Camille Cornand
email: cornand@gate.cnrs.fr

Maria Alejandra

Erazo Diaz

Béatrice Rey

Adam Zylbersztejn
email: zylbersztejn@gate.cnrs.fr

Aurélien Baillon

Béatrice Boulu-Reshef

Sibilla Di Guida

Jan Engelmann

On the robustness of higher order attitudes to ambiguity framing

Keywords: Ambiguity attitudes, Higher order attitudes, Ambiguity aversion, Ambiguity prudence, Ambiguity temperance, Framing D81, C91 Ambiguity attitudes, Higher order attitudes, Ambiguity aversion, Ambiguity prudence, Ambiguity temperance, Framing. JEL codes: D81, C91

In a context-free preference situation, we conduct a laboratory experiment in which we test higher order ambiguity attitudes (order 2, order 3, and order 4) using a simple model with two states of nature (good and bad). We compare ambiguity attitudes when the random variable capturing ambiguity is introduced on the probability associated with the good state of nature versus the bad state of nature. In addition, in the case of order 3, we compare ambiguity attitudes when the random variable capturing ambiguity is presented as two harms (as usual in decision theory) versus one harm and one favor. We first establish the theoretical prediction of a general consistency of ambiguity attitudes. Ceteris paribus, these attitudes should remain invariant with respect to (i) the state of nature to which ambiguity is associated, and to (ii) the type of the change (harm or favor) in the probability to which ambiguity is associated. Our empirical results systematically contradict this formal prediction pointing to the behavioral importance of ambiguity framing.

Introduction

Risk aversion drives behavior in many economic contexts. This behavioral feature is however insufficient. We know since the end of the sixties that higher order risk preferences -prudence and temperance -are just as decisive to explain for example precautionary savings (e.g., [START_REF] Leland | Saving and uncertainty: The precautionary demand for saving[END_REF][START_REF] Sandmo | The effect of uncertainty on saving decisions[END_REF][START_REF] Kimball | Precautionary saving in the small and in the large[END_REF][START_REF] Eeckhoudt | Changes in risk and the demand for saving[END_REF], insurance decisions (e.g., [START_REF] Fei | Precautionary insurance demand with state-dependent background risk[END_REF], prevention (e.g., [START_REF] Eeckhoudt | The impact of prudence on optimal prevention[END_REF][START_REF] Crainich | Self-protection with random costs[END_REF][START_REF] Courbage | Decision thresholds and changes in risk for preventive treatment[END_REF][START_REF] Peter | Optimal self-protection in two periods: On the role of endogenous saving[END_REF] and portfolio choices (e.g., [START_REF] Eeckhoudt | Demand for risky assets and the monotone probability ratio order[END_REF][START_REF] Kimball | Precautionary motives for holding assets[END_REF]. This literature on higher order attitudes has been theoretically extended to situations where probabilities are unknown, i.e., ambiguous situations.1 Ambiguity aversion, prudence and temperance have been found to be key to explain prevention behavior (e.g., [START_REF] Treich | The value of a statistical life under ambiguity aversion[END_REF][START_REF] Berger | Treatment decisions under ambiguity[END_REF][START_REF] Berger | The impact of ambiguity and prudence on prevention decisions[END_REF][START_REF] Baillon | Prudence with respect to ambiguity[END_REF][START_REF] Bleichrodt | The value of a statistical life under changes in ambiguity[END_REF] and saving decisions (e.g., [START_REF] Alary | The effect of ambiguity aversion on insurance and self-protection[END_REF][START_REF] Berger | Precautionary saving and the notion of ambiguity prudence[END_REF] for example.

Many laboratory experiments have provided an experimental counterpart to theoretical works under risk, generally finding strong evidence for risk aversion, risk prudence and, to a lesser extent, for risk temperance (e.g., Deck andSchlesinger, 2010, 2014;[START_REF] Ebert | Joint measurement of risk aversion, prudence, and temperance[END_REF][START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF][START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF][START_REF] Bleichrodt | The reflection effect for higher-order risk preferences[END_REF]). Yet, only very few papers have provided experimental tests of higher order ambiguity attitudes. [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF] consider a pure damage context, i.e., with two states of nature, a good state (no damage) and a bad state (damage). Introducing the hazard capturing ambiguity on the good state of nature, they report ambiguity aversion, ambiguity prudence and, to a lesser extent, ambiguity temperance. Importantly, in the theoretical literature on pure damage the announced probability refers to the occurrence of a damage (bad state). 2 Theoretical models of pure damage that introduce ambiguity follow this presentation format and introduce the ambiguity parameter on the probability associated with the bad state of nature (e.g., [START_REF] Treich | The value of a statistical life under ambiguity aversion[END_REF][START_REF] Snow | Ambiguity aversion and the propensities for self-insurance and self-protection[END_REF][START_REF] Alary | The effect of ambiguity aversion on insurance and self-protection[END_REF][START_REF] Gollier | Optimal insurance design of ambiguous risks[END_REF][START_REF] Bleichrodt | The value of a statistical life under changes in ambiguity[END_REF]. While there is already a large body of experimental evidence documenting framing effects in higher order risk attitudes (see e.g., Deck andSchlesinger, 2010, 2014;Ebert andWiesen, 2011, 2014;[START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF], very little is known about the behavioral effects of framing on ambiguity attitudes.

Our study aims at filling this gap in the literature.

Following a simple two-state model based on [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF], we conduct a contextfree preference laboratory experiment in which we compare ambiguity attitudes -ambiguity aversion (order 2), ambiguity prudence (order 3), and ambiguity temperance (order 4) -when the random variable capturing ambiguity is introduced on the probability associated with the good state of nature versus the bad state of nature. In addition, we compare prudence attitudes when the random variable capturing ambiguity is presented as two harms (as usual in decision theory) versus one harm and one favor. 3Theory points to the general consistency of ambiguity attitudes which, ceteris paribus, should remain invariant (i) with respect to the state of nature to which ambiguity is associated and (ii) with respect to the type of the change (harm and favor) in the probability to which ambiguity is associated. Our empirical results, however, systematically deviate from this formal prediction. Under two harms, moving from the good state to the bad state frame amplifies ambiguity aversion, reduces prudence and enhances temperance. In addition, when ambiguity is presented as one harm and one favor, moving from the good state frame to the bad state frame enhances ambiguity prudence. Finally, we provide evidence that the effect of good versus bad state framing on ambiguity aversion (but not other attitudes) is associated with cognitive skills: its magnitude is substantial for participants with the lowest Cognitive Reflection Test (CRT) score, and progressively fades away as the CRT score increases. For subjects with the highest cognitive skills, we find no evidence of ambiguity attitude distortions due to framing effects.

The remainder of this paper is organized as follows. In the next section, we present the theoretical foundation of our measurements of higher order ambiguity attitudes. Section 3 presents our empirical strategy. Section 4 describes the design of our experiment. The results are presented in Section 5. Section 6 concludes.

Theoretical background and hypotheses

Consider a decision-maker (DM) confronted with two possible states of nature, either bad (henceforth BS) or good (henceforth GS). DM's expected utility writes: pu(R) + (1p)u(R) where u verifies u ′ > 0 and probability p is associated with the good state of nature (R > R).

Following [START_REF] Treich | The value of a statistical life under ambiguity aversion[END_REF], [START_REF] Berger | Treatment decisions under ambiguity[END_REF] and [START_REF] Bleichrodt | The value of a statistical life under changes in ambiguity[END_REF], we introduce ambiguity via a random variable ξ added to p:

(p + ξ)u(R) + (1 -(p + ξ))u(R)
Ambiguity arises because DM does not know the realization ξ of the random variable ξ and thus lacks knowledge of the probability of being in the good state with wealth level R (and symmetrically in the bad state with wealth level R). 4Without ambiguity, the expected utility writes as:

pu(R) + (1 -p)u(R). (1)
Let ξ GS be a random variable capturing ambiguity associated with the good state of nature. We obtain:

(p + ξ GS )u(R) + (1 -p -ξ GS )u(R). (2)
For a rational and ambiguity-averse agent, ξ GS consists of one or two harms ("bad news"). Depending on the specification, it captures different facets of ambiguity attitudes:

❼ ξ GS = ε where ε is a zero-mean random variable (E(ε) = 0) or ξ GS = 0 (degenerated random variable) captures ambiguity aversion (ambiguity of order 2),

❼ ξ GS = [-k, ε; 1 2 , 1 2 ] or ξ GS = [0, -k + ε; 1 2 , 1 2 ]
with k > 0 captures ambiguity prudence (ambiguity of order 3),

❼ ξ GS = [ε 1 , ε2 ; 1 2 , 1 2 ] or ξ GS = [0, ε1 + ε2 ; 1 2 , 1 2 ]
where ε1 and ε2 are zero mean independent random variables, captures ambiguity temperance (ambiguity of order 4).

The intuition is the following. An ambiguity averse DM prefers 0 to the zero mean random variable ε: p ≻ p + ε. In the same way, a rational DM prefers 0 to a certain loss added to p, p ≻ pk, since p is associated with the good state of nature. Thus, ambiguity aversion can be measured by a choice between harm ε or no harm at all (0). To grasp ambiguity prudence, we use a choice between aggregating the two harms or disaggregating them: an ambiguty prudent DM prefers to disaggregate harms by combining -k with 0 and 0 with ε rather than -k with ε and 0 with 0. Analogously, to measure ambiguity temperance, we use a choice between aggregating two harms ([0, ε1 +

ε2 ; 1 2 , 1 2 ]) or disaggregating them ([ε 1 , ε2 ; 1 2 , 1 2
]): a temperate DM prefers to disaggregate harms by combining ε1 with 0 and 0 with ε2 rather than ε1 with ε2 and 0 with 0. For further details, see [START_REF] Eeckhoudt | Putting risk in its proper place[END_REF] and [START_REF] Eeckhoudt | Apportioning of risks via stochastic dominance[END_REF].

Experimental manipulation 1: Testing for attitudes towards ambiguity associated with the bad state of nature. In the context of equation ( 1), denote ξ BS a random variable capturing ambiguity associated with the bad state of nature5 :

(1 -p + ξ BS )u(R) + (p -ξ BS )u(R).
(3)

Expressions ( 1) and ( 3) are identical iff

ξ BS = -ξ GS (4)
For the sake of experimental implementation, we assume that ε and εi (i = 1, 2) are symmetric random variables, so that we obtain: ε = -ε and εi = -ε i . This allows us to formulate the following hypotheses about the consistency of ambiguity attitudes which, ceteris paribus, should remain invariant with respect to the state of nature to which ambiguity is associated:

Hypothesis H1 (consistency of the ambiguity attitudes of order 2 w.r.t. the state of nature).

(p : R, R) ≻ (p + ε : R, R) ⇔ (1 -p : R, R) ≻ (1 -p + ε : R, R).
Hypothesis H2 (consistency of the ambiguity attitudes of order 3 w.r.t. the state of nature).

({p -k, p + ε} : R, R) ≻ ({p, p -k + ε} : R, R) ⇔ ({1 -p + k, 1 -p + ε} : R, R) ≻ ({1 -p, 1 -p + k + ε} : R, R).
Hypothesis H3 (consistency of the ambiguity attitudes of order 4 w.r.t. the state of nature). {p + ε1 , p + ε2 } : R, R) ≻ ({p, p + ε1 + ε2 } : R, R) ⇔ ({1p + ε1 , 1p + ε2 } : R, R) ≻ ({1p, 1p + ε1 + ε2 } : R, R).

Experimental manipulation 2: Testing for prudence in apportioning harms and favors.6 Now, consider ξ GS ( ξ BS ) consisting of one harm ε, and one favor +k (-k) with k > 0.

An ambiguity prudent DM always prefers to disaggregate harms: combine now 0 with 0 and +k (-k) with ε rather than 0 with ε and +k (-k) with 0. This insight provides another way to test for the consistency of the ambiguity attitudes of order 3:

Hypothesis H4 (consistency of the ambiguity attitudes of order 3 w.r.t. the type of the change (harm and favor)).

({p, p + k + ε} : R, R) ≻ ({p + k, p + ε} : R, R) ⇔ ({1 -p, 1 -p -k + ε} : R, R) ≻ ({1 -p -k, 1 -p + ε} : R, R).

Empirical strategy

The theoretical framework along with its testable implications, as laid out is Section 2, is operationalized through a set of lottery choices presented in Table 1. It consists of 55 choice tasks divided into sets of 5 choices. The first 30 choice tasks in Table 1 replicate the tasks implemented by [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF]. Accordingly, the first 15 tasks are risk choice tasks, and each of the sets corresponds to risk orders 2, 3, and 4.

Experimental testbed for H1-H3. Tasks from 16 to 30 are ambiguous choice tasks of order 2, 3, and 4, in which the variable capturing ambiguity is introduced on the probability associated with the good state of nature. Lines 31 to 45 in Table 1, in turn, present 15 ambiguous lotteries in which, contrary to [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF], the random variable capturing ambiguity is introduced on the probability associated with the bad state of nature. These 15 choice tasks represent the exact counterparts of choice tasks 16 to 30, the sole difference being that ambiguity is now introduced on the bad state.

Experimental testbed for H4. The last 10 choice tasks in Table 1 aim at comparing ambiguity prudence choices with ambiguity presented as one harm and one favor in the bad state of nature (lines 46-50) versus ambiguity prudence with ambiguity presented as one harm and one favor on the good state (lines 51-55).

Experimental procedures

We recruited 227 students to participate in a computerized experiment7 conducted at the GATE-Lab in Lyon. According to our criteria of outliers, 18 subjects were removed from the sample because they failed to answer correctly to more than half of the questions in the understanding questionnaire. 8 Hence, our final sample size is 209. Subjects were told that the experiment could last up to 90 minutes, that they would receive 5 e as a participation fee, and they could additionally earn a variable amount depending on random draws and their own decisions. The mean age of subjects is 21 years, 44% are female, and 45% study economics or finance.

The experiment consisted of two parts. In the first part, we elicited risk attitudes in 15 choices. In the second part, we elicited ambiguity attitudes in 40 choices. Each part started with its specific instructions, which were presented on the screen9 . Within each part, the order of the choices was randomized across subjects.

For each lottery in Table 1, we presented the two alternatives for each task graphically, using diagrams of circles representing both options A and B, which were framed as option Left (L) and Right (R), respectively. The position in the screen (left or right) for options A and B was randomized across subjects. Figure 1 displays the screen of task 11 in Table 1. Subjects' task consists in choosing an option between A and B. To make their selection, subjects had to click on one of the texts (Option L, Option R) positioned on the top of the image. As Figure 1 displays, the final outcomes in the risk task, are shown in yellow. These yellow circles only appear on the screen after 10 seconds.

In the ambiguous choices, circles have two different colors representing the good or bad state of nature, the circles are gradually being colored to illustrate the variation of the probabilities, capturing ambiguity. The green color represents the bad state of nature, and the blue color represents the good state of nature. Figure 2 shows the screen of task 31 in Table 1. The arrow10 around the circle in the left panel represents the fact that the circle is progressively colored in green from 0

Amb 26A × 1.5 26B × 1.5 29 Amb ({1/3, 1/3 + [-1/6, 1/6] + [-1/6, 1/6]} : 30, 0) ({1/3 + [-1/6, 1/6], 1/3 + [-1/6, 1/6]} : 30, 0) 30 Amb ({2/3, 2/3 + [-1/6, 1/6] + [-1/6, 1/6]} : 30, 0) ({2/3 + [-1/6, 1/6], 2/3 + [-1/6, 1/6]} : 30, 0) Amb 1 -p + ε : R, R 1 -p : R, R 31
({1/2 + 1/4, 1/2 + [-1/4, +1/4]} : 0, 45) 39 Amb ({1/3, 1/3 + 1/6 + [-1/6, +1/6]} : 0, 30) ({1/3 + 1/6, 1/3 + [-1/6, +1/6]} : 0, 30) 40 Amb ({2/3, 2/3 + 1/6 + [-1/6, +1/6]} : 0, 30) ({2/3 + 1/6, 2/3 + [-1/6, +1/6]} : 0, 30) Amb {1 -p, 1 -p + ε1 + ε2 } : R, R {1 -p + ε1 , 1 -p + ε2 } : R, R 41 Amb ({1/2, 1/2 + [-1/8, 1/8] + [-1/8, 1/8]} : 0, 30) ({1/2 + [-1/8, 1/8], 1/2 + [-1/8, 1/8]} : 0, 30) 42 Amb ({1/2, 1/2 + [-1/8, 1/8] + [-1/8, 1/8]} : 15, 45) ({1/2 + [-1/8, 1/8], 1/2 + [-1/8, 1/8]} : 15, 45) 43 Amb ({1/2, 1/2 + [-1/8, 1/8] + [-1/8, 1/8]} : 0, 45) ({1/2 + [-1/8, 1/8], 1/2 + [-1/8, 1/8]} : 0, 45) 44 Amb ({1/3, 1/3 + [-1/6, 1/6] + [-1/6, 1/6]} : 0, 30) ({1/3 + [-1/6, 1/6], 1/3 + [-1/6, 1/6]} : 0, 30) 45 Amb ({2/3, 2/3 + [-1/6, 1/6] + [-1/6, 1/6]} : 0, 30) ({2/3 + [-1/6, 1/6], 2/3 + [-1/6, 1/6]} : 0, 30) Amb {1 -p -k, 1 -p + ε} : R, R {1 -p, 1 -p -k + ε} : R, R 46 Amb ({1/2 -1/4, 1/2 + [-1/4, +1/4]} : 0, 30) ({1/2, 1/2 -1/4 + [-1/4, +1/4]} : 0, 30) 47 Amb 46A + 15 46B + 15 48 Amb 46A × 1.5 46B × 1.5 49 Amb ({1/3 -1/6, 1/3 + [-1/6, +1/6]} : 0, 30) ({1/3, 1/3 -1/6 + [-1/6, +1/6]} : 0, 30) 50 Amb ({2/3, 2/3 -1/6 + [-1/6, +1/6]} : 0, 30) ({2/3, 2/3 -1/6 + [-1/6, +1/6]} : 0, 30) Amb {p + k, p + ε} : R, R {p, p + k + ε} : R, R 51 Amb ({1/2 + 1/4, 1/2 + [-1/4, +1/4]} : 30, 0) ({1/2, 1/2 + 1/4 + [-1/4, +1/4]} : 30, 0) 52 Amb ({1/2 + 1/4, 1/2 + [-1/4, +1/4]} : 45, 15) ({1/2, 1/2 + 1/4 + [-1/4, +1/4]} : 45, 15) 53 Amb ({1/2 + 1/4, 1/2 + [-1/4, +1/4]} : 45, 0) ({1/2, 1/2 + 1/4 + [-1/4, +1/4]} : 45, 0) 54 Amb ({1/3 + 1/6, 1/3 + [-1/6, +1/6]} : 30, 0) ({1/3, 1/3 + 1/6 + [-1/6, +1/6]} : 30, 0) 55 Amb ({2/3 + 1/6, 2/3 + [-1/6, +1/6]} : 30, 0) ({2/3, 2/3 + 1/6 + [-1/6, +1/6]} : 30, 0)
Note: The probability always refers to the first wealth level.

A zero-mean random variable ε represented by {-a, +a} means that ε is a discrete random variable, e.g., that it takes the value -a with probability 1 2 and the value +a with probability 1 2 . A zero-mean random variable ε represented by [-a, +a] means that ε is a random variable distributed according to a uniform distribution taking values in [-a, +a]. The big circles in Figure 2 represent the aggregate outputs for each option. They appear after 7 seconds and the button to confirm the decision is only available after 12 additional seconds. These features of the tasks are implemented in order to let subjects focus on the aggregated or disaggregated news.

Each choice task corresponds to a sealed envelope that was prepared before the experiment. The content of the envelopes is described to the subjects in the instructions. The envelopes contain all the possible outcomes from choices A and B of each task. For instance, the envelope corresponding to risk task 11 contains two smaller envelopes that represent each of the options A and B. The envelope representing option A has inside eight tags: six tags indicating 15 e, one tag indicating 0 e, and one tag indicating 30 e. Therefore, the tags illustrate that if option A is chosen, the probability of winning 15 e, 0 e, or 30 e is 3/4, 1/8, and 1/8, respectively (as shown in the left panel of Figure 1). The second smaller envelope representing option B contains two tags. One of them indicates 22.5 e and the other one 7.5 e. Therefore, for option B, the probability of winning 22.5 e and 7.5 e is 1/2 (as shown in the right panel of Figure 1).

Similarly, the envelope corresponding to ambiguity task 31 contains two smaller envelopes for options A and B. Inside the envelope for option A, there are five smaller envelopes, each of them representing different probabilities of obtaining the outcome of the bad state of nature. The probabilities are 0, 1/4, 1/2, 3/4, and 1. Each of these envelopes has inside different amounts of tags showing the final payoff. For instance, the envelope with probability 1/4 contains four tags: one indicating 0 e and three tags indicating 30 e. The envelope representing option B contains two tags: one displaying the value 30 e and another one displaying 0 e. This corresponds to the probability of winning the two possible outcomes of this option (as shown in the right panel of Figure 2).

All the 55 envelopes were stored in a box located in the laboratory on the sight of the subjects. The envelopes did not have visible identification of the task they represented. At the beginning of each experimental session, one of the subjects was randomly selected to pick one of the envelopes from the box and was asked to sign it. Subsequently, the envelope was left on the sight of all the subjects. At the end of the experiment, the selected envelope was opened by the subject who signed it to determine the payoffs. As pointed out by [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF], ambiguity opens up the possibility that results are affected by subjects' beliefs. A priori subjects do not have reasons to expect one outcome to be more likely than another. This concern is related to suspicion. Subjects may suspect the experimenter to voluntarily influence the outcome. To avoid suspicion: we manually implemented the above described procedure in front of the subjects. Also, subjects were told that, if they wish, they could open and check all the envelopes at the end of the experiment.

Once subjects finished the choice tasks, they proceeded to answer a battery of questions, including self-evaluation of risk attitudes, cognitive reflection test, and a socio-demographic questionnaire (see Appendix 7.2 for details). 12 5 Results

In this section, we first look at the aggregate attitudes for ambiguity of order 2 (related to aversion), 3 (related to prudence), and 4 (related to temperance). For each of them, we distinguish between two lottery frames: one in which the hazard is associated with the good state (GS) and one in which the hazard is related to the bad state (BS).

Following [START_REF] Deck | Exploring higher order risk effects[END_REF], [START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF] and [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF], for each subject and under each frame we build an individual score equal to the number of choices made in the corresponding set of 5 lotteries (see Table 1) that point to ambiguity aversion/prudence/temperance. As a result, each DM's ambiguity attitudes of a given order are empirically measured by two scores: the BS score and the GS score.

Below, we provide aggregate comparisons of both scores across frames. We find systematic discrepancies between frames, part of which are associated with the DM's cognitive skills. Additional exploratory analyses focusing on attitudes of order 3 also highlight the DM's sensitivity to the way in which news are communicated (i.e., two harms versus one harm and one favor).

On the statistical note, it is important to mention at this point that our empirical investigation is based on count variables. Hence, standard tests developed for continuous variables (e.g., 12 Self-evaluation of risk attitudes involves two standard psychometric tools: General Risk Propensity Scale (GRiPS) and Hexaco Personality Inventory test. This exercise provides exploratory evidence on applicability of these non-incentivized measures in predicting risk attitudes observed in our incentivized lotteries 1-15. We find a meaningful (i.e., negative) correlation between the GRiPS score and risk attitudes of different orders observed in lottery choices: aversion (ρ = -0.252, p < 0.001) and temperance (ρ = -0.197, p = 0.004). Yet, no such correlation is found for risk prudence (ρ = -0.002, p = 0.980). The outcome of Hexaco Personality Inventory test, in turn, has little-to-none predictive power for aversion (ρ = 0.009, p = 0.894), prudence (ρ = 0.122, p = 0.078), or temperance (ρ = 0.062, p = 0.375) risk attitudes. parametric t-test or nonparametric ranksum/signrank test) are not appropriate. Our method of hypothesis testing builds on parametric Poisson regression, the most standard approach to modeling count data (see, e.g., [START_REF] Cameron | Microeconometrics: Methods and Applications[END_REF]. More specifically, for a count variable y i = 0, 1, 2, . . ., a K × 1 vector of regressors x i = (x i1 , x i2 , . . . , x iK ) and a K × 1 vector of its respective coefficients β = (β 1 , β 2 , . . . , β K ) ′ , this model defines the conditional mean for individual i as E(y

i |x i ) = µ i = e x ′ i β .
In what follows, we provide further details on the exact specifications of µ i and the resulting testing procedures.

Testing H1: Ambiguity of order 2

To grasp the degree of ambiguity aversion in our sample, Figures 3a and3b present the distribution of GS scores (lotteries 16 -20, left panel) and BS scores (lotteries 31 -35, right panel), respectively. Our first observation is that both distributions are significantly different from a random choice benchmark distribution (χ 2 test, p < 0.001). This suggests that individuals in our sample do not behave as order 2 neutral DM that systematically display indifference between options throughout a given set of lotteries. 13Looking at Figure 3a, we nonetheless find that the mean GS score is 2.56 (out of 5) pointing to neutrality as it is not significantly different from what would be expected from random choices (p = 0.615). 14 To test this hypothesis, we estimate a constant-only specification of the Poisson model: µ i = e β 0 . It follows that β 0 = ln(µ). Hence, testing H 0 : µ = 2.5 can be operationalized by testing H 0 : β 0 = ln(2.5) ≈ 0.916 through asymptotic z-test.

Moving to the BS scores presented in Figure 3b and using the same statistical testing procedure, we find that the mean score is equal to 2.81 and significantly different from the theoretical benchmark of neutrality of 2.5 (p = 0.004).

Finally, this shift in ambiguity attitudes between BS and GS is found to be statistically significant (p = 0.008). 15 This leads us to rejecting H1 with the first result:

Result 1. Moving from the good state frame to the bad state frame amplifies ambiguity aversion. 5.4 Testing H4: Ambiguity of order 3 with one harm and one favor

Figure 6 presents the distribution of ambiguity choices at order 3 when the random variable capturing ambiguity is presented as one harm and one favor on the bad state (right panel) and on the good state (left panel). The right panel of Figure 6 shows that on average subjects are ambiguity prudent. The mean subject chooses the ambiguity prudent option 2.87 out of 5 times, which is significantly higher than what would be observed if he made random choices (p < 0.001).

The left panel of Figure 6, in turn, points to neutrality: the mean subject chooses the ambiguity prudent option 2.61 out of 5 times, which is not significantly different from what would be observed if he chose randomly (p = 0.301).

Finally, there is a significant difference between the distribution of the left and right panels (p = 0.011). Once again, this shift in mean score is significant (p = 0.007) pointing to the following result that rejects H4: Result 4. When ambiguity is presented as one harm and one favor, moving from the good state frame to the bad state frame enhances ambiguity prudence.

Individual heterogeneity in ambiguity attitudes: Exploring the role of cognitive skills

This section offers an exploratory investigation of the interplay between the observed patterns in the attitudes towards ambiguity and cognitive skills. Although several experimental studies highlight a link between attitudes towards risk and cognitive skills (most notably, subjects with higher cognitive skills are found to exhibit a weaker degree of risk aversion), the available empirical evidence is much scarcer and not supportive of that conclusion for the domain of ambiguity (for an excellent review, see [START_REF] Prokosheva | Comparing decisions under compound risk and ambiguity: The importance of cognitive skills[END_REF], and in particular Table 1.2 therein).

In this part, we add to this literature by asking a novel question: does the effect of framing (i.e., bad state vs. good state frame) on ambiguity attitudes depend on cognitive skills?16 Table 2 breaks down the aggregate outcomes in two dimensions: whether the state of nature to which probabilities are associated is bad or good, and the DM's CRT score (either 0, 1, 2, or 3). Let us first look at ambiguity averse behavior and the case of ambiguity related to the bad state of nature. We cannot reject the hypothesis that the mean attitude is independent from the cognitive skill measure (echoing the aforementioned empirical findings). This changes once we consider the good state of nature for which we reject this hypothesis at the 5% level. We observe a substantial (in terms of mean magnitude) and highly significant (p < 0.001) shift in attitudes among subjects with lowest cognitive skills (CRT score of 0) from aversion towards preference for ambiguity. This effect fades away in a monotone manner as the CRT score improves: it remains significant for those with a relatively low CRT score of 1 (p = 0.022), and completely disappears in participants with higher scores (either 2 or 3, both p > 0.5). The second conclusion that stems from the figures presented in Table 2 is the absence of such patterns for higher order attitudes: prudence and temperance.

Table 3 provides additional econometric support for this result. For each dependent variable of interest we fit a Poisson regression model (denoted M1-M4) in which explanatory indicator variable 1[GS] is interacted with a set of four CRT score indicator variables and furthermore coupled with individual controls related to attitudes towards risk (as elicited in lotteries 1-5, 6-10, and 11-15), gender and academic major. For ambiguity aversion (model M1), we do not reject H T est1 0 stating that lottery choices are not associated with cognitive skills under the bad state frame. Rejecting H T est2 0 , in turn, suggests that such association exists for the good state frame. Finally, the empirical estimates and significance of coefficients α 1 , α 5 , α 6 and α 7 a with the outcomes of testing H T est3 0 , H T est4 0 and H T est5 0 point to the conclusion that the bad-good gap in ambiguity aversion is driven by the DM with low cognitive skills. Echoing the evidence summarized in Table 2, no such evidence is found in models M2-M4. In addition, the sign and significance of coefficient α 8 in model M1 points to the existence of mixed attitudes: the attitudes towards risk are found to be predictive of the attitudes towards ambiguity. 17 17 In Appendix 7.4 we provide further details on risk attitudes observed in our experimental sample. In line with previous literature (see [START_REF] Eeckhoudt | The economics of risk: A (partial) survey. Handbook of Risk Theory: Epistemology, Decision Theory, Ethics, and Social Implications of Risk[END_REF], for a review), subjects are risk averse and risk prudent (as in [START_REF] Deck | Consistency of higher order risk preferences[END_REF][START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF][START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF][START_REF] Haering | Exploring the consistency of higher order risk preferences[END_REF] (see Appendices 7.4.1 and 7.4.2). We also find that they are risk temperate (see Appendix 7.4.3), which is in line with [START_REF] Noussair | Higher order risk attitudes, demographics, and financial decisions[END_REF] and [START_REF] Heinrich | Higher-order risk preferences in social settings[END_REF], but contrasts with [START_REF] Deck | Consistency of higher order risk preferences[END_REF] and [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF] who find risk intemperance and [START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF] who do not observe a significant deviation from neutrality for temperance. Note. In each model, the dependent variable is the number of lottery choices (ranging between 0 and 5) reflecting a given attitude. Explanatory variables: 1[GS] = 1 for good state frame and = 0 otherwise, 1[CRT = X] = 1 for CRT score equal to X and = 0 otherwise, Risk aversion/prudence/temperance take integer values 0-5 corresponding to choices in lotteries 1-5/6-10/11-15, 1[M ale] = 1 for male participants and = 0 for female participants, 1[EconF inance] = 1 for participants majoring in economics or finance and = 0 otherwise. Cluster-robust SE (individual-level clustering, 2 clusters per individual). N = 418 (209 clusters).

Discussion and conclusion

We have proposed a laboratory experiment to test higher order ambiguity attitudes (order 2, order 3, and order 4) using a simple model with two states of nature (good and bad). When the hazard capturing ambiguity is introduced on the good state of nature and in the form of two harms (which is the case considered in [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF]), our experiment shows that subjects are neutral toward ambiguity. These results contrast with those of [START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF] where subjects behave in a prudent manner. We conjecture that these differences may stem from two distinct design features of the present experiment. First, it captures ambiguity in a visual (rather than purely numerical) way. Second, apportionment of harms is presented in two steps (rather than a single step): lotteries are first described in an exhaustive manner (in a raw form that clearly identifies harms) before being presented in a reduced form.

Our experiment also shows that when the hazard capturing ambiguity is framed differently, i.e., introduced on the bad state of nature, results depend on the way news are communicated. When facing two harms, subjects prefer to aggregate them and are thus imprudent, while facing one harm and one favor makes them prefer to disaggregate and be prudent.

Our evidence therefore suggests that the way news about hazards are communicated is not neutral. Communicating such news by presenting probability on the bad state induces more aversion and temperance towards ambiguity, but also weakens prudence. Nonetheless, presenting probability in the bad state may strengthen prudence in the case of one harm and one favor.

Additionally, our results show that preference for ambiguity is higher when the probability is associated with the good state of nature than when it is associated with the bad state of nature among subjects who scored low in the CRT. This difference in ambiguity preferences tends to disappear when CRT scores increase, which reveals that cognitive skills are associated with framing effects on ambiguity aversion. This relationship is not observed for ambiguity prudence or temperance.

We would like to finish by discussing some of the practical implications of our study for communicating with individuals about real-world hazards. When talking about damage (global warming, natural disasters, nuclear accidents, etc.), scientists and experts often announce the frequency of the damage, i.e., the probability associated with the bad state. Whenever such information is presented as a sure reduction (one favor) and a hazard (one harm), our experimental evidence suggests that the behavioral reaction to such communication is enhanced prudence. By contrast, in the case where information could be presented as a sure increase (one harm) and a hazard (one harm), it may be desirable to rephrase the presentation in an equivalent way by focusing on the good state of nature: sure decrease (one harm) and a hazard (one harm).

Appendix

English translation of the instructions

General instructions

This experiment involves 55 choices between two options involving amounts of money and chance. At the end of the experiment, 1 of your choices will be paid. One of you will now randomly draw a sealed envelope containing one of the choices (a number from 1 to 55). 18The envelope will be opened at the end of the experiment and the option that you have chosen in that particular choice will then be resolved and paid for real. If you wish, we can verify the content of the envelope at the end of the experiment. Each choice has an equal chance to be selected. As such, it is in your best interest to make each decision as if it was the one that will be chosen. On top of this payment, you will receive a show-up fee of 5 e, provided that you make all choices and complete a short questionnaire at the end of the experiment.

There will be two sets of choices. In the first set, you will have 15 choices and, in the second one, there will be 40 choices. We will wait for everybody to make the first 15 choices before proceeding to the next 40 choices; you might have to wait a while before the new set of 40 choices appears on your screen. We will provide the instructions corresponding to the 15 and 40 choices at the begging of each set. Before you start making your choices, we ask you to fill an understanding questionnaire about the tasks corresponding to each set. The 55 choices concern two options, called Option L (left) and Option R (right).

First set of 15 choices

In the first set of 15 choices, you have to decide between Option L and Option R, like in the examples19 below. After providing the examples, the payment process is explained in the event that the envelope drawn and signed at the start of the session corresponds to a choice from this first series.

Example 1

As a first (fictitious) example for choices 1 to 15, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. The example 1 presented on the screen, reads as follow.

For option L:

❼ The payoff is 10 e with probability 1 2 ; ❼ The payoff is 40 e with probability 1 2 .

For option R:

The payoff is 25 e for sure.

When you move the mouse over the disk on the screen, the probabilities associated with the payoffs are displayed.

Note that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In the example, the expected payoff is 25 ➾ for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the two options.

Example 2

As a second (fictitious) example for choices 1 to 15, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. ❼ The payoff on the gray area is 20 e with probability 1 2 ; ❼ The payoff on the white area is 20 e with probability 1 2 .

To the left of the intermediate size disk, two small gray disks represent two changes in the payoff of the gray area:

❼ a definite reduction of 10 e (small gray disk at the top);

❼ and a reduction of 10 e with probability 2 3 and an increase of 20 e with probability 1 3 (small gray disk at the bottom).

No small white disk appears to the right of the white payoff of 20 e, which means that this payoff is not modified.

The final result of this process is described by the large yellow disk. The probability of winning 20 e is 1 2 , the probability of winning 30 e is 1 6 , and the probability of winning 0 e is 1 3 . For option R: Let's consider the intermediate size disk at the top right. This disk represents the initial situation:

❼ the payoff on the gray area is 20 e with probability 1 2 ; ❼ the payoff on the white area is 20 e with probability 1 2 .

To the left of the intermediate size disk, a small gray disk represents a change in the payoff of the gray area: ❼ a reduction of 10 e with probability 2 3 and an increase of 20 e with probability 1 3 .

To the right of the intermediate size disk, a small white disk represents a change in the payoff of the white area: ❼ a definite reduction of 10 e.

The final result of this process is described by the large yellow disk. The probability of winning 10 e is 5 6 and the probability of winning 40 e is 1 6 . Remember that when you move the mouse over the large yellow disk, the probabilities associated with the payoffs are displayed.

There is a waiting time for the display of the final results (large yellow disk), so that you can get a good idea of how these final results are composed.

Recall that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In the example, the expected payoff is 15 e for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the two options.

Example 3

As a third and last (fictitious) example for choices 1 to 15, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. ❼ the payoff on the gray area is 30 e with probability 1 2 ; ❼ the payoff on the white area is 30 e with probability 1 2 .

To the left of the intermediate size disk, two small gray disks represent two changes in the payoff of the gray area: ❼ a reduction of 15 e with probability 1 2 and an increase of 15 e with probability 1 2 (small gray disk at the top); ❼ and a reduction of 15 e with probability 1 2 and an increase of 15 e with probability 1 2 (small gray disk at the bottom).

No small white disc appears to the right of the white payoff of 30 e, which means that this payoff is not modified.

The final result of this process is described by the large yellow disk. The probability of winning 30 e is 3 4 , the probability of winning 60 e is 1 8 and the probability of winning 0 e is 1 8 .

For option R: Let's consider the intermediate size disk at the top right. This disk represents the initial situation:

❼ the payoff on the gray area is 30 e with probability 1 2 ; ❼ the payoff on the white area is 30 e with probability 1 2 .

To the left of the intermediate size disk, a small gray disk represents a change in the payoff of the gray area: ❼ a reduction of 15 e with probability 1 2 and an increase of 15 e with probability 1 2 .

To the right of the intermediate size disk, a small white disk represents a change in the payoff of the white area:

❼ a reduction of 15 e with probability 1 2 and an increase of 15 e with probability 1 2 .

The final result of this process is described by the large yellow disk. The probability of winning 45 e is 1 2 and the probability of winning 15 e is 1 2 . Remember that when you move the mouse over the large yellow disk, the probabilities associated with the payoffs are displayed.

Also, there is a waiting time for the display of the final results (large yellow disk), so that you can get a good idea of how these final results are composed.

Recall that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In the example, the expected payoff is 30 e for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the two options. Envelopes for payoffs (set of lotteries from 1 to 15)

Let's consider again the example 3 for choices 1 to 15. We now explain the composition of the corresponding (fictitious) envelope to this choice.

As we previously explained in the screen, an envelope will be used to proceed with the payoff of the lottery. Its composition exactly follows the description of the lottery. In the case of example 3, the envelope contains the following.

Besides containing the number that identifies the lottery, the envelope contains two smaller envelopes, one for each option (L and R).

In the small envelope depicting the situation that is on the left of the screen (option L), there would be 8 tags, six indicating 30 e, one indicating 60 e, and 1 indicating 0 e.

In the other small envelope, the one depicting the situation that is on the right of the example screen (option R), there would be 2 tags, one indicating 45 e, and another one indicating 15 e.

So, the resulting probabilities of yielding a prize (owing to the draw from the envelope) precisely correspond to those reported on the example of the screen in both options.20

Second set of 40 choices

In the second set of 40 choices, you will be asked to make a choice between Option L (left) and Option R (right). For these 40 choices, there are always two types of payoffs: low payoff and high payoff. The probability of low payoff is always indicated in green, and the probability of high payoff is always indicated in blue.

After providing the examples, the payment process is explained in the event that the envelope drawn and signed at the start of the session corresponds to a choice from this second series.

Example 1

As a first (fictitious) example for choices 16 to 55, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. The example 1 presented on the screen, reads as follows.

For option L: Let's consider the small disk at the top right of the large disk. This small disk represents the initial situation:

❼ a low payoff (green color) of 10 e with a probability 1 2 ; ❼ a high payoff (blue color) of 40 e with a probability 1 2 .

+[-1 2 , + 1 2 ] means that the probability associated with the high payoff of 40 e, initially equal to 1 2 , becomes equal to 1 2 (initial probability) + [-1 2 , + 1 2 ]; Also, +[-1 2 , + 1 2 ] means that the value that will be added to modify the initial probability of 1 2 is randomly selected between -1 2 and 1 2 . Each value inside the interval has the same chance to be selected.

Note that +[-1 2 , + 1 2 ] is displayed in blue color to clearly show that it is the probability associated with the high payoff the one modified.

The final result of this process is described on the large disk at the bottom left. The probability of winning the high payoff of 40 e, is between 0 and 1. The fact that there is an equal change for each of the values between 0 and 1 to be selected, is represented by the blue color continuously coloring the disk.

For option R: Let's consider the small disk at the top right of the large disk. This small disk represents the initial situation: ❼ a low payoff (green color) of 10 e with a probability 1 2 ; ❼ a high payoff (blue color) of 40 e with a probability 1 2 .

This initial situation is not affected. The result of the final situation, represented by the large disk, is identical to that of the initial situation.

When you move the mouse over the small disks, the initial probabilities associated with the payoffs are displayed.

There is a waiting time for the display of the final results (large disk), so that you can get a good idea of how these final results are composed.

Note that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In the example, the expected payoff is 25 e for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the options.

Example 2

As a second (fictitious) example for choices 16 to 55, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. The example 2 presented on the screen, reads as follows.

For option L: The 2 branches, 0 or 100%, mean that there is the same probability of being in the scenario described by the top branch of the screen or in the scenario described by the bottom branch of the screen.

Let's first consider the scenario described by the branch at the top of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 0 e with a probability 1 2 ; ❼ a high payoff (blue color) of 45 e with a probability of 1 2 .

There is not value written next to the small disk. Then, the initial situation is not affected. The initial probability of winning the low payoff of 0 e remains equal to 1 2 . The result of the final situation, represented by the large disk, is identical to the initial situation, Let's now consider the scenario described by the branch at the bottom of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 0 e with a probability 1 2 ; ❼ a high payoff (blue color) of 45 e with a probability of + 1 4 .

Note that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In this example, the expected payoff is 16.8 e for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the options. Example 3

As a third (fictitious) example for choices 16 to 55, let's consider the choice between the two options depicted below. The explanation related to this example are presented in the instructions on paper. The example 3 presented on the screen, reads as follows.

For option L: The 2 branches, 0 or 100%, mean that there is the same probability of being in the scenario described by the top branch of the screen or in the scenario described by the bottom branch of the screen.

Let's first consider the scenario described by the branch at the top of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 10 e with probability 2 3 ; ❼ a high payoff (blue color) of 40 e with probability of 1 3 .

There is not value written next to the small disk. Then, the initial situation is not affected. The initial probability of winning the high payoff of 40 e remains equal to 1 3 . The result of the final situation, represented by the large disk, is identical to that of the initial situation.

Let's now consider the scenario described by the branch at the bottom of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 10 e with probability 2 3 ; ❼ a high payoff (blue color) of 40 e with probability of 1 3 .

+[-1 6 , + 1 6 ] and +[-1 6 , + 1 6 ] mean that the probability associated with the high payoff of 40 e, initially equal to 1 3 , becomes equal to 1 3 (initial probability) + [-1 6 , + 1 6 ] + [-1 6 , + 1 6 ]. The probability undergoes two random modifications (of the same type): +[ 1 6 , + 1 6 ]): The first modification, coming from one of the intervals +[-1 6 , + 1 6 ], means that the value that will be added to modify the initial probability of 1 3 , is randomly selected between -1 6 and 1 6 . Each value inside the interval has the same chance to be selected. In line with previous literature (see [START_REF] Eeckhoudt | The economics of risk: A (partial) survey. Handbook of Risk Theory: Epistemology, Decision Theory, Ethics, and Social Implications of Risk[END_REF], for a review), subjects are generally found to be risk averse. On average, subjects choose the risk averse option 4.28 out of 5 times, which is significantly different from the average that would be observed if subjects chose randomly (p < 0.001). Additionally, the observed distribution of risk averse choices is significantly different from what would be observed if subjects chose randomly (χ 2 test, p < 0.001).

Order 3: Prudence

The distribution of risk prudent choices made by the subjects in the experiment is presented in Figure 15. In line with previous research [START_REF] Deck | Consistency of higher order risk preferences[END_REF][START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF][START_REF] Attema | Measuring multivariate risk preferences in the health domain[END_REF][START_REF] Haering | Exploring the consistency of higher order risk preferences[END_REF], we find that subjects are risk prudent. On average, subjects choose the risk prudent option 3.88 out of 5 times, which is significantly different from the average that would be observed if subjects chose randomly (p < 0.001). In addition, the observed distribution of risk prudent choices is significantly different from what would be observed if subjects chose randomly (χ 2 test, p < 0.001).

Figure 1 :

 1 Figure 1: Task 11

Figure 2 :

 2 Figure 2: Task 31

Figure 3 :

 3 Figure 3: Ambiguity of order 2: distribution of individual scores

Figure 4 :Result 3 .

 43 Figure 4: Ambiguity of order 3: distribution of individual scores

Figure 5 :

 5 Figure 5: Ambiguity of order 4: distribution of individual scores

Figure 6 :

 6 Figure 6: Distribution of ambiguity prudent choices with one harm and one favor

Figure 7 :

 7 Figure 7: Image example 1

Figure 8 :

 8 Figure 8: Image example 2

Figure 9 :

 9 Figure 9: Image example 3

Figure 10 :

 10 Figure 10: Image example 1

Figure 11 :

 11 Figure 11: Image example 2

Figure 12 :

 12 Figure 12: Image example 3

  Figure14presents the distribution of risk averse choices subjects made in the experiment. In line with previous literature (see[START_REF] Eeckhoudt | The economics of risk: A (partial) survey. Handbook of Risk Theory: Epistemology, Decision Theory, Ethics, and Social Implications of Risk[END_REF], for a review), subjects are generally found to be risk averse. On average, subjects choose the risk averse option 4.28 out of 5 times, which is significantly different from the average that would be observed if subjects chose randomly (p < 0.001). Additionally, the observed distribution of risk averse choices is significantly different from what would be observed if subjects chose randomly (χ 2 test, p < 0.001).

Figure 14 :

 14 Figure 14: Number of times the risk averse option is chosen

Figure 15 :

 15 Figure 15: Number of times the risk prudent option is chosen

  Figure16displays the distribution of risk temperate choices. As the figure shows, we observe risk temperance. On average, subjects chose the risk temperate option 2.77 out of 5 times, which is significantly above the average that would be observed if subjects make random choices (p = 0.005). Also, the observed distribution of risk temperate choices is significantly different from what would be observed if subjects chose randomly (χ 2 test, p < 0.001). We note that the existing evidence regarding risk temperance is mixed. Our results are in line with[START_REF] Heinrich | Higher-order risk preferences in social settings[END_REF] who find evidence for risk temperance. Contrary,[START_REF] Deck | Consistency of higher order risk preferences[END_REF] and[START_REF] Baillon | Measuring higher order ambiguity preferences[END_REF] find risk intemperance.

Figure 16 :

 16 Figure 16: Number of times the risk temperate option is chosen

Table 1 :

 1 • to 360 • to illustrate ambiguity. 11 Choice tasks

	Task	Dom	Ord	Option A	Option B
		Risk		(p : R 1 , R 2 )	R
	1	Risk		(1/2 : 30, 0)	15
	2	Risk		(1/2 : 45, 15) = 1A + 15	1B + 15
	3	Risk		(1/2 : 45, 0) = 1A × 1.5	1B × 1.5
	4	Risk		(1/3 : 30, 0)	10
	5	Risk		(2/3 : 30, 0)	20
		Risk		(p : R, R -k + ε)	(p : R + ε, R -k)
	6	Risk		(1/2 : 15, 15 -7.5 + [-7.5, +7.5])	(1/2 : 15 + [-7.5, +7.5], 15 -7.5)
	7	Risk		6A + 15 = (1/2 : 30, 30 -7.5 + [-7.5, +7.5])	6B + 15 = (1/2 : 30 + [-7.5, +7.5], 30 -7.5)
	8	Risk		6A × 2 = (1/2 : 30, 30 -15 + [-15, +15])	6B × 2 = (1/2 : 30 + [-15, +15], 30 -15)
	9	Risk		(1/2 : 10, 10 -5 + (1/3 : 10, -5))	(1/2 : 10 + (1/3 : 10, -5) , 10 -5)
	10	Risk		(1/2 : 25, 25 -15 + (2/3 : 5, -10))	(1/2 : 25 + (2/3 : 5, -10) , 25 -15)
		Risk		(p : R, R + ε1 + ε2 )	(p : R + ε1 , R + ε2 )
	11	Risk		(1/2 : 15, 15 + [7.5, -7.5] + [7.5, -7.5])	(1/2 : 15 + [7.5, -7.5], 15 + [7.5, -7.5])
	12	Risk		11A + 15 = (1/2 : 30, 30 + [7.5, -7.5] + [7.5, -7.5])	11B + 15 = (1/2 : 30 + [7.5, -7.5], 30 + [7.5, -7.5])
	13	Risk		11A × 3 2 = (1/2 : 22.5, 22.5 + [11.25, -11.25] + [11.25, -11.25])	11B × 3 2 = (1/2 : 22.5 + [11.25, -11.25], 22.5 + [11.25, -11.25])
	14	Risk		(1/2 : 10, 10 + (1/3 : 10, -5) + (1/3 : 10, -5))	(1/2 : 10 + (1/3 : 10, -5) , 10 + (1/3 : 10, -5))
	15	Risk		(1/2 : 20, 20 + (2/3 : 5, -10) + (2/3 : 5, -10))	(1/2 : 20 + (2/3 : 5, -10) , 20 + (2/3 : 5, -10))
		Amb		p + ε : R, R	p : R, R
	16	Amb		(1/2 + [-1/2, +1/2] : 30, 0)	1A
	17	Amb		16A + 15 = (1/2 + [-1/2, +1/2] : 45, 15)	2A
	18	Amb		(1/2 + {-1/2, +1/2} : 45, 0)	3A
	19	Amb		(1/3 + [-1/3, +1/3] : 30, 0)	4A
	20	Amb		(2/3 + [-1/3, +1/3] : 30, 0)	5A
		Amb		{p, p -k + ε} : R, R	{p -k, p + ε} : R, R
	21	Amb		({1/2, 1/2 -1/4 + [-1/4, +1/4]} : 30, 0)	({1/2 -1/4, 1/2 + [-1/4, +1/4]} : 30, 0)
	22	Amb		21A + 15	21B + 15
	23	Amb		21A × 1.5	21B × 1.5
	24	Amb		({1/3, 1/3 -1/6 + [-1/6, +1/6]} : 30, 0)	({1/3 -1/6, 1/3 + [-1/6, +1/6]} : 30, 0)
	25	Amb		({2/3, 2/3 -1/6 + [-1/6, +1/6]} : 30, 0)	({2/3 -1/6, 2/3 + [-1/6, +1/6]} : 30, 0)
		Amb		{p, p + ε1 + ε2 } : R, R	{p + ε1 , p + ε2 } : R, R
	26	Amb		({1/2, 1/2 + [-1/8, 1/8] + [-1/8, 1/8]} : 30, 0)	({1/2 + [-1/8, 1/8], 1/2 + [-1/8, 1/8]} : 30, 0)
	27	Amb		26A + 15	26B + 15
	28

Table 2 :

 2 Attitudes towards ambiguity and cognitive skills: bad state vs. good state

	CRT score	0 (N = 28) 1 (N = 37) 2 (N = 55) 3 (N = 89)	p
			Aversion
	BS (lotteries 31-35)	3.143	3.000	2.564	2.787	0.429
	GS (lotteries 16-20)	1.893	2.486	2.455	2.854	0.036
	p	<0.001	0.022	0.551	0.611
			Prudence
	BS (lotteries 36-40)	2.036	2.432	2.073	2.483	0.296
	GS (lotteries 21-25)	2.179	2.595	2.273	2.697	0.278
	p	0.576	0.511	0.296	0.172
			Temperance
	BS (lotteries 26-30)	2.286	2.784	2.909	2.685	0.416
	GS (lotteries 41-46)	2.286	2.216	2.345	2.584	0.580
	p	1.000	0.025	0.008	0.474
		Prudence (one harm and one favor)
	BS (lotteries 46-50)	2.321	2.703	2.473	2.753	0.376
	GS (lotteries 51-55)	2.643	3	2.600	3.045	0.545
	p	0.246	0.241	0.492	0.056

Note. For each attitude and each state, figures represent the aggregate mean number of observed choices (ranging between 0 and 5). p-values in the last column (last row of each sub-table) come from Poisson regression χ 2 test (z-test) comparing the distribution of conditional means across CRT scores for a given state (between states for a given CRT score with individual-level clustered SE).

Table 3 :

 3 Attitudes towards ambiguity and cognitive skills: Poisson regression models

	Model: dep. var.	M1: Aversion	M2: Prudence		M3: Temperance		M4: Prudence harm+favor
	Exp. var. (coef.)	estimate	SE	p	estimate	SE	p	estimate	SE	p	estimate	SE	p
	Intercept (α 0 )	0.598	0.222	0.007	0.586	0.230 0.011	0.652	0.185 0.000	0.891	0.179	0.000
	1[GS] (α 1 )	-0.507	0.133	0.000	0.068	0.119 0.567	0.000	0.161 1.000	0.130	0.109	0.236
	1[CRT = 1] (α 2 )	-0.127	0.108	0.240	0.211	0.150 0.160	0.184	0.135 0.174	0.155	0.126	0.219
	1[CRT = 2] (α 3 )	-0.235	0.110	0.033	0.033	0.148 0.822	0.227	0.122 0.062	0.055	0.113	0.626
	1[CRT = 3] (α 4 )	-0.125	0.095	0.187	0.241	0.138 0.082	0.194	0.122 0.111	0.199	0.111	0.073
	1[GS] × 1[CRT = 1] (α 5 )	0.319	0.158	0.043	-0.003	0.153 0.983	-0.228	0.191 0.234 -0.025	0.141	0.857
	1[GS] × 1[CRT = 2] (α 6 )	0.464	0.152	0.002	0.024	0.147 0.869	-0.215	0.181 0.233	-0.079	0.131	0.544
	1[GS] × 1[CRT = 3] (α 7 )	0.531	0.141	0.000	0.015	0.133 0.912	-0.038	0.170 0.821	-0.029	0.121	0.812
	Additional controls
	Risk aversion (α 8 )	0.088	0.038	0.020	0.002	0.035 0.954	0.052	0.027 0.057	0.001	0.027	0.982
	Risk prudence (α 9 )	0.000	0.028	0.990	0.026	0.030 0.403	-0.011	0.024 0.646 -0.023	0.020	0.263
	Risk temperance (α 10 )	0.038	0.025	0.123	0.020	0.027 0.449	0.002	0.022 0.945	0.015	0.019	0.442
	1[M ale] (α 11 )	0.023	0.073	0.756	-0.055	0.078 0.479	-0.078	0.065 0.229	-0.069	0.057	0.227
	1[EconF inance] (α 12 )	0.150	0.069	0.029	-0.087	0.077 0.258	0.057	0.064 0.373	0.049	0.055	0.370
	Additional coefficient tests
	H T est1 0 H T est2 0 H T est3 0 H T est4 0 H T est5 0 Pseudo-R 2 : α 2 = α 3 = α 4 = 0 : α 5 = α 6 = α 7 = 0 : α 1 + α 5 = 0 : α 1 + α 6 = 0 : α 1 + α 7 = 0			0.202 0.001 0.027 0.549 0.610 0.024		0.131 0.996 0.504 0.288 0.173 0.011		0.316 0.155 0.027 0.008 0.470 0.010			0.158 0.918 0.511 0.364 0.481 0.008
	P rob > χ 2			<0.001		0.130		0.008			0.040

While the hazard affects the revenue under risk, under ambiguity, it affects the probability distribution of the states of nature.

In the literature where the damage is non-pecuniary, for example in the Value of a Statistical Life (VSL) literature[START_REF] Dreze | The social utility of a human life[END_REF], the announced probability is the probability of death, and in models of irreplaceable commodity[START_REF] Cook | The demand for insurance and protection: The case of irreplaceable commodities[END_REF], the announced probability is the probability of commodity loss. In the literature where the damage is pecuniary, the announced probability is the probability of the monetary loss.

Following standard nomenclature, an outcome, whether random or not, is called a favor (harm) if it is strictly preferred to (dominated by) zero.

We assume 0 < p + ξ < 1 for all realization of the random variable ξ.

We assume 0 < 1p + ξ BS < 1 for all realizations of the random variable ξBS .

We consider this part of the study as exploratory. Including other orders would substantially increase the duration of the experiment making it lengthy and tedious for the participants which, as a consequence, could likely be detrimental to data quality.

The design and behavioral conjectures have been pre-registered at AsPredicted (#78997). An anonymized version of the AsPredicted pre-registration is available at the following URL: https://aspredicted.org/RK6 4 58.

Our definition criteria of outliers is specified in the preregistration.

The full content of the instructions can be found in Appendix 7.1.

The arrow is included in this paper for illustration purposes. However, it is not part of the screen in the experiment. Instead, the colors move automatically. A video recording of the examples of ambiguity tasks referred to in the instructions is provided at: https://page.hn/w61e8e.

The implementation of the moving proportions in the circles echoes[START_REF] Garcia | Ambiguity and excuse-driven behavior in charitable giving[END_REF].

This result holds for all the distributions presented in the remaining figures referenced in this section: Figure4, Figure5, as well as Figure13inAppendix 7.3.

Note that these ambiguity neutral preferences disappear once we discard task 19, in which the probability of winning was in the interval [0, 2/3] and most subjects (138 out of 209) prefer the ambiguity seeking option. This stands in line with the previous literature[START_REF] Trautmann | Ambiguity attitudes. The Wiley Blackwell Handbook of Judgment and Decision Making[END_REF] point to ambiguity seeking when facing lower likelihoods. As shown in Figure13in Appendix 7.3, the mean subject chooses the ambiguity averse option 2.22 out of 4 times, which is significantly different from the theoretical benchmark of neutrality of 2 (p = 0.028). See[START_REF] Oechssler | A test of mechanical ambiguity[END_REF] for a review of the literature on ambiguity aversion.

This hypothesis is tested through the following conditional mean specification: µ i = e β0+β1×1i[GS] where 1 i [GS] = 1 for GS and = 0 for BS. Data are pooled yielding two scores (BS and GS) per i which requires individual-level SE clustering. Then, the significance (z-)test of interest refers to the average marginal effect (AME, here equal to -0.258) of 1 i [GS] which captures the mean difference between GS and BS.

For a meaningful discussion on the capacity of CRT to predict a wide range of economic behaviors, see Brañas-Garza et al. (2019).

One of the subjects is randomly selected to pick one of the envelopes and sign it.

The written explanation of the examples was given in paper.

At the end of this part of the instructions, subjects proceed with an understanding questionnaire and perform the 15 choices corresponding to this type of lotteries.

Marie Claire Villeval, as well as participants to the 2022 ASFEE conference. We also thank Quentin Thévenet for valuable technical support. This research has benefited from the financial supports of the French National Research Agency (ANR) under the French-German project "ASUR" (ANR-18-FRAL-0 0 08-01 ASUR), the IDEXLYON from Université de Lyon (project INDEPTH) within the Programme Investissements d'Avenir (ANR-16-IDEX-0005) and it was performed within the framework of the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-007) operated by the French National Research Agency. Adam Zylbersztejn acknowledges VISTULA Fellowship.

+ 1 4 and [-1 4 , + 1 4 ] mean that the probability associated with the low payoff of 0 e, initially equal to 1 2 , becomes equal to 1 2 (initial probability) + 1 4 + [-1 4 , + 1 4 ]. The probability undergoes two modifications:

+ 1 4 means that the value of the initial probability is increased by 1 4 with certainty; [-1 4 , + 1 4 ] means that the value that will be added to modify the initial probability of 1 2 is randomly selected between -1 4 and + 1 4 . Each value inside of the interval has the same chance to be selected.

Note that + 1 4 and +[-1 4 , + 1 4 ] are displayed in green color to clearly show that it is the probability associated with the low payoff that is the one modified.

The final result of this process is described on the large disk at the bottom left. The probability of winning the low payoff 0 e, is between 1 2 and 1. The fact that there is an equal chance for each of the values between 1 2 and 1 to be selected, is represented by the green color continuously coloring the disk.

For option R:

The 2 branches, 0 or 100%, mean that there is the same probability of being in the scenario described by the top branch of the screen or in the scenario described by the bottom branch of the screen.

Let's first consider the scenario described by the branch at the top of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 0 e with probability 1 2 ; ❼ a high payoff (blue color) of 45 e with probability of 1 2 .

+ 1 4 means that the probability associated with the low payoff of 0 e, initially equal to 1 2 , becomes 1 2 (initial probability) + 1 4 . Note that + 1 4 is displayed in green color to clearly show that it is the probability associated with the low payoff the one modified.

The final result of this process is described on the large disk at the bottom left: the probability associated with the low payoff of 0 e is equal to 3 4 . Let's now consider the scenario described by the branch at the bottom of the screen. The small disk represents the initial situation: ❼ a low payoff (green color) of 0 e with probability 1 2 ; ❼ a high payoff (blue color) of 45 e with probability 1 2 .

+[-1 4 , + 1 4 ] means that the probability associated with the low payoff of 0 e, initially equal to Also, [-1 4 , + 1 4 ] means that the value that will be added to modify the initial probability is randomly selected between -1 4 and 1 4 . Each value inside the interval has the same chance to be selected.

Note that +[-1 4 , + 1 4 ] is displayed in green color to clearly show that it is the probability associated with the low payoff the one modified.

The final result of this process is described on the large disk at the bottom left. The probability of winning the low payoff of 0 e is between 3 4 and 1. The fact that there is an equal chance that each of the values between 1 4 and 3 4 is selected is represented by the fact that the green color is continuously coloring the disk.

Remember that when you move the mouse over the small disks, the initial probabilities associated with the payoffs are displayed.

Also, there is a waiting time for the display of the final results (large disks), so that you can get a good idea of how these final results are composed.

The second modification, coming from the other interval +[-1 6 , + 1 6 ], means that the value that will be added to modify the probability that has become random and equal to [ 1 6 , + 1 2 ] because of the first modification explained above, is randomly selected between -1 6 and 1 6 . Each value inside the interval has the same chance to be selected;

Note that +[-1 6 , + 1 6 ] and +[-1 6 , + 1 6 ] are displayed in blue color to clearly show that it is the probability associated with the high payoff that is the one modified.

The final result of this process is described on the large disk at the bottom left. The probability of winning the high payoff of 40 e, is between 0 and 2 3 . The fact that there is an equal chance for each of the values between 0 and 2 3 to be selected, is represented by the blue color continuously coloring the disk.

For option R:

The 2 branches, 0 or 100%, mean that there is the same probability of being in the scenario described by the top branch of the screen or in the scenario described by the bottom branch of the screen.

Let's first consider the scenario described by the branch at the top of the screen. The small disk represents the initial situation:

❼ a low payoff (green color) of 10 e with probability 2 3 ; ❼ a high payoff (blue color) of 40 e with probability of 1 3 .

+[-1 6 , + 1 6 ] means that the probability associated with the high payoff of 40 e, initially equal to 1 3 , becomes 1 3 (initial probability) + [-1 6 , + 1 6 ]. Also, +[-1 6 , + 1 6 ] means that the value that will be added to modify the initial probability is randomly selected between -1 6 and 1 6 . Each value inside the interval has the same chance to be selected.

Note that +[-1 6 , + 1 6 ] is displayed in blue color to clearly show that the probability associated with the high payoff is the one modified.

The final result of this process is described on the large disk at the bottom left. The probability of winning the high payoff of 40 e, is between 1 6 and 1 2 . The fact that there is an equal chance that each of the values between 1 6 and 1 2 is selected is represented by the blue color continuously coloring the disk.

Let's now consider the scenario described by the branch at the bottom of the screen. It is identical to the one described by the top branch. Remember that when you move the mouse over the small disks, the initial probabilities associated with the payoffs are displayed.

Also, there is a waiting time for the display of the final results (large disks), so that you can get a good idea of how these final results are composed.

Note that in each choice, the expected payoff (i.e. the amount you would earn on average if you selected the same option over a large number of times) of both options is identical. In this example, the expected payoff is 20 e for both options. However, the potential payoffs, and the chances to win these payoffs, differ between the options. Envelopes for payoffs (set of lotteries from 16 to 55)

Let's consider again the example 3 for choices 16 to 55. We now explain the composition of the corresponding (fictitious) envelope to this choice.

As we previously explained in the screen, an envelope will be used to proceed with the payoff of the lottery. Its composition exactly follows the description of the lottery. In the case of example 3, the envelope contains the following.

Besides containing the number that identifies the lottery, the envelope contains two smaller envelopes, one for each option (L and R).

Inside of the envelope depicting option L, there would be 2 smaller envelopes. One of these, would depict the top branch of option L and the other envelope would depict the bottom branch of option L. Each of these envelopes would contain the following.

❼ The envelope depicting the top branch: there would be three tags, 2 indicating 10 e and 1 indicating 40 e.

❼ The envelope depicting the bottom branch: there would be five smaller envelopes, one for each probability value 0, 1 6 , 2 6 , 3 6 , and 4 6 . Each of these five envelopes would be marked with a tag indicating the randomly drawn probability value and have inside tags indicating the value of the payoff. For examples, if the chosen envelope is marked with a probability value 1 6 , there would be inside six tags, 5 tags indicating 0 e and 1 tag indicating 40 e.

Inside the envelope depicting option R, there would be also 2 smaller envelopes. One would depict the top branch of option R and the other one would depict the bottom branch of option R. The content of each of these 2 envelopes is the following.

❼ The envelope depicting the top branch: there would be five smaller envelopes, one for each probability value 2 12 , 3 12 , 4 12 , 5 12 , 6 12 . Each of these five envelopes would be marked with a tag indicating the randomly drawn probability value and have inside tags indicating the value of the payoff. For example, if the chosen envelope is marked with a probability value 4 12 , which is equal to 1 3 , the envelope would have three tags inside, 2 tags indicating 0 e and 1 tag indicating 40 e.

❼ The envelope depicting the bottom branch: the content would be identical to the one of envelope depicting the top branch.

So, the resulting probabilities of yielding a prize (owing to the draw from the envelope) precisely correspond to those reported on the example of the screen in both options.

Additional invidual measurements 7.2.1 Cognitive Reflection Test (CRT)

We use standard three-item CRT [START_REF] Frederick | Cognitive reflection and decision making[END_REF]. Our measure of cognitive reflection is given by the total number of correct answers (from 0 to 3) to the following set of questions.

1. A bar and a ball cost 1.10 e in total. The bat costs an euro more than the ball. How much does the ball cost? [Correct answer: 5 cents; intuitive answer: 10 cents]

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? [Correct answer: 5 minutes; intuitive answer: 100 minutes]

3. In a lake, there is a patch of lily pads. Every day, the patch double in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? [Correct answer: 47 days; intuitive answer: 24 days] 7.2.2 General Risk Propensity Scale (GRiPS)

We administer the GRiPS developed and validated by [START_REF] Zhang | Development and validation of the general risk propensity scale (grips)[END_REF]. This test involves several statements with a 5-level Likert scale ranging from "strongly disagree" (1) to "strongly agree" (5):

1. Taking risks makes life more fun.