdellaoui et al., 2016):

β = 1 xu -(x) 0 x u -(t)dt (2.19)
The utility function u -(.) is considered to be concave, linear and convex on the interval when β > -0.5, β = -0.5 and β < -0.5 respectively.

Step 3: measuring loss aversion

The third step allows to measure the loss aversion index λ as deĄned in (2.3).

Following Abdellaoui et al. (2007b), the estimation of the loss aversion index can be done using a set of K indifference relationships that involve mixed binary prospects:

ce k ∼ (x k , y k , p k , 1 -p k ) , k = 1, 2, ..., K (2.20)
with y k < 0 < x k . Under CPT these indifferences imply that:

1 (ce k ≥0) + λ1 (ce k <0) u(ce k ) = w + (p k )u(x k ) + λw -(1 -p k )u(y k ) (2.21)
where 1 (.) refers to the indicator function. Denote by D + a dummy variable that takes the value 1 if the certainty equivalent is positive (or zero) and 0 otherwise.

Similarly, D -is a dummy variable that takes value 1 if the certainty equivalent 8 Denote by u -(.) and δ - k (k = 1, 2, ..., K) the estimated utility function and decision weights. The predicted certainty equivalent ce l is solution to the equation u

-(-ce l ) = u -(-x l ) -u -(-y l ) K k=1 δ - k I k l + u -(-y l ).
145

Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components is negative and 0 otherwise. Assuming additive error at the basic utility scale (e k ), the empirical counterpart of equation ( 2.21) then becomes:

D + k + λD - k u(ce k ) = δ + k u(x k ) + λ δ - k u(y k ) + e k ( 2

Key features of the method

This section highlights six features of the method.

Robust to collinearity.

The method controls for the collinearity between utility and weighting functions (e.g. Zeisberger et al., 2012) with smooth parameters. Indeed, these smooth parameters allow to choose optimal combination of curvatures of utility and weighting functions in the gain and loss domains.

Applicability to unknown probabilities. The use of Ąxed effects for estimating decision weights is taken from [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]. The method is then directly applicable to cases of uncertainty where probabilities are unknown. The probability dummy variables can then be replaced by the event dummy variables in equations (2.11) and (2.16). It does not require setting any speciĄc conditions on the event space, and hence can be applied to real-life uncertainty situations (Baillon et al., 2018).9

Error-robust. In contrast to the tradeoff method (Wakker and Deneffe, 1996), the smoothing spline method considers that subjects can make error in their responses. Estimations of prospect theory components result from a minimization of (penalized) sum of squared response errors.

Easy (not cognitively demanding).

The method is a certainty equivalent method. As such, it uses simple questions that involves the lowest possible number of outcomes (i.e., 3): comparisons of certain outcomes and binary lotteries. Hence, the method is less cognitively demanding than the tradeoff method that rely on comparison of two non-degenerate lotteries (e.g. Abdellaoui et al., 2008).

Tractable. The method can be implemented by using optimization programs available in standard statistical software. For example, the ML routine for Stata popularized by Harrison and Rutström (2008) and [START_REF] Moffatt | Experimetrics: Econometrics for experimental economics[END_REF] for parametric risk elicitation can be used to implement the smoothing spline method.

Data-inefficient.

The method builds upon the smoothing spline literature.

Application of the method is more data consuming than parametric methods (e.g. Ahamada and Flachaire, 2010;Green and Silverman, 1993). This method should thus be used in two main cases. First, behavioral studies often elicit few observations per subjects. In such case, the method can be used to derive estimates at the aggregate level by pooling subjects as in TK92. The dataset include 25 subjects. For each subject, only 28 observations are available to derive estimates of utility and weighting function in each domain. The pooled data with 700 (= 25 × 28) observations per domain can then be used to derive estimate at the aggregate level. Second, the method can be performed at the individual level if the number of observations per subjects is sufficiently large as in GW99. For each subject, the dataset include 165 observations to measure utility and weighting functions in the gain domain.

Application

This section applies the smoothing spline method on the data of TK92 and GW99. For each subject, there are 28 values of certainty equivalents for binary lotteries that involve 7 pairs of positive monetary outcomes (0, 50), (0, 100), (0, 200), (0, 400), (50, 100), (50, 150) and (100, 200), and 9 probabilities of getting the higher Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components outcome: 0.01, 0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95 and 0.99. Also, there are, for each subject, 28 values of certainty equivalents for binary lotteries that involve 7 pairs of negative monetary outcomes (0, -50), (0, -100), (0, -200), (0, -400), (-50, -100), (-50, -150) and (-100, -200), and the same list of 9 probabilities as in the gain domain. These probabilities are now associated to losing the higher outcome.

Data

Data for the third step: for mixed prospects, individual data are not available.

We then use the median values reported in Table 6 of TK92. We use the four indifferences on mixed prospects that consist in eliciting the values of gains x to make subjects indifferent between the mixed prospects (x, y; 0.5, 0.5) and 0.

The values of y are -25, -50, -100 and -150.

Data of GW99

Subjects and procedure: GW99 run a computerized experiment with 10 graduate students in psychology. They also use the switching outcome procedure for eliciting certainty equivalents.

Data for the first step (only gain): Each subject has 165 certainty equivalents.

The 165 values of certainty equivalents correspond to binary lotteries that involve 15 pairs of positive monetary outcomes (0, 25), (0, 50), (0, 75), (0, 100), (0, 150), (0, 200), (0, 400), (0, 800), (25, 50), (50, 75), (50, 100), (50, 150), (100, 150), (100, 200) and (150, 200) and 11 probabilities of obtaining the higher outcome: 0.01, 0.05, 0.1, 0.25, 0.4, 0.5, 0.6, 0.75, 0.9, 0.95 and 0.99. For the data of GW, the mean of individual utility function in the gain domain is also concave. 12 The utility curvature is α = 0.623 and is signiĄcantly different from linearity (pvalue < 0.0001).

Results

Utility function

Loss aversion

The estimated value of loss aversion is λ = 1.636 and is signiĄcantly different from loss neutrality (pvalue < 0.0001). Hence, subjects exhibit loss aversion.

Discussion

The estimation result of concave utility function in the gain domain is a very common Ąnding in the literature, irrespective to the approach: parametric (e.g. lŠHaridon and Vieider, 2019, TK92), semi-parametric (e.g. Abdellaoui et al., 2008Abdellaoui et al., , 2011a;;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] and non-parametric (e.g. Wakker and Deneffe, 1996;Abdellaoui et al., 2016;Blavatskyy, 2021, GW99).

In the loss domain, the Ąnding of convex utility function corroborates the original parametric estimates of TK92. Though, the evidences about utility curvature in loss domain are mixed in the literature. Results based on nonparametric methods (e.g. Abdellaoui et al., 2011b;Attema et al., 2018;Abdellaoui et al., 2016;Blavatskyy, 2021;Hajimoladarvish, 2017) tend to provide evidences in favor of a convex utility function. In contrast, results based on semi-parametric methods (e.g. Abdellaoui et al., 2008;Attema et al., 2013Attema et al., , 2016;;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] tend to provide evidences in favor of a concave utility function.

The methods also leads to the rejection of partial reĆection (α + β = 0).

Empirical evidences on partial reĆection in the literature is mixed. The rejection of partial reĆection is consistent with some studies (Abdellaoui et al., 2013(Abdellaoui et al., , 2016;;Attema et al., 2013, 2016, ABL), but not with others (e.g. Abdellaoui, 2000;[START_REF] Andersen | Choice behavior, asset integration and natural reference points[END_REF]Abdellaoui et al., 2007b;Booij and Van de Kuilen, 2009;Harrison and Rutström, 2009;Booij et al., 2010).

The estimation results provide support for identical probability weighting function across domains of OPT and reject the duality condition of RDU.

Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components

Tests of identical probability weightings and duality are scarce in the literature. Abdellaoui (2000) and [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] reject both duality and identical probability weighting functions across domains under risk. Abdellaoui et al. (2005) do not reject duality under uncertainty although they reject identical weighting functions across domains. The observation of identical probability weights and the rejection of duality provide support for OPT.

The method conĄrms loss aversion, with a loss aversion index of λ = 1.636.

This estimate is less than the 2.25 reported by TK92. Our estimate is close to the estimated value of 1.6 that was elicited in both Booij et al. (2010) who use structural estimation techniques, and Abdellaoui et al. (2008) for pooled data.

It is also close to the estimates of λ = 1.8 reported by Pennings and Smidts (2003). The evidence of loss aversion is a very common Ąnding in the literature as only few studies Ąnd evidences for loss seeking (e.g. Abdellaoui et al., 2013;Nilsson et al., 2011).

Conclusion

In sum, this paper introduces a smoothing spline method to elicit the utility function, the weighting function and the loss aversion. The method allows to control for the collinearity between utility and weighting function and can be applied under both risk and uncertainty. Its application on experimental data provides reliable results.

Chapter 2. Smoothing Spline Method for Measuring Prospect Theory Components

2.A Appendix Data of TK92

2.B Appendix Data of GW99

Introduction

Ambiguous situations are pervasive in human decisions. These decisions vary from choosing a place to work to important investment decisions. Subjects decide under ambiguous situations when the objective probabilities of the possible events are unknown [START_REF] Knight | Risk, uncertainty and profit[END_REF]. In contrast, subjects make decisions under risk when the objective probabilities are known.

The standard theory under ambiguity Ű Subjective Expected Utility (SEU) Ű considers that subjects (i) form subjective probabilities or beliefs on events, (ii)

have the same utility function under ambiguity as under risk, and (iii) value lotteries as expected utility over outcomes in which the weights are the beliefs. Ellsberg (1961)Šs paradox showed that people deviate from SEU by exhibiting ambiguity attitudes (aversion or even seeking). Consequently, several models (e.g. Gilboa, 1987;Schmeidler, 1989;Tversky and Kahneman, 1992;[START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF] have been proposed to account for ambiguity attitudes by allowing events weighting function and a difference in utility between risk and ambiguity.

The estimation of the utility and weighting functions Ű as a measure of ambiguity attitudes Ű has been focused on continuous-valued sources of uncertainty, meaning that the universal event is an interval of real numbers (Abdellaoui et al., 2021b;Van De Kuilen and Wakker, 2011). The main advantage of continuousvalued sources of uncertainty is that the concept of exchangeability of events, introduced by Baillon (2008), can be used to build elicitation methods (see subsection 3.2.2). Two events are exchangeable for a decision maker when she is indifferent towards permutations of their outcomes. Exchangeability allows to iteratively partition the universal event in equally likely events. Then, with a set of exchangeable events for which the subjective probabilities are known, elicitation methods can provide a measurement of the utility and weighting functions.

In our daily life, situations of continuous-valued sources of uncertainty are not common compared to situations that involve discrete sources of uncertainty.

A Discrete source of uncertainty refers to any source of uncertainty that takes their values in a discrete set of events, which are not necessarily equally likely.

The universal event is no longer an interval of real numbers. As such, it excludes the possibility of building exchangeable events. Baillon et al. (2018a,b) shed light on this regard by proposing two methods that do not require exchangeable events; therefore, the two methods can be used for discrete sources of uncertainty.

However, these methods rely on restricted parametric assumptions (source independent utility function and the neo-additive weighting function of Chateauneuf et al. ( 2007)), which makes them prone to misspeciĄcation issues.

The objective of this chapter is to develop a method to completely estimate utility function, weighting function and beliefs for discrete sources of uncertainty.

Examples of discrete sources of uncertainty are present in almost all Ąelds of economics. Some of the experimental from game theory include trust, ultimatum, and public good games. In the trust game (e.g. [START_REF] Bohnet | Betrayal aversion: Evidence from brazil, china, oman, switzerland, turkey, and the united states[END_REF], for instance, the universal event of the Trustor is often the union of two unequally likely events; either the Trustee reciprocates or betrays. Also, in the ultimatum game (e.g. [START_REF] Slonim | Learning in high stakes ultimatum games: An experiment in the slovak republic[END_REF], the universal event for the Ąrst mover is the union of two unequally likely events; either the second mover accepts or rejects the money sent by the Ąrst mover. Similarly, in a public good game with two players (e.g. [START_REF] Kosfeld | Institution formation in public goods games[END_REF], the universal event of each player is the union of two unequally likely events; either the other player contributes or the other player

does not contribute.
Further examples of discrete sources of uncertainty can be found in health economics. For instance, the universal event of someone that consumes harmful products (e.g., tobacco, alcohol, unhealthy diets) can be represented as the union

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty of two unequally likely events: either the development of a non-communicable disease or stay healthy (e.g. [START_REF] Bloom | The economic burden of chronic diseases: estimates and projections for china, japan, and south korea[END_REF][START_REF] Mane | Modeling the macroeconomic effects of disease: Extension and application in the context of senegal[END_REF]. This example can also be extended to communicable diseases like AIDS or COVID-19, where people do not know the exact probability of getting sick and they decide whether or not to wear condoms or masks (e.g. [START_REF] Cuddington | Modeling the macroeconomic effects of aids, with an application to tanzania[END_REF][START_REF] Rieger | To wear or not to wear? factors inĆuencing wearing face masks in germany during the covid-19 pandemic[END_REF]. Additionally, transport economics (e.g [START_REF] Guarda | Decreasing fare evasion without Ąnes? a microeconomic analysis[END_REF] and taxation economics (e.g. [START_REF] Dhami | Why do people pay taxes? prospect theory versus expected utility theory[END_REF]Dhami and Hajimoladarvish, 2020) exempliĄed these situations. For instance, subjects who do not pay for the transport tickets face the union of two unequally likely events; being checked or not by controllers. Similarly, subjects who avoid taxes can be caught or not.

To measure these types of discrete sources of uncertainty, we propose a two-stage method in which the parametric assumptions of the utility function and the weighting function are made sequentially. The method allows for source-dependent utility function and any two-parameter weighting function (e.g. Goldstein and Einhorn, 1987;Prelec, 1998;Chateauneuf et al., 2007). Using simulations, we show that the two-stage structure of the method favors robustness to misspeciĄcation issues (see subsection 3.D).

We combine our method and the empirical data from an experiment conducted based on the trust and coordination game, in which subjects make decisions under different sources of uncertainty. Our data allows us to confront the method with two validity tests on (i) symmetric events and (ii) stability of beliefs for unequally likely events of the same sources of uncertainty involved in different decision contexts. Our method successfully passes validity tests, which supports the reliability of the results derived from it.

We provide three mains results. First, we identify which sources of uncertainty are captured by likelihood insensitivity or pessimism. We Ąnd that several forms of uncertainty attitudes operate through variations in the likelihood insensitivity component, the main driver of the inverse S-shaped weighting function in the literature (e.g. Wakker, 2010;Åstebro et al., 2015;Abdellaoui et al., 2011a).

Subjects exhibit more likelihood insensitivity towards sources of uncertainty involving not symmetric events, which indicates that the beliefs formation process of unknown events is cognitively demanding. Second, empirical evidence supports theories that model ambiguity attitudes with the weighting function rather than with the utility function. Finally, our method reveals that subjects exhibit two additional behaviors which are not related to attitudes toward the source of uncertainty: payoff dependence aversion and variety of payoffs seeking.

Payoff dependence aversion refers to the fact that subjects dislike that their own payoffs depend on the preferences of others, this behavior is captured by a more concave utility function. Variety of payoffs seeking means that subjects prefer a greater number of possible payoffs, when such possible payoffs depend on the preference of others, this is captured by a decrease in pessimism.

The remainder of this chapter proceeds as follows. Section 3.2 presents different theoretical approaches to model ambiguity attitudes and existing measurement methods. Section 4.3 presents our elicitation method. Section 4.4

provides a review of related literature about attitudes towards sources of uncertainty in the coordination and trust game. Section 3.5 presents the experimental design. Section 3.6 provides the results. We discuss the results in section 3.7 and, conclude in section 3.8.

Theoretical background

This section presents a general theoretical framework of ambiguity attitudes.

Also, we present existing methods of measuring ambiguity attitudes and beliefs and, the explanation of their limits. and Marinacci (2001), with the assumption that the decision maker can assign subjective probabilities to events, even when she does not maximize SEU (e.g. Ellsberg, 1961, p. 659). The biseparable preference model is a very general ambiguity model (e.g., Attema et al., 2018;Abdellaoui et al., 2021a) because it contains many of the ambiguity models (e.g. Gilboa, 1987;Schmeidler, 1989;[START_REF] Gilboa | Maxmin expected utility with non-unique prior[END_REF]Tversky and Kahneman, 1992) that have been proposed to explain EllsbergŠs paradox as special cases.

Biseparable preferences model

For decisions under risk, the objective probabilities of events are known. as

V (L) = w(p)(u(x) -u(y)) + u(y) (3.2)
with w(.) as the weighting function or source function for risk and u(.) the utility function that captures the attitude toward the outcomes. Both w(.) and u(.)

are strictly increasing functions.

Source-dependent Utility (SDU) models assume identical weighting functions between risk and uncertainty, i.e. W (.) = w(.). Source-dependent weighting (SDW) models assume identical utility functions between risk and uncertainty,

i.e. U (.) = u(.).

Existing methods

Elicitation methods based on exchangeability of events

The main difficulty for measuring the ambiguity model (3.1) resides in how to disentangle the weighting function W (.) from the beliefs P (.) (e.g. Li et al., 2020) (LW, hereafter). The solution proposed in the literature is based on the concept of exchangeability of events (Baillon, 2008). Two events are exchangeable for a decision-maker when she is indifferent towards permutations of their outcomes.

Formally, two events E 1 and E 2 are exchangeable if (x, E 1 , y) ∼ (x, E 2 , y), which implies that such events are equally likely or symmetric: P (E 1 ) = P (E 2 ). If these events are complementary, then the subjective probability associated with each event should be 1 2 , assuming the additivity of P (.). Based on this concept of exchangeability of events, several methods have been proposed (e.g. Abdellaoui et al., 2011a;Van De Kuilen and Wakker, 2011;Abdellaoui et al., 2021a;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]. The common idea underlying these methods is to start by splitting the universal event into two Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty

exchangeable events E 1 and E 2 , such that P (E 1 ) = P (E 2 ) = 1 2 . The following steps of these methods consist of splitting E 1 and E 2 into exchangeable events that will result in 1 4 as the subjective probability. Repeating the procedure allows to construct iteratively a series of exchangeable events that have a subjective probability of 1 2i , with i = 1, 2, ..., n. With the set of exchangeable events for which one knows the subjective probability, these methods can provide a measurement for W (.) and U (.).

The construction of these exchangeable events are only possible for continuousvalued sources of uncertainty, which means that the universal event is an interval of real numbers (Abdellaoui et al., 2021b;Van De Kuilen and Wakker, 2011).

For instance, when the source of uncertainty is the temperature in a town or the stock market index.

Elicitation method not based on exchangeable events

Oppositely, a discrete source of uncertainty comes from a source of uncertainty that takes their values in a discrete set of events which are not necessarily equally likely. Consequently, the universal event is no longer an interval of real numbers, therefore, it is not longer possible to build exchangeable events. In the next subsection, we introduce the indexes of Baillon et al. (2018b) (BW, hereafter), which are applicable to measure ambiguity towards discrete sources of uncertainty.

Belief hedges method of BW

For discrete sources of uncertainty, BW introduced the belief hedges method that consists of evaluating ambiguity attitudes through two indexes. BW assume a minimal degree of richness of the state space Ω, meaning that there should be three nonnull events E 1 = A, E 2 = B and E 3 = C that are mutually exclusive Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty and exhaustive:

E 1 ∪ E 2 ∪ E 3 = Ω and E i ∩ E j = ∅ for i ̸ = j. Denote by E ij
the union E i ∪ E j of two events. We call E i a single event and E ij a composite event. Denote by Ω

* 1 = ¶E 1 , E 2 , E 3 ♢ = ¶A, B,
m E = f [P (E)] (3.3)
The two indexes of the ambiguity function of BW are given by:

b = 1 -(m 1 + m 2 ) a = 3 1 3 -(m 2 -m 1 ) with m 1 = 1 3 [m A + m B + m C ] and m 2 = 1 3 [m AB + m AC + m BC ]
being the averages matching probability for the single and composite events.

The quantity b, called ambiguity aversion index, approximates the elevation of the decision makerŠs ambiguity function. Ambiguity neutrality (i.e. w(.) = W (.)) implies b = 0. A higher value of b is associated with more ambiguity aversion from the pessimism component of the weighting function. The quantity a, called ambiguity-generated insensitivity (a-insensitivity), approximates the Ćatness of the ambiguity function in the middle region. Ambiguity-neutrality implies a = 0. A higher value of a is associated with more ambiguity aversion from the likelihood insensitivity component of the weighting function.

The main purpose of the two indexes of BW is to compare a risky situation with an ambiguous situation. Although, it can be tempting to use these two indexes to compare two different sources of uncertainty, our main point (formulated in Proposition 1) is to stress the fact that this second use of the two indexes could be misleading.

Proposition 1. Consider two different sources of uncertainty 1 and 2 involving each three mutually exclusive and exhaustive events E i , i = 1, 2, 3. Denote by W j (.) and P j (.) the weighting and belief functions for the source of uncertainty j, with j = 1, 2. Assume that subjects have: (A1) the same non-linear weighting functions for the two sources of uncertainty : W 1 (.) = W 2 (.) ≡ W (.) (A2) different beliefs for events : P 1 (.) ̸ = P 2 (.).

Assumptions (A1) and (A2) imply that a

1 ̸ = a 2 and b 1 ̸ = b 2

Proof

Under (A1), we have the same ambiguity function f 1 (.) = f 2 (.) = w -1 [W (.)] ≡ f (.), with w(.) being the probability weighting function for risk. So, there is no difference in ambiguity attitudes in the sense of SDW. The two indexes of BW for the two sources of uncertainty are given by

b i = 1 - 1 3    E∈Ω * 1 f [P i (E)] + E∈Ω * 2 f [P i (E)]    i = 1, 2 a i = 1 -    E∈Ω * 2 f [P i (E)] - E∈Ω * 1 f [P i (E)]    i = 1, 2
Since the ambiguity function f (.) is bijective and P 1 (.) ̸ = P 2 (.), there is no reason, a priori, to expect that b 1 = b 2 and a 1 = a 2 . To illustrate, letŠs consider the following numerical example. For the source of uncertainty 1, assume that E 1 , E 2 , and E 3 are symmetric: P 1 (E 1 ) = P 1 (E 2 ) = P 1 (E 3 ) = 1 3 . For the source 2, assume P 2 (E 1 ) = 1 10 , P 2 (E 2 ) = 1 10 and P 3 (E 3 ) = 8 10 . Also, assume w(p) = p for risk and the non-linear Prelec (1998) The Proposition 1 shows that the indexes of BW may be misleading to learn about the differences in ambiguity functions related to different sources of uncertainty when the distributions of beliefs differ markedly between these sources. Note that the only case where the indexes of BW work perfectly, is when the ambiguity function is linear (Baillon et al., 2021, Theorem 16). This happens when the weighting functions w(.) and W (.) are the speciĄcation of Chateauneuf et al. (2007).

Our point applies in particular to LW, who compare the indexes of BW between uncertainty generated by nature with uncertainty generated by a second player in the trust game. In this case, events are symmetric for nature (e.g Abdellaoui et al., 2011a), while they are asymmetric for the trust game. Hence, the results found by LW might be driven by the beliefs. A second drawback is that the indexes are proposed under the framework of SDW and this does not allow for ambiguity attitudes to be captured by the utility function. Baillon et al. (2018a) proposed a method which releases the assumption of identical utility functions across sources or, in particular, between risk and uncertainty. This method allows to elicit the utility function, the neo-additive weighting function W (P (E)) = sP (E)+c proposed by Chateauneuf et al. (2007), and the beliefs do not require exchangeable events.1 The method consists of using certainty equivalent data of binary lotteries that involve three mutually exclusive and exhaustive events (E 1 , E 2 , E 3 ) and, one composite event (say E 12 ).

Neo-additive method

The neo-additive weighting function and the parametric utility function (e.g. power utility x α ) are speciĄed. The certainty equivalent data can be used in three-stages or one-stage procedure to estimate the utility, the neo-additive weighting function, and the beliefs of each P (E i ), i = 1, 2, 3.

In the three-stages procedure, parametric assumptions are made sequentially.

In the one-stage, the certainty equivalent data associated to one event (say E 1 ) is used to estimate the utility function parameter (say α) and the one event weight (say W (P (E 1 ))), according to the method of Abdellaoui et al. (2008). In the second stage, the certainty equivalent data related to the three remaining events (E 2 , E 3 and E 12 ) are used to compute, in a deterministic way, the three event weights W (P (E 2 )), W (P (E 3 )) and W (P (E 12 )), according to Abdellaoui et al. (2011b). In the third stage, the four event weights from the Ąrst and second stage allow to estimate the two-parameter of the neo-additive weighting function and the three beliefs as follows

c = W (P (E 1 )) + W (P (E 2 )) -W (P (E 12 )) (3.4) s = 3 i=1 W (P (E i )) -3 W (P (E 1 )) + W (P (E 2 )) -W (P (E 12 )) (3.5) P (E i ) = W (P (E i )) -c s for i = 1, 2, 3 (3.6)
In the one-stage procedure, the parametric assumption of the utility and the weighting functions are not made sequentially, but simultaneously. Then, the certainty equivalent data is used in a single step to estimate the utility function parameter (say α), the two-parameters (s and c) of the neo-additive weighting function, and the two beliefs P (E 1 ) and P (E 2 ), knowing that

P (E 3 ) = 1 -P (E 1 ) -P (E 2 ).
Even though the one-stage and three-stages neo-additive methods allow for source-dependent utility function, they might suffer from two drawbacks. The Ąrst drawback applies to both methods. This drawback relies on the fact that the assumption of the neo-additive weighting function may be restrictive to Ąt the data (e.g. Li et al., 2018), probably due to misspeciĄcation issues [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]. Second, in the case of the three-stages approach, the certainty equivalents that are used to compute the event weights in a deterministic way during the second stage contain with response errors. These response errors are not controlled and, then they could bias the future estimates of the event weights in the second stage, as well as generate additional bias in the beliefs of the third stage (Etchart-Vincent, 2004, pg. 221).

Elicitation Method

In this section, we extend the multistage neo-additive method of Baillon et al.

(2018a) to any two-parameter weighting function to elicit beliefs P (.), utility function U (.), and weighting function W (.) for discrete sources of uncertainty. In addition, we show that the two-stage method is more robust to misspeciĄcation than the one-stage method. We keep the same notations as in section 3.2.

Stage 1: Elicitation of utility function and willingness to bet

This stage is based on the all at once method of [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]. The researcher starts by considering a set of m = 3 mutually exclusive and exhaustive nonnull events Ω * 1 = ¶E 1 , E 2 , E 3 ♢. 2 The resulting set of composite events is given by Ω * 2 = ¶E 12 , E 13 , E 23 ♢. Further, the researcher will pick a composite event in Ω * 2 , say E 12 (see also Baillon et al., 2018a). Subsequently, the researcher elicits in an experiment, at least two certainty equivalents for each single event and the chosen composite event

E ∈ Ω * 1 ∪ E 12 ce h k ∽ (x h E , y h E ; E, E c ), h = 1, 2, . . . , N E and N E ≥ 2 (3.7)
with N E being the number of certainty equivalents that involve the event E.

x h E and y h E refer to the outcomes such that x h E > y h E . In total, the number of certainty equivalents elicited is N = E∈Ω * 1 ∪E 12 N E ≥ 2(m + 1) = 8. Now, we denote by ce, x, and y the variables that collect the values ce h E , x h E , and y h E , respectively. Also, we denote by I E a dummy variable that takes the value 1 if the event E occurs and 0 otherwise. Denote δ E ≡ W (P (E)) for

E ∈ Ω *
1 ∪ E 12 . We call δ E the willingness to bet on the event E (Ghirardato and Marinacci, 2001;Abdellaoui et al., 2011a). Also, we assume that the certainty equivalents are observed with additive response error terms e. Assuming that U is invertible, it turns out

ce l = U -1     (U (x l ) -U (y l ))    E∈Ω * 1 ∪E 12 δ E I E l    + U (y l )     + e l (3.8)
Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty where l is the l th line in ce, x, and y. Finally, the Eq. (3.8) is estimated by nonlinear least squares, by giving an explicit functional form for U (and thus for U -1 ). The two-popular utility function are power (eq. 3.16) and exponential (eq. 3.17).

From the estimations results, one gets the parameter(s) of the utility function U (.) and the willingness to bet δ E on the event E ∈ Ω * 1 ∪ E 12 . These willingness to bet correspond to the compound function W (P (.)) evaluated at each single and composite events in the set Ω *

1 ∪ E 12 . This stage allows to reject subjective expected utility theory (that is W (z) = z), if any of the following two equalities is not satisĄed

E∈Ω * 1 δE = 1 and δE 12 = δE 1 + δE 2 (3.9)
The following stage allows to break down the willingness to bet in terms of weighting function W (.), and beliefs P (E) for E ∈ Ω * 1 ∪ E 12 .

Stage 2: Elicitation of weighting function and beliefs

Following Gonzalez and Wu (1999), we assume that the weighting function W (.) is characterized by two parameters η and γ, which correspond to the insensitivity of the decision-maker to likelihood information, and the decision-makerŠs pessimism/optimism, respectively. To make explicit the dependence of the weighting function on η and γ, we write W (.) ≡ W η,γ (.).

With m = 3 single events, we have the following system of 5 equations:

W η,γ (P (E i )) = δE i , i = 1, 2, ..., m = 3 (3.10) W η,γ (P (E 1 ) + P (E 2 )) = δE 12 (3.11) m i=1 P (E i ) = 1 (3.12)
The system of equations (3.10)-(3.12) contains exactly 5 unknown elements:

P (E 1 ), P (E 2 ), P (E 3 ), η, and γ. The Ąrst three equations in (3.10) come from Eq. (3.8). The fourth Eq. in (3.11) comes from Eq. (3.8) and, the fact that the events E 1 and E 2 are mutually exclusive. The last Eq. in (3.12) comes from the fact that the events E 1 ,E 2 , and E 3 are exhaustive. Any two-parameter weighting functions can be speciĄed (see Epper and Fehr-Duda, 2020, for a review) in the system of equations (3.10)-(3.12). The three popular weighting functions in the ambiguity literature are the speciĄcations3 of GE87 (eq. 3.18), P98 (eq. 3.19) and CEG7 (eq. 3.20).

When the estimated decision weights satisfy strict monotonicity4 in the sense that δ E 12 > δ E 1 and δ E 12 > δ E 2 , the system of equations (3.10) -(3.12) could be solved (numerically) to estimate the strictly increasing two-parameter weighting function (i.e. η and γ) and the beliefs P (E 1 ), P (E 2 ), and P (E 3 ).

It is noteworthy to talk about our method when the number m of single events is different from 3. When the number of single events is more than 3, the procedure to apply our method remains unchanged. The beliefs of additional single events can be estimated by using the corresponding number m of the single events in equations (3.10) and (3.12). When the number of single events is m = 2, the Eq. (3.11) collapses from the method because W (1) = 1 by assumption.

In this case, our method does not allow to identify two-parameters weighting

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty function. Instead, it allows to identify one-parameter weighting function (e.g. Tversky and Kahneman, 1992;Prelec, 1998).

Despite the fact that we focus on the presentation of our method on discrete source of uncertainty, it can also apply to continuous-valued sources of uncertainty (see Appendix 3.A for details). In this context, using the subjective probabilities of the three exclusive and exhaustive events E 1 , E 2 , E 3 ⊂ R allows to completely estimate continuous two-parameter distribution like the beta distribution (Abdellaoui et al., 2021a). Consequently, our method covers all types of sources of uncertainty.

Also, the method accommodates both SDU and SDW since we do not require equality of utility or weighting functions between risk and uncertainty. Then, the data allows to discriminate between SDU and SDW.5

Related literature on uncertainty attitudes in experiments

The reminder of the chapter shows how our method can be used to measure beliefs and disentangle crucial forms of uncertainty in trust and coordination games. In this section, we present some of the related literature.

Crucial forms of uncertainty in trust and coordination games

In economic experiments, subjects playing the traditional trust game, face two sources of uncertainty: strategic uncertainty and social ambiguity. First, subjects face strategic uncertainty when the actions of others are uncertain in strategic interactions. Strategic uncertainty confronts individuals with the delicate task of forming beliefs about other individualsŠ decisions [START_REF] Renou | Minimax regret and strategic uncertainty[END_REF].

Second, decision-makers face social ambiguity, when the uncertainty comes from the non-strategic decisions of other individuals. Social ambiguity refers to the fact that subjects treat acts by humans, even in the absence of strategic interactions, differently from acts of nature, which do not involve human agency (LW). Hence, behind any strategic uncertainty, there is also social ambiguity, which might play a role in the decision-making process.

Additionally, social preferences play a role in trust games since players are aware that their actions impact not only their payoffs, but also the payoffs of others [START_REF] Bohnet | Betrayal aversion: Evidence from brazil, china, oman, switzerland, turkey, and the united states[END_REF]

Social ambiguity, strategic uncertainty and betrayal aversion

Under SDU, strategic uncertainty, social ambiguity and betrayal aversion would be captured by the utility function. Contrary, under SDW, they would be captured by the weighting function. The weighting function captures such ambiguous attitudes into two components: optimism/pessimism and likelihood insensitivity (Gonzalez and Wu, 1999). Optimism/pessimism reĆects the extent to which subjects overweight/underweight the beliefs regarding whether the resulting outcome will be beneĄcial for them. On the other hand, likelihood insensitivity refers to subjectsŠ cognitive ability to distinguish between several levels of subjective probabilities or beliefs (e.g. [START_REF] Choi | Probability weighting and cognitive ability[END_REF]. Wakker (2010) refers to optimism/pessimism as a motivational component and, to likelihood insensitivity as a cognitive component in the decision-making process.

BZ develop an experiment, using a version of the trust game, to identify Nevertheless, a vast majority of previous studies (e.g., [START_REF] Heinemann | Measuring strategic uncertainty in coordination games[END_REF][START_REF] Ivanov | Attitudes to ambiguity in one-shot normal-form games: An experimental study[END_REF][START_REF] Bruttel | Measuring strategic-uncertainty attitudes[END_REF], do not control for social ambiguity when they investigate strategic uncertainty. In this chapter, we also aim to identify which components of the utility function and the weighting function (pessimism and likelihood insensitivity) capture the effect of social ambiguity, strategic uncertainty, and betrayal aversion.

Experimental Design

We recruit 174 students to participate in a computerized experiment, which is conducted online. Subjects are invited through the subjects pool of GATE-Lab.

Subjects are told that the experiment could last up to 45 minutes, that they would receive e1.5 as a participation fee and, they could additionally earn a variable amount up to e20. Such additional payment corresponds to a randomly selected outcome of one of the decisions made during the experiment. The mean age of subjects is 21 years and 56.9% are female. Our experiment follows a within-subjects design.

The experiment consists of 5 experimental conditions. Four out of these conditions have two stages, the remaining condition consists only of the second stage. In the Ąrst stage, we implement experimental treatments based on the coordination game and the trust game. In the second stage, we apply a binary decisions task between a safe option and a lottery to elicit beliefs, ambiguity attitudes, and utility functions. 6 The order in which subjects play the Ąve experimental conditions is randomized.

We refer to the blocks containing one or two stages as experimental conditions and to the task implemented in each of the stages as experimental treatments.

The goal of the conditions and treatments is to implement our method experimentally and combine it with empirical data. Each of the experimental conditions allows to elicit ambiguity attitudes linked to different crucial forms of uncertainty.

First stage

We use a within-subjects design along the experiment. In four out of the Ąve conditions of the experiment, the Ąrst stage contains the following experimental treatments: social ambiguity -coordination game (social ambiguity -cg), strategic uncertainty -coordination game (strategic uncertainty -cg), social ambiguitytrust game (social ambiguity -tg), and betrayal ambiguity. In these conditions, 89 subjects play the role of Player 1 and, 85 subjects take the role of Player 2.

Subjects keep their role along the whole experiment. For each condition, new couples formed by Player 1 and Player 2 are randomly re-matched. Subjects are informed that they do not play against the same partner more than once and, they do not receive feedback about the decisions of their counterparts until the end of the experiment.

At the beginning of each condition, Player 1s are informed whether the condition contains one or two stages, speciĄc instructions for each stage are given at the beginning of each stage. Our implemented procedure for incentives allows to avoid hedging issues and it is established as follows. Player 1 received the payoff of one randomly selected decision in either one of the two stages of the four conditions, or one of the decisions made in the remaining treatment (nature). Also, one out of the four decisions done by Player 2, is randomly selected for payoff.

With the social ambiguity -cg treatment, we measure ambiguity attitudes and social ambiguity. Player 1s make a strategic decision between Left (L), Right (R), and Middle (M). On the other hand, Player 2s receive e5 and, their task is to answer where they would prefer to spend this money between an Amazon voucher, a Google Play voucher, and an Apple Store voucher. Player 2s do not know the payoff matrix. As such, Player 2s decide between three possible options that represent their own preferences and are independent of Player 1sŠ decisions. Therefore, Player 1s should not base their decisions on a strategic interaction. However, decisions of Player 2s directly affect Player 1sŠ payoff, which is why Player 1s face social ambiguity. The structure of the payments7

for this treatment is displayed in Table 3.5.1. Treatments social ambiguity -tg and betrayal ambiguity are based on the experimental design of LW. In the social ambiguity -tg treatment (see Figure 4.4.3), Player 1 decides between distrust (D) or trust (T). If Player 1 decides D, she receives a payoff of 10 ECU with certainty. On the other hand, if Player 1 decides T, the payment is determined based on the preferences of Player 2.

Player 2s receive e5 and are asked to decide where they would prefer to spend this money between an Amazon voucher, a Google Play voucher, or an Apple Store voucher. In this treatment, as in the social ambiguity -cg, Player 1s make their decisions facing ambiguity attitudes and social ambiguity.

Second stage: elicitation of beliefs, ambiguity attitudes, and utility function

Only Player 1s perform the second stage of each condition and the remaining condition. The Ąrst stage in every condition is followed by the second stage.

Therefore, Player 1 perform the second stage of each condition immediately after each of the treatments and, only when the task of the second stage is completed, Player 1 moves to the next condition. We elicit Player 1sŠ certainty equivalents through the switching outcome technique (Gonzalez and Wu, 1999;Tversky and Kahneman, 1992) for a list of 12 binary lotteries L = (x, y; E, E c ) that involved Player 2sŠ decisions as events. Such events can be either L, R, and M, or Amazon, Google Play, and Apple Store vouchers, depending on the immediately latest treatment performed by the participant. To infer the certainty equivalent of each lottery L = (x, y; E, E c ), Player 1s are asked to make a series of binary decisions between a lottery and a list of equally spaced safe payoffs, ranged from the maximum value x to the minimum value y of the lottery.

Table 3.5.3 displays an example of the lotteries corresponding to the second stage of the conditions containing the treatments strategic uncertainty -cg and betrayal aversion, where the decisions done by Player 2s correspond to the options L, R, or M . Consider for example, lottery number 1 in the Ąrst set of lotteries in Table 3.5.3. In this case, Player 1 is asked to make eight decisions between a safe outcome and a lottery. Payoffs for the safe option vary from 15 ECU to 8 ECU, while the lottery remains constant.

= L E c 1 = R ∪ M 11.5 ECU 2 15 ECU 8 ECU E 1 = R E c 1 = L ∪ M 11.5 ECU 3 15 ECU 8 ECU E 1 = M E c 1 = L ∪ R 11.5 ECU 4 15 ECU 8 ECU E 1 = L ∪ R E c 1 = M 11.5 ECU Second set of lotteries 5 10 ECU 0 ECU E 1 = L E c 1 = R ∪ M 5 ECU 6 10 ECU 0 ECU E 1 = R E c 1 = L ∪ M 5 ECU 7 10 ECU 0 ECU E 1 = M E c 1 = L ∪ R 5 ECU 8 10 ECU 0 ECU E 1 = L ∪ R E c 1 = M 5 ECU Third set of lotteries 9 15 ECU 0 ECU E 1 = L E c 1 = R ∪ M 7.5 ECU 10 15 ECU 0 ECU E 1 = R E c 1 = L ∪ M 7.5 ECU 11 15 ECU 0 ECU E 1 = M E c 1 = L ∪ R 7.5 ECU 12 15 ECU 0 ECU E 1 = L ∪ R E c 1 = M 7.5 ECU
Finally, Player 1s complete another set of binary decisions which are not linked with any treatment performed before by the participant. Such an additional set of binary decisions is the experimental condition called nature ambiguity. In this task, Player 1s also decide between a safe outcome or a lottery. However, in this case, the outcome of the lottery is determined by nature, which is a randomly equally likely selection between L, R, or M made by the computer. Therefore, this condition allows us to measure only ambiguity attitudes. Example of one of the screens of the task in the nature ambiguity treatment.

Our experimental design allows us to disregard social preferences in this part of the experiment. Therefore, at the stage of elicitation of certainty equivalents, social preferences collapse for Player 1sŠ decision-making process. We use the data of certainty equivalents as input to elicit the utility function, weighting function, and beliefs with our method presented in Section 4.3.

Comparison between treatments and hypotheses

We perform a series of comparisons based on the decisions done by Player 1 in the second stage of the conditions previously presented. The aim of these comparisons between the Ąve treatments is to isolate and capture the effect of social ambiguity, strategic uncertainty, and betrayal aversion. The following are our conjectures.

Comparison between social ambiguity -cg and strategic uncertainty -cg: the condition social ambiguity -cg measures ambiguity

attitudes and social ambiguity. The strategic uncertainty -cg condition,

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty measures ambiguity attitudes, social ambiguity, and strategic uncertainty.9

Hence, with the comparison between these two conditions, we are able to capture the effect of strategic uncertainty.

2. Comparison between nature ambiguity and social ambiguitycg: the nature ambiguity condition captures only ambiguity attitudes and, the social ambiguity -cg condition captures both ambiguity attitudes and social ambiguity. Consequently, the comparison of these two conditions, allows us to capture the effect of social ambiguity under the context of the coordination game.

3. Comparison between nature ambiguity and social ambiguitytg: nature ambiguity condition measures ambiguity attitudes and, social ambiguity -tg measures both ambiguity attitudes and social ambiguity.

Hence, through the comparison of these conditions, we are able to capture the effect of social ambiguity in the context of the trust game.

Comparison between social ambiguity -tg and betrayal aversion:

the condition social ambiguity -tg measures ambiguity attitudes and social ambiguity. The betrayal aversion condition, captures ambiguity attitudes, social ambiguity, strategic uncertainty, and betrayal aversion. Through the comparison of these two treatments we can capture the combined effect of strategic uncertainty and betrayal aversion. In addition, we are able to disentangle the effect of strategic uncertainty and betrayal aversion by controlling for the isolated effect of strategic uncertainty obtained from Based on the previous comparisons, we aim to test the following predictions.11

Hypothesis 1: social ambiguity is captured by pessimism.

Hypothesis 2: strategic uncertainty is captured by likelihood insensitivity.

Hypothesis 3: betrayal aversion is captured by the utility function.

Results

All statistical tests are two-sided z-test computed from median regressions, unless otherwise stated. Inline with the simulation results (see Appendix 3.D), we perform our multi-stage method by assuming sequentially power utility function (Eq. 3.16) in the Ąrst stage and the weighting function of GE87 (Eq. 3.18) in the second stage. First, we estimate the utility and willingness to bet on the events. Second, we estimate beliefs, likelihood insensitivity, and pessimism. The details of individual estimates are provided in Appendix 3.B.

First stage: utility and event weights

In the Ąrst stage, we estimate at the individual level, the utility and willingness to bet on the events. 3.6.2 provides the summary of the estimated values. SEU is true if we cannot reject both that (i) the weights of three mutually exclusive events sum 1 and, (ii) the weight of the composite event is equal to the sum of the weights of the two single events involved in the composition (Eq. 3.9).

Condition (i) cannot be rejected for social ambiguity -cg (pvalue = 0.7642) and it is rejected in all the other treatments (all pvalues < 0.0002). Condition (ii) is systematically rejected in all the treatments (pvalues < 0.0045). Also, a join test of conditions (i) and (ii) leads to a strong rejection in all treatment (all pvalues < 0.0001). Then, subjects violate SEU.

Second stage: beliefs and weighting function

In the second stage, we used the weights of single and composite events from the Ąrst stage (see Figure 3.6.2) to estimate the beliefs (P (.)), likelihood sensitivity (γ), and the pessimism (η) at the individual level. The null hypothesis of equally likely events cannot be rejected for the nature (pvalue = 0.6656) and strategic uncertainty -cg treatments (pvalue = 0.2974). A priori, symmetry of events is expected for the treatment nature.

Beliefs

Similarly, symmetry of events for the strategic uncertainty -cg treatment can be expected, since the coordination game does not have any dominated strategy.

These results provide a Ąrst successful validity test of our method.

On the other hand, symmetry of events is rejected for the social ambiguitycg (pvalue = 0.0061) and social ambiguity -tg (pvalue = 0.0001). In these two conditions, the cumulative distribution function of the beliefs of Player 1

Social ambiguity, strategic uncertainty, and betrayal aversion

Now we turn into the main purpose of the application of this chapter: the identiĄcation of social ambiguity, strategic uncertainty and betrayal aversion through pessimism, likelihood insensitivity and utility curvature. Table 3.6.4

Likelihood insensitivity (γ): the estimates of the likelihood insensitivity for treatments 0, 1 and 3 are 0.633, 0.534, and 0.513, respectively. Likelihood insensitivity is lower in treatment 0 than 1 (p -value= 0.0001). Also, the likelihood insensitivity is lower in treatment 0 than in 3 (pvalue = 0.0037).

Consequently, we conclude that social ambiguity is captured by an increase in likelihood insensitivity. Also, social ambiguity can operate through a decrease in pessimism (treatments 0 vs 1). This partially conĄrms our Hypothesis 1: social ambiguity is capture by pessimism.

Strategic uncertainty

The difference between treatments 1 (social ambiguity -cg) and 2 (strategic uncertainty -cg) corresponds to strategic uncertainty.

Utility curvature (α): the estimate of CRRA parameter for treatments 1 and 2 are 0.876 and 0.988, respectively. The difference in the utility parameters between treatment 1 and 2 is not signiĄcant (p-value= 0.5900, two sided sign test).

Pessimism (η): the estimate of pessimism for treatments 1 and 2 are 0.736 and 0.570, respectively. Pessimism is lower in treatment 1 than in treatment 2 (pvalue < 0.0001).

Likelihood insensitivity (γ): the estimates of likelihood insensitivity

for treatments 1 and 2 are 0.534 and 0.607, respectively. The likelihood insensitivity in treatment 1 is larger than in treatment 2 (p -value= 0.0037).

Consequently, we conclude that strategic uncertainty is captured by a decrease in likelihood insensitivity and by an increase in pessimism. This partially conĄrms our Hypothesis 2: strategic uncertainty is captured by likelihood insensitivity.

Betrayal aversion

The difference between treatments 3 (social ambiguity -tg) and 4 (betrayal Pessimism (η): the estimate of pessimism for treatments 3 and 4 are 0.611 and 0.613, respectively. The difference in pessimism between these treatments is not signiĄcant (p -value= 0.9343).

Likelihood insensitivity (γ): the estimate of likelihood insensitivity for treatments 3 and 4 are 0.513 and 0.524, respectively. The difference in the likelihood insensitivity between treatments 3 and 4 is not signiĄcant (p -value= 0.8122).

In the previously presented results from strategic uncertainty (treatment 1 versus 2), we show that strategic uncertainty decreases likelihood insensitivity. Additionally, the analysis of betrayal aversion shows a non-signiĄcant difference of the likelihood insensitivity between treatments 3 and 4. Nevertheless, given that the comparison between treatments 3 and 4 contains the effect of strategic uncertainty, we should Ąnd different likelihood insensitivities. This opposite result is due to the fact that the effect of strategic uncertainty offsets the betrayal aversion effect, leading to a lack of difference in likelihood insensitivity between treatments 3 and 4. In other words, betrayal aversion and strategic uncertainty are captured by likelihood insensitivity in two opposites directions: strategic uncertainty is captured by a decrease in likelihood insensitivity, while betrayal aversion is captured by an increase in likelihood insensitivity. Consequently, we conclude that betrayal aversion is captured by an increase in likelihood insensitivity. This rejects our Hypothesis 3: betrayal aversion is captured by the utility function.

Dependence payoff aversion and variety of payoff seeking

Besides the previously presented results, we Ąnd that subjects exhibit two additional behaviors which are not related to attitudes toward the source of uncertainty. In this section, we present these Ąndings. Treatments 1 (social ambiguity -cg) and 3 (social ambiguity -tg) have the same source of uncertainty (i.e., preferences of Player 2, which constitutes social ambiguity). Hence, any differences in the utilities and the weighting functions between these two conditions is not due to attitudes toward the underlying source of uncertainty. Instead, the difference between treatments 1 (social ambiguity -cg) and 3 (social ambiguitytg) corresponds to the mixture of dependence payoff aversion and the variety of payoff attitudes.

Utility curvature (α): the estimates of CRRA parameter for treatments 1 and 3 are 0.876 and 0.968, respectively. The difference in the utility parameters between treatments 1 and 3 is signiĄcant (p -value= 0.0165, two sided sign test). Hence, utility is more concave in treatment 1 than in treatment 3.

Pessimism (η): the estimate of pessimism for treatments 1 and 3 are 0.736 and 0.611. The difference between these treatments is signiĄcant (p -value= 0.0001). Therefore, pessimism is lower in treatment 1 than in treatment 3.

Likelihood insensitivity (γ): the estimates of the likelihood insensitivity

for treatments 1 and 3 are 0.534 and 0.513. The difference in the insensitivity between treatments 3 and 4 is not signiĄcant (p -value= 0.5884).

We conclude that the greater concavity of the utility function in treatment 1 compared to treatment 3, represents a payoff dependence aversion. Also, the higher pessimism in treatment 1 compared to treatment 3 constitutes variety of payoff seeking.

Discussion

Experimental discussion

Our method allows to replicate some well known results. First, we conĄrm that the weighting function, in the case of uncertainty, is not an identity function.

Consequently, subjects distort beliefs and then violate the traditional SEU theory (e.g. Abdellaoui et al., 2005Abdellaoui et al., , 2011aAbdellaoui et al., , 2016Abdellaoui et al., , 2021a;;Attema et al., 2018;[START_REF] Li | Trust as a decision under ambiguity[END_REF]Li et al., , 2020;;[START_REF] Tversky | Weighing risk and uncertainty[END_REF][START_REF] Camerer | Violations of the betweenness axiom and nonlinearity in probability[END_REF][START_REF] Bruttel | Measuring strategic-uncertainty attitudes[END_REF]Bleichrodt et al., 2018;Fehr-Duda and Epper, 2012;lŠHaridon and Vieider, 2019). Typically, subjects overweight small subjective probability and underweight intermediate and high subjective probability. Also, we Ąnd that only the weighting function differs across different sources of ambiguity, but not the utility function. This provides support for ambiguity theories based on the weighting function (e.g. Schmeidler, 1989), but not for ambiguity theories based on the utility function (e.g. [START_REF] Klibanoff | A smooth model of decision making under ambiguity[END_REF]. These results are consistent with previous studies (e.g. Abdellaoui et al., 2016;Attema et al., 2018;Abdellaoui et al., 2022;[START_REF] Bruttel | Measuring strategic-uncertainty attitudes[END_REF].

We make two internal validity tests for our method. First, the treatments social ambiguity -cg and social ambiguity -tg involve the same events. Therefore, the distributions of beliefs in these two conditions should be the same. Our method successfully produces this results. Second, the events in the nature treatment are a priori symmetric; as well as the beliefs in the strategic uncertainty -cg treatment, which does not have any dominated strategy. Our method also successfully satisĄes the symmetry test for both nature and strategic uncertainty -cg treatments. Replicating well known results and successfully passing validity tests provide support for our method (Abdellaoui et al., 2008).

We apply our method to measure beliefs towards different discrete sources of uncertainty. One of the remarkable Ąndings in this regard concerns the beliefs about trustworthiness. When subjects trust, they put themselves in a vulnerable situation based upon the belief the other will respond in a positive way [START_REF] Özer | Trust and trustworthiness[END_REF]. As Arrow (1972) wrote Şvirtually every commercial transaction has within itself an element of trust". Because decisions of trust play a major role in social and economic interactions, it becomes important to be able to measure beliefs about trustworthiness, considering that the trustor distorts her own formation of beliefs (weighting functions). We Ąnd that the cumulative distribution function of the beliefs about trustworthiness is Ąrst order stochastically dominated by being betrayed. Most subjects believe that trust is not reciprocated with a mean of subjective beliefs of people being trustworthy equal to 29%.

Regarding our empirical aim of identifying the role of social ambiguity, strategic uncertainty and betrayal attitudes, our method provides the following contributions.

First, we Ąnd that social ambiguity operates mainly through an increase in the likelihood insensitivity. Therefore, subjects prefer social ambiguity over nature ambiguity when there is a small probability of winning, and prefer nature ambiguity over social ambiguity when there is a high probability of winning. The Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty increase in likelihood insensitivity suggests that subjects Ąnd social ambiguity more cognitively demanding compared to nature ambiguity (e.g. Wakker, 2010;[START_REF] Choi | Probability weighting and cognitive ability[END_REF]. Social ambiguity can also operate through a decrease in pessimism compared to nature ambiguity. The fact that subjects are less pessimistic towards ambiguity caused by other humans than ambiguity coming from nature, was pointed out by other studies (e.g. Li et al., 2020;[START_REF] Bolton | Social interaction promotes risk taking in a stag hunt game[END_REF][START_REF] Chark | A neuroimaging study of preference for strategic uncertainty[END_REF]. The decrease in pessimism due to social ambiguity could be explained by the competence hypothesis (Li et al., 2020;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF][START_REF] Fox | Ambiguity aversion, comparative ignorance, and decision context[END_REF]. [START_REF] Fox | Ambiguity aversion and comparative ignorance[END_REF] propose under the competence hypothesis that, subjectsŠ conĄdence is undermined when they contrast their limited knowledge about an event with their superior knowledge about another event. They argue that this contrast between states of knowledge is the predominant source of ambiguity aversion. SubjectsŠ perception of their own knowledge about other humansŠ choices could be higher than their knowledge perception about choices done by nature.

Second, strategic uncertainty also operates, as social ambiguity, through likelihood insensitivity and pessimism, but in opposite directions. Contrary to social ambiguity, strategic uncertainty leads to a decrease in likelihood insensitivity and an increase in pessimism. The difference of likelihood insensitivity supports that subjects prefer social ambiguity over strategic uncertainty for small probabilities of winning and, prefer strategic uncertainty over social ambiguity for high probabilities of winning. These two opposite effects offset. Accordingly, we did not Ąnd a difference of likelihood insensitivity between the treatments nature and strategic uncertainty -cg. This result suggests that subjects tend to exhibit a similar level of likelihood insensitivity towards sources of uncertainty in which events are symmetric (e.g. strategic uncertainty -cg and nature treatments). In contrast, subjects tend to exhibit a high likelihood insensitivity when events are Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty asymmetric, like in our two conditions of social ambiguity (social ambiguity -cg and social ambiguity -tg). This corroborates that beliefs formation process is cognitively demanding.

Third, betrayal aversion also operates through the likelihood insensitivity.

Betrayal aversion increases likelihood insensitivity. We Ąnd that subjects prefer betrayal and social ambiguities over nature ambiguity for small probabilities of winning and prefer nature ambiguity over betrayal and social ambiguities for a high probabilities of winning. Li et al. (2020) do not make a distinction between betrayal aversion and strategic uncertainty. The authors Ąnd that the overall effect of betrayal aversion and strategic uncertainty increases likelihood insensitivity, suggesting that the effect of betrayal aversion is larger than the effect of strategic uncertainty. However, according to our Proposition 1, we should be cautious with the possibility of having a greater effect of betrayal aversion. Indeed, the fact that events are symmetric under nature ambiguity while they are highly asymmetric in the betrayal aversion treatment, can mislead to a difference in likelihood insensitivity measured with the method of Baillon et al. (2018b).

Finally, we identify two main behaviors which are not related to attitudes towards sources of uncertainty. First, the behaviour we call dependence payoff aversion, which represents the fact that subjects dislike situations in which their possible payoffs depend on the preferences of others. This behaviour operates by increasing the concavity of the utility function. Second, the behavior we call variety of payoffs seeking, which proposes that subjects prefer to have more options of possible payoffs, when these payoffs depend on others. The variety of payoffs seeking is captured by a decrease in pessimism for situations that contain more possible payoffs (e.g. social ambiguity -cg) compared to situation containing a lower amount of possible payoffs (e.g. social ambiguity -tg).

Methodological discussion

Our method allows to completely measure the utility function, it is more robust to misspeciĄcation issues, it is easy, and error-robust. Below we discuss these features.

Complete measurement of utility function. Throughout the combination of our method with experimental data, we show the importance of measuring the utility function, which contrasts with previous methods, in which the utility function is not measured (e.g. Baillon et al., 2018b;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]Abdellaoui et al., 2021a). We show that the utility function can capture additional behaviors (e.g. payoff dependence aversion), unrelated to the source of uncertainty. This implies that not measuring the utility function makes more difficult to have a clean empirical measurement of ambiguity attitudes from the existing methods that do not allow the estimation of utility function (e.g. Baillon et al., 2018b;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]Abdellaoui et al., 2021a).

More robust to misspecification. We propose a multistage method instead of one-stage method [START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]. In the Ąrst stage, we only specify utility and estimate events weights non-parametrically. Based on event weights from the Ąrst stage, the method allows to estimate the parameters of any weighting function. Our method thus allows for more Ćexibility in the parametric choices of weighting function in comparison to existing methods (e.g. Baillon et al., 2018bBaillon et al., , 2021Baillon et al., , 2018a) that rely on the neo-additive weighting function of (Chateauneuf et al., 2007).

Easy and error-robust. Our method is based on simple choices that involve the lowest possible number of outcomes (i.e., three). As such, this method is not cognitively demanding -easy -for subjects, compared to methods that are based on exchangeable events or matching probabilities (e.g. Baillon et al., 2018b;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF]Abdellaoui et al., 2021a), in which each choice involves four outcomes [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Abdellaoui et al., 2008). Finally, contrary to previous methods (e.g. Baillon et al., 2018b,a), our method account for response errors that are pervasive in experimental data [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF].

Conclusion

We proposed a two-stage method that clearly measures beliefs and ambiguity attitudes towards discrete sources of uncertainty. Subjects make decisions under these types of uncertain situations in a daily life basis. The method successfully passes validity tests and provides plausible results for trust and coordination games, showing the reliability of the results derived from it. In this chapter, we implement our method to discrete sources of uncertainty; nevertheless, it also applies to continuous-valued sources of uncertainty. Therefore, this method allows to measure beliefs and ambiguity attitudes related to several Ąelds in Economics.

3.A Applicability of the method for continuousvalued sources of uncertainty

This appendix aims to show the validity of our method for continuous-valued sources of uncertainty. Consider the case in which an experimenter aims to measure the distribution of beliefs that a subject holds about a source of uncertainty S that takes its values in an interval I = [s 0 , s 3 ] ⊂ R. The experimenter can proceed through the following three stages.

First stage: utility and event weights. In this step, the experimenter needs to arbitrarily split the universal event I in three exclusive and exhaustive

events E 1 = [s 0 , s 1 ], E 2 = (s 1 , s 2 ] and E 3 = (s 2 , s 3 ] with s 0 < s 1 < s 2 < s 3 .
Hence, we have the composite event 2, 3 (3.13) with F a,b (s 0 ) = 0 and F a,b (s 3 ) = 1. These three equations in (3.13) are summarized in the following two equations

E 12 = [s 0 ,
F a,b (s i ) -F a,b (s i-1 ) = P ([s i-1 , si ]) , i = 1,
F a,b (s 2 ) = P ([s 0 , s1 ]) + P ([s 1 , s2 ]) (3.14) 1 -F a,b (s 1 ) = P ([s 1 , s2 ]) + P ([s 2 , s3 ]) ( 3

3.B Individual estimates

Tables 3.B.1 -3.B.9 give results of our Ąrst stage (α and W (P (.)) and second stage (η, γ and P (.)). Dots in tables mean monotonicity violation and then η, γ and P (.) cannot be estimated.

3.C Experimental instructions

In this appendix we present the instructions we show to Players 2 in the experiment. The order of the presentation of the instructions of each experimental condition is randomized, accordingly to the randomization of the order of the conditions in the experiment. These instructions are translated from the original French instructions.

Beginning instructions

The experiment consists of Ąve (5) parts and will last approximately 45 minutes.

You will receive speciĄc instructions for each part at the beginning of each of them. At the end of the experiment, only one part out of the Ąve will be randomly selected to determine your Ąnal payment. Each of these Ąve parts has the same chance of being randomly selected by the computer. In each part, you make several decisions. If a part is randomly selected for payment, one of the decisions in that part will be randomly selected by the computer. Each decision Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty has the same chance of being drawn at random. Therefore, only one of your decisions will affect your Ąnal payment, but it could be any of your decisions.

Thus, it is in your best interest to make each decision as if it were the one that will be selected for payment.

Payments for your decisions will be expressed in experimental currency units (ECU). Please note that each ECU is equal to 1 euro. For example, 1 ECU = e 1 and 15 ECU = e 15.

Social ambiguity -coordination game

You will now read the instructions for Part 1 of the experiment. Part 1 has two sub-parts. You will receive instructions for each sub-part before you make your decisions in each of them.

First stage

Instructions for the Ąrst sub-part of Part 1

In this part of the experiment, you are randomly paired with another participant, we call this person, Participant 2. You will never be informed of Participant 2Šs identity, nor will Participant 2 ever be informed of your identity.

Your Ąnal payment will depend on your decision and the decision of Participant 2.

Your decision in this section will be to choose an action between Left, Right or Middle. Participant 2 will receive 5 euros. Then, Participant 2 will decide where he/she would prefer to spend these 5 euros between one of the following options: An Amazon voucher, a Google Play voucher or an Apple Store voucher.

You will not be notiĄed of Participant 2Šs decision until you receive payment for

Second stage

Instructions for the second sub-part of Part 1

In the second and Ąnal subpart of this part of the experiment, you will choose between several options. The options will be presented in 12 tables (see an example of the table below). Each row represents one option. For each option, you will be asked to indicate whether you prefer Alternative A or Alternative B.

• Alternative A offers you a safe payment. In each line you will be asked to indicate whether you prefer Alternative A or Alternative B.

Both alternatives are initially displayed in gray. You must click on one of the two alternatives to select it. Your selection will be highlighted in blue.

You can change your selection at any time by clicking on the cell of the desired alternative, before moving on to the next screen. Once you conĄrm your decision, you cannot go back and change your previous decision.

If you select Alternative A for a given row, the computer will mark Alternative A for all previous rows (up to the Ąrst). Similarly, if you select Alternative B for a line, the computer will mark Alternative B for all subsequent lines (up to the last one).

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty

Example

Suppose that the following option is randomly selected for payment: During this task, you will be able to use the back button to re-view the decisions that you and Participant 2 were asked to make in the Ąrst sub-part of this part of the experiment.

Strategic uncertainty -coordination game

You will now read the instructions for Part 2 of the experiment. Part 2 has two sub-parts. You will receive instructions for each sub-part before you make your decisions in each of them.

Second stage

Instructions for the second sub-part of Part 2

In the second and Ąnal sub-part of this part of the experiment, you will choose between several options. The options will be presented in 12 tables (see an example of the table below). Each row represents one option. For each option, you will be asked to indicate whether you prefer Alternative A or Alternative B.

• Alternative A offers you a safe payment. In each line you will be asked to indicate whether you prefer Alternative A or Alternative B.

Both alternatives are initially displayed in gray. You must click on one of the two alternatives to select it. Your selection will be highlighted in blue.

You can change your selection at any time by clicking on the cell of the desired alternative, before moving on to the next screen. Once you conĄrm your decision, you cannot go back and change your previous decision.

If you select Alternative A for a given row, the computer will mark Alternative A for all previous rows (up to the Ąrst). Similarly, if you select Alternative B for a line, the computer will mark Alternative B for all subsequent lines (up to the last one).

First stage

Instructions does not know how his or her decision is associated with your payment.

A numerical example of possible payments for this part of the experiment can be summarized as follows:

• If you choose Left, you receive 30 ECU for sure.

• If you choose Right, your payment depends on the decision of Participant 2, as follows:

-If Participant 2 chooses an Amazon voucher, you receive 45 ECU.

In the second and Ąnal sub-part of this part of the experiment, you will choose between several options. The options will be presented in 12 tables (see an example of the table below). Each row represents one option. For each option, you will be asked to indicate whether you prefer Alternative A or Alternative B.

• Alternative A offers you a safe payment.

• In each line you will be asked to indicate whether you prefer Alternative A or Alternative B.

Both alternatives are initially displayed in gray. You must click on one of the two alternatives to select it. Your selection will be highlighted in blue.

You can change your selection at any time by clicking on the cell of the desired alternative, before moving on to the next screen. Once you conĄrm your decision, you cannot go back and change your previous decision.

If you select Alternative A for a given row, the computer will mark Alternative A for all previous rows (up to the Ąrst). Similarly, if you select Alternative

Second stage

Instructions for the second sub-part of Part 4

In the second and Ąnal sub-part of this part of the experiment, you choose between several options. The options are presented in 12 tables (see an example of the table below). Each row represents an option. For each option, you must indicate whether you prefer Alternative A or Alternative B.

• Alternative A offers you a safe payment. In each line you will be asked to indicate whether you prefer Alternative A or Alternative B.

Both alternatives are initially displayed in gray. You must click on one of the two alternatives to select it. Your selection will be highlighted in blue.

You can change your selection at any time by clicking on the cell of the desired alternative, before moving on to the next screen. Once you conĄrm your decision, you cannot go back and change your previous decision.

If you select Alternative A for a given row, the computer will mark Alternative A for all previous rows (up to the Ąrst). Similarly, if you select Alternative B for a line, the computer will mark Alternative B for all subsequent lines (up to the last one).

Example

Suppose that the following option is randomly selected for payment: During this task, you will be able to use the back button to re-view the decisions that you and Participant 2 were asked to make in the Ąrst sub-part of this part of the experiment.

Nature

In The payment is determined as follows:

-If the computer selects Left, you win 20 ECU.

-If the computer selects Right or Middle, you win 13 ECU.

3.D Comparison of multi-stage and one stage approaches

We propose a multistage method in which the utility function and the probability weighting function are speciĄed sequentially. 13 In this section, we compare our multi-stage approach with the one-stage approach in which the utility and weighting functions are speciĄed simultaneously. To that end we conduct parameter recovery and misspeciĄcation exercises (e.g. Gao et al., 2020;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Nilsson et al., 2011).

Simulated data

We consider six speciĄcations resulting from the combination of two utility functions u(.) and three weighting functions w(.).14

The two utility functions u(.) are P(ower) (Eq. 3.16) and E(xponential) (Eq.

3.17):

U (z) = z α (3.16) u(z) = 1 -exp(-αz) α (3.17)
For the power utility, α < 1 (resp. α > 1) means concavity (resp. convexity) and α = 1 corresponds to the linear case. For the exponential utility, α > 0 (resp. α < 0) means concavity (resp. convexity) and α -→ 0 corresponds to the linear case. To have a common measure of the utility curvature to facilitate comparisons, we adopt the following measure of the utility curvature over the range of outcomes [0, q] [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]Abdellaoui et al., 2016)

β = 1 qu(q) q 0 u(t)dt
with β > 0.5 (resp. β < 0.5) meaning concavity (resp. convexity) and α = 0.5 corresponds to the linear case.

The three weighting functions w(.) are the speciĄcations of GE87 (Eq. 3.18), P98 (Eq. 3.19) and CEG7 (Eq. 3.20)

W (P (E)) = ηP (E) γ ηP (E) γ + (1 -P (E)) γ (3.18) W (P (E)) = exp -η -ln(P (E)) γ (3.19) W (P (E)) = γP (E) + η (3.20)
with γ > 0 , η > 0.

For the speciĄcation of CEG7, the pessimism and insensitivity indexes are

given by 1η -2γ and 1η, respectively (e.g. Abdellaoui et al., 2011a). For the speciĄcation of P98, the parameters η and γ are an index of pessimism and an anti-index of likelihood insensitivity, respectively (Abdellaoui et al., 2021a). For the speciĄcation of GE87, the parameters η and γ are an antiindex of pessimism and an anti-index of likelihood insensitivity, respectively (e.g. Gonzalez and Wu, 1999). 15 Insensitivity makes weighting the function Ćatter in the range of intermediate subjective probability and steeper near the ends.

Hence, the weighting function follows an inverse S-shaped. Pessimism determines the elevation of the weighting function.

The calibration of lotteries follows the outcomes in [START_REF] Li | Trust as a decision under ambiguity[END_REF]Li et al. ( , 2020) ) and the ones from our experiment. We consider 12 lotteries L = (x, y, E, E c ) that results from the combination of three pairs of outcomes (x, y)= (10, 0), (15, 0), and (15, 8) and, four events E = E 1 , E 2 , E 3 and E 12 .

Simulated data 1: P & GE87. We simulate data for 250 (s = 1, 2, ..., 250) hypothetical subjects. For each subject s, we draw the parameters of weighting function η and γ of GE87 from U(0.1, 1.5). We draw the parameter of the power 15 For this specification, the crossing point is given by

W (p * ) = p * = 1 1+η 1 γ-1
and, W (.) is well defined over all the probability range including the boundary W (0) = 0 and W (1) = 1.

utility function α from an uniform distribution U(0.1, 2.1) (e.g. Abdellaoui et al., 2008;Spiliopoulos and Hertwig, 2019). For the beliefs, we draw P (E 1 ) and P (E 2 ) from U (0, 1) and keep only the cases where P (E 1 ) + P (E 2 ) < 1. We derive then P (E 3 ) = 1 -P (E 1 ) -P (E 2 ). Then, the simulated α, η, γ, P (E 1 ), P (E 2 ), and P (E 3 ) are plugged into the RDU formulas to generate noiseless certainty equivalents of the 12 lotteries. In the last step of the data generation process, we draw 12 random values from a normal distribution with expected value 0 and standard deviation σ = 0.25, which we add to the previously generated 12 noiseless certainty equivalents to obtain the noisy ones.

Simulated data 2: P & P98. similar as simulated data 1, but in this case the two-parameter weighting function of P98 is used. We draw η and γ of P98 from U(0.1, 1.5).

Simulated data 3: P & CEG7. Similar as simulated data 2, but in this case the two-parameter weighting function of CEG7 is used. We draw η and γ of CEG7 from U(0, 1). Simulated data 4: E & GE87. Similar as simulated data 1, but in this case the CARA utility function is used. We draw α from U(-0.15, 0.15).

Simulated data 5: E & P98.

Similar as simulated data 2, but in this case the CARA utility function is used. We draw α from U(-0.15, 0.15). Simulated data 6: E & CEG7. Similar as simulated data 3, but in this case the CARA utility function is used. We draw α from U(-0.15, 0.15).

Simulation results

We conduct two types of estimations for each approach by using the six simulated data. In the Ąrst type of estimation, we estimate by using the correct speciĄcation of the utility and weighting functions that are behind the simulated data. This Ąrst type of estimation corresponds to the parameter recovery exercise in which Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty the purpose is to assess the ability of the two approaches to identify the targeted parameters (Murphy and ten Brincke, 2018;Gao et al., 2020;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF].

In the second type of estimation, we make the estimation on each of the simulated data by assuming the 5 other speciĄcations of utility and weighting functions that are not behind the simulated data. This second type of estimation corresponds to the misspeciĄcation exercise in which the purpose is to assess the extend to which a wrong speciĄcation of utility and weighting functions will affect the estimation results (Gao et al., 2020;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF]. Table 3.D.1 provides the average of the squared difference between the true values of the parameters and their estimated values over the 250 hypothetical subjects. Table 3.D.1 shows evidence of the fact that the two-stage approach leads to smaller error than the one-stage approach. These results can be explained by the use of a semi-parametric method in the Ąrst stage of our method. The Ąrst stage provides a semi-parametric estimates in which the utility function is speciĄed and no parametric assumption is made on event weights. Previous simulation results (e.g. [START_REF] Kim | Comparison of semiparametric and parametric methods for estimating copulas[END_REF][START_REF] Mahmoud | Semiparametric single index multi change points model with an application of environmental health study on mortality and temperature[END_REF][START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] have shown that semi-parametric methods are preferable to parametric methods due to misspeciĄcation issues. Furthermore, the two-stage approach based on the power utility function in combination with the two-parameter weighting function of GE87 leads to smaller errors.

Introduction

Principal-agent theory has played a key role in understanding human behavior across disciplines ranging from Ąnance, accounting, strategy and political science to neuroeconomics (Jensen and Meckling, 1976;Lambert, 2001;Miller, 2005;Brocas and Carrillo, 2008;Dranove et al., 2017). In economics, the principal-agent framework is a cornerstone of numerous Ąelds including the theory of incentives.

A central result in this literature is the existence of a tradeoff between providing incentives to foster the effort of risk-averse agents and protecting them against risk (Borch, 1962;Mirrlees, 1974;Holmström, 1979;Shavell, 1979;[START_REF] Milgrom | Economics, Organization and Management[END_REF]Laffont and Martimort, 2002;Bolton and Dewatripont, 2005;Gibbons and Roberts, 2013). The risk-incentives tradeoff (RIT, henceforth) emerges because providing steeper incentives implies strengthening the link between output and rewards which, in turn, exposes agents to a greater risk whenever output is a noisy measure of effort. As a result, the optimal contract typically incorporates a variable pay that incentivizes the agent to exert effort and a Ąxed pay that partially hedges the agent against output shocks. These types of contracts are widespread, and unsurprisingly RIT has been applied to a wide variety of economic settings ranging from sharecropping to medical insurance (e.g., Zeckhauser, 1970;Stiglitz, 1974) and compensation setting in Ąrms (Gibbons and Roberts, 2013).

Although the rationale for RIT is appealing, empirical evidence remains scarce [START_REF] Garen | Executive compensation and principal-agent theory[END_REF]Prendergast, 2002;Lazear and Oyer, 2013). A glimpse of hope has recently come from laboratory studies reporting some evidence for RIT [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF]Chowdhury and Karakostas, 2020).

These lab studies control for possible confounding factors such as organizational hierarchies, implicit incentives or tacit knowledge that are notoriously difficult to control for in the Ąeld. Yet, the size of the effect reported in [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF] and Chowdhury and Karakostas (2020) remains small.1

Furthermore, recent evidence from laboratory experiments by [START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF] is not consistent with RIT since the presence of output risk does not lead agents to demand weaker incentive schemes, that is lower piece rates.

In this paper, we use theory and experiments to investigate whether the limited evidence for RIT is due to a lack of robustness of the underlying theory or to the confounding factors in empirical tests. Principal-agent models are notorious for their lack of tractability (Grossman and Hart, 1983;Rogerson, 1985) which has led researchers to focus on particular speciĄcations such as the LEN (Linear Exponential Normal) model (see [START_REF] Milgrom | Economics, Organization and Management[END_REF]Varian, 1992;Laffont and Martimort, 2002;Bolton and Dewatripont, 2005;Gibbons and Roberts, 2013;Besanko et al., 2017). In this model, the risk-neutral principal proposes the agent a linear contract composed of a Ąxed pay and a share of output. The risk-averse agent who maximizes expected utility (assumed to be exponential) then decides whether to accept the contact or not. In case of acceptance, the agent chooses a level of effort under the agreed-upon contract.

Even though the principal cannot observe the level of effort, she can observe the Ąnal output, which is impacted by an additive (normally distributed) shock.

The classical version of RIT is derived assuming Expected Utility Theory (EUT, henceforth) and the LEN speciĄcation. Although the LEN model has often been discussed and defended by contract theorists on the basis of tractability and realism (Holmstrom and Milgrom, 1987;Diamond, 1998;Laffont and Martimort, 2002;Bolton and Dewatripont, 2005;Carroll, 2015;Holmström, 2017), little is known about the robustness of RIT in non-EUT settings. This led us to study the robustness of RIT to alternative theories that allow for distortions of probabilities (Rank-Dependent Utility theory, RDU, henceforth, Quiggin, 1982) and an explicit preference for skewness (Mean-Variance-Skewness, MVS, henceforth, Spiliopoulos and Hertwig, 2019). Risk attitudes have been traditionally characterized by the curvature of the utility function. However, non-EUT models characterize risk attitudes along different dimensions. For example, overall risk attitudes under RDU stem both from utility risk attitudes (i.e., the curvature of the utility function) and probability risk attitudes (i.e., probability weighting). Under MVS, overall risk attitudes depend on agentsŠ preferences for variance and skewness.

We theoretically show that RIT is pervasive under RDU because it occurs not only when agents are overall risk-averse, but also when they are risk-neutral or risk-seeking. For example, RIT happens when overall risk-neutral or riskseeking agents are moderately risk-seeking in the probability domain (i.e., they moderately overweight probabilities) and risk-averse in the utility domain (i.e., their utility function is concave). This result suggests that RIT might be more widespread than predicted by EUT. Yet, this observation contrasts with the limited empirical evidence for RIT.

In line with existing empirical evidence, MVS provides a setup in which RIT is less pervasive than under EUT. This happens because RIT does not occur for all risk-averse agents (as in EUT) and disappears for those that exhibit a preference for positive skewness. Furthermore, the optimal variable pay (Ąxed pay) can increase (decrease) with risk, which is what we refer to as reversed RIT. This occurs when the distribution of the shock is positively skewed and the aversion to variance is less pronounced than the value of skewness for a risk-averse agent exhibiting MVS preferences.

These theoretical results show that non-EUT models provide reasons for both hope and despair regarding the robustness of RIT. To test the predictions of the alternative theories, we develop a novel experimental testbed for RIT that eliminates confounding factors. We focus on agentsŠ decisions by eliciting the minimum Ąxed pay they are willing to accept for different values of the variable pay. As principals do not make contractual decisions (see, e.g., [START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF] we can discard confounding factors related to their risk attitudes. This design also eliminates any asymmetry of information between the principal and the agent whose preferences are unknown. We also use monetary effort instead of a real-effort task (see e.g., Anderhub et al., 2002;Keser and Willinger, 2007;[START_REF] Gächter | Design a contract: A simple principalagent problem as a classroom experiment[END_REF] to discard other confounding factors often present in experimental data such as social motives and reference points (see [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF], as well as more general ones such as organizational hierarchies, delegation, implicit incentives, tacit knowledge, uncertainty and market dynamics (Jensen and Meckling, 1995;[START_REF] Raith | SpeciĄc knowledge and performance measurement[END_REF]Adams, 2005;[START_REF] Devaro | An empirical analysis of risk, incentives and the delegation of worker authority[END_REF][START_REF] Edmans | Dynamic ceo compensation[END_REF]He et al., 2014). We do not mean to underplay the importance of these factors but rather aim at implementing a testbed for the basic mechanism underlying RIT. It follows that a lack of evidence supporting RIT in our setup would be a deĄnitive blow for the theory.

To ensure that our design can be effectively used to study RIT, we analyze a BareBone (BB, henceforth) principal-agent model.

Our experiment shows that RIT is remarkably robust and more pervasive than predicted by EUT. In line with RDU but in contrast with EUT and MVS, RIT arises even when agents are risk-seeking. This Ąnding has direct implications for various applications of the theory in which agents are risk-seeking, as is the case of executive compensation [START_REF] Garen | Executive compensation and principal-agent theory[END_REF][START_REF] Edmans | The effect of risk on the ceo market[END_REF][START_REF] Edmans | Dynamic ceo compensation[END_REF][START_REF] Edmans | Executive compensation: A survey of theory and evidence[END_REF] and high-pay workers [START_REF] Ma | Portfolio manager compensation in the us mutual fund industry[END_REF]. Risk-seeking is likely to be pervasive in these cases because of selection effects [START_REF] Maccrimmon | Characteristics of risk taking executives[END_REF][START_REF] Brenner | The risk preferences of us executives[END_REF]. Furthermore, executive packages are often positively skewed due to, for example, the use of stock options [START_REF] Edmans | Executive compensation: A survey of theory and evidence[END_REF]. As a result, an agent who appears to be risk-averse when rewarded according to a linear contract might be risk-seeking when facing a skewed compensation package.

The remainder of the paper is organized as follows. Section 4.2 presents the theoretical models for RIT under EUT, RDU and MVS. In Section 4.3, we describe the experimental design. Section 4.4 presents the results of the experiment and Section 4.5 concludes.

Model

Standard setup and predictions

Assumptions

A risk-neutral principal offers a contract to an agent to perform a task. If the agent does not accept the contract, he receives an outside option y 0 . If he accepts the contract, then he has to exert effort e to produce output z = z(e), where z = e + ε and ε is a random variable. Thus, there is a noisy relationship between effort and output. The cost of effort function C(e) is increasing and convex.

The principal observes the level of output, but not the underlying level of effort.

The principal maximizes her revenue θz, where θ > 0 denotes the marginal product of effort. To ease exposition, we consider a binary shock model (see e.g., [START_REF] Milgrom | Economics, Organization and Management[END_REF], which is often used in empirical tests of the theory (see [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF][START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF]. 2 In Appendix 4.C we further show that our predictions qualitatively hold if we consider a continuous shock. In particular, this includes the special case of the normally distributed shock used in the LEN model (see e.g., [START_REF] Milgrom | Economics, Organization and Management[END_REF];

RDU (L) • • • ϵ ϵ 0 ϵ 1 ϵ * u(E[L])
Risk seeking MVS Under MVS, the agent evaluates a lottery according to its mean, variance and skewness (Kraus and Litzenberger, 1976;Spiliopoulos and Hertwig, 2019) as follows:

Risk aversion

M V S(L) = E(L) + a v V (L) + a s S(L) (4.4)
where a v is a parameter that captures attitudes towards variance and a s captures attitudes towards skewness. In line with the LEN framework and with empirical evidence (Kraus and Litzenberger, 1976;Spiliopoulos and Hertwig, 2019), we assume that the agent is averse to variance (i.e., a v < 0) and seeks positive skewness (i.e., a s > 0). Furthermore, in contrast to EUT, we explicitly consider following the literature (e.g., Spiliopoulos and Hertwig, 2019;Mitton and Vorkink, 2007) that a v and a s are unrelated. 6 M V S(L)

• • • ϵ ϵ 0 ϵ 1 ϵ * E[L]
Risk seeking

Risk aversion

RIT predictions

For each theory, we can characterize the optimal contract: the Ąxed pay (α * ) and the variable pay (β * ) proposed by the principal, as well as the optimal level of effort e * provided by the agent. We provide the corresponding proofs in Appendix 4.A. Here, we focus on characterizing the conditions of existence of RIT for the optimal contract. DeĄnition 4 below characterizes RIT and reversed RIT. RIT occurs when increasing the variable pay generates a trade-off between desirable and undesirable consequences, thus forcing the principal to set a compensation contract with an intermediate intensity of incentives (i.e., 0 < β * < 1). On the positive side, increasing the variable pay incentivizes the agent to exert more effort. On the negative side, it increases the level of risk faced by the agent because it makes his pay more sensitive to output shocks.

It follows that under RIT an increase in the magnitude of the output shock requires the principal to set a contract that limits the agentŠs exposure to the shock. This is achieved by decreasing the variable pay while increasing the Ąxed pay to ensure the agent is willing to accept the contract (see DeĄnition 4i).

Definition 4 (RIT and Reversed RIT)

i) RIT corresponds to the case in which the optimal variable pay (Ąxed pay) decreases (increases) in the output shock ϵ for a given p.

ii) Reversed RIT corresponds to the case in which the optimal variable pay (Ąxed pay) increases (decreases) in the output shock ϵ for a given p.

iii) No RIT corresponds to the case in which there is no relationship between the optimal pay and the output shock ϵ for a given p.

Under EUT, RIT always occurs for risk-averse agents (Assumption 3Š Example 2 (Reversed RIT for a risk-averse agent under MVS). We consider (ψ, θ, y 0 ) = (0.5, 1, 20) and a v = -0.0229 and a s = 0.0037 following the estimates provided in Spiliopoulos and Hertwig (2019). In the absence of shock (i.e., ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 19.5. In the presence of a shock (ϵ, p) = (1, 0.32), we obtain β * = 1.02 and α * = 9.70. Thus, the optimal variable (Ąxed) pay in the presence of a shock is larger (smaller) than in the absence of shock implying reversed RIT. Because the expected value of contract (E[L * ]=20.09) is higher than its certainty equivalent (20), the agent is risk-averse for the optimal contract (α * , β * ) = (9.70, 1.02). As a result, we observe reversed RIT for a risk-averse agent.

We summarize our theoretical RIT predictions for EUT, RDU and MVS in

The BareBone model

Our aim is to test RIT in a BareBone (BB) experimental design that is robust to commonly observed deviations from standard Assumptions 1-5. In practice, the risk-neutrality of the principal cannot be ensured so that Assumption 1 does not necessarily hold in the lab. In addition, principals do not know the risk preferences of agents notwithstanding Assumption 4. To alleviate these issues, we adopt an empirical approach that directly elicits the minimum Ąxed pay (α m ) agents are ready to accept given a preset value of the variable pay (β).

This approach allows us to focus on agentsŠ decisions abstracting away from principalsŠ contractual decisions. In our BB approach, RIT can be deĄned as i) The minimum Ąxed pay increases in utility risk-aversion and probability risk-aversion.

ii) Under probability risk-aversion (w(p) < p), the minimum Ąxed pay increases in ϵ.

iii) Under probability risk-seeking (w(p) > p), there exists a threshold r to (β, ϵ) > 0 such that the minimum Ąxed pay increases (decreases) in ϵ if and only if r > r to (β, ϵ).

iv) We have r to (β, ϵ) < r N (β, ϵ), where r N (β, ϵ) is the level of absolute risk aversion such that a probability risk-seeking agent exhibits risk-neutrality for the contract (α m , β).

v) The two thresholds r to (β, ϵ) and r N (β, ϵ) decrease in ϵ and β.

Under MVS, we show that RIT holds whenever the shock is negatively skewed (p ≥ 1/2) in which case the agent is risk-averse (see Proposition 3ii).

Furthermore, Proposition 3iii shows that for a positively skewed shock (p < 1/2), RIT [reversed RIT] holds as long as the value of the ratio , τ := -av as , between aversion to variance and preference for positive skewness is above [below] a certain threshold (τ to (β, ϵ)), that is as long as the agent has a sufficiently high [low] aversion to variance relative to his preference for positive skewness. Given that τ to (β, ϵ) is higher than the ratio (τ N (β, ϵ)) for which an MVS agent exhibits risk-neutrality given the contract (α m , β) (see Proposition 3iv), a risk-averse MVS agent exhibits reversed RIT for anyav as ∈ (τ N (β, ϵ), τ to (β, ϵ)). 7 Finally, 7 From Definition 2 (overall risk attitudes) and equation (4.4) of MVS, the risk premium in MVS is equal to a v V (L) + a s S(L). Risk neutrality corresponds to a v V (L) + a s S(L) = 0 or equivalently to -av as = S(L) V (L) := τ N (β, ϵ) and risk-aversion (risk-seeking) corresponds to a v V (L) + a s S(L) > 0 (< 0) or equivalently to -av as < (>) τ N .

Elicitation of minimum fixed pay

We elicit the minimum Ąxed pay (α m ) an agent is willing to accept given the incentive contract (β), as well as the magnitude and the probability of occurrence of the shock (ϵ, p). We thus elicit α m for various combinations of (β, ϵ, p) based on the following indifference condition:

L(α m ♣β, ϵ, p) ∼ y 0
where y 0 is the riskless outside option and L(.) is the lottery associated with a given incentive contract (β) and a given shock (ϵ, p) as deĄned in (4.1). We vary the triplet (β, ϵ, p) while Ąxing the parameters of the cost of effort function (ψ = 2.5), the marginal product of effort (θ = 100), and the outside option (y 0 = 1, 000). We consider 30 combinations of (β, ϵ, p) ∈ ¶0.3, 0.5, 0.7♢ × ¶3, 4♢ × ¶0.1, 0.25, 0.33, 0.5, 0.75♢. 8 For each combination, we also assume that the agent implements the optimal level of effort e * that maximizes the value of the lottery so that we elicit α m as follows:

L(α m ♣β, ϵ, p; e * ) ∼ y 0

Where e * = βθ 2ψ . In the experiment, we automatically implement the optimal level of effort because it is a trivial decision for the agent. This allows us to focus on the choice of α m . For each combination (β, ϵ, p), we elicit α m using a multiple price list à la Holt and Laury (2002) in which we vary the Ąxed pay of a contract in increments of 50 between 0 and 1,000 for a total of 21 possible values. We set an upper bound equal to the value of the outside option (1,000).9 We do not consider the trivial case of ϵ = 0 for which the task boils down to picking the highest value in a table of numbers.

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well task: 0.1, 0.25, 0.33, 0.50, and 0.75. An answer within a 5% range of the actual frequency is worth 10 cents. Our design simpliĄes previous RIT experiments in two ways. First, we focus on the agentŠs decision to accept or reject a contract that is exogenously set by the experimenter (see e.g., [START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF]. As previously mentioned, this allows us to leave aside issues related to unknown risk preferences and asymmetric information between the agent and the principal. Furthermore, it allows us to discard fairness motives that can affect the principalŠs offer and the agentŠs acceptance decision. As observed in other principal-agent experiments, an equal split of revenues is often a modal response (see e.g., Anderhub et al., 2002;Keser and Willinger, 2007;[START_REF] Gächter | Design a contract: A simple principalagent problem as a classroom experiment[END_REF][START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF].

In contrast to [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF] and [START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF] who implement a real-effort task, our design relies on a monetary measure of effort (as in Chowdhury and Karakostas, 2020). Not using a real-effort task to elicit effort allows us to specify the cost of effort function and focus on the agentŠs acceptance decision. In our design, providing effort consists in making a money transfer at a monetary cost to the agent. The optimal effort decision turns out to be trivial to calculate and is automatically computed by our experimental software. The use of monetary effort allows us to present agentŠs choices in a payoff table (see Figure 4.3.1). We expect this layout contributes to downplaying the role of reference points so that we can center our analysis of RIT under non-EUT models on probability distortions (RDU) and attitudes towards variance and skewness (MVS). 10 In RIT setups using a real-effort task, reference dependence appears to play a role in explaining the impact of output shocks 10 Despite experiencing a potential negative shock, agents were typically not shown any losses. In very few instances, a small loss of 30 appeared in the first row when the shock was 4, the fixed pay was 0, and the variable pay was 0.3. This occurred in less than 0.5% of the cells presented to participants. Furthermore, these rows involved trivial decisions and were not critical switching points between Options A and B.

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well recruited a total of 237 participants from a pool of more than 2,000 students at a major experimental economic laboratory in France. 13 All sessions were conducted online using Qualtrics. The average duration was 23 (29) minutes for the main (survey) sessions. The average earnings for the two sessions were 18.54 euros including a 4 euro Ćat fee paid for completing both sessions. The complete set of instructions is available in Appendix 4.E.

Results

Risk attitude parameters and classification of agents

The aim of this section is twofold. First, we test the assumptions about risk attitude components (utility function, probability weighting function, attitudes towards variance and skewness) underlying our theoretical predictions. That is, we aim to empirically check if agents exhibit a concave utility function (r > 0) under EUT, and overweight small probabilities (w(p) > p) and underweight intermediate and high probabilities (w(p) < p) under RDU. For MVS, we also check whether people exhibit an aversion to variance (a v < 0) and a preference for skewness (a S > 0) (MVS). Second, using these three models we classify participants into types by identifying the speciĄcation that best Ąts their decisions.

Risk attitude parameters

We use agentsŠ switching points in the minimum Ąxed pay elicitation task to determine their certainty equivalents for binary lotteries with various probabilities.

-neutral if ♣α ϵ m -α 0 m ♣< 25, risk-averse if α ϵ m -α 0 m ≥ 25 and risk-seeking if α ϵ m -α 0 m ≤ -25.
We now consider individual-level analyses. An agent exhibits RIT (reversed

RIT) [no RIT] if α 3 m < α 4 m (α 3 m > α 4 m ) [α 3 m = α 4 m ]
, that is the agent asks for a higher (lower) [identical] minimum Ąxed pay for a larger shock. 14 Table 4.4.5 provides an overview of our empirical Ąndings on the occurrence of RIT depending on individual risk attitudes.

Result 1 (RIT and risk attitudes at the individual level): Most risk-averse (50.96%) and risk-seeking (60.04%) agents make choices that are consistent with RIT. Most risk-neutral agents (52.42%) make choices that are consistent with the absence of RIT.

According to the theoretical predictions in Table 4.2.1, the data should be concentrated on the diagonal of Table 4.4.5 under EUT. However, we observe that only 41.29% of the choices are on the diagonal, thus rejecting EUT predictions.

The main deviation from EUT resides in the fact that risk-seeking agents also 14 RIT could also be defined using the differences α 3 mα 0 m and α 4 mα 0 m . Instead, we use the term "risk attitudes" to refer to these differences as they coincide with the existence of a risk premium for contracts (α 3 m , β) and (α 4 m , β).

occurrence of RIT (last three columns).

Result 2 (Shock size, variable pay, individual risk attitudes and RIT)

An increase in the variable pay (β) or the shock size (ϵ) increases the likelihood of risk-aversion while decreasing the likelihood of risk-neutrality and risk-seeking attitudes. In addition, an increase in the variable pay increases the probability of RIT while decreasing the probabilities of No-RIT and Reversed-RIT.

Result 2 corroborates the RDU predictions (Proposition 2v) and contradicts

MVS (Proposition 3v

). This result also contradicts EUT which posits that both RIT and risk attitudes should not be impacted by changes in the variable pay or the shock.

We now turn to the aggregate analysis of risk attitudes and RIT. Figure 4.4.2 plots the average risk premium associated with a given combination of parameters (p, β, ϵ). Across the 30 combinations, participants are risk-averse (i.e., exhibit positive risk premium) in 80% of the cases (24 out of 30 combinations). Yet, participants are risk-seeking (i.e., exhibit a negative risk premium)

for (p, β, ϵ) = (0.1, 0.3, 3), (0.25, 0.3, 3) and (0.33, 0.3, 3), and risk-neutral for (p, β, ϵ) = (0.1, 0.5, 3), (0.1, 0.3, 4) and (0.25, 0.3, 4). In Figure 4.4.3 we show that the difference in minimum Ąxed pay across shocks (α 4 mα 3 m ) is systematically positive pointing to RIT at the aggregate level for all 30 combinations of parameters regardless of risk attitudes. These aggregate results once again provide support for RDU while contradicting MVS and EUT. We summarize these aggregate Ąndings below.

Result 3 (RIT at the aggregate level): RIT holds at the aggregate level for all combinations of parameters. (b) We cannot estimate the margina effect ϵ on RIT because we already exploit the data on both small and large shocks to estimate RIT.

-neutral if ♣α 3 m -α 0 m ♣< 25, risk-averse if α 3 m -α 0 m ≥ 25 and risk-seeking if α 3 m -α 0 m ≤ -25.

Conclusion

This paper studies the tradeoff between risk and incentives (RIT) under alternatives to the standard EUT model: the mean-variance-skewness (MVS) model and the the rank-dependent utility (RDU) model. At a theoretical level, we show that RIT is a robust phenomenon under RDU (and notwithstanding EUT and MVS) because it can be observed even when agents are risk-seeking. By contrast, we show that RIT is less robust under MVS than under EUT since it may not hold even for risk-averse agents, thus offering a possible explanation for the limited empirical support for the tradeoff.

To test the predictions of the three theories, we develop a novel experimental design that eliminates the potential confounds appearing in the existing literature.

To our surprise, we found extensive evidence for RIT. Most strikingly, RIT emerges even in situations in which agents are risk-seeking, which is a distinct prediction of RDU.

Our Ąndings are not only reassuring for the principal-agent theory, but also suggest RIT predictions can be applied to a broader range of situations than originally anticipated. These situations include contractual settings in which agents are likely to exhibit risk-seeking attitudes such as executive compensation [START_REF] Garen | Executive compensation and principal-agent theory[END_REF][START_REF] Edmans | The effect of risk on the ceo market[END_REF][START_REF] Edmans | Dynamic ceo compensation[END_REF][START_REF] Edmans | Executive compensation: A survey of theory and evidence[END_REF][START_REF] Ma | Portfolio manager compensation in the us mutual fund industry[END_REF]. It follows that risk-seeking agents might demand a Ąxed monetary compensation for any additional risk. This novel Ąnding can also have interesting implications in Ąelds like Ąnance and entrepreneurship. For example, Ąnancial advisors might need to craft portfolios with a substantial share of safe assets for clients that are otherwise categorized as risk-tolerant. Furthermore, our Ąndings suggest that, unlike ŞKnightianŤ theory of entrepreneurship [START_REF] Knight | Risk, uncertainty and profit[END_REF]Kihlstrom and Laffont, 1979;Newman, 2007), risk-seeking entrepreneurs might want to share part of the risk associated with new ventures.

Hence, the best response effort function is an increasing function of the variable pay and does not vary with respect to the shock ϵ, the relative risk-aversion coefficient r and the Ąxed pay α. The proof with p = 1/2 is provided in Appendix A2 of [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF]. The optimization problem of the principal is to maximize the expected value of θzy by accounting for agentŠs incentive compatibility constraint (IC) and participation constraint (PC)

                             max α,β θe -(α + βθe) s.t. : e = βθ 2ψ α + βθe -ψe 2 - 1 -p 2p rβ 2 θ 2 ϵ 2 ≃ y 0
The participation constraint is an application of the Pratt (1964) approximation of the risk premium to the agentŠs maximization problem as in [START_REF] Milgrom | Economics, Organization and Management[END_REF]. The optimal linear contact (α * , β * ) of the principal is given by

β * (ϵ, r, ψ, θ) ≃ 1 1 + 2ψrϵ 2 1-p p (4.5) α * (ϵ, r, ψ, θ) ≃ y 0 + 1 2 1 -p p rϵ 2 - 1 2ψ θβ * (ϵ, r, ψ, θ) 2 (4.6)
Furthermore, the expression of β * (ϵ, r, ψ, θ), yields

∂β * (ϵ, r, ψ, θ) ∂ϵ < 0 ∂β * (ϵ, r, ψ, θ) ∂r < 0 For k = 1-p p ϵ,
CE =α + βθe -ψe 2 + w(p) p -1 - β 2 θ 2 ϵ 2 2 1 + w(p) p 1 -2p p × A a α + βθe -ψe 2 + o(ϵ 2 ) (4.8) with A a (z) = - u ′′ (z) u ′′ (z)
being the absolute risk-aversion index evaluated at the outcome z and o(ϵ 2 ) denoting the approximation error.

Lemma 1 provides an approximation of the certainty equivalent. This approximation allows us to generate a closed-form solution for the optimal contract (α * , β * ) by assuming CARA utility function [START_REF] Milgrom | Economics, Organization and Management[END_REF], that is A a (z) = r for all z. Similar to EUT, we capture RIT in RDU in Proposition A2ii below.

Proposition A2 (RIT with RDU). 15 Under A0, A1, A2, A3Š, A3Ť, A4, A5 and assuming RDU agent, for any probability p ∈ (0, 1):

i) Optimal variable pay β * (ϵ, r, w(p), ψ, θ) decreases with r and probability riskaversion whereas the Ąxed pay α * (ϵ, r, w(p), ψ, θ) increases with r and probability risk-aversion.

ii) If the agent exhibits probability risk-aversion, the optimal variable pay β * (ϵ, r, w(p), ψ, θ) decreases with ϵ whereas the optimal Ąxed pay α * (ϵ, r, w(p), ψ, θ) increases with ϵ.

iii) If the agent exhibits probability risk-seeking, there is r to (β * , ϵ) such that for In the Lemma, we derive an equivalent of the Pratt (1964) approximation of risk premium for RDU and use it to provide incentive compatibility and participation constraints as in [START_REF] Milgrom | Economics, Organization and Management[END_REF]. To do so, let us Ąrst consider a binary random variable x = (x + 1-p p σ, xσ; p, 1p) with σ > 0, p ∈ (0, 1) so that E(x) = x and V (x) = 1-p p σ 2 . Under RDU theory, the certainty equivalent (ce) of x satisĄes

u(ce) = w(p) u x + 1 -p p σ -u(x -σ) + u(x -σ) (4.9)
Applying second-order Taylor approximation to the right-hand side (RHS) of expression (4.9) around the expected value x leads to

RHS = u(x) + σ w(p) p -1 u ′ (x) + σ 2 2 1 + w(p) p 1 -2p p u ′′ (x) + o(σ 2 ) (4.10)
with o(σ 2 ) denoting the approximation error such that lim Plugging (4.11) into (4.9) and applying Ąrst-order Taylor approximation on the left-hand side (LHS) of the expression (4.9) around the expected value x yields

LHS = u(x) + aσ 2 u ′ (x) + o(σ 2 ) (4.12)
Since LHS=RHS, according to (4.9), we can identify the unknown slope a a

= 1 σ w(p) p -1 + 1 2 1 + w(p) p 1 -2p p u ′′ (x) u ′ (x) (4.13) Let A a (z) = - u ′′ (z) u ′′ (z)
be the absolute risk-aversion index evaluated at x. Plugging (4.13) in (4.11) gives the Pratt (1964) risk premium π under RDU for the binary

random variable x = (x + 1-p p σ, x -σ; p, 1 -p). π := x -ce = - w(p) p -1 σ + σ 2 2 1 + w(p) p 1 -2p p A a (x) + o(σ 2 ) (4.14)
So that the certainty equivalent is

ce = x + w(p) p -1 σ - σ 2 2 1 + w(p) p 1 -2p p A a (x) + o(σ 2 ) (4.15) Note that RDU becomes EUT if w(p) = p. Since V (x) = 1-p p σ 2 , (4.14) collapses to the usual Pratt (1964) formula π = -1 2 u ′′ (x) u ′ (x) V (x) + o(σ 2 ) whenever w(p) = p.
Relation (4.15) allows us to deĄne the incentive compatibility and participation constraints. In the context of RDU (see Section 4.2), we set x = α + βθeψe 2 and σ = βθϵ so that the certainty equivalent equation (4.15) becomes:

ce =α + βθe -ψe 2 + w(p) p -1 βθϵ - β 2 θ 2 ϵ 2 2 1 + w(p) p 1 -2p p × A a α + βθe -ψe 2 + o(ϵ 2 ) (4.16)
and participation constraint (PC):

                           max α,β π = θe -(α + βθe) s.t. : e = βθ 2ψ α + βθe -ψe 2 + w(p) p -1 βθϵ -β 2 θ 2 ϵ 2 2 1 + w(p) p 1-2p p r ≃ y 0
Like in Proposition A1 above, the participation constraint is an application of the Pratt (1964) approximation of the risk premium. The optimal linear contact (α * , β * ) of the principal is given by

β * (ϵ, r, w(p), ψ, θ) ≃ 1 + 2ψϵ θ w(p) p -1 1 + 2ψrϵ 2 1 + w(p) p 1-2p p (4.17) α * (ϵ, r, w(p), ψ, θ) ≃ y 0 + 1 2 rϵ 2 1 + w(p) p 1 -2p p - 1 2ψ θβ * 2 -

Proof of Proposition A3

From Lemma 1, the absolute risk-aversion index r N (β * , ϵ) that makes an agent with probability risk-seeking behavior to exhibit risk-neutrality (i.e., null risk premium) for the lottery

L * = α * + θβ * e * + 1 -p p ϵ -ψe * 2 , α * + θβ * e * -ϵ -ψe * 2 ; p, 1 -p
associated with the optimal linear contract (α * , β * ) and optimal effort e * is given by

r N (β * , ϵ) ∼ = 2 ϵθβ * δ p -1 1 + δ p 1-2p p with δ = w(p).
From Proposition A2, the threshold r to (β * , ϵ) of the absolute risk-aversion index of an agent with probability risk-seeking behavior that leads the principal to propose an optimal variable pay β * (ϵ, r, δ, ψ, θ) that decreases in ϵ is

r to (β * , ϵ) ∼ = 1 2ϵ w(p) p -1 θ + ϵψ w(p) p -1 1 + w(p) p 1-2p p
Computing the difference between the two thresholds leads to

Sign r N (β * , ϵ) -r to (β * , ϵ) = Sign N β * (ϵ, r, δ, ψ, θ) with N β * (ϵ, r, δ, ψ, θ) = 4 -β * (ϵ, r, δ, ψ, θ) + 4ϵψ θ w(p) p - 1
Recall that ∂β * (ϵ, r, δ, ψ, θ) ∂r < 0 so that we have ∂N β * (ϵ, r, δ, ψ, θ) ∂r < 0.

Furthermore, we have lim

r-→0 N (β * (ϵ, r, δ, ψ, θ)) = 3 + 2ϵψ θ w(p) p -1 .
Hence, for all r > 0 we have N β * (ϵ, r, δ, ψ, θ) > 0 so that r N (β * , ϵ) > r to (β * , ϵ).

QED.

MVS

Before providing the proofs, we Ąrst state and provide some explanations for Propositions A4 and A5.

Proposition A4 (Risk-incentives tradeoff with MVS). Under A0, A1, A2 and A4 and assuming the agent is MVS as speciĄed in (4.4):

i) Optimal variable pay β * (ϵ, a v , a s , ψ, θ) decreases with a v ii) Optimal variable pay β * (ϵ, a v , a s , ψ, θ) increases (resp. decreases) with a s for

p < 1 2 (resp. p > 1 2 ). iii) If p ≥ 1 2 , the optimal variable pay β * (ϵ, a v , a s , ψ, θ) decreases with ϵ. iv) If p < 1 2 , there is g(ϵ, p, a v , a s , ψ, θ) such that if g(ϵ, p, a v , a s , ψ, θ) < 9 4 , then β * (ϵ, a v , a s , ψ, θ) increases with ϵ.
Proposition A4iv shows that the absence of the tradeoff can be expected for p<1/2, i.e., when the lottery

L * = α * + θβ * e * + 1 -p p ϵ -ψe * 2
, α * + θβ * e *ϵψe * 2 ; p, 1p associated with the optimal contract (α * , β * ) is positively skewed. Since risk attitudes are driven by both aversions to variance and preference for positive skewness, it remains unclear if the absence of the tradeoff arises for a risk-seeking or a risk-averse agent. Proposition B5 provides such information.

Proposition A5 (Risk attitudes and absence of tradeoff with MVS):

Consider p<1/2. Hence, we have the following results: i) At the optimal contract, agentŠs risk-aversion (risk-seeking) corresponds to

g(ϵ, p, a v , a s , ψ, θ) > 1 ( g(ϵ, p, a v , a s , ψ, θ) < 1)
ii) If the agent is a risk-seeker, then the optimal variable pay β * (ϵ, p, a v , a s , ψ, θ)

increases with ϵ

iii) For g(ϵ, p, a v , a s , ψ, θ) ∈ 1, 9 4 , the agent exhibits risk-aversion and the optimal variable pay β * increases with ϵ.

According to Proposition A5, if the agent exhibits risk-seeking behavior at the optimal contract proposed by the principal, then the optimal variable pay β * (ϵ, p, a v , a s , ψ, θ) increases with ϵ. However, if the agent is risk-averse at the optimal contract proposed by the principal [i.e., g(ϵ, p, a v , a s , ψ, θ) > 1], the optimal variable pay β * (ϵ, p, a v , a s , ψ, θ) can either increase or decrease with ϵ depending on whether g(ϵ, p, a v , a s , ψ, θ) is greater or smaller than 9 4 .

Remark [RDU vs. MVS]: Proposition A5 echoes Proposition A3 for RDU once we consider, in line with literature (e.g., Gonzalez and Wu, 1999a;Tversky and Wakker, 1995;Gonzalez-Jimenez, 2019;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF], that overweight-ing occurs for p < 1 2 . Then, both MVS and RDU predict the possibility of the absence of tradeoff only for p < 1 2 . However, the MVS and RDU provide different rationale for the absence of tredeoff. The RDU rules out the possibility of the absence of the tradeoff for a risk-averse agent. In contrast, MVS points to the absence of tradeoff for certain risk-averse agent and all risk-seeking agents.

Proof of Proposition A4: Given the linear contract (α, β), the objective function of an MVS agent with a cost function C(e) = ψe 2 is given by

M V S(L) = α + βθe -ψe 2 + a v 1 -p p β 2 θ 2 ϵ 2 + a s 1 -p p 1 -2p p β 3 θ 3 ϵ 3
The Ąrst-order condition of the agentŠs maximization problem leads to the optimal effort function e(β) that increases in the variable pay:

e = βθ 2ψ
The principalŠs optimization problem is to maximizes the expected value of θzy by accounting for the agentŠs incentive compatibility constraint (IC) and participation constraint (PC).

                           max α,β θe -(α + βθe) s.t. : e = βθ 2ψ α + βθe -ψe 2 + a v 1-p p β 2 θ 2 ϵ 2 + a s 1-p p 1-2p p β 3 θ 3 ϵ 3 = y 0 which is equivalent to max β θ 2 β 2ψ + a v 1 -p p β 2 ϵ 2 + a s 1 -2p p 1 -p p β 3 θϵ 3 - β 2 4ψ -y 0
The Ąrst-order condition is given by

1 2ψ + 2a v β 1 -p p ϵ 2 + 3a s 1 -2p p 1 -p p β 2 θϵ 3 - β 2ψ = 0 (4.30)
The second-order condition is given by 2a

v 1 -p p ϵ 2 + 6a s 1 -2p p 1 -p p βθϵ 3 - 1 2ψ < 0 (4.31)
Equation (4.30) implicitly deĄnes the optimal variable pay β * (ϵ, a v , a s , ψ, θ). In particular, it follows that the optimal variable pay in the absence of shock (ϵ = 0) is given by β * (0, a v , a s , ψ, θ) = 1. Moreover, for a s -→ 0 we have lim as-→0

β * (ϵ, a v , a s , ψ, θ) = 1 1 -4a v 1-p p ψϵ 2 < 1 = β * (0, a v , a s , ψ, θ) Also, for p = 1 2 , we have β * (ϵ, a v , 0, ψ, θ) = 1 1 -4a v 1-p p
. For ϵ > 0 and p ̸ = 1 2 , the two possible solutions of (4.30) are given by

β * 1 (ϵ, a v , a s , ψ, θ) = 1 2ψ -2a v 1-p p ϵ 2 - 2a v 1-p p ϵ 2 -1 2ψ 2 -6a s 1-2p p 1-p p θϵ 3 ψ 6a s 1-2p p 1-p p θϵ 3 β * 2 (ϵ, a v , a s , ψ, θ) = 1 2ψ -2a v 1-p p ϵ 2 + 2a v 1-p p ϵ 2 -1 2ψ 2 -6a s 1-2p p 1-p p θϵ 3 ψ 6a s 1-2p p 1-p p θϵ 3 When ϵ > 0, the right solution needs to satisfy the continuity condition lim ϵ-→0 β * i (ϵ, a v , a s , ψ, θ) = 1, i = 1, 2.
Using this continuity condition, and applying lŠHôpitalŠs rule, it follows that the solution is given by β

* 1 (ϵ, a v , a s , ψ, θ): 17 β * (ϵ, a v , a s , ψ, θ) = 1 2ψ -2a v 1-p p ϵ 2 - 2a v 1-p p ϵ 2 -1 2ψ 2 -6a s 1-2p p 1-p p θϵ 3 ψ 6a s 1-2p p 1-p p θϵ 3 (4.32)

Point i and ii)

Implicit function theorem on (4.30) leads to which is greater than the variable pay of 1 corresponding to the absence of noise.

∂β * (ϵ, a v , a s , ψ, θ) ∂a v > 0 Since e * (ϵ, a v , a s , ψ, θ) = θ 2ψ β * (ϵ, a v ,
:= - 6a v 1-p p ϵ 2 1 2ψ -2a v 1-p p ϵ 2 - 2a v 1-p p ϵ 2 -1 2ψ 2 -6a s 1-2p p 1-p p θϵ 3 ψ (4.35) Since e * (ϵ, a v , a s , ψ, θ) = θ 2ψ β * (ϵ, a v ,
QED.

Proof of Proposition A5:

Point i)

At the optimal contract, the agent is risk-averse if

a v +a s 1 -2p p θϵβ * (ϵ, a v , a s , ψ, ϕ) < 0 and risk-seeking if a v + a s 1-2p p θϵβ * (ϵ, a v , a s , ψ, θ) > 0.
Using the expression (4.32) of β * (ϵ, a v , a s , ψ, θ) it follows that the agent is risk-averse at the optimal contract if g(ϵ, a v , a s , ψ, θ) > 1 and risk-seeking if g(ϵ, a v , a s , ψ, θ) < 1.

Point ii)

First, note that for any p < 1 2 and variable pay β > 0 we have

a v + 9 4 a s 1 -2p p βθϵ > a v + a s 1 -2p p βθϵ (4.36)
Second, for any p < 1 2 and a given triplet (α, β, e), the MVS decision maker exhibits risk-seeking for the positively skewed lottery L = α + θβ e +

Point iii)

For any p < 1 2 and an agent who exhibits risk-aversion at the optimal triplet (α * , β * , e * ) with an optimal variable pay β * that increases with ϵ, we have

a v + a s 1 -2p p β * θϵ < 0 < a v + 9 4 a s 1 -2p p β * θϵ (4.40)
Given the expression (4.32) for the optimal variable pay β * (ϵ, a v , a s , ψ, θ), the condition that ensures (4.40) is given by g(ϵ, a v , a s , ψ, θ) ∈ 1, 9 4 (4.41)

with g(ϵ, a v , a s , ψ, θ) given in (4.35).

It follows that when the condition (4.41) holds, the agent exhibits riskaversion at the optimal triplet (α * , β * , e * ) such that variable payβ * increases with ϵ.

shock. 18

Proposition B2 (Consistency of results under RDU):

i) If the agentŠs absolute risk-aversion index does not sufficiently decrease with the outcome, then the optimal variable pay decreases with the probability riskaversion.

ii) If the agent exhibits probability risk-aversion or probability risk-neutrality, then the optimal variable pay decreases with ϵ.

iii) Assume the agent exhibits probability risk-seeking and that for any couple (α, β) the ratio between the average slope of the probability weighting function on the interval (0, p) and the average slope of the probability weighting on the interval (p, 1) are greater than the ratio between the slope of the utility function at the lowest possible value of the reward (net of cost) and the slope of the utility function at the highest possible value (net of cost). Then, the optimal variable pay is greater in the presence of shock than in its absence.

iv) There exist A to a (β) such that if the agent exhibits probability risk-seeking and A a α + β 2 θ 2 4ψ < A to a (β, ϵ) for any couple (α, β), then the optimal variable pay is greater in the presence of the shock than in its absence.

Proposition B3 (Risk attitudes and absence of tradeoff with RDU):

Assume that the agent exhibits probability risk-seeking for a given probability

p. Denote by A N a α * + β * 2 θ 2 4ψ
the absolute risk-aversion index that allows a probability risk-seeking agent to exhibit risk-neutrality for the lottery

L * = α * + θβ * e * + 1-p p ϵ -ψe * 2 , α * + θβ * e * -ϵ -ψe * 2 ; p, 1 -p associated with the optimal linear contract (α * , β * ). Then, A N a α * + β * 2 θ 2 4ψ > A to a (β * , ϵ).
L * is 4. Hence, the agent exhibits risk-seeking behavior at the optimal contract (α * , β * ) = (2.73, 1.17) where the tradeoff between risk and incentives is not observed.

Proof of Proposition B1:

Given the linear contract (α, β), the objective function of a EUT agent with cost function C(e) = ψe 2 is given by

EU (L) = pu(y + ) + (1 -p)u(y -) with u(x) = 1 -exp(-rx) r , y + = α + βθ(e + 1-p p ϵ) -ψe 2 and y -= α + βθ(e - ϵ) -ψe 2
The Ąrst-order condition of the agentŠs maximization problem is given by

(βθ -2ψe)[pu ′ (y + ) + (1 -p)u ′ (y -)] = 0
Since pu ′ (y + ) + (1p)u ′ (y -) ̸ = 0, it turns out that the agentŠs optimal effort function is given by e = βθ 2ψ

Hence, the optimal effort function is an increasing function of the variable pay and does not vary with respect to ϵ, the utility function or the Ąxed pay α.

The optimization problem of the principal is to maximize the expected value of θzy by accounting for the agentŠs incentive compatibility constraint (IC)

and participation constraint (PC):

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

                             max α,β θe -(α + βθe) s.t. : e = βθ 2ψ pu α + βθ e + 1-p p ϵ -ψe 2 + (1 -p)u α + βθ(e -ϵ) -ψe 2 = u(y 0 )
which is equivalent to

                     max α,β θ 2 2ψ β -β 2 ) -α s.t. : pu α + β 2 θ 2 4ψ + 1-p p βθϵ + (1 -p)u α + β 2 θ 2 4ψ -ϵ = y 0
The corresponding Lagrangian function is given by

L(α, β, µ) = θ 2 2ψ β-β 2 )-α+µ pu α+ β 2 θ 2 4ψ + 1 -p p βθϵ +(1-p)u α+ β 2 θ 2 4ψ -ϵ -y 0 Denote by y ′ + = θ 1 -p p ϵ + βθ 2ψ > 0 y ′ -= θ -ϵ + βθ 2ψ > 0
Then, the Ąrst-order conditions are given by

∂L ∂β = θ 2 2ψ 1 -2β) + µ pu ′ (y + )y ′ + + (1 -p)u ′ (y -)y ′ --y 0 = 0 (4.42) ∂L ∂α = -1 + µ pu ′ (y + ) + (1 -p)u ′ (y -) = 0 (4.43) ∂L ∂µ = pu(y + ) + (1 -p)u(y -) -y 0 = 0 (4.44)
with y -= α + β 2 θ 2 4ψβθϵ and y + = α + β 2 θ 2 4ψ + 1-p p βθϵ. Note that from (4.42) and (4.43), it turns out that the optimal variable pay β satisĄes

β = 1 + (1 -p) 2ψϵ θ u ′ (y + ) -u ′ (y -) pu ′ (y + ) + (1 -p)u ′ (y -) ≤ 1 (4.45)
This shows that the optimal variable pay in the absence of shock (ϵ = 0) is 1. Since u ′ (y + ) < u ′ (y -), it also follows that the optimal variable pay in the absence of shock is greater than the optimal variable pay in the presence of shock (ϵ > 0). Let us now derive an even stronger result according to which the optimal variable pay is a decreasing function of ϵ.

Totally differentiating the Ąrst-order conditions (4.42)-(4.44) with respect to β, α, µ and ϵ leads to

Hess(α, β, µ) ×          dβ dϵ dα dϵ dµ dϵ          = -           ∂ 2 β ∂β∂ϵ ∂ 2 β ∂α∂ϵ ∂ 2 β ∂µ∂ϵ           (4.46)
with Hess(α, β, µ) being the Hessian matrix deĄned as follows

Hess(α, β, µ) =            ∂ 2 L ∂β 2 ∂ 2 L ∂α∂β ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂β ∂ 2 L ∂α 2 ∂ 2 L ∂α∂µ ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂µ ∂ 2 L ∂µ 2           
Assuming that there exists at least a local maximum so that the determinant of the Hessian matrix is positive, it follows from (4.46) that Sign dβ dϵ = Sign(E) (4.47)

with Since the utility function is concave, we have u ′ (y + ) < u ′ (y -), A a (y -) > 0 and

E = ∂ 2 L ∂α∂µ ∂ 2 L ∂β∂ϵ ∂ 2 L ∂α∂µ - ∂ 2 L ∂α∂ϵ ∂ 2 L ∂β∂µ - ∂ 2 L ∂µ∂ϵ ∂ 2 L ∂α∂β ∂ 2 L ∂α∂µ - ∂ 2 L ∂α 2 ∂ 2 L ∂β∂µ µ = 1 pu ′ (y + ) + (1 -p)u ′
A a (y + ) > 0 so that E < 0. It turns out that

dβ dϵ < 0
To illustrate this, consider again the expo-power utility function [START_REF] Saha | Expo-power utility: a ŚĆexibleŠ form for absolute and relative risk aversion[END_REF].

The alternative level of utility is given by y 0 = 1exp(-ry γ 0 ) r with y 0 the alternative (outside) outcome. Hence, the Ąrst-order conditions are given by Hence, the optimal effort function is an increasing function of the variable pay and does not vary with respect to ϵ, the utility curvature, the Ąxed pay α or the probability risk attitude captured by w(p).

Point i)

The optimization problem of the principal is to maximize the expected value of θzy by accounting for the agentŠs incentive compatibility constraint (IC)

and participation constraint (PC): with y -= α + β 2 θ 2 4ψβθϵ and y + = α + β 2 θ 2 4ψ + 1-p p βθϵ.

                            
Denote by δ := w(p). Then, totally differentiating the Ąrst-order conditions (4.50)-(4.52) with respect to β, α, µ and δ leads to

Hess(α, β, µ) ×          dβ dδ dα dδ dµ dδ          = -           ∂ 2 β ∂β∂δ ∂ 2 β ∂α∂δ ∂ 2 β ∂µ∂δ           (4.53)
with Hess(α, β, µ) being the Hessian matrix deĄned as follows

Hess(α, β, µ) =            ∂ 2 L ∂β 2 ∂ 2 L ∂α∂β ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂β ∂ 2 L ∂α 2 ∂ 2 L ∂α∂µ ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂µ ∂ 2 L ∂µ 2           
Assuming that there exists at least a local maximum so that the determinant of the Hessian matrix is positive, it follows from (4.53) that Using all the previous derivatives, we have Thus, the optimal variable pay in the absence of shock (ϵ = 0) is 1. Since u ′ (y + ) < u ′ (y -), it also follows under the assumption of probability riskaversion/neutrality (i.e., w(p) ≤ p) that the optimal variable pay in the absence of shock is greater than the optimal variable pay in the presence of shock (ϵ > 0). Now, we show a stronger result according to which the optimal variable pay is indeed a decreasing function of ϵ under the assumption of probability risk-aversion/neutrality. with Hess(α, β, µ) being the Hessian matrix deĄned as follows

Sing dβ dδ = Sign(∆) (4.54) with ∆ = ∂ 2 L ∂α∂µ ∂ 2 L ∂β∂δ ∂ 2 L ∂α∂µ - ∂ 2 L ∂α∂δ ∂ 2 L ∂β∂µ - ∂ 2 L ∂µ∂δ ∂ 2 L ∂α∂β ∂ 2 L ∂α∂µ - ∂ 2 L ∂α 2 ∂ 2 L ∂β∂µ
∂ 2 L ∂β∂δ ∂ 2 L ∂α∂µ - ∂ 2 L ∂α∂δ ∂ 2 L ∂β∂µ = µ θϵ p u ′ (y + )u ′ (y -) > 0 ∂ 2 L ∂α∂β ∂ 2 L ∂α∂µ - ∂ 2 L ∂α 2 ∂ 2 L ∂β∂µ =
Hess(α, β, µ) =            ∂ 2 L ∂β 2 ∂ 2 L ∂α∂β ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂β ∂ 2 L ∂α 2 ∂ 2 L ∂α∂µ ∂ 2 L ∂µ∂β ∂ 2 L ∂α∂µ ∂ 2 L ∂µ 2           
Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

Assuming there exists at least a local maximum so that the determinant of the Hessian matrix is positive, it follows from (4.58) that Sing dβ dϵ = Sign(R) (4.59)

with Since the utility function is increasing and concave, we have 0 < u ′ (y + ) < u ′ (y -), A a (y -) > 0 and A a (y + ) > 0 so that R 2 > 0.

E = ∂ 2 L ∂α∂µ ∂ 2 L ∂β∂ϵ ∂ 2 L ∂α∂µ - ∂ 2 L ∂α∂ϵ ∂ 2 L ∂β∂µ - ∂ 2 L ∂µ∂ϵ ∂ 2 L ∂α∂β ∂ 2 L ∂α∂µ - ∂ 2 L ∂α 2 ∂ 2 L ∂β∂µ µ = 1 pu ′ (y + ) + (1 -p)u ′

Proof of Proposition B3:

From Lemma 1, the absolute risk-aversion index A N a α * + β * 2 θ 2 4ψ that makes an agent with probability risk-seeking behavior to exhibit risk-neutrality (i.e., risk premium equals 0) for the lottery L = α * + β * θ e * + 1-p p ϵψe * 2 , α * + β * θ e *ϵψe * 2 ; p, 1p associated with the optimal linear contract (α * , β * ) and optimal effort e * is given by From Proposition B2, the threshold A to a (β * , ϵ) of the absolute risk-aversion index of an agent with probability risk-seeking behavior that leads the principal to propose an optimal variable pay that is greater in presence of the shock than in its absence is given by > 2A to a β * , ϵ As an example, consider the expo-power utility function [START_REF] Saha | Expo-power utility: a ŚĆexibleŠ form for absolute and relative risk aversion[END_REF]. The alternative level of utility is given by y 0 = 1exp(-ry γ 0 ) r with y 0 being the alternative (outside) outcome. Hence, the Ąrst-order conditions are given by • We set (r, γ) = (0.029, 0.731) as found by Holt and Laury (2002) and (ψ, θ, y 0 ) = (0.5, 1, 4). In the absence of shock (ϵ = 0), the optimal variable Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of a shock (ϵ, p, w(p)) = (1, 0.1, 0.15), the optimal variable pay is β * = 0.95 and the optimal Ąxed pay is α * = 3.37. The optimal variable pay in the absence of a shock is greater than the variable pay in presence of a shock. The expected value is E[L] = 3.83 and the certainty equivalent of L is 4. Hence, the agent exhibits risk-seeking at the optimal contract (α * , β * ) = (3.37, 0.95) where the tradeoff between risk and incentive is observed.

A N a α * + β * 2
• Consider the calibration from the previous example with one change: w(0.1) = 0.2. In the absence of a shock (i.e., ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of a shock (ϵ, p) = (1, 0.1), the optimal variable pay is β * = 1.17 and the optimal Ąxed pay is α * = 2.73. The optimal variable pay in the absence of shock is less than the variable pay in the presence of shock. The expected value is E[L] = 3.41 and the certainty equivalent of L is 4. Hence, the agent exhibits risk-seeking at the optimal contract (α * , β * ) = (2.73, 1.17) where the tradeoff between risk and incentive is not observed.

QED.

4.C -Extension to continuous random shocks

In Section 2, we focus on binary shocks. Herein, we extend this framework to the case of continuous random shocks. We denote by f (.)(F (.)) the probability density (cumulative distribution) function of a continuous random shock ε such that E(ε) = 0 and V (ε). We further assume that the distribution is symmetric around 0, that is f (ϵ) = f (-ϵ) for all ϵ > 0.

Proposition C1 (Consistency of results under RDU): 19 i) Assume that the agent exhibits probability risk-aversion or probability riskneutrality (i.e., w ′ (.) ≥ 0). Hence, the optimal variable pay is greater in the absence of shock than in its presence.

ii) Assume that the agent exhibits probability risk-seeking (i.e., w ′ (.) < 0) or inverse-s-shaped probability weighting (i.e.,∃ a ∈ (0, 1) such that w ′′ (p) < 0 for p ∈ (0, a) and w ′′ (p) > 0 for p ∈ (a, 1) ). Hence the optimal performance can be greater in the presence of shock than in its absence.

Proof of Proposition C1:

Given the linear contract (α, β), the objective function of an expected utility agent with cost function C(e) = ψe 2 is given by Since both u(.) and w(.) are strictly increasing functions, it turns out that the optimal effort function of the agent is given by e = βθ 2ψ

RDU (L) =
19

We focus here on symmetric distributions for which MVS boils down to mean-variance preference. The mean-variance preference corresponds to the RDU with quadratic utility function and probabilistic risk-neutrality (i.e., linear weighting function). In this context, the tradeoff between risk and incentive holds. Consider that the relationship between output and effort is affected by a shock that follows a logistic distribution with mean 0 and variance π 2 3 . The probability density function and cumulative distribution functions are given by f (ϵ) = exp(-ϵ) (1 + exp(-ϵ)) 2 and F (ϵ) 1 1 + exp(-ϵ) For any given contract (α, β), the agentŠs optimal level of effort is given by e = βθ 2ψ

Note that this level of effort does not depend on α. The agent accepts to provide the level of effort if

pu α + β 2 θ 2 4ψ + 1 -p p βθϵ + (1 -p)u α + β 2 θ 2 4ψ -βθϵ ≥ u(y 0 )
Note that the left-hand side of the above inequality is strictly increasing in α.

Hence, there exists a minimum level of Ąxed pay α m such that the participation constraint is binding, that is

Proof of Proposition 2

The rank dependent utility associated with the contract is given by RDU (L) = w(p)u α+βθ e+ 1p p ϵ -ψe 2 +(1-w(p))u α+βθ e-ϵ -ψe 2

For any given contract (α, β), agentŠs optimal level of effort under the accepted contract is

e = βθ 2ψ
Note that this level of effort does not depend on α. The agent accepts to provide a given level of effort if

w(p)u α + β 2 θ 2 4ψ + 1 -p p βθϵ + (1 -w(p))u α + β 2 θ 2 4ψ -βθϵ ≥ u(y 0 )
Note that the left-hand side of the above inequality is strictly increasing in α.

Hence, there exists a minimum level of Ąxed pay α m such that the previous participation constraint is binding, that is It is clear that we should rule out the case r 1 = 0 < r to (β, ϵ). Indeed, for r 1 = 0 we have a linear utility function. Then, to have risk-neutral agent under linear utility function, we should have also δ = w(p) = p. This contradicts our initial assumption δ > p.

Ruling out the case r 1 = 0, it follows that the value of r N (β, ϵ) that allows the equality u(ce) = u(E[L]) for a risk-neutral agent (that compensates probability risk-seeking with utility risk aversion) is such that r N (β, ϵ) > r to (β, ϵ).

Finally, note that this result holds for general utility function (see our Proposition B3).

Point v) A simple derivative of the expression of r to (β, ϵ) shows that this threshold decreases with β and ϵ.

The CARA coefficient for risk-neutrality is implicitly determined by Hence, the agent becomes risk-seeking for a sufficiently small value of variable pay β. In particular, if the agent is risk-neutral or risk-seeking for a contract involving β 1 ; then the agent is risk-seeking for contract involving β 0 < β 1 .

Similarly, r N ϵ dr N (β, ϵ) dϵ = -1 < 0 QED.

Hence, dαm das < 0 if p < 1 2 and dαm das < 0 if p > 1 2 . This means that for p < 1 2 (resp. p > 1 2 ) , the minimum Ąxed pay decreases (resp. increases) with the preference for positively skewed lotteries.

Point ii)

The implicit function theorem applied to (4.77) leads to

dα m dϵ = - 1 -p p β 2 θ 2 ϵ 2a v + 3a s 1 -2p p βθϵ
It follows that for p > 1 2 (negative skewness), we have dαm dϵ > 0.

Point iii)

In contrast, for p < 1 2 (positive skewness), we have Hence, for any couple (a v , a S ) such thata v a s ∈ (τ N (β, ϵ), τ to (β, ϵ)) the agent is risk-averse and the minimum Ąxed pay accepted α decreases with shock.

dα m dϵ > 0 ⇐⇒ - a v a s > τ to (β,

Point v)

Clearly, τ to (β, ϵ) and τ N (β, ϵ) increase with the shock ϵ and β.

QED. Since e * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ) = θ 2ψ β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ), it also follows that ∂e * ∂λ < 0 , ∂e * ∂r + < 0 , ∂e * ∂r -> 0. For given probability p = p, denote by δ + = w + (p) and δ -= w + (1p); then implicit function theorem applied to (4.86) leads to ∂β * ∂δ + > 0 ∂β * ∂δ -< 0 Since e * (α, ϵ, r + , r -, λ, w + , w -, ψ, θ) = θ 2ψ β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ), it also follows that ∂e * ∂δ + > 0 , ∂e * ∂δ -< 0.

Point iii)

From (4.86) we have β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ)

Since e * (α, ϵ, r + , r -, λ, w + , w -, ψ, θ) = θ 2ψ β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ), it also follows that e * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ)

Hence, β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ) and e * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ) are higher in the presence of shock than in its absence if the agent exhibits sufficient probability risk-seeking, moderate utility curvature and loss-aversion.

QED.

4.E H-Experimental instructions

Cumulative Prospect Theory for Risk and Uncertainty: New Measurement Methods and Applications

Yao Thibaut Kpegli

Abstract

The thesis is at the intersection of behavioral economics, experimental economics, and decision theory. Chapters 1, 2, and 3 develop methods to estimate different components of decision models under risk and uncertainty: utility function, weighting function, loss aversion, and beliefs. Applications conĄrm deviations from standard theories (Expected Utility and Subjective Expected Utility) through evidence of weighting function, loss aversion, and ambiguity attitudes. People are more insensitive to likelihood in the presence of asymmetric events than symmetric events, suggesting that belief formation is cognitively demanding. For equal sources of uncertainty, people exhibit payoff dependence aversion and variety of payoffs seeking. Payoff dependence aversion means that people dislike that their own payoffs depend on the preferences of others. This behavior is captured by a more concave utility function. Variety of payoffs seeking means that subjects prefer a greater number of possible payoffs when such possible payoffs depend on the preferences of others. This behavior is captured by more optimism. Chapter 4 studies the existence of the Risk-incentives tradeoff (RIT) under Rank Dependent Utility (RDU) and Mean-Variance-Skewness (MVS). Theoretical analyses show that RIT is remarkably robust under RDU but not under MVS. With data based on a novel experimental design that eliminates confounding factors, chapter 4 provides evidence for RIT even in the case of risk-seeking agents, which is a distinct prediction of RDU. The results provide support for the RIT and suggest that it applies to a broad range of situations, including cases in which agents are risk-seeking (e.g., executive compensation).

Keywords: Risk, uncertainty, beliefs, ambiguity attitudes, experiment.

Résumé

La thèse se situe à lŠintersection de lŠéconomie comportementale, de lŠéconomie expérimentale et de la théorie de la décision. Les chapitres 1, 2 et 3 développent des méthodes pour estimer les différentes composantes des modèles de décision en situation de risque et dŠincertitude : fonction dŠutilité, fonction de pondération, aversion aux pertes et croyances. Les applications conĄrment des déviations par rapport aux théories standard (utilité espérée et utilité espérée subjective) à travers la fonction de pondération, lŠaversion aux pertes et les attitudes dŠambiguïté. Les gens sont plus insensibles à la probabilité en présence dŠévénements asymétriques quŠen présence dŠévénements symétriques, ce qui suggère que la formation des croyances demande des efforts cognitifs. Pour une même source dŠincertitude, les individus font preuve dŠaversion à la dépendance des gains et de préférence pour la variété des gains. LŠaversion à la dépendance des gains signiĄe que les individus nŠaiment pas que leurs gains dépendent des préférences des autres. Ce comportement se traduit par une fonction dŠutilité concave. La préférence pour la variété des gains signiĄe que les individus préfèrent un plus grand nombre de possibilité de gains lorsque les gains dépendent des préférences des autres. Ce comportement se traduit par lŠoptimisme. Le chapitre 4 étudie lŠexistence de lŠarbitrage entre risque et incitations (RIT) dans le cadre de lŠutilité dépendante du rang (RDU) et de la moyenne-variance-skewness (MVS). Les analyses théoriques montrent que le RIT est remarquablement robuste sous RDU mais pas sous MVS. Avec des données basées sur un nouveau modèle expérimental qui élimine les facteurs de confusion, le chapitre 4 fournit des preuves de lŠexistence du RIT même dans le cas où les individus ont des préférences pour le risque, ce qui est une prédiction distincte du RDU. Les résultats conĄrment lŠexistence du RIT et suggèrent quŠil sŠapplique à un large éventail de situations, y compris les cas où les individus ont des préférences pour le risque (par exemple, la rémunération des dirigeants).

Mots Clés : Risque, incertitude, croyances, attitudes face à lŠambiguïté, expérience.

1. 1

 1 Absolute error in estimates: semi-parametric vs. parametric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Semi-parametric and parametric measurements of the probability weighting function (median data from TK92) . . . . . . . . . . 1.3 Semi-parametric and parametric measurements of the probability weighting function (results on pooled data of BFE10) . . . . . 1.B.1 Curvature of the utility function across domains: semi-parametric estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.B.2 Curvature of the utility function under different estimation methods116 1.B.3 Median probability weights under different estimation methods 1.B.4 Distribution of probability weights for gains . . . . . . . . . . . 1.B.5 Distribution of probability weights for losses . . . . . . . . . . 1.C.1 Estimated utility functions . . . . . . . . . . . . . . . . . . . . 1.C.2 Estimated weighting functions . . . . . . . . . . . . . . . . . . 1.D.1 Posterior distributions of curvature and probability weights in the gain domain (world-level) . . . . . . . . . . . . . . . . . . . 1.D.2 Posterior distributions of curvature and probability weights in the loss domain (world-level) . . . . . . . . . . . . . . . . . . . 1.D.3 Estimated weighting functions . . . . . . . . . . . . . . . . . . 1.D.4 Posterior distribution of loss aversion (world-level) . . . . . . . 2.2.1 Spline: exemple of shapes . . . . . . . . . . . . . . . . . . . . . 2.5.1 Utility function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Probability weighting function) . . . . . . . . . . . . . . . . . . 2.B.1 Individual estimates: data of GW99 . . . . . . . . . . . . . . . 3.5.1 Social ambiguity -trust game treatment . . . . . . . . . . . . . 3.5.2 Betrayal aversion treatment . . . . . . . . . . . . . . . . . . . . 3.5.3 Nature ambiguity treatment . . . . . . . . . . . . . . . . . . . 3.6.1 Cumulative distribution of utility curvature . . . . . . . . . . . 3.6.2 Cumulative distribution of event weights . . . . . . . . . . . . 3.6.3 Cumulative distribution of subjective probability (beliefs) . . . 3.6.4 Mean of subjective probability (beliefs) . . . . . . . . . . . . . 3.6.5 Cumulative distribution of pessimism (η) . . . . . . . . . . . . 3.6.6 Cumulative distribution of likelihood insensitivity (γ) . . . . . 3.6.7 weighting function based median of individual estimates . . . . 3.A.1 beliefs of subject A about the IQ score of subject B: probability density (pdf) and cumulative density (cdf) functions. . . . . . . 4.2.1 Valuation of the contract by a probability risk-seeking RDU agent (equation (4.3)) as a function of the shock magnitude. . . 4.2.2 Valuation of the contract by an averse-to-variance (a v < 0) and preference-for-positive-skewness (a s > 0) MVS agent (equation (4.4)) as a function of the shock magnitude when p < 1/2. . . . 4.3.1 Example of the decision screen for (β, ϵ, p) = (0.7, 3, 0.5). In this example, the participant selected Option A for the Ąrst 10 rows (cells A1 to A10 in orange) and selected Option B from the last 11 rows (cells B11 to B21 in orange). Therefore, this participant switched from Option A in row 10 to Option B in row 11. This switching point corresponds to a Ąxed pay between 450 = (10 -1) × 50 and 500 = (11 -1) × 50. . . . . . . . . . . 4.4.1 Estimated probability weights under RDU . . . . . . . . . . . . 4.4.2 Risk premium: α 3 mα 0 m and α 4 mα 0 m . . . . . . . . . . . . . . 4.4.3 RIT and the variation in Ąxed pay α 4 mα 3 m . . . . . . . . . . 4.A.1 Risk attitudes and RIT with RDU (agent exhibits w(p) > p ) . 4.C.1 Density function and probability weighting function . . . . . . 4.C.2 Derivative of probability weighting function . . . . . . . . . . .

  .) the value function, w + (.) the weighting function in the gain domain and w -(.) the probability weighting function in the loss domain. The value function v() is continuous, strictly increasing and satisĄes v(0) = 0. The probability weighting functions are strictly increasing, transform probability from [0, 1] to [0, 1] and satisfy: w -(0) = 1w -(1) = 0 and w + (0) = 1w + (1) = 0.

  step: Elicitation of utility in the loss domain Once p ℓ is known from the Ąrst step, the utility midpoint of any two losses L A and L B for which the utilities are already known can be measured by eliciting just one indifference. By the uniqueness properties of the utility function in prospect theory, we can Ąx a loss L 1 < 0 and set u(L 1 ) = -1 (42) This allows the elicitation of the outcome L 0.5 such that u(L 0.5 ) = -0.5 by looking for the indifference (L 1 , 0; p ℓ , 1p ℓ ) ∼ L 0.5 . This process is used to elicit utility on the interval [L 1 , 0]. For example, by setting L A = 1 and L B = L 0.5 , the outcome L 0.75 such that u(L 0.75 ) = -0.75 can be elicited via the following indifference (L 1 , L 0.5 ; p ℓ , 1p ℓ ) ∼ L 0.75 . Third step: Mirror image technique to link the utility between the two domains In the third step, utility in the loss domain is linked to utility in the gain domain by eliciting three indifferences. The Ąrst indifference consists in taking one of the outcomes that was obtained in the step 2, say L 1 , and elicit ℓ that satisĄes (ℓ, 0; 0.5, 0.5) ∼ L 1 . This indifference implies u(ℓ)w -(0.5) = -1 (43)

  illustrate this point, suppose that we are interested in eliciting utility only over strictly positive outcomes with a power utility function u(z) = z α . For an extremely concave utility function (i.e., α > 0 and α -→ 0) and an extremely concave weighting function (i.e., w + (p k ) = 1 for k = 1, 2, .., K ) along with the PT assumptions w + (0) = 0 and w + (1) = 1, we have ϵ u jk = 0 for all j = 1, 2, ..., n and k = 1, 2, ..., K. For the the aim of estimating u(z) for z ∈ A(z) ≡ ¶$25, $50, $75, $100, $150, $200, $400, $800♢ and the probability weights w + (p) for p ∈ B(p) ≡ ¶0.01, 0.05, 0.10, 0.25, 0.40, 0.50, 0.60, 0.75, 0.90, 0.95, 0.99♢, it follows that the combination of an extremely concave utility function (i.e., u(z) = constant > 0 for z ∈ A(z) and u(0) = 0) and an extremely concave weighting function (i.e., w + (p) = 1 for p ∈ B(p) and w + (0) = 1w + (p) = 0) is the solution of the minimization of least squares estimation.

  0) and v ′ ↓ (0) representing the left and right derivatives of the value function at the reference point. 11 Finally, partial reĆection corresponds to α p = β p and α e = β e in (3.16) and (3.17).

  Equation (1.20) represents the one-parameter weighting function (per domain) of TK92. Equations (1.21) and (1.22) refer to the one-and two-parameter weighting functions ofPrelec (1998), and (1.23)

Chapter 1 .

 1 All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components in Simulation 2 (1) and misspecify the model in the same way as in Simulation 5 (6).

Figure 1 . 1 :

 11 Figure 1.1: Absolute error in estimates: semi-parametric vs. parametric methods (a) Parameter recovery (b) Misspecification

from

  Harrison and Rutström (2009);[START_REF] Eisenberg | Uncovering the structure of self-regulation through data-driven ontology discovery[END_REF]; lŠHaridon andVieider (2019);[START_REF] Pedroni | The risk elicitation puzzle[END_REF];[START_REF] Andersson | Robust inference in risk elicitation tasks[END_REF]; BFE10; GW99 and TK92. The data used inHarrison and Rutström (2009); Eisenberg et al. (2019); Pedroni et al. (2017); Andersson et al. (2020) rely on binary choices, which cannot be used as input in our method. The data of Bruhin et al. (2010); lŠHaridon and Vieider (2019); GW99 and TK92 contain certainty equivalents for binary lotteries. However, the dataset in lŠHaridon and Vieider (2019) does not match our criteria at the individual level because it only contains one certainty equivalent for some of the probability weights. The dataset of GW99 satisĄes our criteria, with an important caveat that it contains observations from only 10 subjects and solely in the gain domain. By contrast, BFE10 [and more speciĄcally, their ŞZurich 03Ť experiment] 26 collected data on 179 subjects in the gain and loss domains with several certainty equivalents per probability, thus Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

  Our semi-parametric estimations show that the utility function is concave in the gain domain: the estimated values of α = 0.525 (TK92) and α = 0.587 (BFE10) are signiĄcantly greater than 0.5 (pvalues < 0.0292). In the loss domain, the estimated values also suggest the utility function is concave because β = -0.483 and β = -0.425 are greater than -0.5 (pvalue < 0.0757 and p-value < 0.0001, respectively). Concavity in the loss domain is in line with the Ąndings of ABL,Attema et al. (2013) and[START_REF] Etchart-Vincent | Monetary incentives in the loss domain and behavior toward risk: An experimental comparison of three reward schemes including real losses[END_REF].

  Figure 4.2.1 presents the estimates of the probability weighting function across domains (labeled Semi-para) for the data of TK92. In the gain domain, probabilistic risk neutrality w + (p) = p is rejected for most probabilities (all pvalues < 0.0258), except for 0.25 (pvalue = 0.0671). Overall, we reject the joint hypothesis of linearity of the probability weighting function over the whole range of probabilities in the gain domain (pvalue < 0.0001). The resulting weighting function is inverse S-shaped because there is overweighting for p ∈ (0, 0.25], and underweighting for p ∈ (0.25, 1). Similar patterns emerge in the loss domain, with overweighting starting for lower probabilities p ∈ (0, 0.1],

  is rejected for 7 probabilities (all pvalues < 0.001), with the exception of probabilities 0.01 and 0.05 (both pvalues > 0.2365). Using a joint test, the duality hypothesis is rejected (pvalue < 0.0001). Data of BFE10. Figure 1.3 presents the estimates of the probability weighting function across domains (once again, labeled Semi-para) for the data of BFE10.

Figure 1 . 2 :

 12 Figure 1.2: Semi-parametric and parametric measurements of the probability weighting function (median data from TK92)

Chapter 1 .

 1 All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect data previously reported by TK92. They run a computerized experiment with 25 graduate students from Berkeley and Stanford with no particular training in decision theory. Each subject participated in three separate one-hour sessions organized over several days, and received $25 for participation.

Figure 1 .

 1 Figure 1.B.1: Curvature of the utility function across domains: semi-parametric estimates

  Figures 1.C.1 and 1.C.2 show the estimates for the utility and probability weighting functions. 39 For the median data, both methods lead to remarkably similar estimates for both the utility and probability weighting functions. Both methods estimate a concave utility function and an inverse S-shaped weighting function with crossover point around p = 0.4. Even though we cannot reject the absence of differences in probability weights between the two methods (all pvalues > 0.3438), the concavity of the utility function is more pronounced in the spline estimation (pvalue = 0.0010).

Figure 1 .

 1 Figure 1.C.2: Estimated weighting functions

Figures 1 .

 1 Figures 1.D.1 and 1.D.2 show the posterior distributions of the world-level utility curvature in the gain and loss domains. The mean of the posterior distribution of the utility curvature in the gain (loss) domain is 0.0085 (-0.0055) suggesting that the utility function is generally concave. In the gain domain, the 95% credible interval is [0.0021, 0.0150] showing that the utility function deviates signiĄcantly from linearity. In the loss domain, linearity of the utility function cannot be rejected at the 5% signiĄcance level with the 95% credible interval of [-0.0165, 0.0057].

Figure 1 .

 1 Figure 1.D.2: Posterior distributions of curvature and probability weights in the loss domain (world-level)

  Figure 1.D.4 shows the posterior distribution of the world-level loss aversion coefficient. The mean of the posterior distribution is 1.785, pointing to a substantial degree of loss aversion. The 95% credible interval of [1.6228, 1.9594] also rejects loss neutrality and points to λ > 1.

Figure 1 .

 1 Figure 1.D.4: Posterior distribution of loss aversion (world-level)

  .7) illustrate this Ćexibility. The spline (2.5) leads to a concave utility function (Figure 2.2.1, panel (a)). The spline (2.6) leads to a convex utility function. The spline (2.7) leads to a concave utility function over two intervals [0, 0.25] and [0.5, 0.75], and also leads to a convex utility over the two intervals [0.25, 0.50] and [0.75, 1].

Figure 2

 2 Figure 2.2.1: Spline: exemple of shapes (a) Spline function (2.5) (b) Spline function (2.6)

k

  the estimates of the utility function u, the probability weights in the gain domain w + (p k ) and the probability weights in the loss domain w -(1p k ) from steps 1 and 2.Estimate of loss aversion index λ ofKöbberling and Wakker (2005) is given by the minimization of the sum of squared error:

  TK92 run a computerized experiment with 25 graduate students from Berkeley and Stanford with no particular training in decision theory. Each subject participated in three separate one-hour sessions organized over several days, and received $25 for participation. Procedure: the data are generated via the switching outcome procedure in which an indifference value is inferred through a list of equally spaced certain outcomes, ranging from the admissible maximum indifference value to the admissible minimum indifference value. Internal consistency of the responses to each prospect is monitored by a computer software to reduce response errors. Data for the first and second steps: all outcomes are expressed in US dollars.

Figure 2 .

 2 Figure 2.5.1 provides the estimated utility functions on both datasets. 10 The pool estimate on the data of TK92 leads to a concave (resp. convex) utility in the gain (resp. loss) domain. The utility curvature in gain domain is α = 0.554 and is signiĄcantly different from linearity (pvalue = 0.0011) 11 , which corresponds to α = 0.5. In the loss domain, the mean of utility curvature in the gain domain is β = -0.517 and is signiĄcantly different from linearity (pvalue = 0.0036) where β = -0.5. Furthermore, partial reĆection is rejected (H 0 : α + β = 0, pvalues = 0.0355).

Figure 2 .Figure 2

 22 Figure 2.5.1: Utility function

Figure 2 . 5 . 2 :

 252 Figure 2.5.2: Probability weighting function) (a) Data of TK92 (b) Data of GW99

Figure 2

 2 Figure 2.B.1: Individual estimates: data of GW99 (a) Subject 1 (b) Subject 2

  Denote by L = (x, y; E, E c ) a binary lottery that gives the outcome x if the event E occurs and y otherwise. E denotes an event of the state space Ω and E c denotes the complement of E in Ω. Outcomes are real numbers. For notational convenience, we assume that x > y ≥ 0. We denote ≽ as the preference relation of the decision-maker over prospects. The relations ≻ and ∼ denote strict preference and indifference, respectively. The preference relation of the decision-maker is represented by the following model that values the prospect L = (x, y; E, E c ) asV (L) = W (P (E))(U (x) -U (y)) + U (y) (3.1)where W (.) is the weighting function or source function for uncertainty(Abdellaoui et al., 2011a), P (E) is the subjective probability or beliefs of E occurring, and U (.) the utility function that captures the attitude toward outcomes. Both W (.) and U (.) are strictly increasing functions.Model (3.1) corresponds to the biseparable preferences model of Ghirardato

  Denote by L = (x, y; p, 1p) a binary lottery that gives outcome x with probability p and y otherwise. The preference relation of the decision-maker is Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty represented by the following model that values the prospect L = (x, y; p, 1p)

2 =

 2 C♢ the set of single events and by Ω * ¶E 12 , E 13 , E 23 ♢ the set of the composite events. BW propose their two indexes in the framework of SDW. The difference in the weighting functions under ambiguity and risk is measured by the ambiguity function f (.) = w -1 [W (.)]. The matching probability m E of an event E is the probability that ensures the following indifference (x, y; E, E c ) ∼ (x, y; m E , 1m E ). Under SDW, the ambiguity function corresponds to the matching probability (Dimmock et al., 2016, Theorem 3.1):

  compound invariance family W (z) = exp(-(-ln(z)) α ) β with α = 0.65 and β = 1.05 (Wakker, 2010, pg. 270) for both treatments 1 and 2. With these values, we have b 1 = 0.11 ̸ = 0.06 = b 2 and a 1 = 0.31 ̸ = 0.19 = a 2 . QED.

  betrayal aversion through the Minimum Acceptable Probability (MAP) related to the utility function. The MAP is the probability for which the Trustor is indifferent between trust and distrust. BZ identify betrayal aversion as the difference in MAP between two treatments: the trust game and the risky dictator game (RDG). In the trust game treatment, if the Trustor decides to trust, the Ąnal payoffs for both Trustor and Trustee are determined by the Trustee. Contrary, in the RDG, if the Trustor trusts, the payoffs for both players are determined by nature. The possible payoffs under both treatments are the same. Their results show that subjects state higher MAPs in the trust game compared to the RDG, which means that subjects are betrayal averse. Quercia (2016) provides an improvement of MAP design and conĄrm betrayal aversion. LW show that the MAP design of BZ does not hold under SDW. The difference in MAP across treatments can be explained by the weighting function Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty and beliefs instead of the utility. The authors use the two indexes of pessimism and likelihood insensitivity provided by BW to disentangle social ambiguity and strategic uncertainty in the trust game. They Ąnd that pessimism is lower when subjects face social ambiguity than when they face nature ambiguity. Also, they Ąnd that strategic uncertainty and betrayal attitudes only have cognitive implications by making subjects more likelihood insensitive in the trust game compared to nature ambiguity. The fact that social ambiguity is captured by the pessimism component of the weighting function in the trust game, suggests that social ambiguity plays a major role in strategic uncertain situations. Therefore, it is important to control for social ambiguity when studying strategic uncertainty.

Figure

  Figure 3.5.1: Social ambiguity -trust game treatment Player 1

Figure

  Figure 3.5.2: Betrayal aversion treatment Player 1

  Figure 3.5.3: Nature ambiguity.

Utility curvature :

 curvature Figure 3.6.1 displays the cumulative distributions of the utility curvature and

Figure 3

 3 Figure 3.6.1: Cumulative distribution of utility curvature (a) Coordination game (b) Trust game

Figure

  Figure 3.6.2: Cumulative distribution of event weights (a) Nature (b) Social ambiguity -cg

Figure 3

 3 Figure 3.6.3 displays the cumulative distributions of beliefs and Figure 3.6.4 plots the mean of the estimated values.

Figure

  Figure 3.6.3: Cumulative distribution of subjective probability (beliefs) (a) Nature (b) Social ambiguity -cg

  aversion) corresponds to the mixture of strategic uncertainty and what BZ called betrayal aversion. Utility curvature (α): the estimate of CRRA parameter for treatments 3 and 4 are both 0.968. The difference in the utility parameters between treatment 3 and 4 is not signiĄcant (p -value= 0.5203, two sided sign test).

  s 2 ]. Applying the stage 1 of our method, presented in section 4.3 allows us to estimate the utility function and the four event weights: δE for E = E 1 , E 2 , E 3 , E 12 . Second stage: weighting and beliefs of single events. Applying the second stage presented in section 4.3 allows us to break down the estimated events weights δE into the weighting function (i.e. δ, γ) and the beliefs of the single events P ([s 0 , s 1 ]), P ([s 1 , s 2 ]), and P ([s 2 , s 3 ]). Third stage: density and cumulative distribution over the range [0, 1]. This stage complements the two stages presented in section 4.3 because S is a continuous-valued sources of uncertainty. The interval I can be re-scaled to be in the range Ĩ = [0, 1]: s = s-s 0 s 3 -s 0 ∈ [0, 1] for s ∈ I = [s 0 , s 3 ]. At this stage, a two-parameter speciĄcation of the distribution is needed. A common and Ćexible distribution is the beta distribution B(a, b) with param-Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty eters a, b. Denote by F a,b (.) the cumulative distribution function of the beta distribution. We then have the following three equations

2 3

 2 .15) Solving (numerically) the system of the two equations (3.14) and (3.15) provides the estimation of the distribution of beliefs (i.e. a, b). For illustration purposes, lets consider that an experimenter aims to elicit the beliefs of a subject A about the IQ score of a subject B. The IQ score belongs to [0, 1], with high values meaning a high IQ score. After applying stages 1 and 2 with E 1 = [0, 0.25], E 2 = [0.25, 0.50] and E 3 = [0.5, 1], the experimenter Ąnds the following: P [0, 0.25] = 0.1, P [0.25, 0.5] = 0.7 and P [0.5, 1] = 0.2. Then, the equations (3.14) and (3.15) of the third stage corresponds to F a,b 9. Solving these two equations, provides â = 6.62 and b = 9.95. The density and cumulative functions are provided in Ągure 3.A.1.

Figure 3 .

 3 Figure 3.A.1: beliefs of subject A about the IQ score of subject B: probability density (pdf) and cumulative density (cdf) functions.

Chapter 3 .

 3 Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty this experiment. The values below are numerical examples of how Participant 2Šs decision affects your payment. • If you choose Left and Participant 2 chooses an Amazon voucher, you will receive 30 ECU. • If you choose Left and Participant 2 chooses a Google Play voucher, you will receive 20 ECU. • If you choose Left and Participant 2 chooses an Apple Store voucher, you will receive 16 ECU. • If you choose Right and Participant 2 chooses an Amazon voucher, you will receive 16 ECU. • If you choose Right and Participant 2 chooses a Google Play voucher, you will receive 30 ECU. • If you choose Right and Participant 2 chooses an Apple Store voucher, you receive 20 ECU. • If you choose Middle and Participant 2 chooses an Amazon voucher, you will receive 20 ECU. • If you choose Middle and Participant 2 chooses a Google Play voucher, you will receive 16 ECU. • If you choose Middle and Participant 2 chooses an Apple Store voucher, you will receive 30 ECU. Your possible payments (in ECU), depending on your decision and the decision of Participant

•

  If you select Alternative B for this line, you can earn 30 ECU or 16 ECU. Your payment depends on the decision of Participant 2 that you were associated with in sub-part 1 of this part of the experiment (the most recent task you completed). Payment is determined as follows: -If Participant 2 chooses an Amazon voucher, you earn 30 ECU. -If Participant 2 chooses either a Google Play voucher or an Apple Store voucher, you earn 16 ECU.

  for the Ąrst sub-part of Part 3 In this part of the experiment, you are again randomly paired with another participant. We call this new person Participant 2. However, this Participant 2 is a different person than the ones you were paired with in the previous parts of the experiment. You will never be informed of Participant 2Šs identity, nor will Participant 2 be informed of your identity. Depending on your decision, your payment may or may not depend on Participant 2Šs decision. Your decision in this sub-section will be to choose an action between the Left or Right possibilities. Participant 2 receives 5 euros. Participant 2 decides where he or she would prefer to spend the 5 euros between one of the following options: an Amazon voucher, a Google Play voucher or an Apple Store voucher. You will not be informed of Participant 2Šs decision until the end of the experiment. If you chose Left, you will receive a sure payment, and Participant 2Šs decision does not affect your payment. If you choose Right, your payment is determined by Participant 2Šs decision. Participant 2 knows that your payment may or may not depend on their decision. However, Participant 2

  25 ECU if Participant 2 chooses Left in the first sub-part of this part of the experiment or 18 ECU if Participant 2 chooses Right or Middle • If you select Alternative A for this line, you earn 23 ECU. • If you select Alternative B for this line, you can earn 25 ECU or 18 ECU. Your payment depends on the decision done by the Participant 2Šs that you were associated with in sub-part 1 of this part of the experiment (the most recent task you completed). Payment is determined as follows: -If Participant 2 chooses Left, you earn 25 ECU. -If Participant 2 chooses Right or Middle, you earn 18 ECU.

Figure 4

 4 Figure 4.2.1: Valuation of the contract by a probability risk-seeking RDU agent (equation (4.3)) as a function of the shock magnitude.

Figure 4 . 2 . 2 :

 422 Figure 4.2.2: Valuation of the contract by an averse-to-variance (a v < 0) and preference-for-positive-skewness (a s > 0) MVS agent (equation (4.4)) as a function of the shock magnitude when p < 1/2.

  and Reversed RIT in the BB model). RIT (Reversed RIT) [No RIT] corresponds to the case in which the minimum Ąxed pay accepted by an agent (α m ) increases (decreases) [does not change] in the output shock ϵ for given values of β and p. Propositions A2-A3 (Appendix 4.A), B2-B3 (Appendix 4.B) and C1 (Appendix 4.C). Proposition 3. (RIT with RDU).

Figure

Figure 4 m

 4 Figure 4.4.2: Risk premium: α 3 mα 0 m and α 4 mα 0 m

  Figure 4.A.1: Risk attitudes and RIT with RDU (agent exhibits w(p) > p )

  furthermore impose the following linear form for the certainty equivalent with an unknown slope a ce = x + aσ 2 + o(σ 2 ) (4.11) Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

  (y -) (4.48)∂ 2 L ∂β∂ϵ = µθ(1p) u ′ (y + )u ′ (y -) + β u ′′ (y + )y ′ +u ′′ (y -)y ′ -∂ 2 L ∂µ∂α = pu ′ (y + ) + (1p)u ′ (y -) > 0 ∂ 2 L ∂α∂ϵ = µβθ(1p) u ′′ (y + )u ′′ (y -) > 0 ∂ 2 L ∂µ∂β = py ′ + u ′ (y + ) + (1p)y ′ -u ′ (y -) > 0 ∂ 2 L ∂µ∂ϵ = βθ(1p) u ′ (y + )u ′ (y -) < 0 ∂ 2 L ∂α∂β = µ py ′ + u ′′ (y + ) + (1p)y ′ -u ′′ (y -) ≤ 0 ∂ 2 L ∂α 2 = µ pu ′′ (y + ) + (1p)u ′′ (y -) ≤ 0Using all the previous derivatives, we have p) u ′ (y+ )u ′ (y -)µ θ 2 βϵ p (1p)× u ′ (y + )u ′ (y -) (1p)A a (y + ) + pA a (y -) ∂β∂µ = µθϵ(1p)u ′ (y + )u ′ (y -) A a (y -) -A a (y + ) ≥ 0 E = θ(1-p) u ′ (y + )-u ′ (y -)1µ -µβθϵu ′ (y + )u ′ (y -) u ′ (y + )A a (y -)+ 1p p u ′ (y -)A a (y + )

  Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

  u α + βθ e + 1-p p ϵψe 2 + (1w(p))u α + βθ(eϵ)ψe 2 = y 0 u α + β 2 θ 2 4ψ + 1-p p βθϵ + (1w(p))u α + β 2 θ 2 4ψϵ = y 0The corresponding Lagrangian function is given byL(α, β, µ) ) + µ w(p)u ′ (y + )y ′ + + (1w(p))u ′ (y -)y ′ -µ w(p)u ′ (y + ) + (1w(p))u ′ (y -) = 0 (4.51) ∂L ∂µ = w(p)u(y + ) + (1w(p))u(y -)y 0 = 0 (4.52)

  ′ (y + ) + (1p)u ′ (y -) (4.55)∂ 2 L ∂β∂δ = µ u ′ (y + )y ′ +u ′ (y -)y ′ -) ∂ 2 L ∂µ∂α = w(p)u ′ (y + ) + (1w(p))u ′ (y -) > 0 ∂ 2 L ∂α∂δ = µ u ′ (y + )u ′ (y -) > 0 ∂ 2 L ∂µ∂β = w(p)y ′ + u ′ (y + ) + (1w(p))y ′ -u ′ (y -) > 0 ∂ 2 L ∂µ∂δ = u(y + )u(y -) > 0 ∂ 2 L ∂α∂β = µ w(p)y ′ + u ′′ (y + ) + (1w(p))y ′ -u ′′ (y -) ≤ 0 ∂ 2 L ∂α 2 = µ w(p)u ′′ (y + ) + (1w(p))u ′′ (y -) ≤ 0

  y + )u ′ (y -)w(p)(1-w(p)) A a (y -)-A a (y + ) y + )u ′ (y -) w(p)u ′ (y + ) + (1w(p))u ′ (y -) w(p)(1w(p)) u(y + )u(y -) A a (y -) -A a (y + ) (4.56) Clearly ∆ > 0 (i.e., dβ dδ > 0 ) under CARA utility assumption, indicating that the approximation errors in Arrow-Pratt risk premium does not alter the results established in previous propositions. For utility function such that A a (y -) -A a (y + ) -→ 0, we have ∆ > 0 (i.e., 50) and (4.51), it turns out that the optimal variable pay β * satisĄes y + ) -(1w(p))u ′ (y -) w(p)u ′ (y + ) + (1w(p))u ′ (y -)

2 =

 2 y + )-(1-w(p))u ′ (y -)+β w(p) 1p p u ′′ (y + )y ′ + -(1-w(p))u ′′ (y -)y ′ -∂ 2 L ∂µ∂α = w(p)u ′ (y + ) + (1w(p))u ′ (y -) > 0 ∂ 2 L ∂α∂ϵ = µβθ w(p) 1p p u ′′ (y + ) -(1w(p))u ′′ (y -) ∂ 2 L ∂µ∂β = w(p)y ′ + u ′ (y + ) + (1w(p))y ′ -u ′ (y -) > 0 ∂ 2 L ∂µ∂ϵ = βθ w(p) 1p p u ′ (y + ) -(1w(p))u ′ (y -) < 0 ∂ 2 L ∂α∂β = µ w(p)y ′ + u ′′ (y + ) + (1w(p))y ′ -u ′′ (y -) ≤ 0 ∂ 2 L ∂α 2 = µ w(p)u ′′ (y + ) + (1w(p))u ′′ (y -) ≤ 0Using all the previous derivatives, we have y+ ) -(1w(p))u ′ (y -) )(1w(p))u ′ (y + )u ′ (y -)× 1p p A a (y + ) + A a (y -) )(1-w(p))u ′ (y + )u ′ (y -) A a (y -)-A a (y + ) ≥ 0 Hence R = R 1 -R 2 with R 1 = θ w(p)u ′ (y + ) + (1w(p))u ′ (y -) w(p) 1p p u ′ (y + ) -(1w(p))u ′ (y -) R βθ 2 ϵu ′ (y + )u ′ (y -) y + )A a (y -) + 1w(p) p u ′ (y -)A a (y + )w(p)u ′ (y + ) + (1w(p))u ′ (y -)

  a β * , ϵ , then A N a α * + β * 2 θ 2 4ψ

  βθ(e + ϵ)ψe 2 d[1w(1 -F (ϵ))] (4.67) Noting that d[1w(1 -F (ϵ))] = w ′ (1 -F (ϵ))f (ϵ)dϵ, the derivative of (4.67)with respect to e is given by(βθ -2ψe) +∞ -∞ u ′ α + βθ(e + ϵ)ψe 2 w ′ (1 -F (ϵ))f (ϵ)dϵ = 0

  ′ (1 -F (ϵ))f (ϵ)dϵy 0 = 0 (4.70) From (4.68) and (4.69), it follows that the optimal variable pay β * satisĄes w′ (1 -F (ϵ))f (ϵ)dϵ +∞ -∞ u α + β 2 θ 2 4ψ + βθϵ w ′ (1 -F (ϵ))f (ϵ)dϵ (4.71) It follows directly from (4.71) that if w ′ (.) ≥ 0, then +∞ -∞ ϵu α+ β 2 θ 2 4ψ +βθϵ w ′ (1-F (ϵ))f (ϵ)dϵ < 0 so that β < 1.Point ii) From (4.71), it follows directly

Figure 4 .

 4 Figure 4.C.1: Density function and probability weighting function

Figure

  Figure 4.C.2: Derivative of probability weighting function

F

  (α m , ϵ) := pu(e) + (1p)u α m + β 2 θ 2 4ψ βθϵu(y 0 ) = 0 (4.73) Denote by y * -= α + β 2 θ 2 4ψβθϵ and y * + = α + β 2 θ 2 4ψ + 1-p p βθϵ, then y * -< y * + . Point i) The minimum Ąxed pay increases with the utility (outcome)-risk aversion Assuming CARA utility function, equation (4.73) becomes F (α m , ϵ) :=pexpr α m + β

F

  (α m , ϵ) :=w(p)u α + β The minimum Ąxed pay increases with the utility risk-aversion and probability risk-aversion Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and WellThe implicit function theorem applied to (4.75) directly leads todα m dδ = -u(y * + )u(y * -) w(p)u ′ (y * + ) + (1w(p))u ′ (y * -)pay increases with probability risk-aversion.Point ii) Implicit function theorem leads todα m dϵ = βθ (1w(p))u ′ (y * -) -w(p) p u ′ (y * + ) w(p)u ′ (y * + ) + (1w(p))u ′ (y * -)It follows that for w(p) ≤ p (underweighting), we have directly dαm dϵ > 0. Similarly, we Ąnd that dαm dβ > 0 under the assumption that e = βθ 2ψ > ϵ.Point iii) In contrast, assuming w(p) > p (overweighting), we have m+ β 2 θ 2 4ψβθϵ u ′ α m + β 2 θ 2 4ψ + 1-p p βθϵFor CARA utility function this implies thatdα m dϵ < 0 ⇐⇒ r < r to (β, ϵ)with the threshold deĄned as r to (β, ϵ) rank dependent utility theory, the minimum accepted α decreases with the shock size if we have substantial overweighting and moderate utility curvature.Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

F

  (r N , β) := δexprβθ 1p p ϵ + (1δ)exp[rβθϵ] -

  (β, ϵ), is risk-neutral ifa v a s = τ N (β, ϵ) and risk-seeking ifa v a s < τ N (β, ϵ).Clearly, τ N (β, ϵ) = Sto (β, ϵ).

  (-r -β * θϵ) 1 + expr + β * θ 1-p p ϵ ⇐⇒ β * > 1 = lim ϵ-→0

  (-r -β * θϵ) 1 + expr + β * θ 1

Table 1 :

 1 Four-Step Elicitation Procedure ofAbdellaoui et al. (2007b)

	Assessed quantity	Indifference	Under CPT	Choice variables 0.5
	Step 1
		Loss domain
	x 1

*

Elicitation of p ℓ and pg such thtat w -(p ℓ ) = 0.5 and w + (pg) =

Table 11

 11

	, pp. 263-264). Second, eliciting indifference values on bounded intervals
	allows us to use a standard switching outcome procedure (Booij and Van de
	Kuilen, 2009).

Table 1

 1

	.1: Summary of literature on semi-parametric and non-parametric
	methods

Table 1 .

 1 2: Simulation results

	Simulation n •	Estimation	Data generation	Average absolute value of error
	u()		w()	u()	w()	α	w +	β	w -
				Parameter recovery
	Power	our method	Power	our method	0.0059 0.0087 0.0058 0.0084
	Expo	our method	Expo	our method	0.0050 0.0069 0.0051 0.0068
	Power	TK92	Power	TK92	0.0043 0.0024 0.0034 0.0023
	Power	P98-I	Power	P98-I	0.0023 0.0019 0.0024 0.0018
	Power	P98-II	Power	P98-II	0.0062 0.0053 0.0059 0.0054
	Power	GE87	Power	GE87	0.0082 0.0076 0.0083 0.0079
	Expo	TK92	Expo	TK92	0.0048 0.0023 0.0042 0.0023
	Expo	P98-I	Expo	P98-I	0.0031	0.0018 0.0028 0.0018
	Expo	P98-II	Expo	P98-II	0.0078 0.0090 0.0050 0.0041
	Expo	GE87	Expo	GE87	0.0048 0.0035 0.0047 0.0033
				Misspecifcation
	Power	our method	Expo	TK92	0.0703 0.0180 0.0710	0.0179
	Power	our method	Expo	P98-I	0.0672 0.0194 0.0690 0.0203
	Power	our method	Expo	P98-II	0.0611 0.0198 0.0624 0.0200
	Power	our method	Expo	GE87	0.0667 0.0184 0.0681 0.0188
	Expo	our method	Power	TK92	0.0688 0.0776 0.0663	0.0741
	Expo	our method	Power	P98-I	0.0695 0.0724 0.0699 0.0753
	Expo	our method	Power	P98-II	0.0707 0.0709	0.0713 0.0704
	Expo	our method	Power	GE87	0.0678 0.0761 0.0679 0.0734
	Power	TK92	Expo	our method	0.1143 0.1046 0.1170	0.1072
	Power	P98-I	Expo	our method	0.0972 0.0948 0.1004 0.0961
	Power	P98-II	Expo	our method	0.0690 0.0597	0.0676 0.0581
	Power	GE87	Expo	our method	0.0714 0.0576 0.0696 0.0554
	Expo	TK92	Power	our method	0.1127 0.1168 0.1017	0.1047
	Expo	P98-I	Power	our method	0.0974 0.0934 0.0992 0.0944
	Expo	P98-II	Power	our method	0.0683 0.0957 0.0713 0.0983
	Expo	GE87	Power	our method	0.0691 0.0917 0.0720 0.0938

Table 1 .

 1 3: Goodness of Ąt across methods: data of TK92

	Parametric	Semi-parametric

Table 1 .

 1 5: Curvature of the utility function and loss aversion

		Data of TK92	Data of BFE10
		Semi-parametric Parametric	Semi-parametric Parametric
	Curvature Gain (α)	0.525	0.544	0.587	0.589
	Curvature Loss (β)	-0.483	-0.525	-0.425	-0.428
	Loss aversion (λ)	1.751	1.730	-	-

1.5.3.3 Probability weighting function

Data of TK92.

  is rejected for 4 probabil-Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components ities (0.01, 0.05, 0.25 and 0.90; all pvalues < 0.0241), but not for others (0.1, 0.5, 0.75, 0.95 and 0.99; all pvalues > 0.1192). For tail probabilities (p = 0.01 and 0.05 in this dataset), 29 overweighting is more pronounced in the gain than in the loss domain (pvalues < 0.0121). Using a joint test, the hypothesis of identical probability weights across domains is rejected (pvalue < 0.0001).

Table 1 .

 1 6: Summary of the discussion

		Data of TK92	Data of BFE10	Consistent
		Semi-para	Para	Semi-para	Para	Semi-para Para
	Curvature Gain (α)	Concave	Concave	Concave Concave	Yes	Yes
	Curvature Loss (β)	Concave	Convex	Concave Concave	Yes	No
	Loss aversion (λ) No	No	No	No	Yes	Yes

RDU: w

+ (p) = 1w -(1p)

Table 1 .

 1 2. A.1: Results of the Ąrst step Table 1.A.2: Results of the second step (loss aversion)

			Power utility		Exponential utility
		No constraint	Constraint αp = βp
				Gain domain
	αp or αe	0.904 * * *	(0.0366)	0.976 * * *	(0.0314)	0.00158 *	(0.000886)
	α	0.525 * * *	(0.0101)	0.506 * * *	(0.00804)	0.552 * * *	(0.0290)
	w(0.01)	0.0471 * * *	(0.00828)	0.0369 * * *	(0.00659)	0.0445 * * *	(0.00874)
	w(0.05)	0.167 * * *	(0.0138)	0.157 * * *	(0.0138)	0.164 * * *	(0.0143)
	w(0.10)	0.134 * * *	(0.0151)	0.115 * * *	(0.0133)	0.124 * * *	(0.0142)
	w(0.25)	0.276 * * *	(0.0142)	0.263 * * *	(0.0142)	0.274 * * *	(0.0155)
	w(0.50)	0.410 * * *	(0.0139)	0.388 * * *	(0.0123)	0.409 * * *	(0.0176)
	w(0.75)	0.571 * * *	(0.0137)	0.558 * * *	(0.0140)	0.573 * * *	(0.0164)
	w(0.90)	0.686 * * *	(0.0137)	0.666 * * *	(0.0133)	0.692 * * *	(0.0201)
	w(0.95)	0.791 * * *	(0.0124)	0.783 * * *	(0.0133)	0.793 * * *	(0.0139)
	w(0.99)	0.947 * * *	(0.00474)	0.943 * * *	(0.00520)	0.957 * * *	(0.00834)
				Loss domain
	βp or βe	1.069 * * *	(0.0485)	0.976 * * *	(0.0314)	-0.00154 *	(0.000931)
	β	-0.483 * * *	(0.0113)	-0.506 * * *	(0.00804)	-0.449 * * *	(0.0305)
	w(0.01)	0.0244 * * *	(0.00570)	0.0337 * * *	(0.00646)	0.0236 * * *	(0.00551)
	w(0.05)	0.0924 * * *	(0.0127)	0.103 * * *	(0.0136)	0.0932 * * *	(0.0123)
	w(0.10)	0.105 * * *	(0.0139)	0.127 * * *	(0.0135)	0.106 * * *	(0.0126)
	w(0.25)	0.207 * * *	(0.0140)	0.222 * * *	(0.0140)	0.205 * * *	(0.0141)
	w(0.50)	0.414 * * *	(0.0152)	0.440 * * *	(0.0123)	0.405 * * *	(0.0186)
	w(0.75)	0.602 * * *	(0.0146)	0.617 * * *	(0.0139)	0.595 * * *	(0.0168)
	w(0.90)	0.758 * * *	(0.0139)	0.776 * * *	(0.0121)	0.746 * * *	(0.0190)
	w(0.95)	0.810 * * *	(0.0133)	0.819 * * *	(0.0132)	0.805 * * *	(0.0148)
	w(0.99)	0.947 * * *	(0.00552)	0.951 * * *	(0.00512)	0.936 * * *	(0.0107)
	Log Likelihood	-120.9645	-124.5477	-121.9931
	N	56		56		56
	AIC criterion	283.9289	289.0954	285.9861
	CV	3.149	3.344	3.305
	Standard errors in parentheses.

* p < 0.1, * * p < 0.05, * * * p < 0.01 to test the significance of coefficients.

3 Semi-parametric results under identical probabil- ity weighting function and duality assumptions

This appendix show results under the constraints of identical probability weighting (OPT) and duality assumption (RDU). Tables 1.A.3 and 1.A.4 present these results.

Table 1 .

 1 A.3: Results under identical probability weighting assumption

		Power utility	Exponential utility
	αp or αe	0.904 * * *	(0.0456)	0.00132	(0.000942)
	α	0.525 * * *	(0.0126)	0.544 * * *	(0.0310)
	βp or βe	1.053 * * *	(0.0523)	-0.00127	(0.000961)
	β	-0.487 * * *	(0.0124)	-0.458 * * *	(0.0316)
	w + (0.01) = w -(0.01) 0.0326 * * *	(0.00760)	0.0301 * * *	(0.00700)
	w + (0.05) = w -(0.05)	0.128 * * *	(0.0148)	0.125 * * *	(0.0137)
	w + (0.10) = w -(0.10)	0.119 * * *	(0.0161)	0.114 * * *	(0.0137)
	w + (0.25) = w -(0.25)	0.242 * * *	(0.0156)	0.237 * * *	(0.0153)
	w + (0.50) = w -(0.50)	0.414 * * *	(0.0159)	0.408 * * *	(0.0184)
	w + (0.75) = w -(0.75)	0.587 * * *	(0.0155)	0.584 * * *	(0.0169)
	w + (0.90) = w -(0.90)	0.720 * * *	(0.0151)	0.715 * * *	(0.0200)
	w + (0.95) = w -(0.95)	0.801 * * *	(0.0141)	0.799 * * *	(0.0147)
	w + (0.99) = w -(0.99)	0.947 * * *	(0.00557)	0.950 * * *	(0.00962)
	N	56		56
	Log Likelihood	-145.7	-143.2982
	AIC	315.400	310.596
	CV	3.456	3.518
	Standard errors in parentheses

* p < 0.1, * * p < 0.05, * * * p < 0.01 Table

1

.A.4: Results under duality assumption w + (p) = 1w -(1p) * p < 0.1, * * p < 0.05, * * * p < 0.01 to test the significance of coefficients.

  Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

Table 1 .

 1 A.5 presents the results of the Ąrst step that simultaneously estimates the utility and probability weighting functions in the full domain. Table1.A.6 summarizes the estimates of loss aversion using parameters of the utility and probability weighting functions from the Ąrst step as inputs.

	Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit
	Prospect Theory Components

Table 1 .

 1 A.5: Results of the Ąrst step: power utility (without constraint)

		TK92		P98-I		P98-II	GE87
					Gain domain
	αp	0.839 * * *	(0.0280)	0.772 * * *	(0.0224)	0.834 * * *	(0.0510)	0.907 * * *	(0.0537)
	α	0.544 * * *	(0.00828)	0.564 * * *	(0.00715)	0.545 * * *	(0.0151)	0.524 * * *	(0.0148)
	a	0.643 * * *	(0.0120)	0.589 * * *	(0.0158)	0.590 * * *	(0.0145)	0.620 * * *	(0.0181)
	b							1.076 * * *	(0.0557)	0.693 * * *	(0.0555)
					Loss domain
	βp	0.906 * * *	(0.0286)	0.867 * * *	(0.0254)	1.042 * * *	(0.0709)	1.058 * * *	(0.0641)
	β	-0.525 * * *	(0.00786)	-0.536 * * *	(0.00729)	-0.490 * * *	(0.0170) -0.486 * * *	(0.0151)
	a	0.704 * * *	(0.0153)	0.676 * * *	(0.0205)	0.680 * * *	(0.0189)	0.708 * * *	(0.0239)
	b							1.186 * * *	(0.0687)	0.673 * * *	(0.0568)
	N	56			56				56	56
	Log Likelihood	-155.124		-158.499		-153.315	-152.111
	AIC	320		327		321	318
	CV	3.41		3.57		3.52	3.87
	Standard errors in parentheses
	Observations	TK92	P98-I	P98-II	GE87
			1		1.618		1.429	1.143	1.405
			2		1.318		1.156	0.846	1.066
			3		1.258		1.082	0.734	0.979
			4		1.253		1.058	0.732	0.961
			5		1.146		0.980	0.630	0.841
			6		1.127		0.920	0.580	0.822
			Median		1.256 1.070	0.773	0.970

* p < 0.1, * * p < 0.05, * * * p < 0.01 Table 1.A.6: Results of the second step for Loss aversion: power utility (without constraint)

Table 1 .

 1 A.7: Results of the Ąrst step: power utility (with constraint)

		TK92		P98-I		P98-II	GE87
					Gain domain
	αp	0.872 * * *	(0.0204)	0.814 * * *	(0.0179)	0.926 * * *	(0.0441)	0.978 * * *	(0.0425)
	α	0.534 * * *	(0.00582)	0.551 * * *	(0.00544)	0.519 * * *	(0.0119)	0.506 * * *	(0.0109)
	a	0.635 * * *	(0.0110)	0.583 * * *	(0.0165)	0.590 * * *	(0.0152)	0.629 * * *	(0.0185)
	b							1.166 * * *	(0.0511)	0.632 * * *	(0.0422)
					Loss domain
	βp	0.872 * * *	(0.0204)	0.814 * * *	(0.0179)	0.926 * * *	(0.0441)	0.978 * * *	(0.0425)
	β	-0.534 * * *	(0.00582)	-0.551 * * *	(0.00544)	-0.519 * * *	(0.0119) -0.506 * * *	(0.0109)
	a	0.713 * * *	(0.0152)	0.681 * * *	(0.0225)	0.678 * * *	(0.0199)	0.695 * * *	(0.0226)
	b							1.083 * * *	(0.0474)	0.741 * * *	(0.0477)
	N	56			56				56	56
	Log Likelihood	-156.499		-162.219		-156.147	-153.719
	AIC	321		332		324	319
	CV	3.313		3.586		3.426	3.589
	Standard errors in parentheses
	Observations	TK92	P98-I	P98-II	GE87
			1		2.040	2.009	2.076	2.175
			2		1.738	1.723	1.764	1.844
			3		1.730	1.723	1.743	1.808
			4		1.730	1.716	1.743	1.808
			5		1.668	1.637	1.704	1.792
			6		1.615	1.616	1.496	1.674
			Median		1.730 1.720	1.743	1.808

* p < 0.1, * * p < 0.05, * * * p < 0.01 Table 1.A.8: Results of the second step for Loss aversion: power utility (with constraint) * p < 0.1, * * p < 0.05, * * * p < 0.01 Table 1.A.10: Results of the second step for Loss aversion: exponential utility

1 One step estimation of utility function, weighting function and loss aversion

	Table 1.A.11: Simultaneous estimation of utility function, weighting function
			and loss aversion
			Power utility		Exponential utility
		No constraint	Constraint αp = βp
				Gain domain
	αp or αe	0.896 * * *	(0.0433)	0.991 * * *	(0.0326)	-0.000209	(0.000922)
	α	0.528 * * *	(0.0121)	0.502 * * *	(0.0082)	0.493 * * *	(0.0307)
	w + (0.01)	0.0484 * * *	(0.00990)	0.0351 * * *	(0.00696)	0.0327 * * *	(0.00798)
	w + (0.05)	0.169 * * *	(0.0161)	0.155 * * *	(0.0154)	0.152 * * *	(0.0162)
	w + (0.10)	0.136 * * *	(0.0179)	0.111 * * *	(0.0143)	0.107 * * *	(0.0149)
	w + (0.25)	0.278 * * *	(0.0166)	0.260 * * *	(0.0158)	0.256 * * *	(0.0175)
	w + (0.50)	0.412 * * *	(0.0165)	0.384 * * *	(0.0134)	0.377 * * *	(0.0185)
	w + (0.75)	0.572 * * *	(0.0160)	0.555 * * *	(0.0157)	0.551 * * *	(0.0190)
	w + (0.90)	0.688 * * *	(0.0161)	0.662 * * *	(0.0147)	0.656 * * *	(0.0227)
	w + (0.95)	0.792 * * *	(0.0143)	0.781 * * *	(0.0150)	0.778 * * *	(0.0172)
	w + (0.99)	0.948 * * *	(0.00547)	0.943 * * *	(0.00590)	0.940 * * *	(0.0111)
				Loss domain
	βp or βe	1.045 * * *	(0.0383)	0.991 * * *	(0.0326)	-0.000979	(0.000723)
	β	-0.489	(0.0092)	0.502 * * *	(0.0082)	-0.467 * * *	(0.0239)
	w + (0.01)	0.0265 * * *	(0.00496)	0.0320 * * *	(0.00554)	0.0262 * * *	(0.00470)
	w + (0.05)	0.0950 * * *	(0.0107)	0.101 * * *	(0.0113)	0.0957 * * *	(0.0100)
	w + (0.10)	0.110 * * *	(0.0117)	0.123 * * *	(0.0120)	0.111 * * *	(0.0103)
	w + (0.25)	0.211 * * *	(0.0116)	0.220 * * *	(0.0119)	0.210 * * *	(0.0114)
	w + (0.50)	0.421 * * *	(0.0123)	0.436 * * *	(0.0115)	0.415 * * *	(0.0146)
	w + (0.75)	0.606 * * *	(0.0120)	0.615 * * *	(0.0119)	0.602 * * *	(0.0132)
	w + (0.90)	0.762 * * *	(0.0112)	0.773 * * *	(0.0107)	0.755 * * *	(0.0147)
	w + (0.95)	0.812 * * *	(0.0109)	0.818 * * *	(0.0110)	0.809 * * *	(0.0116)
	w + (0.99)	0.948 * * *	(0.00448)	0.950 * * *	(0.00436)	0.941 * * *	(0.00800)
				Mixed Prospect
	λ	0.864 * * *	(0.242)	1.688 * * *	(0.0786)	1.684 * * *	(0.135)
	Log Likelihood	-131.289	-134.001	-133.181
	N	60		60		60
	AIC criterion	308.5782	312.0025	312.3615
	Standard errors in parentheses
	* p < 0.1, * * p < 0.05, * * * p < 0.01

Table 1 .

 1 B.1: Semi-parametric estimation results: pooled data

		Power	Exponential
			Gain domain
	Utility
	parameter	1.031 * * *	(0.0216)	0.00710 * * *	(0.0007)
	w + (0.05)	0.191 * * *	(0.0062)	0.220 * * *	(0.0063)
	w + (0.10)	0.250 * * *	(0.0083)	0.276 * * *	(0.0068)
	w + (0.25)	0.330 * * *	(0.0076)	0.359 * * *	(0.0067)
	w + (0.50)	0.450 * * *	(0.0067)	0.501 * * *	(0.0060)
	w + (0.75)	0.606 * * *	(0.0074)	0.635 * * *	(0.0064)
	w + (0.90)	0.734 * * *	(0.0075)	0.756 * * *	(0.0063)
	w + (0.95)	0.779 * * *	(0.0069)	0.799 * * *	(0.0061)
			Loss domain
	Utility
	parameter	1.088 * * *	(0.0259) -0.00606 * * *	(0.00072)
	w -(0.05)	0.193 * * *	(0.0077)	0.195 * * *	(0.0065)
	w -(0.10)	0.263 * * *	(0.0091)	0.270 * * *	(0.0068)
	w -(0.25)	0.374 * * *	(0.0083)	0.373 * * *	(0.0068)
	w -(0.50)	0.518 * * *	(0.0073)	0.501 * * *	(0.0063)
	w -(0.75)	0.659 * * *	(0.0078)	0.655 * * *	(0.0069)
	w -(0.90)	0.730 * * *	(0.0082)	0.730 * * *	(0.0071)
	w -(0.95)	0.798 * * *	(0.0064)	0.787 * * *	(0.0065)
	N	8906	8906
	Log Likelihood	-25191.14	-25103.62
	AIC	50776.27	50601.23
	CV	4.400	4.2494
	Standard errors in parentheses

* p < 0.1, * * p < 0.05, * * * p < 0.01

Table 1 .

 1 B.2: Parametric estimation results with power utility: pooled data

		TK92	P98-I	P98-II	GE87
			Gain domain
	αp	1.219 * * *	(0.0152)	1.102 * * *	(0.0115)	1.043 * * *	(0.0212)	1.040 * * *	(0.0215)
	α	0.451 * * *	(0.00310)	0.476 * * *	(0.00260)	0.490 * * *	(0.00509)	0.490 * * *	(0.00517)
	a	0.591 * * *	(0.00490)	0.484 * * *	(0.00704)	0.483 * * *	(0.00705)	0.870 * * *	(0.0196)
	b					0.955 * * *	(0.0140)	0.475 * * *	(0.00708)
			Loss domain
	βp	1.442 * * *	(0.0186)	1.330 * * *	(0.0148)	1.091 * * *	(0.0257)	1.078 * * *	(0.0249)
	β	-0.409 * * *	(0.00312)	-0.429 * * *	(0.00274)	-0.478 * * *	(0.00587)	-0.481 * * *	(0.00577)
	a	0.616 * * *	(0.00539)	0.508 * * *	(0.00792)	0.506 * * *	(0.00799)	1.049 * * *	(0.0256)
	b					0.853 * * *	(0.0147)	0.477 * * *	(0.00753)
	N	8906	8906	8906	8906
	Log Likelihood	-25369	-25305	-25260	-25227
	AIC	51110	50980	50895	50828
	CV	10.005	4.381	4.383	4.375
	Standard errors in parentheses

* p < 0.1, * * p < 0.05, * * * p < 0.01

Table 1 .

 1 B.3: Parametric estimation results with exponential utility: pooled data

		TK92	P98-I	P98-II	GE87
					Gain domain
	αe	-0.0008	(0.0005)	0.0004	(0.0004)	0.0066 * * *	(0.0007)	0.0072 * * *	(0.0007)
	α	0.490 * * *	(0.0065)	0.505 * * *	(0.0056)	0.581 * * *	(0.0079)	0.589 * * *	(0.0079)
	a	0.345 * * *	(0.0132)	0.502 * * *	(0.0071)	0.492 * * *	(0.0072)	1.025 * * *	(0.0163)
	b					0.862 * * *	(0.0093)	0.467 * * *	(0.0068)
			Loss domain
	βe	-0.0171 * * *	(0.0009) -0.0144 * * *	(0.0007) -0.0068 * * *	(0.0007) -0.0059 * * *	(0.00069)
	β	-0.307 * * *	(0.0087)	-0.333 * * *	(0.0067)	-0.416 * * *	(0.0083)	-0.428 * * *	(0.0083)
	a	0.319 * * *	(0.0146)	0.492 * * *	(0.0077)	0.490 * * *	(0.0079)	1.015 * * *	(0.0167)
	b					0.871 * * *	(0.0096)	0.468 * * *	(0.0071)
	N	8906		8906		8906		8906
	Log Likelihood	-25560	-25348	-25165	-25131
	AIC	51489	51066	50704	50636
	CV	4.842		4.512		4.304		4.253
	Standard errors in parentheses

* p < 0.1, * * p < 0.05, * * * p < 0.01 Standard errors in parentheses * p < 0.1, * * p < 0.05, * * * p < 0.01

  distributions of the curvature coefficients obtained under both methods are

	plotted in Figures 1.B.1 and 1.B.2, and their median values are summarized in
	Table 1.B.4.
	Figures 1.B.1 and 1.B.2 indicate that, regardless of the estimation method
	at hand, the dominant pattern is the concavity of the utility function in both
	domains.

Table 1

 1 Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components values < 0.0361) for the probabilities p = 0.05 and p = 0.10, and insigniĄcant for the remaining probabilities (all pvalues > 0.1347). This suggests that Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components

	there is stronger overweighting of probabilities p = 0.05 and p = 0.10 in the
	gain domain than in the loss domain. Hence, subjects are more optimistic
	about very small probabilities of gaining money than they are pessimistic about
	very small probabilities of losing money. In contrast, identical probability
	weighting cannot be refuted with the parametric method because the median
	differences in the estimated probability weights are systematically insigniĄcant
	(all p -values > 0.7651). 36
	Table 1.B.4: Median estimates for exponential utility function using individual
			estimates
		Semi-parametric		Parametric
		Median	IQR †	Median	IQR †
	Utility parameter (Gain)	0.0068	[-0.0001; 0.0152]	0.0081	[0.0007; 0.0136]
	Utility parameter (Loss)	-0.0065	[-0.0172; 0.0001]	-0.0063	[-0.0158; 0.0002]
	AIC ⋆		50,601.23		50,636.95
	CV ⋆		4.2494		4.2529

.B.5 and Figure 1.B.3 summarize the estimated probability weighting functions across domains. In addition, Figures 1.B.4 and 1.B.5 show the underlying distributions of individual estimates. Both methods consistently point to an inverse S-shaped weighting function across domains, with a crossover point around p = 0.5. We cannot reject H 0 : w + (0.5) = 0.5 or H 0 : w -(0.5) = 0.5 with pvalues > 0.3698.

Focusing on our semi-parametric method, we report that the median difference in probability weights between gains and losses is positive and signiĄcant (p -† IQR stands for interquartile range ⋆ AIC and CV are from pooled estimates

36

The same results hold on the median differences of the parameters of the probability weighting function (all pvalues > 0.6539)

Table 1 .

 1 B.5: Median values of probability weights based on individual estimates

		Semi-parametric	Parametric
		Median	IQR +	Median +	IQR +
				Gain
	w + (0.05)	0.2156	0.1339; 0.3181	0.2033	0.1273; 0.3030
	w + (0.10)	0.2666	0.1826; 0.3696	0.2670	0.1865; 0.3502
	w + (0.25)	0.3635	0.2763; 0.4521	0.3741	0.2971; 0.4437
	w + (0.50)	0.5046	0.4196; 0.5660	0.5163	0.4302; 0.5685
	w + (0.75)	0.6441	0.5251; 0.7269	0.6323	0.5549; 0.7067
	w + (0.90)	0.7900	0.6524; 0.8720	0.7315	0.6420; 0.8135
	w + (0.95)	0.8038	0.6916; 0.8961	0.7991	0.6999; 0.8698
				Loss
	w -(0.05)	0.1886	0.0908; 0.3026	0.1922	0.1184; 0.3000
	w -(0.10)	0.2545	0.1490; 0.3811	0.2521	0.1729; 0.3688
	w -(0.25)	0.3628	0.2750; 0.4907	0.3663	0.2906; 0.4568
	w -(0.50)	0.5076	0.4289; 0.5724	0.4928	0.4345; 0.5645
	w -(0.75)	0.6506	0.5770; 0.7392	0.6398	0.5463; 0.6967
	w -(0.90)	0.7488	0.6340; 0.8353	0.7560	0.6564; 0.8118
	w -(0.95)	0.7922	0.6867; 0.8821	0.8131	0.7123; 0.8721
	IQR stands for interquartile range

+ For parametric methods, the median weights and IQR are computed from the individual probability weighting function

to

  Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components estimate world-level and country-level utility functions, probability weights and loss aversion. To perform HBPE, we rewrite equation (1.10) to account for the fact that risk preferences are elicited at the country level (c):

  Chapter 1. All at Once! A Comprehensive and Tractable Semi-Parametric Method to Elicit Prospect Theory Components aversion λ c follows a log-normal distribution LN (λ, σ 2 λ ) with λ ⇝ N (0, 10) and σ 2 λ ⇝ IG(0.001, 0.001).

Table 1 .

 1 ). The crossover point is around 3/8 in both domains as the estimated mean lies within the 95% credible interval (see Table1.D.1). D.1: Posterior statistics for utility and probability weights

	We have overweighting

* MCSE stands for Monte Carlo standard errors

2.2.1 Prospect theory for binary lottery

	For notational convenience, let x > y ≥ 0 (x < y ≤ 0) for non-mixed prospects involving only gains (losses). For mixed prospects (i.e., involving both gains
	and losses), outcomes are denoted with an asterisk and y

Consider a binary lottery L = (x, y; p, 1p) yielding outcome x with probability p and outcome y with probability 1p, both outcomes being real numbers. 3 * < 0 < x * . ≿ is a preference relation over prospects with ≻ (∼) denoting strict preference (indifference). Preferences are represented by CPT with a probability weighting function w i and a value function v as deĄned in equation (

2

.1) for non-mixed prospects and in equation (

2

.2) for the mixed ones:

Table 2 .

 2 A.1 provides detailed results on the data of TK92. We use three internal knots that correspond to quartile of the certainty equivalent (e.g. Ahamada and Flachaire, 2010).

	Table 2.A.1: Individual estimates: data of TK92
	Domain	Gain (i = +)	Loss (i = -)
		Utility function
	a i 2	-0.00207	-0.00220
	a i 3	0.00000662	0.00000898
	b i 1	-0.0000120	-0.00000555
	b i 2	0.0000152	0.00000192
	b i 3	-0.00000901	-0.00000510
		Probability weights
	w i (0.01)	0.0501	0.0378
	w i (0.05)	0.133	0.104
	w i (0.10)	0.183	0.154
	w i (0.25)	0.263	0.252
	w i (0.50)	0.415	0.407
	w i (0.75)	0.515	0.545
	w i (0.90)	0.661	0.675
	w i (0.95)	0.707	0.714
	w i (0.99)	0.916	0.919
	Order of the spline and smooth parameter
	Q +	3	3
	ρ +	9500	9000
	CV	12.42	11.49
	N	700	700

Table 2 .

 2 B.1 provides detailed results for the 10 subjects in the data of GW99. We use three internal knots that correspond to quartile of the certainty equivalent (e.g. Ahamada and Flachaire, 2010).

Table 2 .

 2 B.1: Individual estimates: data of GW99

	Subject	1	2	3	4	5	6	7	8	9	10
					Utility function
	a + 2	-0.0005	-0.0007	-0.0006	-0.0045	-0.0219	-0.0008	-0.0067	-0.0067	-0.0053 -0.0085
	a + 3	-5 × 10 -6	-7 × 10 -4
	b + 1	7 × 10 -6	4 × 10 -6	-1.1 × 10 -4	-1.3 × 10 -7	0.0205	-3.4 × 10 -4	9.5 × 10 -4	2.2 × 10 -4	0.0013	0.0038
	b + 2	-6 × 10 -7	6.3 × 10 -6	1.3 × 10 -4	5.25 × 10 -5	0.0008	-0.0001	0.0041	0.0022	0.0020	0.0035
	b + 3	-1.2 × 10 -6	-3 × 10 -6	0.0004	0.0040	0.0006	0.0009	0.0015	0.0040	0.0020	0.0012
					Probability weights
	w + (0.01)	0.137	0.0801	0.340	0.101	0.233	0.0682	0.127	0.162	0.0282	0.196
	w + (0.05)	0.177	0.155	0.334	0.175	0.281	0.113	0.237	0.171	0.0689	0.225
	w + (0.10)	0.175	0.286	0.428	0.130	0.365	0.133	0.230	0.187	0.121	0.301
	w + (0.25)	0.240	0.347	0.485	0.220	0.451	0.280	0.256	0.283	0.245	0.404
	w + (0.40)	0.255	0.420	0.507	0.154	0.517	0.440	0.281	0.242	0.359	0.489
	w + (0.50)	0.195	0.434	0.523	0.183	0.535	0.542	0.246	0.221	0.442	0.523
	w + (0.60)	0.315	0.448	0.489	0.178	0.574	0.623	0.319	0.355	0.492	0.595
	w + (0.75)	0.406	0.537	0.563	0.217	0.597	0.795	0.352	0.396	0.687	0.576
	w + (0.90)	0.415	0.779	0.640	0.238	0.681	0.931	0.443	0.434	0.799	0.699
	w + (0.95)	0.458	0.885	0.612	0.255	0.730	0.960	0.466	0.469	0.896	0.737
	w + (0.99)	0.661	0.819	0.738	0.362	0.850	0.968	0.530	0.620	0.769	0.866
				Order of the spline and smooth parameter
	Q +	3	3	2	2	2	2	2	2	2	2
	ρ +	10000	9000	5100	1700	1	3000	600	3500	500	400
	CV	14.00	14.02	18.63	7.23	14.49	6.24	8.31	11.86	11.68	14.01
	N	165	165	165	165	165	165	165	165	165	165

Table 3

 3 Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty game, and is known by both Player 1 and Player 2, is shown in Table3.5.2.

	.5.1: Social ambiguity -coordination game treatment.
			Player 2
			Amazon Google Play Apple Store
	Player 1	Left Right	15, 5 8, 5	10, 5 15, 5	8, 5 10, 5
		Middle	10, 5	8, 5	15, 5
	To measure strategic uncertainty, we implement the strategic uncertainty -
	cg treatment. The matrix of the game, which follows a traditional coordination

Table 3

 3

			.5.3: Binary lotteries
	No. of lottery	x	y	E	E c	Midpoint of outcome
						lotteries
	First set of lotteries
	1	15 ECU 8 ECU	E 1

  Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty comparison 1. 10 Consequently, comparing social ambiguity -tg and betrayal aversion, allows to measure the effect of betrayal aversion.

	5. Comparison between social ambiguity -cg and social ambiguity
	-tg: these two conditions have the same source of uncertainty, but differ
	in two aspects. First, in the social ambiguity -cg treatment, Player 1
	does not have the possibility to make her payoffs independent from the
	preferences of Player 2. Contrary, the social ambiguity -tg offers this
	possibility. We call such difference dependence payoff attitudes. Second, in
	the social ambiguity -cg, Player 1 has multiple options of payoffs (8, 10,

and 15) associated to each preference (i.e. Amazon, Google Play and Apple Store) of Player 2. Opposite, the social ambiguity -tg treatment does not offer such variety of payoff to Player 1 associated to each preference of Player 2. We call this second difference variety of payoff attitudes.

  Table 3.6.1 provides the summary of the estimated values.

	The median utility curvatures are 0.930, 0.876, 0.988, 0.968 and 0.968 for nature
	ambiguity, social ambiguity -cg, strategic uncertainty -cg, social ambiguity -
			Table 3.6.1: Utility function
		Nature	Social ambiguity-cg	Strategic	Social ambiguity-tg	Betrayal
				uncertainty		aversion
	Median (α)	0.930	0.876	0.988	0.968	0.968
	IQR	[0.771, 1.196]	[0.625, 1.042]	[0.760, 1.194]	[0.790,1.259]	[0.750 ,1.248]
			IQR: interquartile range

tg, and betrayal aversion, respectively. These values are less than 1, the utility functions are concave in all treatments. Also, the utility curvature in the social ambiguity -cg is signiĄcantly different from linear (pvalues < 0.0001). Contrary, for the other treatments, we cannot reject null hypothesis of linear utility (all pvalues > 0.0733).

Table 3 .

 3 6.2: Median of event weights or willingness to bet and M mean Amazon, Google Play and Apple Store in social ambiguity-cg and social ambiguity-tg L, R and M mean Reciprocate, No hurt strategy and Betray in Betrayal aversion L, R and M mean Left, Right and Middle in nature and strategic uncertainty-cg

		Nature	Social ambiguity-cg	Strategic	Social ambiguity-tg	Betrayal
				uncertainty-cg		aversion
	W (P (L))	0.305	0.382	0.284	0.341	0.288
		[0.235 , 0.369]	[0.258, 0.478]	[0.203, 0.364]	[0.231, 0.432]	[0.204, 0.383]
	W (P (R))	0.292	0.303	0.274	0.247	0.282
		[0.212, 0.292]	[0.206, 0.399]	[0.202, 0.361]	[0.158, 0.354]	[0.211, 0.361]
	W (P (M ))	0.291	0.325	0.273	0.289	0.323
		[0.212 , 0.364]	[0.219 , 0.392]	[0.187 , 0.369]	[0.185 , 0.370]	[0.235 , 0.390]
	W (P (L ∪ R))	0.467	0.549	0.474	0.454	0.459
		[0.337 , 0.600]	[0.421 , 0.660]	[0.363 , 0.586]	[0.367 , 0.625]	[0.335 , 0.540]
	Interquartile ranges are in [.]
	L, R

Table 3

 3

				.B.1: Individual estimate: nature
	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	1	0.549	0.494	0.447	0.491	0.643	1.319	0.525	0.361	0.283	0.356
	2	0.754	0.386	0.365	0.365	0.672	1.086	0.874	0.348	0.326	0.326
	3	1.399	0.235	0.235	0.235	0.337	0.395	0.365	0.333	0.333	0.333
	4	0.821	0.385	0.385	0.385	0.601	0.973	0.633	0.333	0.333	0.333
	5	3.180	0.113	0.113	0.206	0.113	.	.	.	.	.
	6	0.554	0.369	0.369	0.415	0.553	0.937	0.584	0.309	0.309	0.382
									Continued on next page

Table 3 .B.2 -continued from previous page

 3

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	7	2.088	0.153	0.134	0.141	0.153	0.172	0.006	1	0	0
	8	0.891	0.229	0.229	0.229	0.483	0.527	0.828	0.333	0.333	0.333
	9	0.768	0.243	0.317	0.288	0.358	0.476	0.253	0.176	0.478	0.346
		1.589	0.174	0.151	0.209	0.197	0.254	0.216	0.294	0.164	0.542
		0.729	0.305	0.327	0.259	0.327	.	.	.	.	.
		0.580	0.540	0.540	0.540	0.774	2.002	0.772	0.333	0.333	0.333
		1.392	0.299	0.270	0.309	0.299	0.437	0.013	0.136	0	0.864
		1.471	0.200	0.220	0.184	0.282	0.297	0.237	0.322	0.441	0.237
		0.912	0.324	0.308	0.308	0.515	0.687	0.592	0.352	0.324	0.324
		0.867	0.368	0.368	0.368	0.650	1.038	0.837	0.333	0.333	0.333
		1.524	0.371	0.467	0.393	0.467	0.753	0.010	0	1	0
		1.129	0.348	0.295	0.361	0.388	0.599	0.237	0.380	0.180	0.440
		3.523	0.145	0.098	0.020	0.230	.	.	.	.	.
		0.713	0.410	0.431	0.463	0.683	1.364	0.825	0.306	0.329	0.364
		3.675	0	0	0.032	0.001	0.006	3.379	0.114	0.267	0.618
		0.744	0.306	0.374	0.374	0.580	0.907	0.743	0.275	0.363	0.363
		1.064	0.256	0.296	0.296	0.533	0.692	0.815	0.298	0.351	0.351
		1.478	0.050	0.067	0.058	0.067	0.067	0.018	0	0.984	0.016
		0.654	0.291	0.317	0.317	0.613	0.857	0.943	0.314	0.343	0.343
		1.493	0.323	0.323	0.323	0.458	0.635	0.413	0.333	0.333	0.333
		1.196	0.284	0.235	0.214	0.289	.	.	.	.	.
		0.772	0.424	0.406	0.406	0.702	1.268	0.857	0.347	0.327	0.327
		0.695	0.319	0.273	0.273	0.363	0.463	0.182	0.519	0.241	0.241
		1.108	0.269	0.246	0.246	0.640	0.762	1.164	0.349	0.325	0.325
		1.490	0.133	0.113	0.094	0.284	0.203	0.671	0.398	0.333	0.269
		0.532	0.412	0.412	0.442	0.580	1.046	0.517	0.316	0.316	0.368
		0.858	0.293	0.293	0.293	0.447	0.578	0.483	0.333	0.333	0.333
		1.269	0.235	0.217	0.261	0.292	0.382	0.288	0.320	0.247	0.433
		0.809	0.287	0.297	0.269	0.413	0.509	0.355	0.340	0.373	0.286
		0.783	0.238	0.238	0.238	0.485	0.542	0.797	0.333	0.333	0.333
		1.228	0.070	0.070	0.070	0.334	0.194	1.364	0.333	0.333	0.333
		1.122	0.214	0.201	0.201	0.214	.	.	.	.	.
		0.667	0.285	0.236	0.236	0.471	0.525	0.635	0.394	0.303	0.303
		0.934	0.316	0.300	0.300	0.534	0.701	0.673	0.350	0.325	0.325
		0.537	0.550	0.550	0.550	0.749	1.911	0.644	0.333	0.333	0.333
		0.775	0.373	0.373	0.377	0.373	.	.	.	.	.
		0.790	0.076	0.076	0.076	0.076	.	.	.	.	.
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
									Continued on next page

Table 3 .B.2 -continued from previous page

 3

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
		0.832	0.354	0.326	0.326	0.644	0.936	0.890	0.354	0.323	0.323
		1.018	0.273	0.273	0.273	0.465	0.571	0.607	0.333	0.333	0.333
		0.411	0.455	0.546	0.464	0.861	2.320	1.276	0.310	0.374	0.316
		1.962	0.307	0.142	0.129	0.482	0.371	0.671	0.566	0.231	0.203
		0.472	0.369	0.369	0.369	0.634	1.005	0.783	0.333	0.333	0.333
		0.742	0.254	0.207	0.207	0.391	0.409	0.517	0.411	0.294	0.294
		1.042	0.348	0.348	0.348	0.669	1.038	0.963	0.333	0.333	0.333
		1.101	0.272	0.248	0.248	0.419	0.488	0.505	0.370	0.315	0.315
		1.080	0.322	0.322	0.322	0.443	0.615	0.371	0.333	0.333	0.333
		0.992	0.246	0.044	0.203	0.878	1.353	3.052	0.385	0.248	0.367
		1.158	0.313	0.313	0.294	0.600	0.791	0.839	0.341	0.341	0.318
		0.799	0.386	0.394	0.385	0.394	.	.	.	.	.
		2.902	0.077	0.077	0.077	0.141	0.117	0.489	0.333	0.333	0.333
		1.334	0.212	0.212	0.212	0.422	0.444	0.719	0.333	0.333	0.333
		1.334	0.188	0.164	0.164	0.262	0.264	0.343	0.405	0.298	0.298
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
		0.871	0.306	0.292	0.163	0.335	.	.	.	.	.
		0.281	0.486	0.524	0.620	0.524	1.338	0.016	0	0	1
		0.699	0.348	0.205	0.394	0.659	1.120	1.294	0.360	0.243	0.396
		0.843	0.377	0.444	0.350	0.582	0.864	0.425	0.301	0.453	0.246
		1.018	0.273	0.273	0.273	0.465	0.571	0.607	0.333	0.333	0.333
		0.379	0.515	0.414	0.414	0.556	0.941	0.180	0.662	0.169	0.169
		3.812	0.210	0.146	0.402	0.210	0.423	0.043	0	0	1
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
		0.930	0.216	0.258	0.258	0.430	0.512	0.666	0.284	0.358	0.358
		1.081	0.349	0.349	0.349	0.503	0.735	0.459	0.333	0.333	0.333
		2.117	0.092	0.092	0.092	0.274	0.195	0.949	0.333	0.333	0.333
		0.891	0.372	0.372	0.291	0.426	.	.	.	.	.
		1.204	0.218	0.243	0.243	0.386	0.450	0.550	0.296	0.352	0.352
		1.018	0.273	0.273	0.273	0.465	0.571	0.607	0.333	0.333	0.333
		0.802	0.074	0.074	0.074	0.965	1.492	4.225	0.333	0.333	0.333
		1.172	0.451	0.176	0.201	0.533	0.536	0.463	0.715	0.121	0.164
		0.929	0.281	0.281	0.281	0.536	0.672	0.781	0.333	0.333	0.333
		0.963	0.373	0.333	0.333	0.400	0.577	0.111	0.571	0.215	0.215
		0.926	0.346	0.286	0.346	0.471	0.686	0.500	0.373	0.255	0.373
		1.799	0.083	0.083	0.044	0.437	0.188	1.368	0.369	0.369	0.262
		0.821	0.385	0.385	0.385	0.601	0.973	0.633	0.333	0.333	0.333
		0.949	0.323	0.266	0.217	0.497	0.523	0.501	0.455	0.325	0.220
									Continued on next page

Table 3 .B.2 -continued from previous page

 3

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	83 0.039	0.889	0.889	0.889	0.976	18.021 1.170	0.333	0.333	0.333
	84 0.941	0.291	0.291	0.291	0.430	0.556	0.441	0.333	0.333	0.333
	85 0.702	0.396	0.368	0.270	0.481	.	.	.	.	.
	86 0.762	0.246	0.246	0.246	0.468	0.535	0.716	0.333	0.333	0.333
	87 1.074	0.300	0.300	0.364	0.300	.	.	.	.	.
	88 0.952	0.332	0.310	0.310	0.489	0.656	0.496	0.363	0.318	0.318
	89 0.967	0.047	0.047	0.047	0.117	0.081	0.710	0.333	0.333	0.333

Table 3 .

 3 B.3: Individual estimate: social ambiguity-cg

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	1	1.316	0.440	0.168	0.168	0.643	0.604	0.826	0.579	0.210	0.210
	2	0.739	0.465	0.379	0.362	0.654	1.035	0.615	0.429	0.298	0.273
	3	0.172	0.845	0.807	0.784	0.908	5.999	0.449	0.446	0.308	0.246
	4	0.718	0.831	0.350	0.288	0.885	1.761	0.379	0.938	0.042	0.020
	5	0.466	0.617	0.594	0.637	0.759	2.349	0.547	0.334	0.296	0.370
	6	1.212	0.158	0.509	0.105	0.653	0.470	0.398	0.091	0.879	0.030
	7	0.479	0.605	0.605	0.630	0.605	.	.	.	.	.
	8	0.857	0.303	0.283	0.283	0.456	0.575	0.497	0.363	0.319	0.319
	9	1.084	0.211	0.160	0.250	0.211	0.299	0.033	0.034	0	0.966
		0.966	0.346	0.332	0.273	0.346	.	.	.	.	.
		0.563	0.438	0.409	0.452	0.540	0.984	0.361	0.344	0.275	0.381
		0.680	0.438	0.295	0.392	0.686	1.186	0.977	0.395	0.256	0.349
		0.976	0.434	0.346	0.360	0.434	.	.	.	.	.
		0.808	0.391	0.449	0.431	0.501	0.872	0.237	0.215	0.429	0.356
		0.732	0.333	0.361	0.379	0.498	0.777	0.479	0.284	0.340	0.376
		0.932	0.605	0.146	0.397	0.747	1.396	1.147	0.520	0.138	0.342
		0.910	0.631	0.419	0.338	0.631	.	.	.	.	.
		1.194	0.259	0.208	0.225	0.259	0.318	0.010	1	0	0
		2.219	0.159	0.139	0.159	0.410	0.363	1.013	0.345	0.309	0.345
		1.017	0.230	0.164	0.374	0.363	0.584	0.774	0.296	0.196	0.507
		0.376	0.298	0.298	0.108	0.298	.	.	.	.	.
		0.613	0.526	0.311	0.436	0.610	1.100	0.528	0.505	0.156	0.339
		0.541	0.860	0.484	0.443	0.912	2.877	0.543	0.801	0.113	0.086
		0.330	0.563	0.563	0.603	0.731	2.031	0.576	0.312	0.312	0.376
		0.816	0.360	0.360	0.360	0.549	0.828	0.556	0.333	0.333	0.333
		3.255	0.089	0.089	0.089	0.189	0.151	0.630	0.333	0.333	0.333
		0.932	0.451	0.267	0.301	0.498	0.653	0.294	0.685	0.120	0.195
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
		0.429	0.522	0.485	0.651	0.608	1.698	0.409	0.253	0.191	0.556
		1.131	0.374	0.224	0.361	0.565	0.855	0.861	0.397	0.221	0.381
		0.865	0.451	0.297	0.325	0.534	0.742	0.386	0.566	0.189	0.245
		1.348	0.150	0.209	0.185	0.520	0.496	1.168	0.292	0.369	0.339
		2.204	0.015	0.055	0.049	0.055	0.055	0.095	0	0.664	0.336
		0.875	0.304	0.303	0.321	0.530	0.731	0.704	0.326	0.324	0.350
		0.922	0.219	0.219	0.219	0.219	0.280	0	0.002	0.104	0.894
		0.533	0.412	0.361	0.353	0.441	0.656	0.108	0.645	0.202	0.152
		0.352	0.537	0.512	0.598	0.698	1.855	0.599	0.313	0.278	0.409
		1.529	0.384	0.184	0.151	0.569	0.483	0.631	0.600	0.231	0.169
									Continued on next page

Table 3 .

 3 B.7: Individual estimate: social ambiguity-tg

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	1	1.757	0.270	0.079	0.018	0.391	.	.	.	.	.
	2	1.266	0.245	0.203	0.203	0.597	0.615	1.149	0.365	0.318	0.318
	3	0.806	0.397	0.302	0.302	0.619	0.838	0.741	0.419	0.291	0.291
	4	28.42	0	0	0	0	0	0.506	1	0	0
	5	0.745	0.643	0.643	0.701	0.643	.	.	.	.	.
	6	1.296	0.092	0.092	0.525	0.113	0.375	0.327	0.018	0.018	0.965
	7	1.620	0.219	0.203	0.191	0.540	0.527	1.037	0.353	0.331	0.316
	8	0.677	0.379	0.379	0.379	0.530	0.831	0.442	0.333	0.333	0.333
	9	1.972	0.098	0.052	0.052	0.118	0.085	0.240	0.732	0.134	0.134
		1.259	0.299	0.386	0.286	0.386	.	.	.	.	.
		0.604	0.367	0.367	0.298	0.367	.	.	.	.	.
		0.909	0.365	0.285	0.332	0.402	0.577	0.226	0.498	0.163	0.339
		0.394	0.315	0.300	0.300	0.315	.	.	.	.	.
		1.353	0.245	0.263	0.289	0.396	0.516	0.513	0.288	0.327	0.385
		0.876	0.296	0.321	0.321	0.528	0.726	0.676	0.308	0.346	0.346
		1.231	0.504	0.212	0.293	0.504	0.649	0.042	1	0	0
		1.516	0.431	0.247	0.175	0.431	.	.	.	.	.
		0.746	0.376	0.376	0.376	0.376	0.602	0	0.286	0.286	0.428
		0.986	0.392	0.392	0.384	0.558	0.887	0.474	0.339	0.339	0.323
		1.832	0.146	0.042	0.378	0.251	0.451	1.211	0.310	0.128	0.561
		0.644	0.107	0.134	0.107	0.134	.	.	.	.	.
		0.902	0.417	0.239	0.311	0.417	0.569	0.035	0.999	0	0.001
		0.968	0.661	0.205	0.205	0.810	1.047	0.861	0.673	0.164	0.164
		0.468	0.541	0.541	0.523	0.670	1.494	0.375	0.347	0.347	0.305
		1.977	0.108	0.108	0.108	0.333	0.245	1.021	0.333	0.333	0.333
		1.984	0.179	0.179	0.179	0.367	0.356	0.705	0.333	0.333	0.333
		0.794	0.328	0.352	0.352	0.454	0.673	0.359	0.289	0.356	0.356
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
		0.634	0.491	0.400	0.371	0.734	1.279	0.808	0.414	0.308	0.278
		1.226	0.717	0.213	0.215	0.902	1.584	1.303	0.589	0.205	0.206
		0.699	0.420	0.409	0.433	0.478	0.836	0.201	0.331	0.280	0.389
		0.920	0.308	0.348	0.370	0.557	0.858	0.723	0.287	0.342	0.371
		0.610	0.380	0.191	0.563	0.409	0.943	0.483	0.291	0.054	0.655
		1.077	0.239	0.294	0.255	0.406	0.483	0.438	0.272	0.416	0.312
		1.095	0.660	0.554	0.657	0.660	1.930	0.037	0.548	0	0.452
		0.800	0.231	0.186	0.186	0.434	0.419	0.736	0.390	0.305	0.305
		0.954	0.286	0.092	0.209	0.286	0.325	0.083	0.926	0	0.074
		1.290	0.222	0.139	0.162	0.260	0.260	0.288	0.580	0.158	0.262
									Continued on next page

Table 3 .

 3 B.9: Individual estimate: betrayal aversion-tg

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	1	1.284	0.287	0.187	0.187	0.407	0.397	0.488	0.507	0.246	0.246
	2	1.025	0.286	0.268	0.277	0.673	0.888	1.214	0.342	0.325	0.333
	3	0.906	0.353	0.353	0.353	0.544	0.807	0.562	0.333	0.333	0.333
	4	0.660	0.332	0.332	0.812	0.474	1.975	0.704	0.124	0.124	0.753
	5	1.006	0.485	0.426	0.437	0.519	0.915	0.154	0.544	0.203	0.253
	6	0.835	0.248	0.219	0.415	0.304	0.557	0.382	0.203	0.143	0.654
	7	1.417	0.171	0.186	0.163	0.367	0.336	0.679	0.328	0.363	0.309
	8	0.770	0.309	0.288	0.288	0.465	0.592	0.500	0.364	0.318	0.318
	9	2.036	0.056	0.056	0.025	0.056	.	.	.	.	.
		0.968	0.291	0.329	0.345	0.329	0.508	0.016	0	0.097	0.903
		0.561	0.494	0.395	0.439	0.516	0.914	0.166	0.600	0.117	0.283
		0.668	0.369	0.439	0.390	0.683	1.173	0.816	0.299	0.378	0.322
		1.929	0.207	0.140	0.103	0.222	.	.	.	.	.
		0.962	0.274	0.220	0.247	0.544	0.625	0.931	0.368	0.298	0.334
		1.056	0.219	0.261	0.300	0.419	0.557	0.664	0.262	0.335	0.403
		1.522	0.152	0.071	0.310	0.174	0.308	0.524	0.262	0.065	0.673
		1.295	0.383	0.316	0.277	0.517	0.640	0.396	0.481	0.305	0.214
		0.939	0.341	0.368	0.368	0.368	0.583	0.010	0	0.500	0.500
		2.736	0.083	0.083	0.099	0.282	0.208	1.109	0.320	0.320	0.361
		0.561	0.577	0.337	0.285	0.623	.	.	.	.	.
		0.017	0	0	0.096	0.946	1.361	15.938 0.270	0.270	0.460
		0.706	0.252	0.335	0.517	0.335	0.734	0.039	0	0	1
		1.366	0.472	0.501	0.147	0.832	.	.	.	.	.
		0.764	0.409	0.409	0.409	0.547	0.915	0.403	0.333	0.333	0.333
		0.712	0.719	0.385	0.385	0.719	.	.	.	.	.
		1.493	0.323	0.323	0.323	0.458	0.635	0.413	0.333	0.333	0.333
		1.017	0.355	0.273	0.313	0.481	0.650	0.503	0.419	0.251	0.330
		0.884	0.361	0.361	0.361	0.670	1.071	0.922	0.333	0.333	0.333
		0.752	0.215	0.215	0.382	0.215	.	.	.	.	.
		1.365	0.346	0.189	0.366	0.352	0.560	0.245	0.442	0.028	0.531
		1.076	0.255	0.255	0.354	0.400	0.604	0.575	0.271	0.271	0.458
		1.035	0.225	0.251	0.328	0.331	0.492	0.414	0.219	0.285	0.496
		1.054	0.176	0.284	0.124	0.284	.	.	.	.	.
		0.940	0.314	0.298	0.338	0.424	0.613	0.402	0.326	0.285	0.389
		1.319	0.454	0.483	0.483	0.483	0.933	0.010	0	0.500	0.500
		0.573	0.331	0.223	0.327	0.518	0.723	0.797	0.383	0.239	0.378
		1.439	0.072	0.100	0.100	0.256	0.196	0.981	0.279	0.360	0.360
		1.216	0.266	0.240	0.240	0.342	0.406	0.291	0.403	0.298	0.298
									Continued on next page

Table 3 .B.10 -continued from previous page

 3

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
		0.729	0.211	0.211	0.181	0.469	0.442	0.812	0.350	0.350	0.299
		0.976	0.329	0.286	0.326	0.494	0.688	0.589	0.360	0.285	0.355
		5.298	0.007	0.007	0.007	0.081	0.025	1.800	0.333	0.333	0.333
		0.511	0.684	0.679	0.658	0.684	.	.	.	.	.
		0.790	0.076	0.076	0.076	0.076	.	.	.	.	.
		1.094	0.371	0.292	0.292	0.452	0.583	0.309	0.511	0.245	0.245
		0.307	0.094	0.442	0.389	0.720	1.279	1.639	0.178	0.428	0.395
		1.103	0.755	0.755	0.953	0.755	.	.	.	.	.
		0.883	0.543	0.645	0.507	0.734	1.682	0.329	0.258	0.559	0.183
		1.123	0.042	0.347	0.779	0.347	1.366	0.207	0	0.010	0.990
		0.533	0.448	0.381	0.455	0.577	1.067	0.504	0.368	0.251	0.381
		0.723	0.201	0.285	0.251	0.488	0.566	0.798	0.266	0.393	0.342
		0.802	0.074	0.074	0.965	0.074	.	.	.	.	.
		1.371	0.266	0.182	0.218	0.529	0.561	0.978	0.391	0.280	0.329
		1.665	0.301	0.252	0.252	0.301	.	.	.	.	.
		0.802	0.074	0.074	0.965	0.074	.	.	.	.	.
		0.922	0.347	0.347	0.347	0.724	1.181	1.150	0.333	0.333	0.333
		1.395	0.244	0.253	0.253	0.545	0.636	0.933	0.326	0.337	0.337
		1.537	0.477	0.477	0.323	0.477	.	.	.	.	.
		0.799	0.348	0.485	0.453	0.485	0.883	0.040	0	0.835	0.165
		1.208	0.217	0.217	0.255	0.531	0.623	1.064	0.319	0.319	0.363
		43.27	0	0	0	0	0	1.002	0.021	0.021	0.958
		1.317	0.059	0.324	0.368	0.324	0.528	0.157	0	0.350	0.650
		0.654	0.280	0.259	0.239	0.451	0.508	0.538	0.378	0.332	0.290
		0.851	0.392	0.288	0.191	0.889	1.376	1.777	0.395	0.334	0.270
		0.891	0.084	0.084	0.356	0.319	0.509	1.510	0.243	0.243	0.514
		0.993	0.280	0.280	0.320	0.406	0.568	0.448	0.301	0.301	0.398
		0.721	0.494	0.485	0.511	0.589	1.222	0.308	0.325	0.301	0.375
		0.739	0.540	0.540	0.836	0.540	.	.	.	.	.
		1.013	0.298	0.316	0.316	0.636	0.899	0.998	0.320	0.340	0.340
		0.580	0.204	0.204	0.403	0.484	0.796	1.127	0.268	0.268	0.464
		1.330	0.213	0.238	0.213	0.395	0.421	0.565	0.315	0.371	0.315
		1.227	0.212	0.212	0.212	0.449	0.469	0.797	0.333	0.333	0.333
		0.657	0.345	0.458	0.345	0.526	0.766	0.278	0.207	0.586	0.207
		0.768	0.459	0.288	0.288	0.459	.	.	.	.	.
		1.669	0.072	0.084	0.118	0.084	0.111	0.017	0	0	1
		0.075	0.094	0.094	0.945	0.094	.	.	.	.	.
		0.716	0.396	0.361	0.318	0.511	0.699	0.310	0.450	0.336	0.214
									Continued on next page

Table 3 .B.10 -continued from previous page

 3

	id	α	W (P (L))	W (P (R))	W (P (M ))	W (P (L ∪ R))	η	γ	P (L)	P (R)	P (M )
	77 1.039	0.388	0.242	0.166	0.488	.	.	.	.	.
	78 1.045	0.288	0.275	0.365	0.346	0.552	0.287	0.252	0.212	0.536
	79 0.750	0.296	0.346	0.281	0.434	0.547	0.291	0.288	0.472	0.240
	80 2.908	0.195	0.174	0.174	0.375	0.356	0.686	0.364	0.318	0.318
	81 0.968	0.286	0.372	0.205	0.430	.	.	.	.	.
	82 1.297	0.198	0.198	0.235	0.683	0.814	1.613	0.323	0.323	0.354
	83 0.790	0.076	0.076	0.076	0.076	.	.	.	.	.
	84 0.753	0.467	0.497	0.471	0.497	.	.	.	.	.
	85 1.168	0.227	0.253	0.350	0.271	0.447	0.178	0.087	0.173	0.740
	86 0.359	0.507	0.484	0.484	0.507	.	.	.	.	.
	87 1.037	0.413	0.413	0.462	0.413	.	.	.	.	.
	88 0.861	0.563	0.282	0.323	0.563	.	.	.	.	.
	89 0.643	0.328	0.262	0.262	0.382	0.468	0.224	0.549	0.225	0.225

  2, are summarized in the table below. Note that Participant 2 is informed that his or her choice will affect you, but he or she does not know in what direction. This means that Participant 2 does not know how your payment changes based on his or her decision.

	Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of
	Uncertainty
			Participant 2
			Amazon voucher Google Play voucher Apple Store voucher
	Your	Left	30	20	16
	decision	Right	16	30	20
		Middle	20	16	30
			Participant 2
			Amazon voucher Google Play voucher Apple Store voucher
	Your	Left	30	20	16
	decision	Right	16	30	20
		Middle	20	16	30

Example

Suppose you decide to choose the Right action and Participant 2 prefers to spend his or her 5 euros on a Google Play voucher (remember that you will not be informed of Participant 2Šs decision until you receive the payment for the experiment). The table below shows in orange the payment (in ECU) you will get in this scenario. If this decision is chosen at random for the payment, you earn 30 ECU.

•

  Alternative B offers you a variable payment that depends on the decision made by Participant 2 in the Ąrst sub-part of this part of the experiment. This means that the payment you can receive varies depending on what Participant 2 decided between an Amazon

voucher, a Google Play voucher, or an Apple Store voucher.

  This alternative changes from table to table, but it is the same for all rows in a given table. Example of a table with payments (in ECU):

	Safe payment of alternative A	Alternative A	Alternative B	Variable payment of alternative B
	30	A1	B1	You get 30 ECU if Participant 2
	28	A2	B2	chooses an Amazon voucher in the
	26	A3	B3	first sub-part of this part of the
	24	A4	B4	experiment or 16 ECU if Participant
	22	A5	B5	2 chooses a Google Play voucher or
	20	A6	B6	an Apple Store voucher
	18	A7	B7
	16	A8	B8

•

  Alternative B offers you a variable payment that depends on the decision made by Participant 2 in the Ąrst sub-part of this part of the experiment. This means that the payment you can receive varies depending on what Participant 2 decided between the Left, Right or Middle actions. This alternative changes from table totable, but it is the same for all rows in a given table. Example of a table with payments (in ECU):

	Safe payment of alternative A	Alternative A	Alternative B	Variable payment of alternative B
	7	A1	B1	You get 7 ECU if Participant 2
	6.5	A2	B2	chooses an Left in the first sub-part of
	6	A3	B3	this part of the experiment or 3.5
	5.5	A4	B4	ECU if Participant 2 chooses a Right
	5	A5	B5	or Middle
	4.5	A6	B6
	4	A7	B7
	3.5	A8	B8

voucher, a Google Play voucher, or an Apple Store voucher

  Alternative B offers you a variable payment that depends on the decision made by Participant 2 in the Ąrst sub-part of this part of the experiment. This means that the payment you can receive varies depending on what Participant 2 decided between an Amazon

	Safe payment of alternative A	Alternative A	Alternative B	Variable payment of alternative B
	45	A1	B1	You get 45 ECU if Participant 2
	42	A2	B2	chooses an Amazon voucher in the
	39	A3	B3	first sub-part of this part of the
	36	A4	B4	experiment or 24 ECU if Participant
	33	A5	B5	2 chooses a Google Play or an
	30	A6	B6	Apple Store voucher
	27	A7	B7
	24	A8	B8

. Alternative B changes from table to table, but it is the same for all rows in a given table. Example of a table with payments (in ECU):

•

  Alternative B offers you a variable payment that depends on the decision made by Participant 2 in the Ąrst sub-part of this part of the experiment.

	Safe payment of alternative A	Alternative A	Alternative B	Variable payment of alternative B
	25	A1	B1	You get 25 ECU if Participant 2
	24	A2	B2	chooses Left in the first sub-part of
	23	A3	B3	this part of the experiment or 18
	22	A4	B4	ECU if Participant 2 chooses Right
	21	A5	B5	or Middle
	20	A6	B6
	19	A7	B7
	18	A8	B8

This means that the payment you can receive varies depending on what Participant 2 decided between Left, Right or Middle actions. Example of a table with payments (in ECU):

  this part of the experiment, you must choose between several options. The options are presented in 12 tables (see an example of the table below). Each Each option has an equal chance of being drawn. Alternative B changes from table to table, but is the same for all rows in a given table.Example of a table with payments (in ECU):Both alternatives are initially displayed in gray. You must click on one of the two alternatives to select it. Your selection will be highlighted in blue. Your payment depends on which option the computer randomly selects. Remember that each option has the same chance of being drawn.

	to the last one).
	row represents an option. For each option, you must indicate whether you prefer Example
	Alternative A or Alternative B.
	Suppose that the following option is randomly selected for payment:
	• Alternative A offers you a safe payment. Safe payment of alternative A Alternative A Alternative B	Variable payment of alternative B
	20	A1	B1		You get 20 ECU if the computer
	randomly chooses Left or 13 ECU if • Alternative B offers you a variable payment that depends on a random the computer randomly chooses Right
	selection made by the computer. The computer chooses one of three or Middle
	Alternative A options: Left, Right or Middle. Safe payment of alternative A Alternative B • If you select Alternative A for this line, you win 20 ECU. Variable payment of alternative B
	20	A1	B1	You get 20 ECU if the computer
	19	A2	B2	randomly chooses Left or 13 ECU if
	18	A3	B3	the computer randomly chooses Right
	17	A4	B4	or Middle
	16	A5	B5
	15	A6	B6
	14	A7	B7
	13	A8	B8
	In each line you will be asked to indicate whether you prefer Alternative A
	or Alternative B.
	You can change your selection at any time by clicking on the cell of the desired
	alternative, before moving on to the next screen. Once you conĄrm your decision,
	you cannot go back and change your previous decision.
	If you select Alternative A for a given row, the computer will mark Alterna-
	tive A for all previous rows (up to the Ąrst). Similarly, if you select Alternative

• If you select Alternative B for this line, you can win 20 ECU or 13 ECU.

Table 3 .

 3 D.1: Result of parameter recovery and misspeciĄcation excercises

	Specification		one-stage			two-stage
	u()	w()	u()	w()	P ()	pool	u()	w()	P ()	pool
					Parameter recovery
	E	CEG87	0.0010	0.0012	0.0096	0.0039	0.0010	0.0012	0.0096	0.0039
	E	GE87	0.0011	0.0012	0.0022	0.0015	0.0011	0.0012	0.0020	0.0014
	E	PR98	0.0011	0.0012	0.0031	0.0018	0.0011	0.0013	0.0028	0.0017
	P	CEG87	0.0005	0.0007	0.0189	0.0067	0.0005	0.0007	0.0189	0.0067
	P	GE87	0.0004	0.0006	0.0034	0.0015	0.0004	0.0006	0.0026	0.0012
	P	PR98	0.0005	0.0007	0.0037	0.0016	0.0005	0.0007	0.0033	0.0015
	pool	pool	0.0008 0.0009 0.0068 0.0028	0.0008 0.0009 0.0065 0.0027
					Misspecification
	E	CEG7	0.0028	0.0069	0.0070	0.0056	0.0029	0.0070	0.0070	0.0056
	E	GE87	0.0023	0.0064	0.0100	0.0062	0.0026	0.0059	0.0083	0.0056
	E	P98	0.0026	0.0070	0.0076	0.0057	0.0028	0.0067	0.0070	0.0055
	P	CEG7	0.0017	0.0018	0.0061	0.0032	0.0017	0.0018	0.0061	0.0032
	P	GE87	0.0018	0.0021	0.0090	0.0043	0.0018	0.0019	0.0076	0.0038
	P	P98	0.0019	0.0020	0.0074	0.0038	0.0019	0.0018	0.0068	0.0035
	pool	pool	0.0022 0.0044 0.0078 0.0048	0.0023 0.0042 0.0071 0.0045
				Parameter recovery and Misspecification
	E	CEG7	0.0019	0.0040	0.0083	0.0048	0.0020	0.0041	0.0083	0.0048
	E	GE87	0.0017	0.0038	0.0061	0.0038	0.0018	0.0035	0.0052	0.0035
	E	P98	0.0018	0.0041	0.0053	0.0038	0.0019	0.0040	0.0049	0.0036
	P	CEG7	0.0011	0.0012	0.0125	0.0049	0.0011	0.0012	0.0125	0.0049
	P	GE87	0.0011	0.0014	0.0062	0.0029	0.0011	0.0012	0.0051	0.0025
	P	P98	0.0012	0.0013	0.0056	0.0027	0.0012	0.0012	0.0051	0.0025
	pool	pool	0.0015 0.0026 0.0073 0.0038	0.0015 0.0026 0.0068 0.0036

  ). For risk-neutral agents, there is no RIT because Ąxed pay and variable pay do not vary with the shock (see DeĄnition 4iii). For risk-seeking agents, RIT is reversed because the optimal variable (Ąxed) pay increases (decreases) with the shock size. These results are standard in the LEN model (see e.g.,[START_REF] Milgrom | Economics, Organization and Management[END_REF]. We nonetheless provide the details of the proofs in Appendix 4.A

	Figure 4.2.2) and a risk-averse agent, the valuation (4.4) is decreasing in the
	shock size. In that case, the principal must offer the agent a higher Ąxed pay
	to maintain utility equal to the outside option (y 0 ). As in EUT, this situation
	corresponds to RIT.
	However, for an intermediate shock (see ϵ * < ϵ < ϵ 1 in Figure 4.2.2) the value
	function of the risk-averse MVS agent is increasing in the shock size. This implies
	that the principal can offer the agent a lower Ąxed pay while maintaining his level
	of utility equal to the outside option. Unlike EUT, this situation corresponds to
	reversed RIT for a risk-averse agent. Finally, in line with EUT, a risk-seeking
	MVS agent systematically exhibits reversed RIT given that his valuation in
	(4.4) is increasing in the shock magnitude. Example 2 provides a numerical
	illustration of a situation in which reversed RIT occurs for a risk-averse agent
	under MVS.
	(Proposition A1) for the case of a binary shock and for the case of a general utility
	function and a continuous shock (see Propositions B1 to B3 and Proposition C1
	in Appendices 4.B and 4.C).
	Under RDU, RIT is even more pervasive than under EUT. As in EUT, it
	occurs whenever agents are risk-averse (see Proposition A2 in Appendix 4.A)
	given that the valuation function (4.3) is decreasing in the shock magnitude. In
	contrast to EUT, it can also occur when agents are risk-neutral or risk-seeking
	(see Proposition A3 in Appendix 4.A). In the case of a risk-seeking agent who
	overweighs probabilities, the value of the contract in (4.3) increases with Ąxed
	pay α (irrespective of risk attitudes) and is inverse J-shaped with respect to
	the shock size (see Figure 4.2.1). For a small shock (ϵ 0 in Figure 4.2.1), (4.3)
	is increasing in the shock magnitude. In that case, the principal can offer the

Table 4 .

 4 2.1. For each type of agent risk attitudes, we report the three theories Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well (EUT, MVS and RDU) for which RIT is present (left column), absent (middle column), or reversed (right column). -MVS † With Assumption 3', an agent cannot be risk-seeking under EUT. To consider risk-seeking agents, we need to consider convex utility giving rise to reversed RIT.

	Table 4.2.1: RIT and risk-attitudes
	AgentŠs risk attitudes	RIT	No RIT	Reversed RIT
	Risk-averse	RDU-EUT-MVS	MVS	MVS
	Risk-neutral	RDU	RDU-EUT-MVS	MVS
	Risk-seeking	RDU	RDU	RDU-EUT †

Table 4 .

 4 4.2: Utility curvature and probability weighting under RDU

	Number (%)	Underweighting	Overweighting	Total
	Probability		p = 0.1
	Concave	8	201	209
		(3.38%)	(84.81%)	(88.19%)
	Convex	14	14	28
		(5.91%)	(5.91%)	(11.81%)
	Total	22	215	237
		(9.28%)	(90.72%)	(100%)
	Probability		p = 0.25
	Concave	46	163	209
		(19.41%)	(68.78%)	(88.19%)
	Convex	13	15	28
		(5.49%)	(6.33%)	(11.81%)
	Total	59	178	237
		(24.89%)	(75.11%)	(100%)
	Probability		p = 0.33
	Concave	76	133	209
		(32.07%)	(56.12%)	(88.19%)
	Convex	19	9	28
		(8.02%)	(3.80%)	(11.81%)
	Total	95	142	237
		(40.08%)	(59.92%)	(100%)
	Probability		p = 0.50
	Concave	126	83	209
		(53.16%)	(35.02%)	(88.19%)
	Convex	19	9	28
		(8.02%)	(3.80%)	(11.81%)
	Total	145	92	237
		(61.18%)	(38.82%)	(100%)
	Probability		p = 0.75
	Concave	164	45	209
		(69.20%)	(18.99%)	(88.19%)
	Convex	27	1	28
		(11.39%)	(0.42%)	(11.81%)
	Total	191	46	237
		(80.59%)	(19.41%)	(100%)

Table 4 .

 4 4.3: Attitudes towards variance and skewness under MVS Note that α ϵ m is measured as a midpoint of a range of values that is equal to 50, which is 5% of the outside option value. To account for this imprecision in our measure, we classify a participant as risk

	Number (%)	Aversion to skewness	Preference for skewness	Total
	Preference for variance	40	4	44
		(16.88%)	(1.69%)	(18.57%)
	Aversion to variance	6	187	193
		(2.53%)	(78.90%)	(81.43%)
	Total	46	191	237
		(19.41 %)	(80.59 %)	(100%)

Table 4 .

 4 4.5: RIT and risk attitudes To account for the estimation inaccuracy due to the use of midpoint of the range of possible values of α 3

	Risk attitudes	RIT	RIT : α 3 m < α 4 m	No RIT : α 3 m = α 4 m	Reversed RIT : α 3 m > α 4 m	Total
			905	474	397	1776
	Risk-averse † : α 0 m < α 3 m
			50.96 %	26.69 %	22.35 %	%
			231	379	113	723
	Risk-neutral † : α 0 m = α 3 m
			31.95 %	52.42 %	15.63 %	%
			634	238	184	1056
	Risk-seeking † : α 0 m > α 3 m
			60.04 %	22.54 %	17.42 %	%
			1770	1091	694	3555
	Total
			49.79 %	30.69 %	19.52 %	%
	†

m (see Figure

4

.3.1), we classify subject as risk

Table 4

 4

			.4.6: Ordered logit (average marginal effects) (a)
			Risk attitudes			RIT
		Aversion	Neutral	Seeking	RIT	No-RIT	Reversed RIT
	β	0.882 * * *	-0.163 * * *	-0.719 * * *	0.376 * * *	-0.137 * * *	-0.239 * * *
	ϵ	0.0675 * * *	-0.0126 * * *	-0.0549 * * *	(b)	(b)	(b)
	p	-0.020	0.004	0.016	-0.185 * * *	0.068 * * *	0.118

* * * * p < 0.1, * * p < 0.05, * * * p < 0.01 for the significance of coefficient tests. (a) Control variables include: numeracy test score, cognitive reflection test score, gender, and age.

  Assumption 3Ť yields r[α + βθk] < 1. Using e > ϵ and e = βθ

	below:					2ψ , we
	have Lemma 1 Under RDU, maximizing the objective function of the agent amounts 1 -p p rϵ 2 -1 < 0 (4.7) 2ψ Inequalities (4.7) and (4.6) jointly imply to maximizing his certainty equivalent CE:
	sign	∂α * (ϵ, r, ψ, θ) ∂t	= -sign	∂β * (ϵ, r, ψ, θ) ∂t	for t = ϵ, r, ψ, θ
	In particular	∂α * (ϵ, r, ψ, θ) ∂ϵ	> 0
			∂α * (ϵ, r, ψ, θ) ∂r	> 0
	Remark: expressions (4.5) and (4.6) also hold for risk-neutral agent (r = 0)
			e =	θ 2ψ	β * (ϵ, r, ψ, θ)
	It turns out that the partial derivatives negatives as ∂α * (ϵ, r, ψ, θ) ∂ϵ and ∂α * (ϵ, r, ψ, θ) ∂e * (ϵ, r, ψ, θ) ∂ϵ ∂r are negative. QED. and ∂e * (ϵ, r, ψ, θ) ∂r	are
	RDU
	Before showing the proofs, let us Ąrst state and provide some explanations of
	Lemma 1 and Propositions A2 and A3. Under RDU, we derive our Ąrst lemma

and risk-seeking agent (r < 0) as long as the second-order condition obtained from the derivative of the Ąrst-order condition A1.2 is negative. Hence, for a risk-neutral agent the Ąxed pay and performance do not vary with ϵ. For a risk-seeking agent, we have reversed RIT. Also, note that the agentŠs optimal level of effort e * is given by

  Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well Since β * (ϵ, r, w(p), ψ, θ) is positive, the relation 4.20 and the expression of Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well risk-neutral for probability p (i.e., w(p) ≤ p), (4.27) and (4.28) jointly imply

	For p < 1 2 , we have 1-2p p Also, we have Chapter 4. β * (ϵ, r, w(p), ψ, θ) jointly imply that > 0. Take k = 1 + 2ψϵ θ w(p) β * (ϵ,r,δ,ψ,θ) ϵ p Assumption 3Ť that 1 -rθβ * (ϵ, r, w(p), ψ, θ)k = 1 -rθϵ 1-2p 1-2p p > 0. It follows from (4.21) p > 0. Thus, for p < 1 2 we have ∂β * (ϵ, r, δ, ψ, θ) ∂δ also positive. This means that for all p ∂β * (ϵ, r, δ, ψ, θ) ∂ϵ < 0 ∂e * (ϵ, r, δ, ψ, θ) ∂r < 0 as ∂β * (ϵ, r, δ, ψ, θ) < 0 ∂r -1 > 0 Plugging (4.20) and (4.21) into the relation (4.19) yields ∂β * (ϵ, r, δ, ψ, θ) ∂δ > 0 (4.25) while (4.23) implies ∂α * (ϵ, r, δ, ψ, θ) ∂e * (ϵ, r, δ, ψ, θ) ∂δ > 0 as ∂β * (ϵ, r, δ, ψ, θ) > 0 ∂δ > 0 ∂ϵ QED.
	∂β * (ϵ, r, w(p), ψ, θ) From (4.23) and (4.25), it also follows that Recall that e * (ϵ, r, δ, ψ, θ) = θ 2ψ β * (ϵ, r, δ, ψ, θ). Hence, < 0 ∂r For k = 1 + w(p) p 1-2p p ϵ > 0, we have under Assumption 3Ť that r[α + βθk] < 1. ∂α * (ϵ, r, δ, ψ, θ) ∂δ < 0 (4.26) ∂e * (ϵ, r, δ, ψ, θ) ∂ϵ < 0 as ∂β * (ϵ, r, δ, ψ, θ) ∂ϵ < 0
	Then, for e > ϵ and e = βθ 2ψ , we have that Note that probability risk-aversion corresponds to a lower level of δ. Hence, Point iii) Assume that the agent exhibits probability risk-seeking for probability
	Expression (4.17) implies that Sign ∂β * ∂r = -Sign 1 + w(p) p Note that 1 + w(p) p 1-2p p = w(p) 1-p 1 -2p p p 2 + 1 -w(p). Since w(p) < 1, it turns out × Sign 1 + 2ψϵ θ w(p) p -1 (4.19) that 1 + w(p) p 1 -2p p > 0 (4.20) This means that the certainty equivalent ce (4.16) decreases in r (or equivalently, the risk premium increases in r). 16 16 See also Theorem 6.1 in Eeckhoudt and Laeven (2015). rϵ 2 1 + w(p) p 1 -2p p -1 2ψ < 0 (4.22) Relations (4.22) and (4.18) imply for t = ϵ, r, w(p), ψ, θ that sign ∂α * (ϵ, r, w(p), ψ, θ) ∂t = -sign ∂β * (ϵ, r, w(p), ψ, θ) ∂t (4.23) In particular, we have that p (i.e., w(p) > p). Hence, (4.27) and (4.28) jointly imply the following equivalence r < r to (β * , ϵ) ⇐⇒ ∂β * (ϵ, r, ψ, θ) ∂ϵ > 0 equations (4.25) and (4.26) mean that the optimal variable pay β Point ii) It follows from the expression (4.17) that Sign with r to (β * , ϵ) ∼ = 1 w(p) p -1 (4.29) 2ϵ θ + ϵψ w(p) 1 + w(p) p 1-2p p p -1 ∂β * (ϵ, r, δ, ψ, θ) ∂ϵ = Sign A(ϵ, r, δ, ψ, θ) (4.27) From (4.23), it also follows that ∂α * (ϵ, r, ψ, θ) > 0 ∂r with r < r to (β * , ϵ) ⇐⇒ ∂α * (ϵ, r, δ, ψ, θ) < 0 ∂ϵ From (4.17), we have that Sign ∂β * (ϵ, r, δ, ψ, θ) ∂δ = Sign 1 -rθϵ 1 -2p p + 2ψrϵ 2 1 -p p (4.24) with δ = w(p). Let us now consider (4.24) under two cases: p ≥ 1 2 and p < 1 For p ≥ 1 2 , we have 1-2p p ≤ 0 so that ∂β * (ϵ, r, δ, ψ, θ) ∂δ is positive. From 4.20 and 4.22 we have respectively that 1 + w(p) p 1-2p p w(p) p 1-2p p > 0 . If the agent exhibits probability risk-aversion or probability > 0 and 1 -2ψrϵ 2 1 + 2 . A(ϵ, r, δ, ψ, θ) =θ -1 w(p) p -1 1 -2ψrϵ 2 1 + w(p) p 1 -2p p -2rϵ 1 + w(p) p (4.28) Since e * (ϵ, r, δ, ψ, θ) = θ 2ψ β * (ϵ, r, δ, ψ, θ), we have 1 -2p p r < r to (β * , ϵ) ⇐⇒ ∂e * (ϵ, r, ψ, θ) ∂ϵ > 0
	305

ϵθβ * (4.18) * (ϵ, r, δ, ψ, θ) decreases as the probability risk-aversion increases while the Ąxed pay α * (ϵ, r, δ, ψ, θ) increases as the probability risk-aversion increases.

  a s , ψ, θ), it also follows (ϵ, a v , a s , ψ, θ) given in (4.32) Since a v < 0, a s > 0 and β * (ϵ, a v , a s , ψ, θ) >

	Point iii)
	From (4.30), the implicit function theorem implies that
	Sign	∂β * ∂ϵ	= Sign 4a v + 9a s	1 -2p p	θϵβ * (ϵ, a v , a s , ψ, θ)	(4.33)
	∂e * ∂a v ∂β * ∂a v > 0 and > 0. Note that ∂e * mean that the optimal variable pay and optimal effort both decrease as the the aversion to variance corresponds to a v < 0. Hence, ∂a v > 0 with β 0 it turns out that for p ≥ 1 2 we have ∂β * < 0. ∂ϵ Since e * (ϵ, a v , a s , ψ, θ) = θ 2ψ β * (ϵ, a v , a s , ψ, θ), it also follows that ∂e * < 0 for ∂ϵ p ≥ 1 2 .
	aversion to variance increases. Point iv)
	Also, from (4.30) we have that For p < 1 2 , either ∂β * ∂ϵ < 0 and ∂β * ∂ϵ > 0 are possible according to (4.33) and
	∂β * ∂a s (4.32). Replacing (4.32) in (4.33), it turns out that > 0 if p < 1 2 (positive skewness)
		∂β * ∂a s ∂β * < 0 ∂ϵ > 0 ⇐⇒ g(ϵ, a v , a s , ψ, θ) < if p > 1 2 (negative skewness) 9 4	(4.34)
	The optimal variable pay increases as the preference for positive skewness with
	increases if p < 1 2 (i.e., positive skewness) and decreases as the preference for g(ϵ, a
	positive skewness increases if p > 1 2 (i.e., negative skewness). Also, because e * (ϵ, a v , a s , ψ, θ) = θ 2ψ β ∂e * ∂a s > 0 if p < 1 2 (positive skewness)
		∂e * ∂a s	< 0	if	p >	1 2	(negative skewness)

* (ϵ, a v , a s , ψ, θ), it also follows * v , a s , ψ, θ)

  Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

	a s , ψ, θ), it also follows that when p < 1 2 we
	have ∂e * ∂ϵ > 0 if and only if g(ϵ, a v , a s , ψ, θ) < 9 4 .
	For example, Spiliopoulos and Hertwig (2019) estimate a v = -0.0229 and a s = 0.0037. Using these estimated values and setting (ϵ, p, ψ, θ) =
	(0.7, 0.1, 0.5, 1), the condition (4.34) holds. This yields β * (ϵ, a v , a s , ψ, θ) = 1.12

  Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well Given the expression (4.32) of the optimal variable pay β * (ϵ, a v , a s , ψ, θ), the = α * +θβ * e * + 1-p p ϵ -ψe * 2 , α * +θβ * e *ϵψe * 2 ; p, 1p . Being risk-seeking means that (4.39) holds, and hence from (4.36) we have that the optimal variable pay satisĄes a v + a s

	condition (4.38) holds iff
	g(ϵ, a v , a s , ψ, θ) < 1	(4.39)
	with g(ϵ, a v , a s , ψ, θ) given in (4.35).
	It follows that when the condition (4.38) holds, the agent exhibits risk-seeking
	for the positively skewed lottery L 1 -2p p Hence, we have ∂β * ∂ϵ > 0 and in particular β	β * θϵ > 0.
	a v + a s	1 -2p p	βθϵ > 0	(4.37)
	Hence, if the agent exhibits risk-seeking for the optimal triplet (α we should have a v + a s 1 -2p p β * θϵ > 0	(4.38)

1-p p ϵψe 2 , α + θβ eϵψe 2 ; p, 1p if * , β * , e * ), then * * (ϵ, a v , a s , ψ, θ) > 1 for ϵ > 0.

  Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well Also note that the agentŠs probability risk-aversion/risk-neutrality (i.e., w(p) ≤ p) implies R 1 < 0. It follows that in the presence of probability risk-aversion/risk-neutrality we have R < 0 and hence Then, the Ąrst-order Taylor approximation of H(ϵ) around ϵ = 0 gives Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

	H(ϵ) 4ψ u ′ α + β 2 θ 2	=	w(p) p	-1 -1 +	w(p) p	1 -2p p	βθϵA a α +	β 2 θ 2 4ψ	(4.63)
	dβ dϵ with o(ϵ) the approximation error which is such that lim < 0 Point iii) ϵ-→0 turns out that H(ϵ) > 0 if the following condition holds	=	o(ϵ) ϵ	= 0. It
	Recall that the equation (4.57) satisĄes by the optimal variable pay
		β = 1 + A a α + β 2 θ 2 2ψϵ θ 4ψ	w(p) < βθϵ 1 -p p 1 1 + u ′ (y ≤ 1 w(p) p -1 w(p) p p 1 -2p := A to a (β, ϵ)	(4.62) (4.64)
	It follows directly that QED.
					β > 1			⇐⇒	w(p) p 1-w(p) 1-p	>	u ′ (y -) u ′ (y + )	> 1
	Point iv)
	From (4.62), it follows directly
	β > 1	⇐⇒	w(p)	1 -p p	u ′ α+βθ	1 -p p	ϵ+	β 2 θ 2 4ψ	-(1-w(p))u ′ α+	β 2 θ 2 4ψ	-βθϵ
	Denote by
	H(ϵ) = w(p)	1 -p p	u ′ α +βθ	1 -p p	ϵ+	β 2 θ 2 4ψ	-(1-w(p))u ′ α +	β 2 θ 2 4ψ	-βθϵ +o(ϵ)

+ ) -(1w(p))u ′ (y -) w(p)u ′ (y + ) + (1w(p))u ′ (y -)

Table 4 .

 4 C.3: Mean of minimum Ąxed pay across treatments † * p < 0.05, * * * p < 0.01 for the significance of coefficient tests. † Mean of fixed pay are computed from regression analyses by allowing heteroscedasticity due to observable individual characteristics (i.e., numeracy skills, cognitive skills, gender and age). ‡ According to the calibration (ψ, θ) = (2.5, 100), the effort is computed as e = βθ 2ψ .356Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well4.C.

	p	β	e ‡	α m 3	α 4 m	α m 4 -α m 3	α 0 m	α 3 m -α 0 m	α 4 m -α 0 m
	0.1	0.30 3.00 881.3 * * *	901.4 * * *	20.05 * * *	910.00	-28.69 * * *	-8.64
	0.1	0.50 5.00 750.0 * * *	792.9 * * *	42.95 * * *	750.00	-0.01	42.95 * * *
	0.1	0.70 7.00 579.9 * * *	628.3 * * *	48.49 * * *	510.00	69.86 * * *	118.3 * * *
	0.25 0.30 3.00 898.1 * * *	917.6 * * *	19.44 * * *	910.00	-11.87 * * *	7.57
	0.25 0.50 5.00 761.7 * * *	803.5 * * *	41.82 * * *	750.00	11.67**	53.49 * * *
	0.25 0.70 7.00 582.2 * * *	626.8 * * *	44.57 * * *	510.00	72.22 * * *	116.8 * * *
	0.33 0.30 3.00 900.5 * * *	926.1 * * *	25.65 * * *	910.00	-9.505 * * *	16.14 * * *
	0.33 0.50 5.00 780.6 * * *	795.1 * * *	14.56 * * *	750.00	30.55 * * *	45.11 * * *
	0.33 0.70 7.00 590.7 * * *	620.2 * * *	29.44 * * *	510.00	80.72 * * *	110.2 * * *
	0.5	0.30 3.00 919.6 * * *	926.2 * * *	6.597**	910.00	9.642 * * *	16.24 * * *
	0.5	0.50 5.00 773.6 * * *	786.4 * * *	12.82 * * *	750.00	23.63 * * *	36.45 * * *
	0.5	0.70 7.00 582.5 * * *	615.4 * * *	32.86 * * *	510.00	72.53 * * *	105.4 * * *
	0.75 0.30 3.00 916.7 * * *	937.6 * * *	20.94 * * *	910.00	6.685 * * *	27.63 * * *
	0.75 0.50 5.00 778.4 * * *	789.5 * * *	11.02**	750.00	28.45 * * *	39.47 * * *
	0.75 0.70 7.00 571.5 * * *	607.3 * * *	35.87 * * *	510.00	61.45 * * *	97.32 * *

*

Standard errors in parentheses. * p < 0.1, *

3 -Decomposition of the cells in Table 4.4.5 accord- ing to the estimated risk attitudesTable 4 .

 4 C.4: RIT, risk-attitudes, and curvature of utility under EUT Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

		Number	Percentage
		RIT and Risk-averse
	Concave	14	1.55 %
	Convex	891	98.45%
	Total	905	100%
		RIT and Risk-neutral
		Number	Percentage
	Concave	23	9.96%
	Convex	208	90.04%
	Total	231	100%
		RIT and Risk-seeking
		Number	Percentage
	Concave	229	36.12%
	Convex	405	63.88%
	Total	634	100%
		No-RIT and Risk-averse
		Number	Percentage
	Concave	24	5.06%
	Convex	450	94.94%
	Total	474	100%
		No-RIT and Risk-neutral
		Number	Percentage
	Concave	40	10.55%
	Convex	339	89.45%
	Total	379	100%
		No-RIT and Risk-seeking
		Number	Percentage
	Concave	112	47.06%
	Convex	126	52.94%
	Total	238	100%
		Reversed RIT and Risk-averse
			Continued on next page
			357

Table 4 .C.5 -continued from previous page

 4

		Number	Percentage
		Number	Percentage
	Concave	77	19.40%
	Convex	320	80.60%
	Total	397	100%
		Reversed RIT and Risk-neutral
		Number	Percentage
	Concave	32	28.32%
	Convex	81	71.68%
	Total	113	100%
		Reversed RIT and Risk-seeking
		Number	Percentage
	Concave	124	67.39%
	Convex	60	32.61%
	Total	184	100%

Table 4 .

 4 C.6: RIT, risk attitudes, utility curvature and probability weighting under RDU

	Number (%)	Underweighting	Overweighting	Total
		RIT and Risk-averse
	Convex	18	0	18
		(1.99 %)	(0.00 %)	(1.99%)
	Concave	404	484	887
		(44.64%)	(53.37%)	(98.01%)
	Total	422	483	905
		(46.63%)	(53.37%)	(100%)
		RIT and Risk-neutral
	Convex	8	2	10
		(19.41%)	(68.78%)	(4.33%)
	Concave	105	116	221
		(45.45%)	(50.22%)	(95.67%)
	Total	113	118	231
		(48.92%)	(51.08%)	(100%)
			Continued on next page

Table 4 .C.7 -continued from previous page

 4 Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

	Number (%)	Underweighting	Overweighting	Total
		RIT and Risk-seeking
	Convex	72	51	123
		(11.36%)	(8.04%)	(19.40%)
	Concave	123	388	511
		(19.40%)	(61.20%)	(80.60%)
	Total	195	439	634
		(40.08%)	(69.24%)	(100%)
		No RIT and Risk-averse
	Convex	48	2	50
		(10.13%)	(0.42%)	(10.55%)
	Concave	206	218	424
		(43.46%)	(45.99%)	(89.45%)
	Total	254	220	474
		(53.59%)	(46.41%)	(100%)
		No RIT and Risk-neutral
	Convex	46	8	54
		(12.14%)	(2.11%)	(14.25%)
	Concave	180	145	325
		(47.49%)	(38.26%)	(85.75%)
	Total	226	153	379
		(59.63%)	(40.37%)	(100%)
		No RIT and Risk-seeking
	Convex	28	31	59
		(11.76%)	(13.03%)	(24.79%)
	Concave	31	148	179
		(13.03%)	(62.18%)	(75.21%)
	Total	59	179	238
		(24.79%)	(75.21%)	(100%)
		Reversed RIT and Risk-aversion
	Convex	30	9	39
		(7.56%)	(2.27%)	(9.82%)
	Concave	168	190	358
		(42.32%)	(47.86%)	(90.18%)
	Total	198	199	397
		(49.87%)	(50.13%)	(100%)
		Reversed RIT and Risk-neutral
	Convex	10	6	16
			Continued on next page
				359

Table 4 .C.7 -continued from previous page

 4

	Number (%)	Underweighting	Overweighting	Total
		(8.85%)	(5.31 %)	(14.16%)
	Concave	29	68	97
		(25.66%)	(60.18%)	(85.84%)
	Total	39	74	113
		(34.51%)	(65.49%)	(100%)
		Reversed RIT and Risk-seeking
	Convex	16	35	51
		(8.7%)	(19.02%)	(27.72%)
	Concave	14	119	133
		(7.61%)	(64.67%)	(72.28%)
	Total	30	154	184
		(16.3%)	(83.7%)	(100%)

Table 4 .

 4 C.8: RIT, risk-attitudes, and preference/aversion for skewness and variance under MVS

	Number (%)	Aversion for skewness	Preference for skewness	Total
		RIT and Risk-averse
	Preference for variance	13	1	14
		(1.44%)	(0.11%)	(1.55%)
	Aversion to variance	20	871	891
		(2.21%)	(96.24 %)	(98.45%)
	Total	33	872	905
		(3.65%)	(96.35 %)	(100.00 %)
		RIT and Risk-neutral
	Preference for variance	19	1	20
		(8.23%)	(0.43%)	(8.66%)
	Aversion to variance	3	208	211
		(1.30%)	(90.04%)	(91.34%)
	Total	22	209	231
		(9.52%)	(90.48%)	(100.00%)
		RIT and Risk-seeking
	Preference for variance	213	13	226
			Continued on next page

Table 4 .C.9 -continued from previous page

 4 Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well Equation (4.86) deĄnes the optimal variable pay β * (α, β, ϵ, r + , r -, λ, w + , w -, ψ, θ).The optimal variable pay in the absence of shock (ϵ = 0) is given by lim

	Number (%)	Aversion for skewness	Preference for skewness	Total
		(33.60%)	(2.05%)	(35.65%)
	Aversion for variance	17	391	408
		(2.68%)	(61.67%)	(64.35%)
	Total	230	404	634
		(36.28%)	(63.72%)	(100.00%)
		No RIT and Risk-averse
	Preference for variance	21	4	25
		(4.43%)	(0.84%)	(5.27%)
	Aversion for variance	10	439	449
		(2.11%)	(92.62%)	(94.73%)
	Total	31	443	474
		(6.54%)	(93.46%)	(100.00%)
		No RIT and Risk-neutral
	Preference for variance	27	11	38
		(7.12%)	(2.90%)	(10.03%)
	Aversion for variance	9	332	341
		(2.37%)	(87.60%)	(89.97%)
	Total	36	343	379
		(9.50%)	(90.50%)	(100.00%)
		No RIT and Risk-seeking
	Preference for variance	100	11	111
		(42.02%)	(4.62%)	(46.64%)
	Aversion for variance	10	117	127
		(4.20%)	(49.16%)	(53.36%)
	Total	110	128	238
		(46.22%)	(53.78%)	(100.00%)
		Reversed RIT and Risk-aversion
	Preference for variance	66	6	72
		(16.62%)	(1.51%)	(18.14%)
	Aversion for variance	12	313	325
		(3.02%)	(78.84%)	(81.86%)
	Total	78	319	397
		(19.65%)	(80.35%)	(100.00%)
		Reversed RIT and Risk-neutral
	Preference for variance	29	2	31
		(25.66%)	(1.77%)	(27.43%)
	Aversion for variance	4	78	82
			Continued on next page

(Abdellaoui et al.,

2008) use the term "efficient" instead. Since this property only relates to the physical resources (such as time and money) used for data collection, we coin the term data-efficiency to avoid confusion with the (unrelated) statistical notion of the efficiency of an estimator.

Indeed, we remark that u(x 1 ) = u(x 0 ) + u(x 2 ) 2

Blavatskyy (2021) proposes a similar procedure.

As pointed out byAbdellaoui (2000Abdellaoui ( , pp. 1509Abdellaoui ( -1510)), testing for duality with parametric methods and non-parametric methods based on the tradeoff approach requires using the specific probability weighting function ofGoldstein and Einhorn (1987) and[START_REF] Lattimore | The inĆuence of probability on risky choice: A parametric examination[END_REF].

This is because under the tradeoff method subjects need to compare two binary lotteries, while other methods based on certainty equivalent elicitation only ask subjects to compare a certain amount with a binary lottery. Thus, the tradeoff method requires processing more information which makes it relatively cognitively demanding.

A method related toAbdellaoui et al. (2011c) is the source method ofAbdellaoui et al. (2011a) that allows for eliciting the source function and the utility function under the biseparable preference model ofGhirardato and Marinacci (2001). An additional assumption is that decision makers can assign subjective probabilities (i.e., beliefs) to events even when they do not maximize subjective expected utility. Another method proposed by[START_REF] Bertani | Fast and simple adaptive elicitations: Experimental test for probability weighting[END_REF] elicits the probability weighting function. However, this method is restrictive because it is only valid for the dual theory ofYaari (1987) in which the utility function is assumed to be linear.

Note that this issue also applies to the parametric methods in TK92 andFehr-Duda et al. (2006) which estimate the utility function in the gain and loss domains in two separate steps. Other parametric methods do not suffer from this problem(Harrison and Rutström, 2008;[START_REF] Post | Deal or no deal? decision making under risk in a large-payoff game show[END_REF][START_REF] Tanaka | Risk and time preferences: Linking experimental and household survey data from vietnam[END_REF].

With our one-step procedure, it is also possible to test whether the probability weighting functions are the same in the gain and loss domains. Outside the framework of CPT, our method also allows for testing the duality of the probability weighting function under RDU.

This notation is related to decision under risk. In the case of decisions under uncertainty, one would simply replace p and 1p by E and E c respectively. E denotes an event in a state space Ω and E c denotes its complement in Ω. In that case, L = (x, y; E, E c ) is a binary prospect that gives outcome x if E occurs, and y otherwise.

CPT makes no explicit link between weighting functions w + (.) and w -(.) which makes it more general than OPT in which w + (p) = w -(p), or RDU that includes the duality condition w + (p) = 1w -(1p).

Note, however, that our method is compatible with any utility function.

Regardless of the exact definition, it is always the case that loss aversion (loss seeking) corresponds to λ > 1(λ < 1), whereas λ = 1 captures loss neutrality.

Note that having the same outcomes for each probability (x i j,k = x i j and y i j,k = y i j for all k) allows for an immediate test of the monotonicity of preferences by checking if certainty equivalents increase with probabilities for given pairs of outcomes (x i j , y i j ). This choice of outcomes could also reduce the cognitive burden of the task.

In Appendix 1.D, we also provide an illustration of how to apply Bayesian techniques with our semi-parametric method using data from l'Haridon and Vieider (2019).

A third source of heteroscedasticity may arise at the aggregate level (pooled data) when the variance of response errors differs across individuals (e.g.Harrison and Rutström, 2008, 2009; l'Haridon and Vieider, 2019, BFE10)

For clarity of exposition, we use underscores to refer to the loss domain so that:δ - k ≡ δ - K-k+1 = w -(1p k ) for k = 1,2, ..., K.

A standard deviation of 0.025 × ♣xy♣ implies response errors of +/-$20 around the true certainty equivalent value, which seems large given the range of lottery outcomes [0, $400].

Note that the scale of the exponential utility parameter depends on the scale of the outcomes used in the lotteries. Taking into account the midpoint of the outcome range [0, 400], the index of absolute risk aversion generated by a power utility function with a parameter in the range (0, 2) is approximately equal to the index generated by an exponential utility with a parameter in the range -1 200 , 1 200 . In the simulation exercise, we allow for a wider range for that parameter: (-0.01, 0.01).

For these specifications, identical probability weighting across domains corresponds to a = c and b = d. Note that an appealing property of the GE87 specification is that it allows for a straightforward test of duality(Abdellaoui, 2000) by checking whether a = c and b = 1 d .

The authors also conducted two other experiments, but these datasets do not match our criteria at the individual level since they include only one certainty equivalent for some of the probability weights.

Tail probabilities are typically considered to be equal to 5% or less (see[START_REF] Barron | Small feedback-based decisions and their limited correspondence to description-based decisions[END_REF][START_REF] Erev | On the weighting of rare events and the economics of small decisions[END_REF][START_REF] Erev | From anomalies to forecasts: Toward a descriptive model of decisions under risk, under ambiguity, and from experience[END_REF][START_REF] Hertwig | Decisions from experience and the effect of rare events in risky choice[END_REF][START_REF] Corgnet | Tail events, emotions and risk taking[END_REF].

See Appendix 1.A.4 for detailed information about parametric specifications and results.

We also report similar results in Table1.5.

Even though we reject both duality and identical probability weights, in Appendix 1.A.3 we show how to impose such constraints in our method.

In addition, this pattern also holds for the Bayesian estimations reported in Appendix 1.D.

They assume a stochastic reference point and the absence of any probability distortions, whereas we assume a fixed reference point and probability distortions.

The corresponding median values are 0.0081 and -0.0063. The respective statistical tests all yield pvalues < 0.0001. In addition, the differences between the median estimates obtained through both methods are small and insignificant for gains (pvalue = 0.6539) as well as for losses (pvalue = 0.8812).

The use of linear spline to approximate the utility function meshes well with the observation that utility is quasi-linear over a small range of outcomes (seeWakker and Deneffe, 1996; Bleichrodt and Pinto, 2000;[START_REF] Rabin | Risk aversion and expected-utility theory: A calibration theorem[END_REF] Fehr-Duda et al., 2006).

To compare the utility function over the range [$0, $800] from these two estimations, we convert them into a common scale, so that u($800) = 1 and u(0) = 0.

Figure 1.D.1: Posterior distributions of curvature and probability weights in the gain domain (world-level)

For the utility function, Power and Exponential are popularly used and sometime a mixture of them called Expo-Power[START_REF] Saha | Expo-power utility: a ŚĆexibleŠ form for absolute and relative risk aversion[END_REF] Holt and Laury,

2002;Abdellaoui et al., 2007a). For the distortion of probability, one-parameter (e.g. TK92) and two-parameter(Prelec, 1998; Goldstein and Einhorn, 1987; Chateauneuf et al., 2007; Abdellaoui et al., 2010) weighting functions (for a review, see table 4 ofEpper and Fehr-Duda, 2020) are available.

See the appendix of Fehr-Duda and Epper (2012) for a similar procedure.

This notation is related to decision under risk. In the case of decisions under uncertainty, one would simply replace p and 1p by E and E c respectively. E denotes an event in a state space Ω and E c denotes its complement in Ω. In that case, L = (x, y; E, E c ) is a binary prospect that gives outcome x if E occurs, and y otherwise.

Note that we do not require monotonicity of the utility and weighting functions as is commonly done in the literature (see e.g. GW99, p. 147).

As in gain domain, the method does not require monotonicity of the utility and weighting functions as is commonly done in the literature (see e.g. GW99, p. 147).

Previous methods require that the universal event is an interval of real numbers (e.g. temperature in a town) which is most suitable to deal with artificial uncertainty situations that can be created in the laboratory(Van De Kuilen and Wakker, 2011).

Similar procedure is proposed byGutierrez and Kemel (2021, study C), but they keep the assumption of same utility for all sources of uncertainty.

We cover the cases of m ̸ =

in the subsection 3.3.2.

We refer toGoldstein and Einhorn (1987) as GE87.

Monotonicity at the aggregate level (e.g. pooled data, mean data and median data) will naturally hold. But, at the individual level this condition might not be satisfied.

In Appendix 3.D we present the parameter recovery and misspecification exercises (e.g.Gao et al., 2020;[START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] Nilsson et al., 2011) with the aim of comparing our multi-stage approach with the one-stage approach.

The complete instructions can be found in the Appendix 3.C

The exchange rate is such that 1 ECU = 1 euro.

The image is presented in English for illustration purposes. However, the experiment was conducted in French.

Following Li et al. (2020), social preferences collapse in the second stage of the conditions. Therefore, social preferences are not considered in these comparisons.

Comparison 1 refers to the difference found between the treatments social ambiguity -cg and strategic uncertainty -cg.

This design and behavioral conjectures have been pre-register at AsPredicted (#71020).

The smaller η is, the higher is the level of pessimism. The smaller γ is, the higher is the level of likelihood insensitivity.

B for a line, the computer will mark Alternative B for all subsequent lines (up to the last one).

In case of continuous valued source of uncertainty, we also allow to specify the distribution of beliefs only in the third stage (see Appendix 3.A).

The vast majority of specifications in ambiguity studies rely on one of these six combination of utility and weighting functions (e.g.Li et al., 2018;[START_REF] Gutierrez | Preference and belief: Ambiguity and competence in choice under uncertainty[END_REF].

[START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF] report a 8.1% (Cohen's d = 0.34) decrease in the piece rate value in their noise treatment as compared to a noise-free baseline. A similar decrease (12.7%) if also found by Chowdhury and Karakostas (2020), notwithstanding their EUT-based prediction of a one-third decrease.

[START_REF] Milgrom | Economics, Organization and Management[END_REF] andLaffont and Martimort (2002) derive fundamental results in the theory of incentives using a model with binary shocks.

Under EUT, a v and a s are linked via utility function. The third order Taylor approximation of the certainty equivalent incorporates attitudes towards variance and skewness due to the second and third derivatives of the utility function.

The lower bound of 0 does not appear to be restrictive in our experiment as only 1.1% of the decisions revealed a minimum fixed pay that is less or equal to 0.

See Appendix 4.D for an analysis of RIT under reference-dependence.

Extensive data confirming this claim is also available from the authors upon request.

This is 20% less than the pre-registered target number (300) due to lower response rate than expected. This can be explained by the sudden increase in COVID cases at that specific time and location. Only 4 participants dropped out between the main experimental task and the survey sessions.

In Appendix 4.D, we extend proposition A2 to the case of the prospect theory agent exhibiting loss aversion and reference-dependence.

Another way to find the right solution is to plug the two possible solutions into the second-order condition (4.31) to see that it is solely satisfied by (4.32).

Note that the problem does not have an explicit analytic solution. Our solution is numerical.

Acknowledgments

1.B Appendix for the application to the data of BFE10

In this appendix, we provide details on individual and pooled results based on the data of BFE10. We also allow for heteroscedastic errors, as discussed in Section 1.3.2.2.

1.B.0.1 Pooled data

Tables 1.B.1 and 1.B.2 summarize our pooled data estimates that we present in the main text.

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty about Player 2 choosing an Amazon voucher Ąrst order stochastically dominates the Player 1Šs beliefs about Player 2 choosing a Google Play and an Apple Store voucher. In the social ambiguity -cg treatment, Player 1 thinks that Player 2 chooses to spend money in Amazon, Google Play and Apple Store vouchers with probability 40.6%, 28.1% and 31.3%, respectively. In the treatment social ambiguity -tg, Player 1 believes that Player 2 chooses to spend money in Amazon, Google Play and Apple Store vouchers with probability 43.4%, 25.7% and 30.9%, respectively. Join test leads to the conclusion that the distribution of beliefs are the same in these two social ambiguity treatments (pvalue = 0.7106). This result provides a second successful validity test of our method. In fact, these two social ambiguity treatments involve the same events. Therefore, the beliefs in these two different ambiguity situations should remain the same.

Symmetry of events is also rejected for betrayal aversion (p-value =0.0051).

The cumulative distribution function of the beliefs about the fact that the Trustee will follow the ŞbetrayŤ strategy Ąrst order stochastically dominates the strategies of Şno hurtŤ and ŞreciprocateŤ. We Ąnd that Player 1 (Trustor) thinks that Player 2 (Trustee) reciprocates, adopts a no hurt strategy, and betrays with probability 29.3%, 29.7% and 41.0%, respectively.

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty results of the weighting function: pessimism (η) and likelihood insensitivity (γ).

The usual pattern of over-weighting of small likelihoods and under-weighting intermediate and high likelihoods is reproduced. The cross-over points are 0.210, 0.345, 0.193, 0.266, and 0.264 in the treatments nature, social ambiguity -cg, strategic uncertainty -cg, social ambiguity -tg, and betrayal aversion, respectively. presents the results of the estimation of the utility curvature (α), pessimism (η)

and likelihood insensitivity (γ).

Social ambiguity

The difference between treatments 0 (nature) and 1 (social ambiguity -cg),

as well as the difference between the treatments 0 (nature) and 3 (social ambiguity -tg) corresponds to what Li et al. (2020) called social ambiguity.

Utility curvature (α): the estimates of the CRRA parameter for treatments 0, 1 and 3 are 0.93, 0.876, and 0.968, respectively. The difference in the utility parameters between treatments 0 and 1 is not signiĄcant (p -value=0.332, two sided sign test). This is also the case for the difference between the treatments 0 and 3 (p -value= 1, two sided sign test).

Pessimism (η): the estimates of pessimism for treatments 0, 1, and 3 are 0.615, 0.736, and 0.611, respectively. Pessimism is lower in treatment 1 in treatment 0 (p -value= 0.0030). Pessimism is the same in treatment 3 and in treatment 0 (p -value= 0.9245).

Chapter 3. Measuring Beliefs and Ambiguity Attitudes Towards Discrete Sources of Uncertainty Van De Kuilen, G. and Wakker, P. P. (2011). The midweight method to measure attitudes toward risk and ambiguity. Management Science, 57(3).

Wakker, P. P. (2010). Prospect theory: For risk and ambiguity. Cambridge university press.

First stage

Instructions for the Ąrst sub-part of Part 2

In this part of the experiment, you are again randomly paired with another participant. We call this new person Participant 2. However, this Participant 2 is a different person than the one you were paired with in the previous part of the experiment. You will never be informed of Participant 2Šs identity, nor will Participant 2 be informed of your identity. Your Ąnal payment will depend on your decision and the decision of Participant 2. During this task, you will be able to use the back button to re-view the decisions that you and Participant 2 were asked to make in the Ąrst sub-part of this part of the experiment.

You and

Social ambiguity -trust game

You will now read the instructions for Part 3 of the experiment. Part 3 has two sub-parts. You will receive instructions for each sub-part before you make your decisions in each of them.

-

Second stage

Instructions for the second sub-part of Part 3

Example

Suppose that the following option is randomly selected for payment: During this task, you will be able to use the back button to re-view the decisions that you and Participant 2 were asked to make in the Ąrst sub-part of this part of the experiment.

Betrayal aversion

You will now read the instructions for Part 4 of the experiment. Part 4 has two sub-parts. You will receive instructions for each sub-part before you make your decisions in each of them.

First stage

Instructions Bolton and Dewatripont, 2005). Below, we outline our assumptions.

Assumption 0 (A0: Binary shock). The shock ε is a binary random variable deĄned as ε = (ϵ, 1-p p ϵ; 1p, p), ϵ ≥ 0 and p ∈ (0, 1] so that E(ε) = 0 and V (ε) = 1-p p ϵ 2 . 3 Assumption 1 (A1: Risk-neutral principal). The principal is risk-neutral and maximizes the expected payoff. for r ̸ = 0 and u(x) = x for r = 0.

With Assumption 3, we deĄne utility risk attitudes in terms of the shape of the utility function. By contrast, the overall risk attitudes of the agent depend on his overall valuation of the contract which is only partly captured by the utility function. We deĄne utility risk attitudes and overall risk attitudes as follows.

Definition 1 (Utility risk attitudes). Utility risk-aversion [risk-neutrality]

(risk-seeking) corresponds to a concave, r > 0 [linear, r = 0] (convex, r < 0) utility function. We assume linear contracts because they are theoretical tractable and empirically relevant (e.g., Holmström, 2017). Also, we do not require β ∈ [0, 1] (e.g., [START_REF] Milgrom | Economics, Organization and Management[END_REF]Laffont and Martimort, 2002). Principal could then set β > 1 and α < 0, especially for risk-seeking agent.

where utility risk-aversion (utility risk-seeking) does not necessarily imply overall risk-aversion (risk-seeking).

In this paper, we consider the standard assumption of a concave utility function (i.e., Assumption 3Š which is equivalent to Assumption 3 with r > 0), unless stated otherwise.

Assumption 3' (A3': Utility risk-averse agent). The agent is utility risk-averse in the sense of r > 0.

For the sake of concision, hereafter we use the term risk attitudes (riskaversion, risk-seeking or risk-neutrality) to refer to overall risk attitudes. Assumption 3" (A3": Relative risk-aversion). The relative risk-aversion index evaluated at x is less than 1, that is -

Assumption 4 (A4: Public knowledge of the agent's risk-attitudes).

The principal knows the agentŠs risk attitudes.

Assumption 5 (A5: Quadratic cost). The cost of effort function is:

Given these assumptions, the compensation associated with the contract, which is the random wage net of the cost of effort, can be described as a lottery L:

where the Ąrst three moments (i.e., mean E, variance V and skewness S) are:

Note that varying ϵ does not affect the expected value of the lottery (E(L)) but impacts variance (V (L)) and skewness (S(L)). By contrast, varying the Ąxed Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well pay (α) impacts the expected value of the lottery without affecting the other two moments. Finally, the variable pay (β) impacts all three moments.

Model specification under EUT, RDU and MVS

We Ąrst determine how the agent evaluates lottery L based on three different speciĄcations: EUT, RDU and MVS.

EUT

Under EUT, the agent values the contract by its expected utility:

In this model, ∂EU (L) ∂ϵ < 0 as long as the utility function u(.) is concave. 5

RDU

Under RDU, the agent distorts probabilities using a probability weighting function w(p), which is a strictly increasing function from [0, 1] to [0, 1] with w(0) = 0 and w(1) = 1. Hence, risk attitudes not only stem from utility curvature (as in EUT), but also from probability weighting. Below, we deĄne probability risk-aversion, risk-neutrality and risk-seeking.

Definition 3 (Probability risk attitudes). Under RDU, an agent exhibits probability risk-aversion [risk-neutrality] (risk-seeking) for a speciĄc probability

5 By contrast, (2) increases with (is unaffected by) the shock magnitude if r < 0 (r = 0).

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

The agentŠs valuation of the contract then becomes:

For a probability risk-seeking agent, the valuation of the contract in (4.3) has an inverted J-shape with respect to ϵ (see Figure 4.2.1). To grasp the intuition, consider an arbitrarily small shock (ϵ 0 ) making the agent approximately utility risk-neutral, thus exhibiting a linear utility function. In that case ∂RDU (L)

Hence, for that level of shock the agent is necessarily risk-seeking because he is both utility risk-neutral and probability risk-seeking. That is, for small shocks, risk attitudes are driven by probability weighting rather than by the curvature of the utility function. As shown in Figure 4.2.1, the valuation of the lottery (RDU (L)) at ϵ 0 is above the utility of the expected value of the lottery (u(E[L])), implying a negative risk premium and hence a risk-seeking agent.

However, as the shock increases in magnitude, utility risk-aversion increases up to a point in which utility risk-aversion exactly offsets probability riskseeking, making the agent risk-neutral. This level of shock (denoted ϵ 1 in Figure 4.2.1) corresponds to a null risk premium associated with the contract lottery (i.e., RDU (L) = u(E[L])). Between ϵ 0 and ϵ 1 , there is also a level of shock (denoted ϵ * in Figure 4.2.1) for which the negative effect of increasing the shock magnitude due to utility risk-aversion is exactly equal to the positive effect due to probability risk-seeking. For shocks greater than ϵ 1 , a probability risk-seeking

An MVS agent exhibits risk-aversion for any negatively-skewed lottery. For positively-skewed lotteries, he is risk-seeking (risk-averse) [risk-neutral]

p βθϵ. For any p ≥ 1/2, we have τ N (β, ϵ) ≤ 0 so that the agent is systematically risk-averse sinceav as > 0. In the presence of aversion to variance (a v < 0) and preference for positive skewness (a s > 0), the valuation function (4.4) is J-shaped with respect to ϵ when p < 1/2 (see Figure 4.2.2). The intuition behind Figure 4.2.2 follows from the fact that for small (large) levels of the shock, the variance of L is larger (smaller) than its skewness. Hence, for a small level of shock (say ϵ 0 ), the agent is necessarily risk-averse since the aversion to variance outbalances the preference for positive skewness. This gives rise to a positive risk premium: the valuation of the lottery (MVS(L)) lies below its expected value. For a sufficiently high level of shock magnitude (∀ϵ > ϵ 1 in Figure 4.2.2), the agent necessarily exhibits risk-seeking since the preference for positive skewness outbalances the aversion to variance. At some level of the shock (denoted ϵ 1 in Figure 4.2.2), the two effects cancel out so that the agent is risk-neutral with a null risk premium (MVS(L)=E[L]). Finally, Figure 4.2.2 also features a level of shock ϵ * for which the negative effect of increasing the shock magnitude due to aversion to variance is exactly equal to the positive effect of increasing the shock magnitude due to the preference for positive skewness. agent a lower Ąxed pay while keeping his utility equal to the outside option (y 0 ). As in EUT, this situation corresponds to reversed RIT. However, for an intermediate shock (ϵ * < ϵ < ϵ 1 in Figure 4.2.1), the risk-seeking RDU agentŠs contract valuation is decreasing in the shock. This implies that the principal needs to offer the agent a higher Ąxed pay to keep his level of utility constant in response to a larger shock. Hence, for a risk-seeking RDU agent RIT emerges at an intermediate shock level. Example 1 provides a numerical illustration of RIT for a (moderately) risk-seeking RDU agent.

Example 1 (RIT for a risk-seeking agent under RDU). We consider r = 0.1, (ψ, θ, y 0 ) = (0.5, 1, 4). In the absence of shock (i.e., ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of a shock (ϵ = 1) and given a RDU agent who overweights probability 0.1 such that w(p) = 0.15, we obtain β * = 0.76 and α * = 3.64. Thus, the optimal variable (Ąxed) pay in the presence of a shock is smaller (larger) than in its absence, which means RIT holds. Because the expected value of the contract (E(L * ) = 3.93) is lower than its certainty equivalent (which is equal to the outside option y 0 = 4 due to the participation constraint), the agent is risk-seeking for the optimal contract (α * , β * )=(3.64,0.76). As a result, RIT is observed for a risk-seeking agent.

Under MVS, we show that RIT may not hold when agents are risk-averse which stands in stark contrast with EUT predictions. In particular, when the shock is positively skewed (p < 1/2), RIT may not hold (and may even reverse) for risk-averse agents who value positive skewness (see Appendix 4.A, Propositions A4 and A5). This happens because the MVS-based valuation of the contract in (4.4) increases with Ąxed pay α (irrespective of risk attitudes) and is J-shaped in the shock magnitude (see Figure 4.2.2). For a small shock (ϵ 0 in Under EUT, we can derive Proposition 1 stating that a risk-averse agent demands a higher Ąxed pay when the shock magnitude increases, giving rise to RIT. This behavior boils down to an enhanced demand for insurance when facing greater risk. Proposition 1ii states that RIT holds under EUT for risk-averse agents in line with Table 4.2 i) The minimum Ąxed pay increases in utility risk-aversion.

ii) For risk-averse agents, the minimum Ąxed pay increases in ϵ and β.

Under RDU, we show that RIT holds whenever the agent is probability riskaverse (see Proposition 2ii). By Assumption 3Š, this implies that the agent is also overall risk-averse. Furthermore, Proposition 2iii states that RIT also holds for a probability risk-seeking agent as long as the index of absolute risk aversion (r) is above a certain threshold (r to (β, ϵ)) such that his level of utility risk-aversion is sufficiently high. Interestingly, this threshold is lower than the value of the index of absolute risk aversion (r N (β, ϵ)) for which a probability risk-seeking agent exhibits risk-neutrality given the contract (α m , β) (see Proposition 2iv). As a result, for any value of the index of absolute risk aversion r ∈ (r to (β, ϵ), r N (β, ϵ)), the agent is risk-seeking and exhibits RIT. Finally, Proposition 2v implies that the agent is more likely to exhibit RIT when the shock magnitude and the variable pay are large. It also implies that the agent is more likely to exhibit risk-seeking attitudes for a small shock and a low variable pay. In Appendix 4.C.1, we provide the proof of Proposition 2 and show its connection with Proposition 3v implies that a MVS agent is more likely to exhibit reversed RIT when the shock magnitude and the variable pay are large. It also implies that the agent is more likely to exhibit risk-seeking attitudes for a high shock magnitude and a high level of variable pay. In Appendix 4.C.1, we provide the proof for Proposition 3 and show its connection with Propositions A4-A5 (Appendix 4.A).

Proposition 4. (RIT under MVS).

i) The minimum Ąxed pay increases in the aversion to variance a v . In addition, if

p βθϵ, where τ N (β, ϵ) is the level ofav as such that the agent exhibits risk-neutrality for the contract (α m , β).

v) The two thresholds τ to (β, ϵ) and τ N (β, ϵ) increase in ϵ and β.

Propositions 2, 3 and 4 show that our BB model can be used to study RIT.

Predictions in Table 4.2.1 thus carry on to the BB model. The next section provides details of the experimental test of the BB model predictions.

Experimental design

In line with the BB model, we study RIT using the minimum Ąxed pay (α m ) accepted by the agent.

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

provides an example of a decision screen for the combination (β, ϵ, p)=(0.7,3,0.5),

where Option A corresponds to the sure payoff associated with the outside option and Option B represents all the possible payments associated with lottery L(α i ♣0.7, 3, 0.5; e * ). The value of Ąxed pay is such that α i = (i -1) × 50, where i is the row number between 1 and 21. For (β, ϵ, p)=(0.7,3,0.5), we have that e * = βθ 2ψ = 14. Thus, for row i = 1, Option B displays the two possible payments associated with L(0♣0.7, 3, 0.5; 14): 280 if the shock is negative and 700 otherwise.

The likelihood of a given payment is visually represented by the frequency of cells in which it appears. Different amounts appear in different colors to facilitate the reading of the table. In total, participants face 30 tables, each corresponding to a different combination of (β, ϵ, p). All amounts in tables are in euro cents. To avoid hedging issues (Charness et al., 2016), one of the 30 tables is selected at random for payment upon a successful completion of the experiment.

For each table, participants pick a single row corresponding to their switching point, i.e., the point beyond which they prefer Option B over Option A. Participants cannot select multiple switching points. In the example presented in ). This implies that the minimum Ąxed pay (α m ) the participant is willing to accept for this contract is in the interval (450,500). In that example, we estimate α m to be the midpoint of the interval, that is 475 (e.g., Abdellaoui et al., 2008a;Gonzalez and Wu, 1999a).

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well on effort [START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF][START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF]. 11 The rationale is that people may exert higher effort in the presence of a shock than in its absence in order to offset any potential monetary loss. In our monetary effort design, this simple mechanism does not apply because the agent cannot hedge against monetary losses by increasing monetary effort. Indeed, monetary effort implies a monetary cost and thus perceived as a loss by the agent. This argument also reĆects the fact that the increase in effort due to output shock observed in real-effort tasks (Sloof and Van Praag, 2010;[START_REF] Corgnet | Revisiting the trade-off between risk and incentives: The shocking effect of random shocks?[END_REF][START_REF] Dohmen | Reference points and the tradeoff between risk and incentives[END_REF] is not observed when monetary effort is used (Chowdhury and Karakostas, 2020). 12

Preliminary survey session

Two days before completing the main experimental task (as discussed in Section 4.3.1), participants completed a series of individual tests and questionnaires. This preliminary set of tasks includes a numeracy test (Schwartz et al., 1997;Cokely et al., 2012), a probability weighting elicitation task [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] for the relevant set of probabilities (i.e., 0.1, 0.25, 0.33, 0.5 and 0.75), probability training mimicking the setup used in the main experimental session, loss aversion measurement (Brink and Rankin, 2013), risk attitude measurement (Holt and Laury, 2002) and a 7-item modiĄed version of the cognitive reĆection test (Frederick, 2005;Toplak et al., 2014).

Procedure

The design has been approved by the local ethical committee at the GATE research institute and pre-registered on the AsPredicted website (#82616). We Tversky and Kahneman, 1992a;Gonzalez and Wu, 1999a;Bleichrodt and Pinto, 2000;[START_REF] Bruhin | Risk and rationality: Uncovering heterogeneity in probability distortion[END_REF]. Under MVS, the coefficients of attitudes towards variance and skewness are a v = -0.00097 (p-value < 0.001) and a s = 4.8 × 10 -7

(p-value < 0.001). These estimates indicate an aversion to variance and a preference for positive skewness and once again stand in line with previous studies (e.g., Spiliopoulos and Hertwig, 2019).

Overall, the basic assumptions underlying our three models are validated by our experimental data on certainty equivalents. On average, participants exhibit a positive CARA coefficient that is below 1 (for both EUT and RDU), an inverse S-shaped probability weighting function (for RDU), and an aversion to variance and a preference for skewness (for MVS).

RIT and risk attitudes

We start by estimating model-free risk attitudes for all participants. To do that, we compare the Ąxed pay in the absence of shock α 0 m = 1000 -

with the Ąxed pay α 3 m (α 4 m ) elicited for shock ϵ = 3 (ϵ = 4). This procedure is used because the minimum value of the Ąxed pay an agent is ready to accept in the absence of shock is equal to the minimum Ąxed pay a risk-neutral agent make choices that are consistent with RIT (ŚRisk-seeking and RITŠ cell) Ű a pattern predicted by RDU, but not by MVS.

We then split the cells in Table 4 In particular, we focus on the decomposition of the most populated cell in Table 6 (ŚRisk-seeking and RITŠ). Not surprisingly, most choices (63.88%) in this cell are characterized by a convex utility function under EUT (see Table 4.C.4).

Since agents characterized by a convex utility function should exhibit reversed RIT, our Ąndings are incompatible with EUT (see Proposition 1).

Under MVS, most choices in the ŚRisk-seeking and RITŠ cell show an aversion to variance and a preference for positive skewness (61.67%, see Table 4.C.8).

However, agents should not exhibit RIT in this case under MVS (see Proposition

3).

Under RDU, most choices in the ŚRisk-seeking and RITŠ cell are characterized by a concave utility coupled with overweighting of probabilities (61.20%, see Table 4.C.6). This pattern is consistent with RDU, which predicts that RIT is observed for risk-seeking agents when they exhibit utility risk-aversion and probability risk-seeking (see Propositions 2iii and 4iv, and Example 1). The alternative pattern of risk-seeking attitudes in which agents exhibit utility riskseeking and probability risk-aversion (probability risk-seeking) characterizes only 11.36% (8.04%) of the choices in the ŚRisk-seeking and RITŠ cell.

A direct implication of Propositions 2v and 3v is that RDU and MVS have opposite predictions regarding the relationship between the shock magnitude, the variable pay, RIT and risk-attitudes. To test these predictions, we estimate an ordered logit model (see Table 4.4.6) to assess the effect of the shock magnitude (ϵ) and the variable pay (β) on risk attitudes (Ąrst three columns) and the Proposition A1 (Risk-incentives tradeoff with EUT): Under A0, A1, A2, A3Š, A4 and A5 EUT, optimal variable pay β * (ϵ, r, ψ, θ) decreases with ϵ whereas optimal Ąxed pay α * (ϵ, r, ψ, θ) increases with ϵ.

Proof of Proposition A1

Given the linear contract (α, β), the objective function of an expected utility agent with a cost function C(e) = ψe 2 is given by

with u(y) = 1exp(-ry) r , y + = α + βθ(e + 1-p p ϵ)ψe 2 and y -= α + βθ eϵψe 2 .

The Ąrst-order condition of the agentŠs maximization problem is given by

it turns out that the best response effort function of the agent is given by e = βθ 2ψ

optimal Ąxed pay α * (ϵ, r, w(p), ψ, θ) decreases with ϵ.

Proposition A2iii points to the absence of RIT under probability risk-seeking.

Furthermore, Proposition A3 provides results on RIT under general risk attitudes which are a combination of utility curvature and probability risk attitudes.

Proposition A3 (Risk attitudes and absence of tradeoff with RDU):

Assume that the agent exhibits probability risk-seeking for a given probability p. Let r N (β * , ϵ) be the absolute risk-aversion index that allows probability risk-seeking agent to exhibit risk-neutrality for the lottery

associated with the optimal linear contract (α * , β * ). Then, r N (β * , ϵ) > r to (β * , ϵ). principal facing a risk-averse agent who exhibits probability risk-seeking proposes an optimal variable pay β * (ϵ, r, w(p), ψ, θ) that decreases in ϵ. In addition, a principal facing an agent who exhibits probability risk-seeking with absolute risk-aversion index r ∈ (r to (β * , ϵ), r N (β * , ϵ)) also proposes an optimal variable pay β * (ϵ, r, w(p), ψ, θ) that decreases in ϵ. Note that such agent exhibits riskseeking behavior since r < r N (β * , ϵ). Finally, the principal only proposes an optimal variable pay β * (ϵ, r, w(p), ψ, θ) that increases in ϵ when facing an agent that exhibits probability risk-seeking with absolute risk-aversion index r ∈ (0, r to (β * , ϵ)). Unlike EUT, RIT under RDU depends on the probability of the binary shock and becomes more pervasive because it now applies to risk-seeking agent (on top of risk-averse agent, as in EUT).

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well QED.

Proof of Proposition A2

Given the linear contract (α, β), the objective function of a RDU agent with cost function C(e) = ψe 2 is given by

The Ąrst-order condition of the agentŠs maximization problem is given by

it turns out that the optimal effort function of the agent is given by e = βθ 2ψ

Hence, the optimal effort function is increasing in the variable pay and does not vary with respect to chock ϵ, the relative risk-aversion coefficient r, the Ąxed pay α and probability risk attitude w(p).

Point i)

The optimization problem of the principal is to maximize the expected value of θzy by accounting for the agentŠs incentive compatibility constraint (IC)

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well QED.

4.B Extension to general utility function

In section 4.2, we employ the LEN model with CARA utility function. This approach provides a closed-form solution to the principal-agent problem through the Arrow-Pratt approximation of the risk premium. We investigate the robustness of the results under general utility speciĄcation for EUT and RDU. Denote by A a (z) = -u ′′ (x) u ′ (x) the absolute risk-aversion index evaluated at x. We provide the followings results and their proofs.

Proposition B1 (Consistency of results under EUT): the optimal variable pay is a decreasing function of ϵ.

Proposition B1 shows that the tradeoff between risk and incentives in EUT framework does not depend on the utility function speciĄcations and is not driven by approximation errors in the Arrow-Pratt risk premium.

Example 3 (An illustration of Proposition B1 using expo-power utility function):

To illustrate this point, we consider the expo-power utility function [START_REF] Saha | Expo-power utility: a ŚĆexibleŠ form for absolute and relative risk aversion[END_REF]

r of which CARA (CRRA) is a special case when r = 1 (r -→ 0). The alternative level of utility is given by y 0 = 1-exp(-ry γ 0 ) r with y 0 being the alternative (outside) outcome. We set (r, γ) = (0.029, 0.731) as found by Holt and Laury (2002) and (ψ, θ, y 0 ) = (0.5, 1, 4). In the absence of shock (ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5.

In the presence of shock (ϵ, p) = (1, 0.5), the optimal variable pay is β * = 0.89 and the optimal Ąxed pay is α * = 3.65. Hence, the optimal variable pay in absence of shock is indeed greater than the variable pay in the presence of the found by Holt and Laury (2002). We set (ψ, θ, y 0 ) = (0.5, 1, 4), with y 0 being the alternative (outside) outcome. In the absence of shock (ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of shock, we set (ϵ, p, w(p)) = (1, 0.1, 0.15). The optimal variable pay is β * = 0.95 and the optimal Ąxed pay is α * = 3.37. The optimal variable pay in the absence of shock is greater than the variable pay in the presence of shock. We have the expected value E[L * ] = 3.83 and the certainty equivalent of L * is 4. Hence, the agent exhibits risk-seeking at the optimal contract (α * , β * ) = (3.37, 0.95) where the tradeoff between risk and incentives is observed.

Example 5 (An illustration of Proposition 12: absence of tradeoff with risk-seeking agent using expo-power utility function): Consider the parameter calibration from example 4 with the only change being w(0.1) = 0.2.

In the absence of a shock (ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of a shock, the optimal variable pay is β * = 1.17 and the optimal Ąxed pay is α * = 2.73. The optimal variable pay in the absence of a shock is less than the variable pay in presence of a shock.

We have that the expected value is E[L * ] = 3.41 and the certainty equivalent of We set (r, γ) = (0.029, 0.731) as in by Holt and Laury (2002) and (ψ, θ, y 0 ) = (0.5, 1, 4). In the absence of a shock (ϵ = 0), the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. In the presence of a shock, we set (ϵ, p) = (1, 0.5). The optimal variable pay is β * = 0.92 and the optimal Ąxed pay is α * = 3.61. The optimal variable pay in the absence of shock is thus greater than the variable pay in the presence of a shock.

QED.

Proof of Proposition B2

Given the linear contract (α, β), the objective function of an expected utility agent with cost function C(e) = ψe 2 is given by

The Ąrst-order condition of the agentŠs maximization problem is given by Hence, the optimal effort function is an increasing function of the variable pay and does not vary with respect to ϵ, the utility curvature r, the Ąxed pay α or the probability risk attitude captured by w(p).

Point i)

The principalŠs optimization problem is to maximize the expected value of θzy by accounting for the agentŠs incentive compatibility constraint (IC) and participation constraint (PC):

The corresponding Lagrangian function is given by

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

Assume that the weighting function is the following linear combination of logarithmic and quadratic functions

with δ, a and b being parameters of the weighting function. The derivative of w(p) is given by Furthermore, assume a quadratic utility function u(x) = xrx 2 that is concave (i.e., r > 0).

• Case 1 (presence of tradeoff with risk-seeking agent): we set (r, ψ, θ, y 0 ) = (0.02, 0.5, 0.5, 4), with y 0 being the alternative (outside)

then the optimal contract is given by the following system of two equations

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

The optimal variable pay is β * = 0.93 and the optimal Ąxed pay is α * = 3.87.

Recall that in the absence of a shock, the optimal variable pay is β * = 1 and the optimal Ąxed pay is α * = 3.5. Hence, the optimal variable pay in the absence of a shock is greater than the variable pay in presence of the shock. The expected value is E[L] = 3.97 and the certainty equivalent of L is 4. Hence, the agent exhibits risk-seeking at the optimal contract (α * , β * ) = (3.87, 0.93) where the tradeoff between risk and incentive is observed.

• Case 2 (absence of tradeoff with risk-seeking agent): Consider the calibration of parameters as before with the only change that r = 0.01. The optimal variable pay is β * = 1.08 and the optimal Ąxed pay is α * = 3.81.

The optimal variable pay in the absence of shock is less than the variable pay in the presence of shock. We have the expected value E[L] = 3.95 and the certainty equivalent of L is 4. Hence, the agent exhibits risk-seeking at the optimal contract (α * , β * ) = (3.95, 1.08) where the tradeoff between risk and incentive is not observed.

QED.

Denote by v(x) = xexp(-rx) a utility function over [0, ∞) with rx < 1 (Assumption A3Ť). The certainty equivalent y * 0 of the lottery (y * + , y * -; p, 1p) according to the utility function v(.) is implicitly deĄned by

Take the absolute risk aversion index of v(.),

Hence the utility function v(.) is associated to higher risk-aversion index than u(.). By the Pratt (1964) approximation, we have y * 0 < y 0 . Hence,

This yields dα m dr > 0.

Point ii) The implicit function theorem yields

Under Assumption A3Š, the minimum α increases with the shock size. Note that if the utility function is instead convex and the second-order condition resulting from the second derivative of EU (L) with respect to the effort is negative, then dα m dϵ < 0. Take the absolute risk-aversion index of v(.),

Hence the utility function v(.) is associated with a higher risk-aversion index than u(.). By Lemma 1 (Appendix 4.A) that provides the equivalent of Pratt Also, for a given probability p = p with δ := w(p), the minimum Ąxed pay decreases with degree of probability overweighting.

Point iv)

The certainty equivalent ce = y 0 of a lottery L = α m + βθ(e + 1-p p ϵ), α m + βθ(eϵ); p, 1p with e = βθ 2ψ is given by

The utility of the expected value

4ψ is given by

DeĄne by g(.) the following differential function over [0, ∞)

The absolute risk-aversion r N (β) for which the equality u(ce) = u(E[L]) holds for a risk-neutral agent is implicitly deĄned by

Note that g(.) has the following three characteristics: (i) g( .) is convex on [0, ∞);

(ii) g(0) = 1 and lim r-→+∞ g(r) = +∞; (iii) g(.) attains its minimum exactly at the thresholds of the tradeoff r to (β, ϵ)

These three characteristics yield two solutions for r N (β, ϵ) : r 1 = 0 < r to (β, ϵ)

and r 2 > r to (β, ϵ).

Proof of Proposition 3:

The mean-variance-skewness preference associated with the contract is given by

For any accepted contract (α, β), the agantŠs optimal level of effort is given by

Note that this level of effort does not depend on α. The agent agrees to provide the level of effort if

Note that the left hand side of the above inequality is strictly increasing in α. Hence, there is a minimum level of Ąxed pay α m such that the previous participation constraint is binding, that is

Point i) The minimum Ąxed pay increases with the aversion to variance From (4.77), we have

Hence, when a v decreases (i.e., high aversion to variance), then the minimum Ąxed pay increases. Moreover

4.C.2 Elicitation of risk preferences

We consider binary lotteries denoted by L = (x, y; p, 1p), with x being the outcome that occurs with probability p, and y being the outcome that occurs with probability 1p. We use 15 lotteries presented in the Table 4.C.1. They are a combination of 5 probabilities (p 1 , p 2 , p 3 , p 4 , p 5 ) = (0.1, 0.25, 0.33, 0.50, 0.75) and three couples of outcomes ¶(100, 0), (100, 50), (50, 0)♢. The elicit certainty equivalents for each lotteries using the switching outcomes technique (e.g., Tversky and Kahneman, 1992a;Gonzalez and Wu, 1999a;Abdellaoui et al., 2008a).

In addition, we utilize the 30 values of the minimum Ąxed pay elicited in the main experiment to obtain further certainty equivalent data. The insight is that the outside option of 1000 is the certainty equivalent of the lottery

We then have in total 45 certainty equivalent data points per individual such that each of the 5 probabilities is presented in 9 binary lotteries. We use this dataset to estimate the parameters of EUT, RDU and MVS at the individual level.

RDU and EUT

For RDU, we follow the procedure developed in [START_REF] Kpegli | All at once! a comprehensive and tractable semi-parametric method to elicit prospect theory components[END_REF] to estimate probability weights. Denote by ce, x and y respectively the values of certainty equivalent, the high outcome x and the small outcome y. Also, denote by I k the dummy variable for the probability p k , that is a variable that takes value 1 if probability is equal to p k and 0 otherwise. We assume CARA utility function where e is the error term, l is the lth line in ce, x, y and e; r the CARA coefficient and w(p k ) = δ k for k = 1, 2, ..., 5. We assume that the error term is normally distributed with mean 0 and heteroscedastic variance σ l = σ♣x ly l ♣.

We then estimate r, δ k and σ by maximum likelihood method.

For the special case of EUT, we assume δ k = p k and estimate only r and σ.

MVS

Under MVS, certainty equivalents satisfy the following empirical equation

with ce, E, V and S denoting respectively values of certainty equivalent, mean, variance and skewness associated with each of the 45 lotteries.

We assume that the error term is normally distributed with mean 0 and heteroscedastic variance σ l = σ♣x ly l ♣. We then estimate a v , a s and σ by maximum likelihood method.

4.D Prospect theory (PT) analysis

In this section, we assume the agent is endowed with a reference-dependence utility function of the following form:

where x is the absolute outcome, R the reference point and v(.) is a value function à la Tversky and Kahneman (1992a), that is assumed to be continuous and strictly increasing with v(0)=0. Following Tversky and Kahneman (1992a) and Abdellaoui et al. (2008a), we specify the value function as follows:

where λ > 0 is the loss-aversion index and u(.) is the basic utility function.

Following the LEN model, we assume the following exponential utility function:

with r + and r -representing the index of absolute risk-aversion in the gain and loss domains respectively.

We denote the probability weighting function in the gain (x ≥ R) and loss (x < R) domains by w + (.) and w -(.). We refer to probability risk-aversion

[risk-seeking] as the case in which w + (p) ≤ p and w

We consider the mixed lottery L = (x 1 , x 2 ; p, 1p) with x 1 ≥ R ≥ x 2 , which is valued as follows:

In our principal-agent setup (see Section 2.1), the agent is facing a lottery with

x 1 = α + βθ e + 1-p p ϵψe 2 and x 2 = α + βθ eϵψe 2 . We further assume that the agentŠs reference point is given by the expected value of the lottery: R = α + βθeψe 2 (4.83)

The risk-free reference point ensures that the agent is systematically in the gain domain when the random shock yields a positive outcome (i.e., 1-p p ϵ) and in the loss domain otherwise (i.e., -ϵ). Based on the previous assumptions, the following proposition summarizes the results regarding the optimal behavior of the agent and the principal.

Proposition E1 (Risk-incentives tradeoff with PT). Under A0, A1, A2, A3Š, A3Ť, A4, A5 and assuming a PT agent as speciĄed in (4.82): i) For a given contract (α, β), the optimal level of effort increases with θ, decreases with ψ and does not depend on the Ąxed pay α, the utility curvature in loss domain r -, the utility curvature in gain domain r + , the loss aversion index λ and the shock.

ii) β * (ϵ, r, ψ, θ) and e * (ϵ, r, ψ, θ) decrease with λ, r + (the utility risk-aversion in the gain domain), probability risk-aversion (i.e., overweighting in loss domain and underweighting in gain domain), while increases with r -(the utility riskaversion in the loss domain).

iii) β * (ϵ, r, ψ, θ) and e * (ϵ, r, ψ, θ) are higher in the presence of shock than in its absence if the agent exhibits sufficient probability risk-seeking, moderate utility curvature and loss-aversion.

Proof of Proposition E1:

Given the linear contract (α, β), the objective function of agent with a cost functionC(e) = ψe 2 is given by

Chapter 4. Against the Odds! The Tradeoff Between Risk and Incentives is Alive and Well

The Ąrst-order condition of the agentŠs maximization problem is given by The Ąrst-order conditions are given by

General conclusion

This thesis contributes to the behavioral and experimental economics literature on risk, beliefs, ambiguity, and incentives. In particular, this thesis (i) proposes new methods to measure utility function, weighting function, loss aversion, and beliefs and (ii) derives new theoretical insights in the realm of incentive setting along with the development of a new experimental paradigm.

Chapter 1 establishes a comprehensive semi-parametric method that satisĄes the four desirable properties of parametric methods: tractable, data-efficient, error-robust, and easy. Chapter 2 provides a nonparametric version of the semi-parametric method in Chapter 1 with a smoothing spline approximation of the utility function. Applications of both chapters 1 and 2 on existing data involving risk conĄrm deviations from expected utility theory through evidence of probability weighting function and loss aversion.

Chapter 3 extends the semi-parametric method in Chapter 1 to measure beliefs and ambiguity attitudes towards discrete sources of uncertainty. Subjects make decisions under these types of uncertain situations in a daily life basis. In various applications, the method successfully passes validity tests and provides insightful results in the case of trust and coordination games. This chapter Ąnds deviations from subjective expected utility theory through evidence of ambiguity attitudes. The chapter highlights that people are more insensitive to likelihood in the presence of asymmetric events than symmetric events, suggesting that This behavior is captured by more optimism.

Chapter 4 studies the existence of the Risk-incentives tradeoff (RIT) under Rank Dependent Utility (RDU) and Mean-Variance-Skewness (MVS). Theoretical analyses show that RIT is remarkably robust under RDU but not under MVS.

With data based on a novel experimental design that eliminates confounding factors, Chapter 4 provides evidence for RIT even in the case of risk-seeking agents, which is a distinct prediction of RDU. The results provide support for the RIT and suggest that it applies to a broad range of situations, including cases in which agents are risk-seeking (e.g., executive compensation).

Naturally, the thesis has both limitations and potential extensions for future research. The following are some of the limitations and possible extensions.

The methods in Chapters 1 and 2 can be extended in several directions.

Even though these methods do not readily apply to the context envisioned by Kőszegi and Rabin (2007), one can speculate on a possible procedure combining Köszegi and RabinŠs approach and these methods. This procedure could start by introducing probability weighting functions in Kőszegi and Rabin (2007) following the work of Baillon et al. (2020). Also, these methods can be used to extend the popular elicitation technique of Holt and Laury (2002) to the case in which probabilities are distorted, following the approach of [START_REF] Abdellaoui | Risk aversion elicitation: reconciling tractability and bias minimization[END_REF].

General conclusion

Following Li et al. (2020), Chapter 3 conducted two-stage experiments. In the Ąrst stage, people played trust games. In the second stage, the Ąrst movers in the Ąrst stage made decisions in which the sources of uncertainty were the unknown decisions of the second movers in the Ąrst stage. Two important identiĄcation assumptions are behind the experimental results of Li et al. (2020) as well as the ones in Chapter 3. The Ąrst is that social preferences collapse in the second stage. This is quite reasonable because in the second stage, the decisions of the Ąrst movers will have no impact on the Ąrst-stage payoffs of the Ąrst movers.

The second assumption is that betrayal attitudes continue to play a role in the second stage. But, this second assumption is less trivial. Indeed, it is possible that betrayal attitudes collapse in the second stage as it is the case of social preferences. This could explain why these two papers do not Ąnd evidence in favor of betrayal aversion. A future avenue of research is to quantify the role of betrayal aversion in trust decisions under the more reasonable assumption that betrayal aversion collapses in the second stage of the experimental design.