Keywords: Array-based clustering algorithm, distance measure, integer mathematical programming, task assignment, team building

A team building approach for competency development.

I. INTRODUCTION

The concept of multidisciplinary team building is one of the key aspects of problem solving in many product development projects. Specialists from various disciplines (e.g. micro robotics, electronics, thermodynamics, manufacturing, mechanics, image processing, optics) are gathered to develop a new product. Team building is a crucial issue in project management. Design teams are multidisciplinary groups composed of members representing many engineering disciplines. Three main elements have to be defined: task, team member and attribute. Design tasks correspond to a group of tasks to be performed in a design project. Team members from different departments possess different competencies, characterized by a set of attributes required in performing the tasks. Attributes can be viewed as soft competency or technical competency. Fig. 1 gives the global overview of our team building approach.

Furthermore, for the task assignment, we suppose that the competencies of an individual increase during task realization and can decrease with time. Therefore, task assignment implies the increase in the competency development of an individual. So, the problem of preserving an intra-domain expert is considered here in order to maintain a team member in his or her expert domain. The literature about team building does not provide the solution reinforcing the intra-domain expert competencies, or dealing with the competency dynamics.

II. REVIEW OF RELATED LITERATURE

Numerous works about team building deal with psychological and sociological competencies (personality types, leadership, communication skills, decision-making ability etc.) [START_REF] Fitzpatrick | Forming effective worker teams with multi-functional skill requirements[END_REF], [START_REF] Chen | Modeling team member characteristics for the formation of a multifunctional team[END_REF], [START_REF] Acuna | Assigning people to roles in software projects[END_REF]. Technical competency is the most common attribute found in team building literature in order to characterize tasks and team members [START_REF] Fitzpatrick | Forming effective worker teams with multi-functional skill requirements[END_REF], [START_REF] Zakarian | Forming teams: an analytical approach[END_REF], [START_REF] Tseng | Novel approach to multi-functional project team formation[END_REF], [START_REF] Hadj-Hamou | Cooperative design : A framework for a competency-based approach[END_REF], [START_REF] Tsai | Human resource selection for software development projects using Taguchi's parameter design[END_REF]. De Korvin et al. [START_REF] De Korvin | Utilizing fuzzy compatibility of skill sets for team selection in multi-phase projects[END_REF], emphasized that the important factor in selecting human resources to a team are the technical competencies. These competencies will be required to implement the various activities for each phase of the project. Moreover, competencies can be considered from two points of view. First, a competency is required to carry out a task and second, a competency is possessed by one (or several) team member(s).

Boucher et al. [START_REF] Boucher | Formalisation and use of competencies for industrial performance optimisation: A survey[END_REF] indicated that competencies can be seen from three distinct views: static, functional and evolutionary. This paper focuses on the functional point of view. This concerns the mechanisms of competency mobilization in a work context where the goal is to make an efficient use of available competencies.

There are various solutions to be found in the research literature. Chen and Lin [START_REF] Chen | Modeling team member characteristics for the formation of a multifunctional team[END_REF], [START_REF] Chen | An Integrated Methodological Framework for Project Task Coordination and Team Organization in Concurrent Engineering[END_REF] proposed an integrated methodological framework in team member assignment. The objective of their work was to develop a framework for project task coordination and team organization from the concurrent engineering perspective in order to assign the right tasks to the right team members. In other words, it is important to know the competencies and performance levels of a team member possesses before being able to allocate tasks to him or her. Zarakian and Kusiak [START_REF] Zakarian | Forming teams: an analytical approach[END_REF] emphasized the importance of multifunctional teams in product development. The proposed method is based on QFD and AHP method. De Korvin et al. [START_REF] De Korvin | Utilizing fuzzy compatibility of skill sets for team selection in multi-phase projects[END_REF] developed a personnel selection model for a multiple phase project. The "fuzzy compatibility" method has been used to select the potential team A team building approach for competency development O. Hlaoittinun, E. Bonjour, M. Dulmet Laboratoire d'Automatique de Besançon, Université de Franche-Comté, Besançon, France members for each project phase. Braha [START_REF] Braha | Partitioning tasks to product development teams[END_REF] presented a mathematical formulation for the problem. Two main issues are addressed by the model: 1) how to specify task dependencies, and 2) how to optimally partition the tasks between a number of teams. Gronau et al. [START_REF] Gronau | Approach for requirement oriented team building in industrial processes[END_REF] developed an algorithm to propose a team composition for a specific task by analyzing the knowledge and skills of the employees. This method is based on the Knowledge Modelling and Description Language (KMDL).

The clustering method presented in this paper is based on one of the array-based clustering algorithms: ROC (Rank Order Clustering). A common characteristic of this algorithm is that all the methods consecutively reorder the rows and columns according to an index until the diagonal blocks are formed. The array-based clustering approach is one of the group technology algorithms. Originally, this approach was applied in manufacturing cell formation in order to group similar part families on dedicated clusters of machines. Examples of array-based clustering algorithms are as follows: ROC (Rank Order Clustering) [START_REF] King | Machine-Component Group Formation in Group Technology[END_REF], ROC2 [START_REF] King | An interactive dataclustering algorithm[END_REF], DCA (Direct Cluster Algorithm) [START_REF] Chan | Direct clustering algorithm for group formation in cellular manufacture[END_REF], BEA (Bond-Energy Algorithm) [START_REF] Mccormick | Problem decomposition and data reorganization by a clustering technique[END_REF], [START_REF] Mak | An adaptive genetic algorithm for manufacturing cell formation[END_REF]. Chu and Tsai [START_REF] Chu | A comparison of three arraybased clustering techniques for manufacturing cell formation[END_REF] outlined a procedure for evaluating alternatives in clustering algorithms under different measuring criteria.

This paper is composed of two parts. First, we will explain our approach in team building. Second, we will illustrate our approach through a simple example.

III. TEAM BUILDING APPROACH

Our approach comprises three main steps (Fig. 2). First, we explain how to generate a task/team member incidence matrix. Second, we present an algorithm in order to group task/team members into families using an array-based clustering algorithm. Third, we show how to solve the task assignment problem by means of an integer linear programming model. The model is based on the attributes of engineering characteristics (task/ team member incidence matrix).

1) Generate a task/ team member incidence matrix: The approach given here is based on incidence matrices. An incidence matrix is a matrix that shows the relationships between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. Below are the five sub-steps to this method: Fig. 2. Multidisciplinary team building approach -Identify relevant attributes to characterize both task and team member. (In this paper, we consider attributes as technical competencies or disciplines). -For each task, evaluate the demanded performance for each attribute (for instance, we obtain the taskdiscipline matrix (Table I)) -For each team member, evaluate the performance level for each technical competency. (for instance, we obtain the team member-discipline matrix (Table II)) -Define a compatibility indicator between task and team member (matching process). -Generate a task/ team member incidence matrix. (see the result in Table III, we obtain the task / team member matrix)

The compatibility indicator (similarity measure) is one of the methods often used in team selection. This method finds the shortest distance between two skill sets so as to find the appropriate candidate. Numerous distance measures have been mentioned in research literature -for example, the Hamming distance used in the personnel selection [START_REF] Canos | Some fuzzy models for human resource management[END_REF]. What is missing in these existing measure methods is that they haven't taken into account the difference between the positive and negative gap values. The positive gap value can be viewed as an overcompetency level, and the negative gap value as an undercompetency level. Therefore, we propose a compatibility indicator as a distance measure that will be helpful to find the best correlation value between two sets of disciplines (task and team member).

Let T, D and P denote ordinary non empty sets. Let R 1 be a relation from T to D and let R 2 be a relation from D to P. Then (R 1 ° R 2 ) is a relation from T to P. Let us denote T={t 1,…, t p }, D={ d 1,…, d q }, P={p 1,…, p r }, ( , ) , ( , )

k l k l l m l m R t d R R d p R = = .

<

indicates that the acquired discipline level of a team member is higher than the task requirement, whereas undercompetency )

kl lm R R >
indicates that it is lower. For our approach, we consider only the undercompetency case and propose the following function to calculate a compatibility indicator.

1 2 1 1 1 2 1 max(0, ) ( )( , ) 1 kl lm kl q l k m q l R R R R R t p = = °= - - ∑ ∑ (1)
Composition value (R 1 ° R 2 ) can be interpreted as the strength indicator of such a relational chain. It can be demonstrated that this value is included between 0 and 1.

2) Identify task/ team member families by clustering method: The ROC clustering algorithm will transform the task/team member incidence matrix into task/team member families. Algorithm principle:

Step 1: For each row of the task/team member matrix, calculate the decimal weight.

Step 2: Sort rows of the matrix in decreasing order of the corresponding decimal weights.

Step 3: Repeat the preceding two steps, for each column.

Step 4: Repeat the preceding three steps until the position of each element in each row and column does not change.

A weight for each row k and column m is calculated as follows, where r is the number of team members and p is the number of tasks.

Weight for row k:

∑ = - r m m r km R 1 2 (2) Weight for column m: ∑ = - p k k p km R 1 2 (3)
This array-based clustering method is simple to apply to task/team member matrices. However, it requires visual inspection of the output to determine the composition of the diagonal block formation. We adopt here the Group Density Index proposed by [START_REF] Tseng | Novel approach to multi-functional project team formation[END_REF]. This index will be used to identify the potential groups after using the ROC algorithm. The variables ' p and ' r refer respectively to the number of rows and the number of columns in potential groups. The following function will be used to calculate the GDI index:

3)

Assign tasks to team members: The task assignment contains a group of tasks (The mission requirement) to be accomplished. The mission requirement is defined here as a set of tasks { p t t ...

IV. ILLUSTRATIVE EXAMPLE 1) Problem Formalization:

A design project with seven tasks requiring ten disciplines is used as an illustrative example. The design department has eleven members; and each member knows a set of different disciplines and related performance levels. We may consider a set of tasks, a set of team members, and a set of disciplines. The value 1 kl R is defined as the demanded performance level of the discipline required by the task. The value 2 lm R is defined as the performance level of the discipline possessed by the team member. The compatibility indicator can be interpreted as the performance level of a team member (p) to achieve a task (t). These values are calculated from [START_REF] Fitzpatrick | Forming effective worker teams with multi-functional skill requirements[END_REF].

T3 0 0 0 0 0,9 0 0 0 1 0 T4 1 0 0,1 0 0 0 0 0,7 0 0 T5 0 0,8 0 1 0 0 0,4 0 0 0,1 T6 0,2 0,3 0 0 0 1 0 0,8 0 0 T7 0 0 0,1 0,1 1 0 0 0 1 0,7
From a task/team member incidence matrix, team members and tasks will be grouped in diagonal blocks by the ROC algorithm. The algorithm furnishes three blocks separately: the first block associates five team members and three tasks; the second associates three team members and two tasks, and the third associates three team members and two tasks.

3) Task assignment: Considering the assignment of the team members as a binary variable, the team can be selected by minimizing the project cost. The objective is to find one suitable candidate for each task which minimizes the total salary cost of the project.

To consider the minimization of salary cost, a simple objective function is defined. The cost function depends on the individual salary multiplied by the compatibility indicator. For our hypothesis, we suppose that the lower the compatibility indicator, the longer the duration of the execution of a task-as long as the compatibility indicator is not lower than the lower bound (a). For each diagonal block in Table III, to formulate the model, the following notation is introduced: k = index for multidisciplinary tasks m = index for team members p = number of multidisciplinary tasks r = number of team members S m = salary of team members R km = compatibility indicator of each couple of task and team members C km = adjusting factor of each couple of task and team members. (This adjusting factor will be applied to salary rate in order to obtain a new appropriate salary value).

   = 0 1 km x
The objective function of the model (( 5)-( 9)) minimizes the total salary cost of the multidisciplinary teams. Constraint [START_REF] Hadj-Hamou | Cooperative design : A framework for a competency-based approach[END_REF] demonstrates that each task will be assigned to only team member. Constraint [START_REF] Tsai | Human resource selection for software development projects using Taguchi's parameter design[END_REF] demonstrates that a team member could not be allocated more than one task. Constraint [START_REF] De Korvin | Utilizing fuzzy compatibility of skill sets for team selection in multi-phase projects[END_REF] demonstrates that the selected team member needs to have a compatibility indicator of at least 0.6. The value of 0.6 represents the lower bound (a) of the compatibility indicator.

For each diagonal block, we obtained the following result, shown in Table IV. When the variable assignment is equal to 1, the team member is selected for the team, if the variable is equal to 0, the team member is not selected. In this example, the team member 1, 10 and 3 are selected for the team for the diagonal block I. The optimization is modeled and calculated by the optimization toolbox in Matlab.

IV. CONCLUSION

This paper has presented a team building approach and it is illustrated through an example. This promising approach differs from the use of existing team building approaches, because it has adopted a clustering algorithm to enable tasks and team members to be grouped into families (departments). The advantage of using the clustering method in the selection process is that it allows the team members to perform tasks in order to maintain their expert domains.

Our study in team member selection can be divided into two sections. At the beginning, we presented the method for competency structuring. The solution encourages the representation of attributes in order to characterize tasks (or team members) and it encourages demonstrating the potential diagonal block of tasks and team members. In the second part, we have studied the task assignment problem in design projects. An integer programming model is formulated to solve the multidisciplinary task assignment problem.

The objective of competency structuring is to give a global image of competency to the project managers and to help them to deal with a multi-functional team building problem. In team building literature, team members belong to a functional department. But in the innovative product design, new jobs (that is, new tasks) may appear; some team members may leave or be recruited by the company. So departments may evolve. No existing If a team member m belongs to the team that is responsible for disciplinary task k Otherwise G2 G3 G1 method has been proposed to link task-and team-member evolution.

In this paper, some limitations should be noted, as they might lead to interesting perspectives in future research. 1) We have not considered the workload (man-hours) of team members and multi-project task assignment. .If a team member is assigned to multiple teams, the member's availability will need to be taken into account. We consider here the single-project team building. Each team member works full time for a single project.

2) In the context of competency dynamics and development, the performance level of team members will gradually increase during the project execution and continuously decrease if they are not assigned to the task corresponding to this competency for some time. The question that we might ask is: how can we apply this approach to support the global competency development of a company?

Fig. 1 .

 1 Fig. 1. Multidisciplinary team building in design project

task k and the team member m 1 klR

 1 Level of discipline l required by the task k, 2 lm R Level of discipline l acquired by the team member m.

  integer programming model is formulated to solve the task assignment problem.

2 ) 1 kl R and 2 lmR

 212 Results from the clustering algorithm: Integrating the relation , we will obtain the task/team member incidence matrix km R presented in TableIII. This matrix contains the value of compatibility indicator km R .

TABLE I TASK

 I

				-DISCIPLINE INCIDENCE MATRICE
		D1	D2	D3	D4	D5	D6	D7	D8	D9	D10
	T1	0,8	0	0,2	0	0	0,9	0	1	0	0
	T2	0	1	0,7	1	0	0	0,6	0	0,3	²0

TABLE II TEAM

 II

				MEMBER-DISCIPLINE INCIDENCE MATRICE
		P1	P2	P3	P4	P5	P6	P7	P8	P9	P10 P11
	D1	0,7	0	1	0,4	0	0,2	0	0,6	0	0,7	0,4

TABLE III TASK

 III

		/ TEAM MEMBER INCIDENCE MATRIX	(	km R =	kl R	lm R	)
		P3	P10	P8	P1	P11	P4	P5		P7	P6	P2	P9
	T4	0,94	0,75	0,72	0,72	0,55	0,28	0,17		0,11		0,11	0	0
	T1	0,86	0,74	0,68	0,69	0,51	0,21	0,14		0,1		0,07	0	0
	T6	0,78	0,65	0,73	0,65	0,56	0,22	0,22		0,17		0,09	0,1	0
	T2	0,08	0,11	0,08	0,08	0,06	0,5	0,58		0,64		0,25	0,4	0,25
	T5	0	0,04	0,17	0,13	0,09	0,65	0,65		0,7		0,22	0,4	0,22
	T7	0,03	0,1	0,07	0,03	0,03	0,07	0,07		0,07		0,76	0,7	0,63
	T3	0	0,05	0	0	0,05		0		0		0,84	0,6	0,63
	S p 1500 1400 1450 1300 1200 1300 1270 1410 1380 1360 1300

TABLE IV EXAMPLE RESULT

 IVRESULT


CERMSEM Alain Chateauneuf

Michèle Cohen

Four notions of

Keywords: Location-independent risk, monotone increase in risk, rank-dependent expected utility. JEL classification: D81

come

Introduction

Second degree Stochastic Dominance (SSD) and its equal-mean version Mean-Preserving Increase in Risk (MPIR) play a leading role in the modeling of increasing risk in Economics, just as First degree Stochastic Dominance (FSD) plays a leading role in modeling increasing random wealth. None of these orders provides clear-cut answers to all questions of interest that have been posed, but they do provide a general framework in which these questions can be studied by developing more stringent types of FSD or SSD.

For example (see Rothschild & Stiglitz [36]), if a portfolio based on one safe and one risky asset is subjected to a MPIR of the risky asset, expected utility of the optimal portfolio decreases, but the fraction invested optimally in the risky asset does not necessarily decrease, as intuition would dictate. There is a vast literature dedicated to the discovery of special types of MPIR or special types of concave utility that coherently rank these fractions.

As a second example (see Ross [START_REF] Ross | Some stronger measures of risk aversion in the small and the large with applications[END_REF]), an expected utility maximizer will always be willing to buy full insurance accepted by a less risk averse one (this is in fact the Pratt postulate for less risk averse), but this is not necessarily so for partial insurance that reduces risk in the MPIR sense, as, again, intuition would suggest. Some stronger forms of MPIR (Bickel & Lehmann [START_REF] Bickel | Descriptive statistics for non-parametric models, III. Dispersion[END_REF][START_REF] Bickel | Descriptive statistics for non-parametric models, IV. Spread[END_REF], Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF], Landsberger & Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF][START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF]) perform coherently on this problem, as sketched a few paragraphs below.

These stochastic orders, some of their properties and some of their roles in the Expected Utility (EU) and in the Rank-dependent Expected Utility (RDEU) setups, are the subject matter of this paper.

The common link of the orders under study is that they are generated as successive applications of mean-preserving or mean-reducing single crossing operations between distribution functions: as shown by Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF], SSD is the uniformly integrable transitive closure of this simple, intuitive operation. Why intuitive? Because FSD is more intuitive than SSD, and single crossing means that for some 0 < p < 1, the restrictions of F and G to a common lower (upper) p-quantile make F dominate G (G dominate F ) by FSD: mass on the left (right) is transferred further to the left (right).

• Single crossing. G single crosses F if for some x 0 ∈ [-∞, ∞], G(x) ≥ F (x) for

x < x 0 and G(x) ≤ F (x) for x > x 0 . Equivalently, if the horizontal distance G -1 (v) -F -1 (v) between F and G, as a function on (0, 1), is non-positive below some v 0 ∈ [0, 1] and non-negative thereafter. As shown by Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF] and Machina & Pratt [START_REF] Machina | Increasing risk: some direct constructions[END_REF], if applied in a mean-preserving way, this type of stretch generates mean-preserving increase in risk (MPIR). If applied in a mean-preserving or mean-reducing way, it generates second degree dominance.

• Monotone single crossing. G is a monotone single crossing of F if the horizontal distance G -1 (v)-F -1 (v) between F and G is a non-decreasing function on the interval (0, 1). Equivalently, if G single crosses every horizontal shift of F . This is the notion of dispersion introduced by Bickel & Lehmann [START_REF] Bickel | Descriptive statistics for non-parametric models, III. Dispersion[END_REF][START_REF] Bickel | Descriptive statistics for non-parametric models, IV. Spread[END_REF] and the notion of monotone increase in risk (MIR) introduced by Quiggin [START_REF] Quiggin | A theory of anticipated utility[END_REF] for the mean-preserving case.

• Left-monotone single crossing. G is a left-monotone single crossing of F if G is a single crossing of F such that G -1 (v) -F -1 (v) is non-decreasing in every interval (of the type (0, v 0 )) where it is non-positive. Equivalently, if G single crosses every horizontal shift of F towards the left. As shown by Landsberger & Meilijson [START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF], if applied in a mean-preserving way this type of stretch generates left-monotone increase in risk (LMIR), that Jewitt defined under the name of location-independent risk, without the requirement of mean preservation.

• Right-monotone single crossing. G is a right-monotone single crossing of F if G is a single crossing of F such that G -1 (v) -F -1 (v) is non-decreasing in every interval (of the type (v 0 , 1)) where it is non-negative. Equivalently, if G single crosses every horizontal shift of F towards the right. This mirror image of Jewitt's order generator gives rise to a similar stochastic order, dual location-independent risk or (under equal means) right-monotone increase in risk. (Y is a right-monotone increase in risk (RMIR) with respect to X iff -Y is a left-monotone increase in risk with respect to -X.)

Marshall & Olkin [START_REF] Marshall | Inequalities: theory of majorization and its applications[END_REF] and Moyes [START_REF] Moyes | Inequality reducing and inequality preserving transformations of incomes: symmetric and individualistic transformations[END_REF] have developed the Relative Differentials Ordering (RDO) as a relative measure of income inequality: the logarithms of RDO-ordered random variables are Bickel & Lehmann ordered (one is a monotone single crossing of the other). RDO is a special case of dual location-independent risk (Chateauneuf & Moyes [START_REF] Chateauneuf | Inequality measurement and the weakening of the transfer principle[END_REF]). More generally, we prove in Theorem 1 that if Y is a right-monotone (left-monotone) single crossing of X then g(Y ) is a right-monotone (left-monotone) single crossing of g(X) for every g that is increasing and convex (concave).

Monotone single crossing is a transitive relation, so it defines directly a stochastic order. The other three are not, so they generate stochastic orders by uniformly integrable transitive closure. What this means is that if F dominates G ( by SSD, MPIR, MIR, LMIR or RMIR) then G is the limit of a uniformly integrable sequence of distributions starting with F and converging to G, such that each is obtained from the previous one by the corresponding single-crossing operation (applied mean preservedly or mean reducingly in the case of SSD and mean preservedly for each of the other four orders), and conversely. Note: uniform integrability is automatic if F , G and all interim distributions are supported by a finite interval.

Single crossings of horizontal shifts can play a decisive role in Insurance, as they order premia for partial insurance by the Arrow-Pratt index: let the uninsured position Y be replaced by an insured position X -C. If their distributions single cross, so will those of u(Y ) and u(X -C) for non-decreasing u. If the insurance contract is welcome under u, second order stochastic dominance yields the same preference inequality between Eη(u(Y )) and Eη(u(X -C)) for η non-decreasing and concave, that is, for a utility function v(•) = η(u(•)) Arrow-Pratt more risk averse than u.

As pointed out earlier (Ross [START_REF] Ross | Some stronger measures of risk aversion in the small and the large with applications[END_REF]), regular single crossing is not enough: even if G single crosses F , it need not single cross all horizontal shifts of F . Thus, unlike MPIR, Bickel & Lehmann dispersion is well suited to the EU study of insurance premia under general non-decreasing utilities (where the premium C may be positive or negative), while Jewitt's location-independent risk fits the needs under concave utilities, since only non-negative premia C are extracted from EU risk averters.

So far we have been working within the expected utility (EU) model, under which a decision maker is characterized by a utility function u. This non-decreasing function, often assumed to be continuous, becomes also concave when the decision maker is postulated to be averse to risk. Under the rank-dependent expected utility (RDEU) model, a decision maker is characterized by two functions, a utility function u (that plays the role of utility under certainty) in conjunction with a probability-transformation function f : [0, 1] → [0, 1] that is non-decreasing and satisfies f (0) = 0, f (1) = 1. Such a decision maker prefers a random variable X to another, Y , if and only if V (X) > V (Y ), where the Quiggin-Yaari functional V (see Quiggin [START_REF] Quiggin | A theory of anticipated utility[END_REF], Yaari [START_REF] Yaari | The dual theory of choice under risk[END_REF]) is given by

V (Z) = V u,f (Z) = - ∞ -∞ u(x)df (P (Z > x)) (1)
A winning probability p is "perceived" by the decision maker as f (p), in the sense that expected utility is computed as if the winning probability was f (p) rather than p. If the transformation function f is the identity function f (p) ≡ p, then V (Z) = V u,I (Z) is the expected utility E[u(Z)] of the random variable. The Yaari functional (see Yaari [START_REF] Yaari | The dual theory of choice under risk[END_REF]) is the special case V (Z) = V I,f (Z), in which the utility under certainty is the identity function u(x) ≡ x. If both transformation and utility are identity functions, then V (Z) = V I,I (Z) is simply the expected value E[Z] of the random variable.

In the EU model, aversion to risk is a rather clear-cut notion: The decision maker always prefers the expectation of a random variable to the random variable itself. This property is fully characterized by concavity of the utility function u. The question can now be reversed as to what are the pairs (X, Y ) of random variables with the same finite mean for which every risk averse decision maker prefers X to Y , leading to the characterization of MPIR.

The situation in the RDEU model is less clear-cut due to the interaction of the utility function and the probability transformation function in implying risk aversion. The universal preference of expected value to the random variable itself for all concave utilities u is modeled by transformation functions f that display pessimism, i.e., are majorized by the identity function (Yaari [42], Quiggin [START_REF] Quiggin | A theory of anticipated utility[END_REF][START_REF] Quiggin | Increasing risk : another definition[END_REF]). However, the universal preference (under concave utilities) for mean-preserving decreases in risk requires the stricter condition that f display increasing pessimism, i.e., that f be convex (Chew,Karni & Safra [10]). Quiggin [START_REF] Quiggin | Increasing risk : another definition[END_REF] showed that the (stronger) Bickel & Lehmann [START_REF] Bickel | Descriptive statistics for non-parametric models, III. Dispersion[END_REF][START_REF] Bickel | Descriptive statistics for non-parametric models, IV. Spread[END_REF] dispersion order, or equivalently Quiggin's [START_REF] Quiggin | A theory of anticipated utility[END_REF][START_REF] Quiggin | Increasing risk : another definition[END_REF] monotone increase in risk, is the weakest order that coherently ranks the preferences (between pairs of random variables with the same finite mean) of decision makers with concave utility function and probability transformation function majorized by the identity function.

We will rederive these two classes of probability transformation function, pessimism and increasing pessimism, ordered by inclusion, and derive two intermediate classes, the starshaped probability transformation functions (Lemma 4), by postulating aversion to different types of risk. These two classes display increasing relative pessimism with respect to a sure loss or with respect to a sure gain.

Diverse attitudes to different types of MPIR lie at the core of the departure of the RDEU model from the EU model. According to the von-Neumann & Morgenstern Independence Axiom (see [START_REF] Von Neumann | Theory of games and economic behavior[END_REF]), a decision maker that prefers a random variable X * to a random variable Y * will also prefer X (a mixture of X * and another random variable W by the flip of some coin) to Y (the mixture of Y * and W by the flip of the same coin), whatever be the distributions of the outcome of the coin and of the random variable W .

Consider two equal-mean dichotomous random variables X * , with values x 1 < x 2 , and Y * , with values y 1 ≤ x 1 and y 2 ≥ x 2 . The (risk averse) preference of X * to Y * is quite natural. The question of (von-Neumann & Morgenstern) Independence axiom consistency is whether this preference is so sharp that it is preserved even when the spread pair (x 1 , x 2 ) is only part of the distribution -the other part, left unchanged, being represented by W . In the absence of this powerful axiom, it need not be. The attitude of RDEU decision makers may vary according to the relative location of the other atoms of W in relation to the values (x 1 , x 2 ). As will be seen in Lemma 4, if W is supported by the interval [x 1 , x 2 ] then RDEU preference for X over Y (i.e., aversion to spread of extremes) only requires f to be majorized by the identity function while if W is allowed to have an arbitrary atom on each of the two rays (-∞, x 1 ) and (x 2 , ∞), the global preference of X to Y (aversion to middle spread) requires f to be convex. The presence of an arbitrary atom of W in (-∞, x 1 ) forces f to be star shaped at 0, the model for aversion to spread of gains, while its presence in (x 2 , ∞) forces f to be star-shaped at 1, modeling aversion to spread of losses. (g is said to be star-shaped at µ if (g(x) -g(µ))/(x -µ) is monotone (non-decreasing, in this application) on R -{µ}).

We thus have four conditions on the probability transformation function that model aversion to particular types of mean-preserving spread in the RDEU set-up, by DM's with concave utility function. These types of increase in risk, so far illustrated only by some simple canonical scenarios, correspond to the four types of single-crossing operation introduced earlier (Theorems 2, 3, 4 and 5.)

The first part of this paper (Section 2) is dedicated to the four stochastic orders under study. The second part (composed of sections 3 and 4) focuses on the probability transformation function of RDEU DM's, postulating their utility functions to be concave. The authors study in [START_REF] Chateauneuf | More pessimism than greediness : a characterization of monotone risk aversion in the Rank Dependent Expected Utility model[END_REF] a departure from concavity of u, and characterize the pairs (u, f ) of RDEU agents averse to monotone increase in risk: as long as f displays strong enough pessimism in some sense, u can even be convex.

To stay at the conceptual level, we choose to avoid heavy measure-theoretic machinery: formal treatment will be restricted to random variables with compact support and to continuous probability transformation functions.

An introduction to the four relevant stochastic orders

This section summarizes properties of mean-preserving increase in risk MPIR (see Hardy, Littlewood & Polya [START_REF] Hardy | Inequalities[END_REF], Strassen [START_REF] Strassen | The existence of probability measures with given marginals[END_REF], Rothschild & Stiglitz [START_REF] Rothschild | Increasing Risk: I. A definition[END_REF], Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF], Lorenz [START_REF] Lorenz | Methods of measuring concentration of wealth[END_REF]), monotone increase in risk (see Bickel & Lehmann [2,[START_REF] Bickel | Descriptive statistics for non-parametric models, IV. Spread[END_REF], Quiggin [START_REF] Quiggin | A theory of anticipated utility[END_REF], Cohen [START_REF] Cohen | Risk aversion concepts in expected-and non-expected-utility models[END_REF]), left-monotone increase in risk (or location-independent risk, see Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF], Landsberger & Meilijson [START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF]) and a dual form of the latter, right-monotone increase in risk.

For each stochastic order we present, following the summary of its relevant basic properties, an indication of its role as a generalization of one of the canonical spreads ( 17)- [START_REF] Karni | Search Theory: the case of search with uncertain recall[END_REF].

The focus on attitude to increase in risk by RDEU DM's should leave out of this study distribution changes that do not preserve the mean. However, many of the other applications of the stochastic orders under study benefit from relaxing the assumption of equal means. We will try to accomodate both.

Given a distribution function H, one of the most common ways to define a probability space and on it a random variable Z distributed H, is to let the probability space be the interval (0, 1) endowed with its Borel σ-field and on it a uniformly distributed random variable U . Defining on the unit interval [0, 1) the quantile function or generalized inverse H -1 of the (not necessarily invertible) distribution function H by

H -1 (p) = inf{x|H(x) > p} (2)
completed as H -1 (1) = sup{x|H(x) < 1}, let Z = H -1 (U ). The elementary fact that Z is distributed H will be used repeatedly. A less elementary fact is the uniqueness of H -1 for this purpose: if g : (0, 1) → R is non-decreasing and g(U ) ∼ H, then g(U ) = H -1 (U ) a.s. This construction of a random variable with given distribution function can be extended to the construction of a pair of comonotone random variables (i.e., non-decreasing functions of some other random variable) with given marginal distributions F and G as F -1 (U ), G -1 (U ), and to that of a corresponding pair of anti-thetic random variables (hedged against each other) as

F -1 (U ), G -1 (1 -U ).
Define two integrated distribution functions

Ψ H (x) = ∞ x (1 -H(t))dt = E[max(Z -x, 0)] (3)
Φ H (x) = x -∞ H(t)dt = Ψ H (x) + x -E[Z] . (4)
Each of these functions is convex and non-negative, with slope never steeper than 1. Φ increases from zero to become asymptotic at ∞ to the linear function x -E[Z] and Ψ decreases to zero from being asymptotic towards -∞ to E[Z] -x.

Second-degree dominance and mean-preserving increase in risk

This order, first studied by Hardy, Littlewood and Polya [START_REF] Hardy | Inequalities[END_REF] and generalized by Strassen [START_REF] Strassen | The existence of probability measures with given marginals[END_REF], has been well known in Economics since Rothschild & Stiglitz [START_REF] Rothschild | Increasing Risk: I. A definition[END_REF]. It was earlier studied by Hadar & Russel [START_REF] Hadar | Rules for ordering uncertain prospects[END_REF] and Hanoch & Levy [START_REF] Hanoch | The efficiency analysis of choices involving risk[END_REF] as Second Degree Dominance, the case of possibly unequal means.

Given two distributions F and G with equal finite means, G is said to represent a Meanpreserving Increase in Risk (MPIR) with respect to

F if Φ G (x) ≥ Φ F (x) for all x ∈ R, or equivalently, if Ψ G (x) ≥ Ψ F (x)
for all x ∈ R. While these two conditions are equivalent in the case of equal means, Second Degree Dominance of G by F is defined by the first condition and Convex Dominance of F by G (common in the Reliability Engineering literature) is defined by the second condition.

Let p ∈ (0, 1) and consider the linear function with slope p tangent to the convex function Φ. This linear function can be seen to intersect the x-axis at the conditional expectation

L H (p) = E[H -1 (U )|U < p] = p 0 H -1 (q)dq p (5)
of Z on its lower p-quantile. For the case where H is the distribution of a non-negative random variable (income), the function pL H (p) E[Z] was introduced by Lorenz [START_REF] Lorenz | Methods of measuring concentration of wealth[END_REF] as a relative measure of inequality. It is geometrically obvious that Φ G majorizes Φ F if and only if L F majorizes L G . This equivalent characterization of Second Degree Dominance is the well known Lorenz-function characterization of MPIR.

Besides these integral characterizations, MPIR has also been characterized in terms of a noise or Martingale condition (the existence of a probability space and on it two random variables X and Y with respective marginal distributions F and G, such that E(Y |X) = X almost surely) and by its relation to (EU) risk aversion (the weak preference of F to G by all EU decision makers with concave utility functions).

It follows from the work of Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF] and Machina & Pratt [START_REF] Machina | Increasing risk: some direct constructions[END_REF] that for two distribution functions F and G with bounded support and equal means, G is a MPIR of F if and only if there is a sequence (F i ) of distribution functions with uniformly bounded supports such that (i) lim F i (x) = G(x) at all continuity points x of G and (ii) for each i ≥ 1, F i is a single crossing spread of F i-1 , where F 0 = F .

Dispersion and monotone increase in risk

The dispersive order, introduced and first studied by Bickel & Lehmann [START_REF] Bickel | Descriptive statistics for non-parametric models, III. Dispersion[END_REF][START_REF] Bickel | Descriptive statistics for non-parametric models, IV. Spread[END_REF] in the context of Non-parametric methods in Statistics, without requiring the two distributions to have equal nor even existing means, was re-defined by Quiggin in its mean-preserving version, and applied to choice between lotteries in the RDEU model. Landsberger & Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF][START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF] applied it to the design and study of efficient insurance contracts in the EU model.

(i) G is more dispersed than F (F less dispersed than G) if for every 0 < p < q < 1, the length G -1 (q) -G -1 (p) of the interval between the p-and the q-quantiles of G (weakly) exceeds the corresponding length for F . Equivalently, as defined in the Introduction,

(ii) G is more dispersed than F if the horizontal distance D(p) = F -1 (p) -G -1 (p) (6)
between F and G is a non-increasing function on the interval (0, 1). Further equivalently, (iii) G is more dispersed than F if on some probability space there are two comonotone random variables X and Z such that X ∼ F and X + Z ∼ G.

If G is more dispersed than F , and if G and F have the same means, G will be said to be a monotone increase in risk with respect to F .

Dispersion is location-free, i. e., it is preserved if each random variable is shifted by the addition of an arbitrary constant. So, if F and G (with finite means and G more dispersed than F ) are shifted so as to equate their means, G represents a monotone increase in risk with respect to F . The middle definition implies the single-crossing property between F and G (in fact, a stronger property holds: any horizontal shift of F single crosses G). Hence, if the mean of G does not exceed that of F , F dominates G by SSD, according to Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF].

Landsberger & Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] proved in the EU context that if the non-decreasing utility function v displays Arrow-Pratt more risk aversion than the non-decreasing utility function u, then a decision maker with utility function v is willing to buy any partial insurance contract that makes the insured position less dispersed than the uninsured position, as soon as a decision maker with utility function u is willing to buy it. As is well known, Pratt [START_REF] Pratt | Risk Aversion in the small and large[END_REF] introduced the notion of more risk averse by precisely this requirement, applied to full insurance. Ross [START_REF] Ross | Some stronger measures of risk aversion in the small and the large with applications[END_REF] showed that mean-preserving increase in risk is not strong enough to yield this result. Monotone reduction in risk is not a strong assumption to make on insurance contracts: Landsberger and Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] proved (again, in the EU context) that by the third definition of dispersion and the statement of the next sub-section, whenever an insurance contract does not reduce risk in the monotone way, there is another contract, preferred to the first by both parties, that does. Thus, monotone reduction in risk is a consequence of Pareto optimality:

The role of comonotonicity in Pareto-optimal allocations

Landsberger & Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF] showed that for any two integrable random variables X and Z, there are two other random variables X * and Z * defined on the same probability space, such that (i) X * + Z * = X + Z almost surely, (ii) X is a MPIR with respect to X * , (iii) Z is a MPIR with respect to Z * and (iv) X * and Z * are comonotone.

This statement means that every non-comonotone way in which two risk averse partners split a random endowment Y as Y = X + Z (or syndicate, see Wilson [START_REF]The theory of syndicates[END_REF]), can be replaced by a different contingent claim, weakly preferred by each, that is comonotone, a desirable property: if the partnership does better, no one of the partners does worse. Insurance contracts are examples of such partnerships, between insured and insurer.

The role of comonotonicity and monotone increase in risk on ordering premia

As pointed out earlier, if (in fact, only if) Y is monotonically riskier than X, then for every (negative, zero or positive premium) C, the distribution of Y single crosses the distribution of X -C. But then the distribution of u(Y ) single crosses that of u(X -C) for all nondecreasing u. From this it follows that if an EU maximizer with utility function u is willing to pay C for the replacement of the uninsured position Y by the insured position X then so does another with utility function v, that is Arrow-Pratt more risk averse than the first.

Observe that the value of C to achieve the inequality Eu(Y ) ≤ Eu(X -C) for a nondecreasing function u can be positive or negative, since such functions cover risk averters as well as risk seekers.

However, if attention is restricted to risk averse DM's, then only positive C are of interest, since it is not needed that G single cross every horizontal shift of F ; it is enough that G single cross all horizontal shifts of F towards the left. This is why the Jewitt locationindependent risk order, studied in the next section, is strong enough to be respected by premia paid by risk averse EU maximizers.

Location-independent risk and its dual version, left-monotone and right-monotone increase in risk

Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF] defined as location-independent risk the following inequality between the integrated cumulative distribution functions, without the requirement of equal means.

Definition 1 G is location-independent riskier than F if Φ G (G -1 (q)) ≥ Φ F (F -1 (q)) for every q ∈ (0, 1).
If G is location-independent riskier than F and the two have the same finite mean, G will be said to be left-monotone riskier than F .

Analogously, Definition 2 introduces a mirror-image order via the corresponding inequality between the integrated de-cumulative distribution functions.

Definition 2 G is dual location-independent riskier than F if Ψ G (G -1 (q)) ≥ Ψ F (F -1 (q))
for every q ∈ (0, 1).

If G is dual location-independent riskier than F and the two have the same finite mean, G will be said to be right-monotone riskier than F .

Observe that Φ H is a convex function, supported at the point H -1 (q) by a linear tangent with slope q. Φ G (G -1 (q)) ≥ Φ F (F -1 (q)) means that at whichever pair of points Φ G and Φ F have the same slope, Φ G is higher. This is the same as saying that at whichever pair of points Φ G and Φ F have the same height, Φ G has smaller slope. Since the same holds for Ψ, Lemma 1 (Landsberger & Meilijson [START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF]).

G is location-independent riskier than F if and only if the horizontal distance

Φ -1 G (x) -Φ -1 F (x) between the integrated cumulative distribution functions is non-decreasing in x ∈ R + .
G is dual location-independent riskier than F if and only if the horizontal distance

Ψ -1 G (x) -Ψ -1 F (x) between the integrated de-cumulative distribution functions is non-increasing in x ∈ R + .
Of course, under equality of means, the same conditions characterize left and right monotone increase in risk.

The role of left monotonicity in ordering premia

Location-independent risk, weaker than Bickel & Lehmann dispersion, was introduced by Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF] in an EU context of Insurance. Just as the insurance premium for a monotone reduction in risk is ordered by the Arrow-Pratt index of risk aversion in the class of nondecreasing utility functions, the insurance premium for a left-monotone reduction in risk is ordered by the index of risk aversion in the class of concave non-decreasing utility functions. In other words, as proved by Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF], if (the uninsured position) G is left-monotone riskier than (the insured position) F and some risk averse EU maximizer is willing to pay a given premium for the replacement of G by F , so is any such decision maker that is Arrow-Pratt more risk averse than the first. Furthermore, (Jewitt [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF]) left-monotone increase in risk is the weakest order with this property, just as (Landsberger & Meilijson [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF]) monotone increase in risk is the weakest order with this property in the class of all nondecreasing utility functions. Since it has already been explained above that left-monotone single crossings order premia as claimed, the picture will be complete if left-monotone increase in risk is identified as the uniformly integrable transitive closure of left-monotone single crossings. That is, phrasing somewhat informally a result in Landsberger & Meilijson [START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF],

Lemma 2 G is location-independent (dual location-independent, left-monotone, right-monotone) riskier than F if it is possible to obtain G as the limit of a sequence of distributions F i such that F 0 = F and for each i ≥ 1, F i left (right, mean-preservedly left, mean-preservedly right) monotonically single crosses F i-1 .

Proofs can be found in Landsberger & Meilijson [START_REF] Landsberger | The generating process and an extension of Jewitt's location independent risk concept[END_REF]. We only show here that if G is a left-monotone single crossing of F then G is left-monotone riskier than F , by expressing

Φ F (F -1 (p)) = F -1 (p) -∞ F (t)dt = p 0 qdF -1 (q) = pF -1 (p) - p 0 F -1 (q)dq (7) to obtain Φ G (G -1 (p)) -Φ F (F -1 (p)) = p 0 [(G -1 (p) -F -1 (p)) -(G -1 (q) -F -1 (q))dq (8) = p[G -1 (p) -F -1 (p) + L F (p) -L G (p)] , (9)
that must be shown to be non-negative. Now observe that for p below the single crossing the integrand in the RHS of ( 8) is non-negative, while for p above the single crossing, ( 9) is nonnegative because it is the sum of two non-negative terms -recall the Lorenz characterization of second degree dominance and the fact that single crossing (whether left-monotone or not), implies second degree dominance.

Since the sum Φ -1

H (x) -Φ -1 F (x) = (Φ -1 H (x) -Φ -1 G (x)) + (Φ -1 G (x) -Φ -1 F (x)
) of nondecreasing functions is non-decreasing, left-monotone increase in risk is transitive. This shows that any G generated from F by a sequence of left-monotone single crossings (with Φ-functions converging from below, a property related to uniform integrability), is leftmonotone riskier than F .

Monotone transformations of random variables ordered by the various notions of dispersion

It is clear that the single crossing property is preserved by increasing transformations. It is just as clear that concave increasing transformations preserve second-degree dominance and convex increasing transformations preserve convex dominance.

Let Y be Bickel & Lehmann more dispersed than X. We will see that if g is increasing and concave, then g(Y ) is location-independent riskier than g(X), and if g is increasing and convex, then g(Y ) is riskier than g(X) in the sense of the mirror image of this order. In particular, if a constant C is adjusted in each of the two cases so that Eg(Y ) = Eg(X -C), then g(Y ) is accordingly a left or right monotone increase in risk with respect to g(X -C). More generally, Theorem 1 (i) If Y is location-independent riskier than X and g : R → R is increasing and concave, then g(Y ) is location-independent riskier than g(X).

(ii) If Y is a dual location-independent riskier than X and g : R → R is increasing and convex, then g(Y ) is dual location-independent riskier than g(X).

Proof. By Lemma 2, it is enough to study the effect of these monotone transformations on left or right monotone single crossings. To see (i), observe that the definition of leftmonotone single crossing implies that on U ∼ U (0, 1) as probability space, it is possible to

express Y = F -1 Y (U ) as a sum of X = F -1 X (U ) and left comonotone noise as F -1 Y (U ) = F -1 X (U ) + Z(U )
, where Z single crosses zero and is non-decreasing while negative. Since g(F -1 Y (U )) and F -1 g(Y ) (U ) are identically distributed non-decreasing functions of U , they must be a.s. equal. Consider g(Y ) = g(X) + [g(X + Z) -g(X))] and let u < v. Then as long as Z(u) ≤ 0, g(X(v) + Z(v)) -g(X(v)) ≥ g(X(v) + Z(u)) -g(X(v)) because g and Z are non-decreasing. Furthermore, on this range g(X(v)

+ Z(u)) -g(X(v)) ≥ g(X(u) + Z(u)) -g(X(u)) because g is concave. Hence, F -1 g(Y ) (U ) -F -1 g(X) (U ) = Z * (U ) is non-decreasing while negative, because the equality Z * (U ) = g(X(U ) + Z(U )) -g(X(U ))
holds while negative. The behavior beyond the crossing is covered by the fact that the single crossing property is preserved under increasing transformations. Thus, Y is a left-monotone single crossing of X. The proof of (ii) is analogous.

Corollary 1 (Chateauneuf & Moyes [START_REF] Chateauneuf | Inequality measurement and the weakening of the transfer principle[END_REF]) Let F and G be distributions of positive random variables (with the same mean), and let G display more inequality in the RDO sense (Marshall & Olkin [29], Moyes [START_REF] Moyes | Inequality reducing and inequality preserving transformations of incomes: symmetric and individualistic transformations[END_REF]). Then G is dual location-independent (respectively, right-monotone) riskier than F .

Proof. Let X ∼ F and Y ∼ G. The notion of RDO means that for any 0 < p < q < 1, the relative p q -quantile F -1 (p) F -1 (q) of F within its q-quantile weakly exceeds the corresponding expression for G. This means that log(Y ) is Bickel & Lehmann more dispersed than log(X). Thus, X and Y are the result of applying the increasing convex transformation g(x) = exp(x) to two random variables ordered by dual location-independent risk. The converse to Corollary 1 is wrong: since the property of being a left or right monotone single crossing is not transitive, RDO does not coincide with dual location-independent risk.

Deductible insurance left monotonically dominates any other

Arrow [START_REF] Arrow | Essays in the theory of risk bearing[END_REF] proved that for a given premium (based on the expected value of the loss), insurance contracts with deductible are optimal for EU risk averse agents.

This result was generalized by Gollier & Schlesinger [START_REF] Gollier | Arrow's theorem on the optimality of deductibles: a stochastic dominance approach[END_REF] to agents averse to MPIR, independently of their decision model under risk.

Vergnaud [START_REF] Vergnaud | Analysis of risk in a non-expected utility framework and application to the optimality of the deductible[END_REF] extended this result further: the insured position induced by any contract with the given premium is left-monotone riskier than the insured position induced by the corresponding contract with deductible.

Furthermore, in a result complementing Jewitt's claim on the behavior of the Arrow-Pratt index, Chateauneuf, Cohen and Vergnaud [START_REF] Chateauneuf | Left-monotone reduction of risk, deductible and call[END_REF] showed that aversion to left-monotone risk is the weakest notion of risk aversion under which deductible insurance contracts are optimal.

The role of left and right monotone increase in risk in a classical search problem

The Chow & Robbins search problem [START_REF] Chow | On optimal stopping rules[END_REF] (see also Karni & Schwartz [20,[START_REF] Karni | Two theorems on optimal stopping with backward solicitation[END_REF]) considers a seller of one good that is given by re-callable potential buyers successive offers X 1 , X 2 , . . ., assumed to be independent and identically F -distributed non-negative random variables.

Each such offer incurs a cost C for the seller so selling after n offers will yield a net revenue N R F,C,n = max 1≤i≤n X i -nC. The optimal stopping time, maximizing E[N R F,C,τ ] over all stopping times τ , is known to be of a control limit form, that dictates selling at the first offer not below the threshold b = Ψ -1

F (C) solving Ψ F (b) = C (see (3)
). The optimal threshold Ψ -1 F (C) turns out to be the optimal expected revenue as well. Thus, the expected overpayment over the threshold exactly pays for sampling. Indeed, restricting attention to such control-limit stopping rules

τ b , E[N R F,C,τ b ] = E[X|X ≥ b] -C/(1 -F (b-)),
the first term being the expected price received and the second term being the expected sampling cost, expressed in terms of the expectation 1/(1 -F (b-)) = 1/P (X > b) of the (geometrically distributed) number of offers sampled. Now,

E[N R F,C,τ ] = E[X|X ≥ b] - C P (X ≥ b) = E[XI X≥b ] -C 1 -F (b-) = b + E[max(X -b, 0)] -C 1 -F (b-) = b + Ψ(b) -C 1 -F (b-) (10)
from which it is not hard to get the result

sup τ E[N R F,C,τ ] = E[X|X ≥ Ψ -1 F (C)] - C 1 -F (Ψ -1 F (C)-) = Ψ -1 F (C) (11)
The Chow & Robbins seller's search problem [START_REF] Chow | On optimal stopping rules[END_REF] can be turned over into the problem of a shopper that gets independent F -distributed price quotes Y i at a cost C each, and waits optimally for a price not exceeding the control limit

d = Φ -1 F (C) that satisfies Φ F (d) = C (see (4)). The optimal total cost is inf τ E[T C F,C,τ ] = E[Y |Y ≤ Φ -1 F (C)] + C F (Φ -1 F (C)) = Φ -1 F (C) . (12)
Suppose that the price quotes the buyer gets are distributed G rather than F . The G's under which the buyer is better off than under F for every C > 0 are those such that for every C > 0, the point Φ -1 G (C) where Φ G reaches height C is to the left of the corresponding point Φ -1 F (C). In other words, Φ G ≥ Φ F , or F must dominate G by SSD. If F and G have the same expectation, this means that G must be a MPIR of F . This makes sense: dispersion helps the patient buyer to benefit from the heavier lower tail. Intuition further dictates the rationale that in a more dispersed market, it pays to the buyer to sample longer in expectance of lower offers. After all, the least dispersed case, the "distribution" of a fixed price, dictates buying at the first quote. This rationale would mean that for all C > 0, the expected sample size 1/(1 -G(Φ -1 G (C)-)) under G should not be less than the corresponding expected sample size under F . I.e., the slope 1 -G(Φ -1 G (C)) should be less than or equal to the slope 1 -F (Φ -1 F (C)) for all C > 0, or, the horizontal distance function Φ -1 G (C) -Φ -1 F (C) should be non-decreasing. While SSD or MPIR are not strong enough to yield this inequality, by Lemma 1 this condition says precisely that G is location-independent riskier than F . Summarizing for the case of equal means,

• If the distribution G of the asking prices is a MPIR of F , the buyer's optimal expected total cost is lower under G, for all C > 0.

• If G is a left-monotone increase in risk with respect to F , then under G the buyer's optimal expected total cost is lower while the expected sampling cost is higher than under F , for all C > 0.

This means that in a left-monotone more dispersed market, the sellers will get a much lower revenue, since besides making the buyer better off, they have to subsidize the buyer's higher sampling cost.

Going back to the seller's problem, it should be clear that it admits such an analysis too. Summarizing the results for the case of equal means,

• If the distribution G of the price offers is a MPIR of F , the seller's optimal expected revenue is higher under G, for all C > 0.

• If G is a right-monotone increase in risk with respect to F , the seller's optimal expected revenue and sampling cost are higher under G, for all C > 0.

This means that in a right-monotone more dispersed market, the buyer will pay a much higher price, since besides making the seller better off, the buyer has to subsidize the seller's higher sampling costs.

Attitudes to increase in risk in the RDEU model

For convenience of exposition, distributions D(X) = (x 1 , p 1 ; x 2 , p 2 ; . . . ; x k-1 , p k-1 ; x k , p k ) of random variables X with finite support, tacitly assumed to satisfy

x 1 < x 2 • • • < x k , will be represented in the equivalent form D(X) = (x 1 , 1 -q 1 ; x 2 , q 1 -q 2 ; . . . ; x k-1 , q k-2 -q k-1 ; x k , q k-1 ) . ( 13
)
The original expression is in terms of the probabilities P (X = x i ) = p i of the atoms while ( 13) is a representation in terms of the right-tail probabilities P (X > x i ) = q i , with p i = q i-1 -q i under the convention q 0 = 1, q k = 0. For such a discrete distribution, the expected utility and the Quiggin-Yaari functional (see [START_REF] Arrow | Essays in the theory of risk bearing[END_REF]) may be expressed as

E[u(X)] = -u(x)dF * (x) = u(x 1 ) + q 1 [u(x 2 ) -u(x 1 )] +q 2 [u(x 3 ) -u(x 2 )] + • • • + q k-1 [u(x k ) -u(x k-1 ] (14)
and

V (X) = -u(x)df (F * (x)) = u(x 1 ) + f (q 1 )[u(x 2 ) -u(x 1 )] +f (q 2 )[u(x 3 ) -u(x 2 )] + • • • + f (q k-1 )[u(x k ) -u(x k-1 ] (15)
respectively, where F * (x) = P (X > x) is the survival function of the random variable X. The function F (x) = 1 -F * (x) = P (X ≤ x) is its distribution function.

If the distribution of X is perturbed by mean-preservedly spreading out two atoms strictly preserving the order of all atoms, the resulting distribution (of a random variable Y) may be expressed in the form (compare with ( 13))

D(Y ) = (x 1 , 1 -q 1 ; . . . ; x i-1 , q i-2 -q i-1 ; x i -ǫ[q j-1 -q j ], q i-1 -q i ; x i+1 , q i -q i+1 ;
. . . ; x j-1 , q j-2 -q j-1 ; x j + ǫ[q i-1 -q i ], q j-1 -q j ; x j+1 , q j -q j+1 ; . . . ; x k , q k-1 ) . [START_REF] Hanoch | The efficiency analysis of choices involving risk[END_REF] This type of spread keeps all probabilities unchanged. It also keeps all except two of the atoms of the distribution of X unchanged. The two exceptional atoms (x i < x j ) are set further apart by the precise multiples of a sufficiently small arbitrary ǫ that will keep the expected value constant, without disrupting the order of the atoms. Such a spread can be conveniently parameterized by specifying ǫ, i and j. The analysis, to be performed in Section 3.1, will be restricted to the four canonical examples (see ( 17)-( 20)), that are respective special cases of the four types of single-crossing operations defined in the Introduction.

The fact that each of the stochastic orders under consideration is generated by a singlecrossing operation between distribution functions, has far reaching implications for analysis. As formalized by the following Lemma, in order to study the effect of each of these four types of increase in risk on the Quiggin-Yaari functional under general concave utility functions u and general transformation functions f , it is enough to study their effect on the Yaari functional, i.e., the Quiggin-Yaari functional with linear utility under certainty u(x) = x.

Lemma 3 If the distribution G of a random variable Y is a mean-preserving or meanreducing single crossing spread of the distribution F of a random variable X and f is a probability transformation function such that V

I,f (X) ≥ V I,f (Y ), then V u,f (X) ≥ V u,f (Y )
for all concave non-decreasing utility-on-certainty functions u.

Proof: If the distribution G single crosses the distribution F , then the distribution 1 -f (G * ) single crosses the distribution 1 -f (F * ). Hence, if 1 -f (F * ) has expectation bigger or equal to that of 1 -f (G * ) (i.e., if the Yaari functionals satisfy the corresponding inequality), then the former dominates the latter by second degree dominance (see Diamond & Stiglitz [START_REF] Diamond | Increases in risk and risk aversion[END_REF]). But then, the preference of F by decision makers with linear utility extends automatically to its preference by decision makers with concave utility.

In other words, since the Quiggin-Yaari functional is in essence Expected Utility with respect to the transformation-function-transformed distributions, the introduction of concave utilities under certainty will automatically preserve the corresponding inequalities satisfied by linear utilities.

Lemma 3 and all the discussion so far deal with different types of transformation function, asking which type models some specific kind of aversion to risk, by DM's with concave utility functions. The authors have characterized in [START_REF] Chateauneuf | More pessimism than greediness : a characterization of monotone risk aversion in the Rank Dependent Expected Utility model[END_REF] the pairs u, f with possibly non-concave u for which the DM is averse to monotone increase in risk. The relaxation of concavity under the Jewitt orders is the subject of our current research.

RDEU decision maakers' aversion to middle spread, spread of extremes, spread of losses and spread of gains

Returning to the four canonical examples of mean-preserving spread announced earlier and motivated by Lemma 3, each is now analyzed in terms of the Yaari functional. As an example, k = 4, i = 2 and j = 3 means that the distribution is supported by four atoms, of which the middle two are subjected to a mean-preserving spread. For each canonical example, the difference V I,f (Y ) -V I,f (X) is displayed. Aversion to a particular type of mean-preserving spread means that V (Y ) -V (X) is non-positive for all choices of the distribution, in the corresponding canonical class. It is always assumed that ǫ > 0 is small enough, so that the order of the atoms is preserved.

1. Canonical middle spread: k = 4, i = 2 and j = 3. Then

V I,f (Y ) -V I,f (X) = ǫ(q 1 -q 3 )[f (q 2 ) - q 2 -q 3 q 1 -q 3 f (q 1 ) - q 1 -q 2 q 1 -q 3 f (q 3 )] . (17)
2. Canonical extreme spread: k = 3, i = 1 and j = 3. Then

V I,f (Y ) -V I,f (X) = ǫ(1 -q 1 )q 2 [ f (q 2 ) q 2 - 1 -f (q 1 ) 1 -q 1 ] . (18)
3. Canonical spread of losses: k = 3, i = 1 and j = 2. Then

V I,f (Y ) -V I,f (X) = ǫ(1 -q 1 )(1 -q 2 )[ 1 -f (q 2 ) 1 -q 2 - 1 -f (q 1 ) 1 -q 1 ] . ( 19
)
4. Canonical spread of gains: k = 3, i = 2 and j = 3. Then

V I,f (Y ) -V I,f (X) = ǫq 1 q 2 [ f (q 2 ) q 2 - f (q 1 ) q 1 ] . (20)

Thus,

Lemma 4 (i) The Yaari functional displays aversion to canonical middle spread [START_REF] Hardy | Inequalities[END_REF] if and only if the transformation function f is convex.

(ii) The Yaari functional displays aversion to canonical extreme spread [START_REF] Jewitt | Risk aversion and the choice between risky prospects: the preservation of comparative statics results[END_REF] if and only if the transformation function f is majorized by the identity function.

(iii) The Yaari functional displays aversion to canonical spread of losses [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF] if and only if the transformation function f is majorized by the identity function and star-shaped at 1.

(iv) The Yaari functional displays aversion to canonical spread of gains [START_REF] Karni | Search Theory: the case of search with uncertain recall[END_REF] if and only if the transformation function f is majorized by the identity function and star-shaped at 0.

As defined in the Introduction, Definition 3 A function g defined on some interval I is said to be star-shaped at µ ∈ I if (g(x) -g(µ))/(x -µ) is monotone on I -{µ}.

For a probability transformation function g majorized by the identity function and starshaped at µ = 0 or 1, (g(x) -g(µ))/(x -µ) is monotone non-decreasing.

Proof of Lemma 4:

In each of the four cases we must characterize the class of transformation functions f for which V I,f (Y ) -V I,f (X) is non-positive for all choices of the distribution parameters. Equivalently, the f 's for which the expression in brackets in the corresponding RHS is non-positive. In expression [START_REF] Hardy | Inequalities[END_REF] this is readily seen to be the class of convex functions. In expression [START_REF] Jewitt | Risk aversion and the choice between risky prospects: the preservation of comparative statics results[END_REF], one direction is clear: if f is majorized by the diagonal then f (q) q ≤ 1 ≤ 1-f (q) 1-q . As for the other direction, due to the monotonicity of f , if f is not majorized by the diagonal, then there are necessarily 0 < q 2 < q 1 < 1 with f (q i ) > q i . Since f (q 2 )

q 2 > 1 > 1-f (q 1 )
1-q 1 , this decision maker strictly prefers Y to X. The remaining two expressions in brackets in [START_REF] Jewitt | Choosing between risky prospects: the characterisation of comparative static results and location independent risk[END_REF] and [START_REF] Karni | Search Theory: the case of search with uncertain recall[END_REF] are non-positive throughout if and only if the corresponding f is star-shaped at the corresponding point, by definition.

Star-shaped probability transformation functions: EU and RDEU, lotteries and insurance

Star-shapedness of f at 0 and at 1 can be given a related but different role, that sharpens the edge between EU and RDEU. Let a risk averse EU-maximizer with utility function u be offered full insurance for a single loss L that occurs with probability 1 -p to her house worth W . The fair premium is P f = (1 -p)L and the supremal premium that can be extracted from the agent is

P s = W -u -1 (u(W-L) (1-p) + u(W) p
) so the load factor the agent can withstand is

LF EU = P s P f = W -u -1 (u(W ) -(1 -p)(u(W ) -u(W -L))) (1 -p)L = u(W ) -u(W -L) L u -1 (u(W )) -u -1 (u(W ) -(1 -p)(u(W ) -u(W -L))) (1 -p)(u(W ) -u(W -L) = u(W ) -u(W -L) L u -1 (v) -u -1 (v -∆) ∆ , (21)
where v = u(W ) and ∆ = (1 -p)(u(W ) -u(W -L)). Since u is concave, u -1 is convex and the load factor is a non-decreasing function of p. Although a change in p is a first rather than a second degree change, EU risk aversion has a clear cut effect: risk averse EU maximizers are willing to pay relatively more for a rarer loss event.

After studying this issue purely in terms of the utility u, we now study it purely in terms of the transformation function f , by its effect on the Yaari index. Skipping straightforward details, the load factor is now LF Y aari = 1-f (p) 1-p . Thus, the load factor is a non-decreasing function of p if and only if f is star-shaped at 1. Among the RDEU agents with linear utility, those with transformation function star-shaped at 1 are precisely the ones that display the unanimous attitude of EU risk averters on this issue.

A related question (but notably less important) involving lotteries instead of insurance is the attitude of EU risk averters to a lottery with price C and probability p of winning a single prize K. The fair value of K is of course K f = C p , and the infimal prize for participation of the EU agent is

K i = C + u -1 ( u(0)-u(-C)(1-p) p ).
Similarly to the analysis for insurance, the top-prize load factor K i K f can be seen to be a non-increasing function of p: EU risk averters unanimously demand relatively higher prizes as compensation for lower winning probabilities.

The corresponding top-prize load factor under the Yaari index is p f (p) , that is nonincreasing if and only if the transformation function f is star-shaped at 0.

Combining these diverse attitudes to lottery loading or spread of gains and premium loading or spread of losses, we can see that although RDEU pessimism is more flexible than EU risk aversion, this flexibility is distinctly normative.

Some further remarks

Landsberger & Meilijson [START_REF] Landsberger | Lotteries, Insurance and Star-shaped utility functions[END_REF] introduced (in the EU framework) a notion of selective risk aversion that is modeled by star-shaped utility functions. Instead of the property of decreasing marginal utility displayed by concavity, star-shape yields decreasing average utility, as measured from the point of reference µ.

It may be of interest to notice the geometrical fact that the transformation functions that are star-shaped at both 0 and 1 are those expressible as pointwise minima of convex transformation functions. Since this class strictly contains the convex functions, aversion to both left-monotone and right-monotone increase in risk does not imply aversion to MPIR. Perhaps there is an interesting stochastic order hidden behind this fifth class.

In the framework of inequality measurement Chateauneuf [START_REF] Chateauneuf | Decreasing inequality : an approach through non-additive models[END_REF] proved that the Yaari [START_REF] Yaari | A controversial proposal concerning inequality measurement[END_REF] index of inequality measurement respects RDO if and only if f is majorized by the identity function and is star-shaped at 0. This is consistent with Corollary 1. In this context, for every income random variable X and any f star shaped at 0, the Yaari index can be interpreted as the average level of income after (rank-dependent) progressive income taxation.

Formulation of the main results

Once the four special classes above of transformation functions have been obtained from basic principles and justified intuitively, it becomes a natural question of interest to study the stochastic orders that model the choices of decision makers with such transformation functions. In other words, the task is to discover the different restrictive notions of meanpreserving increase in risk, of which the four canonical mean-preserving spreads ( 17)-( 20) are special cases. The main results are formulated in the four theorems below. Their proofs appear in the next section.

The rationale behind the three different statements in each of the following theorems, is that the first statement identifies the type of transformation function of a given DM with linear utility, from the DM's attitude to a specific type of mean-preserving increase in risk; the second statement qualifies the attitude to the various kinds of mean-preserving increase in risk of a DM with concave utility function, in terms of the DM's type of transformation function; the third statement identifies the type of dominance relating two given distributions with equal finite means, from unanimous preference pattern between the two by all DM's with linear utility function and a certain type of transformation function.

Thus, in each theorem, statements (i) and (ii) deal with an individual DM with concave utility and characterize the DM's type in terms of aversion to the various kinds of increase in risk, while statements (ii) and (iii) deal with a particular pair of distributions with equal means and characterize the type of increase in risk connecting these two distributions, in terms of the unanimous preference for one over the other by all DM's of a given type.

Theorem 2 Mean-preserving increase in risk and convex transformation function (i) (Chew,Karni & Safra [10], Yaari [START_REF] Yaari | The dual theory of choice under risk[END_REF]) Let a RDEU decision maker with linear utility function be averse to mean-preserving increase in risk. Then the decision maker's transformation function f is convex. (ii) (Chew, Karni & Safra [START_REF] Chew | Risk aversion in the theory of expected utility with Rank Dependent preferences[END_REF]) Let the transformation function be convex and the utility function be concave (e.g., linear). Then the RDEU decision maker is averse to meanpreserving increase in risk. (iii) (Chateauneuf [5], Yaari [START_REF] Yaari | Risk aversion without diminishing marginal utility[END_REF]) Let F and G be two distributions with the same finite mean, such that all RDEU decision makers with linear utility function and convex probability transformation function prefer F to G. Then G represents a mean-preserving increase in risk with respect to F . Theorem 3 Monotone increase in risk and transformation function below the identity function (i) (Quiggin [33]) Let a RDEU decision maker with linear utility function be averse to monotone increase in risk. Then the decision maker's transformation function f is bounded from above by the identity function. (ii) (Quiggin [START_REF] Quiggin | Increasing risk : another definition[END_REF]) Let the transformation function be bounded from above by the identity function, and the utility function be concave (e.g., linear). Then the RDEU decision maker is averse to monotone increase in risk. (iii) Let F and G be two distributions with the same finite mean, such that all RDEU decision makers with linear utility function and probability transformation function bounded from above by the identity function prefer F to G. Then G represents a monotone increase in risk with respect to F . Theorem 4 Left-monotone increase in risk and transformation function below the identity function, star-shaped at 1

(i)
Let a RDEU decision maker with linear utility function be averse to left-monotone increase in risk. Then the decision maker's transformation function f is bounded from above by the identity function and is star-shaped at 1. (ii) Let the transformation function be bounded from above by the identity function and star-shaped at 1, and the utility function be concave (e.g., linear). Then the RDEU decision maker is averse to left-monotone increase in risk. (iii) Let F and G be two distributions with the same finite mean, such that all RDEU decision makers with linear utility function and probability transformation function bounded from above by the identity function and star-shaped at 1 prefer F to G. Then G represents a left-monotone increase in risk with respect to F . Theorem 5 Right-monotone increase in risk and transformation function below the identity function, star-shaped at 0 (i) Let a RDEU decision maker with linear utility function be averse to right-monotone increase in risk. Then the RDEU decision maker's transformation function f is bounded from above by the identity function and is star-shaped at 0. (ii) Let the transformation function be bounded from above by the identity function and star-shaped at 0, and let the utility function be concave (e.g., linear). Then the decision maker is averse to right-monotone increase in risk. (iii) Let F and G be two distributions with the same finite mean, such that all RDEU decision makers with linear utility function and probability transformation function bounded from above by the identity function and star-shaped at 0 prefer F to G. Then G represents a right-monotone increase in risk with respect to F .

The four statements (ii) of Theorems 2, 3, 4 and 5 follow directly from Lemma 3 for the case of concave utility functions, provided these statements hold for linear utility functions, the case to be proven in the sequel.

Since the four classes of transformation function were constructed as necessary conditions for the proper preferences of decision makers when subjected to each of the canonical spreads ( 17)- [START_REF] Karni | Search Theory: the case of search with uncertain recall[END_REF], each stochastic order turns out to provide a characterization of choice by RDEU decision makers with linear utility and transformation function in the corresponding class, as claimed by Theorems 2, 3, 4 and 5.

Mean-preserving increase in risk

In order to prove Theorem 2 (ii), consider the following representations ( 22) and ( 23) of the Yaari functional (substitute u(z) = z in (1)) of a H-distributed random variable Z. Representation [START_REF] Landsberger | Demand for Risky Financial Assets: A Portfolio Analysis[END_REF] follows easily from first principles, the definition of the Stjeltjes integral as a limit of the integral of discretizations of the integrand. Representation ( 23) is meaningful for convex f . While the formula of integration by parts holds for all p 0 , ( 23) is obtained by substituting p 0 = 1.

V I,f (Z) = - ∞ -∞ xdf (H * (x)) = 1 0 (H * ) -1 (p)df (p) (22)
= f ′ (p) p p 0 (H * ) -1 (q)dq| 1 0 - 1 0 [ p p 0 (H * ) -1 (q)dq]df ′ (p) = f ′ (p) p p 0 (H * ) -1 (q)dq| 1 0 + 1 0 [ 1-p 1-p 0 H -1 (r)dr]df ′ (p) = f ′ (0)E[Z] + 1 0 L H (1 -p)(1 -p)df ′ (p) . (23)
It follows from [START_REF] Landsberger | Lotteries, Insurance and Star-shaped utility functions[END_REF]

that if f is convex and Y ∼ G is a MPIR with respect to X ∼ F , then V I,f (X) ≥ V I,f (Y ). This is so because E[X] = E[Y ], (1 -p)df ′ (p)
is a non-negative measure and the integrands are pointwise ordered in the direction needed, by the Lorenztype characterization of SSD following (5). This proves Theorem 2 (ii).

To prove Theorem 2 (iii), suppose that G does not represent a mean-preserving increase in risk with respect to F . A convex transformation function f must be displayed under which the Yaari functional shows strict preference for G. Under the assumption above, the distance L F (•) -L G (•) between the L-functions (see 5) fails to be non-negative. Let N ⊆ (0, 1) be the set of p on which L F (1 -p) -L G (1 -p) is strictly negative. Since N is a non-empty open set, it has positive Lebesgue measure. Motivated by representation [START_REF] Landsberger | Lotteries, Insurance and Star-shaped utility functions[END_REF], we want to build a convex transformation function f whose derivative f ′ increases only on N . With this in mind, let f : [0, 1] → [0, 1] be defined by

f (0) = 0 ; f ′ (q) = Leb(N ∩ [0, q])/ 1 0 Leb(N ∩ [0, p])dp , q ∈ (0, 1) . ( 24
)
where Leb stands for Lebesgue measure. The derivative of f measures the local density of the set N , so it increases only on N . Since f ′ is a properly normalized non-negative and non-decreasing function, f is a convex probability transformation function. By [START_REF] Landsberger | Lotteries, Insurance and Star-shaped utility functions[END_REF],

V I,f (X) -V I,f (Y ) = 1 0 [L F (1 -p) -L G (1 -p)](1 -p)df ′ (p) = N [L F (1 -p) -L G (1 -p)](1 -p)df ′ (p) < 0 . (25)
Hence, G is strictly preferred to F . This proves Theorem 2 (iii).

Remark. The simplifying assumption of continuity of f can be seen at play here, since a discontinuous f would render the RHS of ( 22) unstably dependent on which version of (H * ) -1 is used -left continuous, right continuous or otherwise.

Monotone increase in risk

Consider the representation [START_REF] Landsberger | Demand for Risky Financial Assets: A Portfolio Analysis[END_REF] of the Yaari functional of a H-distributed random variable and the notation [START_REF] Chateauneuf | A review of some results related to comonotonicity[END_REF] for the horizontal distance between distribution functions. If Y ∼ G represents a monotone increase in risk with respect to

X ∼ F then V I,f (X) -V I,f (Y ) = 1 0 D(1 -p)df (p)
is the expectation of a non-decreasing function of a f -distributed random variable. Since f is assumed to be majorized by the identity function, this "distribution" df (p) dominates the uniform distribution dp on (0, 1) by first degree stochastic dominance. Hence, an EU decision maker with utility function D prefers a f -distributed random variable to a uniformly distributed one. In other words,

V I,f (X) -V I,f (Y ) = 1 0 D(1 -p)df (p) ≥ 1 0 D(1 -p)dp = 1 0 [F -1 (1 -p) -G -1 (1 -p)]dp = E[X] -E[Y ] = 0 , (26)
thus proving Theorem 3 (ii).

To prove Theorem 3 (iii), suppose that G does not represent a monotone increase in risk with respect to F . We have to display a transformation function f supported from above by the identity function, under which the Yaari functional shows strict preference for G. Under the current assumption, the function D is not non-decreasing. Keeping in mind [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF], there is a distribution function f supported by the unit interval, that first-degree dominates the uniform distribution (i.e., f (q) ≤ q for all q ∈ (0, 1)), such that a EU-maximizer decision maker with utility function D strictly prefers the uniform distribution to the distribution f . This f can be taken to be continuous and strictly increasing, to conform with the properties of probability transformation functions. Since this provides a strict reversal of the inequality in [START_REF] Lehmann | Testing Statistical Hypotheses[END_REF], a transformation function majorized by the identity has been constructed, under which G is strictly preferred to F . This proves Theorem 3 (iii).

Left-monotone increase in risk

The proof of Theorem 4 is entirely analogous to the proof of Theorem 5 for right-monotone increase in risk. Since the latter has never been dealt with before, it is chosen as the setup for the proof.

Right-monotone increase in risk and transformation functions starshaped at zero

Clearly, this stochastic order is characterized by pointwise inequality between distances of Ψ G and Ψ F , just as left-monotone increase in risk is characterized by the corresponding inequality for the Φ functions (see [START_REF] Chateauneuf | Left-monotone reduction of risk, deductible and call[END_REF]).

If the transformation function f is below the diagonal and star-shaped at 0, then f (p) p is non-decreasing. Since it equals 1 at p = 1, consider the distribution function ν of a random variable U with values in the interval [0, 1] given by P

(U ≤ p) = ν(p) = f (p) p for p ∈ (0, 1], with P (U = 0) = ν(0) = lim p↓0 f (p) p . Under this notation, Lemma 5 For a bounded random variable Z distributed H, V I,f (Z) = E(Z) - 1 0 Ψ H (H -1 (1 -q))dν(q) (27)
Remark: It should be intuitively clear why is a representation like this useful: Since (see [START_REF] Chateauneuf | Left-monotone reduction of risk, deductible and call[END_REF]) right-monotone increase in risk is characterized by pointwise inequality between functions like Ψ H (H -1 (1 -q)), integration with respect to a positive measure will preserve the inequality, i.e., it will transfer it to the Yaari index.

Although we discovered representation [START_REF] Lorenz | Methods of measuring concentration of wealth[END_REF] of V I,f by a number of steps of integration by parts, we opt for proving it otherwise.

Proof of Lemma 5: The proof will be divided into two steps. In the second step the Lemma will be proved for random variables supported by finite sets. In the first step it will be shown that for arbitrary ǫ > 0, if P (|Z -Z| < ǫ) = 1, then each of the three functionals in [START_REF] Lorenz | Methods of measuring concentration of wealth[END_REF] changes by at most 2ǫ when Z is replaced by Z. This is enough because every random variable (say, Z) with bounded support can be uniformly ǫ-approximated by a random variable (call it Z) with finite support. Simply, let Z/ǫ be the integer part of Z/ǫ.

Step 1. The fact that |V I,f (Z) -V I,f ( Z)| and |E(Z) -E( Z)| are less than ǫ are immediate consequences of each of V I,f and E being (i) monotone with respect to First Degree Dominance and (ii) linear with respect to addition of constants. Details are skipped, except the proof for the integral term. Since |Z -Z| < ǫ a.s., |H -1 (1 -q) -H-1 (1 -q)| < ǫ for each q ∈ (0, 1) .

(

) 28
Since Ψ H (x) = E(Z -x) + = E(max(Z, x)) -x, subtract x from each term in E(max( Z, x + ǫ)) -ǫ = E(max( Z -ǫ, x)) ≤ E max(Z, x)

≤ E(max( Z + ǫ, x)) = E(max( Z, x -ǫ)) + ǫ [START_REF] Marshall | Inequalities: theory of majorization and its applications[END_REF] to obtain that for all x, Ψ H (x + ǫ) ≤ Ψ H (x) ≤ Ψ H (x -ǫ). The third and last preparatory step needed is inferring from the representation Ψ H (x) = ∞ x (1-H(t))dt that the derivative of Ψ is in (-1, 0). Indeed, Ψ H ( H-1 (1 -q)) -2ǫ ≤ Ψ H ( H-1 (1 -q) + 2ǫ) ≤ Ψ H ( H-1 (1 -q) + ǫ)

≤ Ψ H (H -1 (1 -q)) ≤ Ψ H ( H-1 (1 -q) -ǫ) ≤ Ψ H ( H-1 (1 -q) -2ǫ) ≤ Ψ H ( H-1 (1 -q)) + 2ǫ , (30)
finishing Step 1.

Step 2. Proof for the discrete case, by induction on the number of atoms of the distribution.

If the distribution has two atoms, express it as (x 1 , p 1 ; x 2 , p 2 ). Then E(Z) -V I,f (Z) = (x 1 + p 2 (x 2 -x 1 )) -(x 1 + f (p 2 )(x 2 -x 1 )) = (p 2 -f (p 2 ))(x 2 -x 1 ). Evaluation of the integral yields 1 0 Ψ H (H -1 (1 -q))dν(q) = Ψ(x 2 )ν(0, p 2 ) + Ψ(x 1 )ν(p 2 , 1) = Ψ(x 1 )ν(p 2 , 1) = (x 2 -x 1 )p 2 ν(p 2 , 1) .

(

) 31
The two are equal if and only if ν(p 2 , 1) = 1 -f (p 2 )/p 2 , or ν(0, p 2 ) = f (p 2 )/p 2 , that holds by definition of the probability measure ν.

The inductive step consists of replacing the distribution H = (x 1 , p 1 ; x 2 , p 2 ; . . . ; x k , p k ) of Z (with x 1 < x 2 < • • • < x k and k ≥ 3) by H of Z obtained from H by replacing x 1 by x 2 . In other words, P ( Z = x i ) = P (Z = x i ) = p i for 3 ≤ i ≤ k and P ( Z = x 2 ) = p 1 + p 2 . Since the induction hypothesis postulates that [START_REF] Lorenz | Methods of measuring concentration of wealth[END_REF] holds for Z, it remains to show that the difference

(E(Z) -V I,f (Z)) -(E( Z) -V I,f ( Z)) = (E(Z) -E( Z)) -(V I,f (Z) -V I,f ( Z)) = -(x 2 -x 1 )p 1 + (x 2 -x 1 )(1 -f (1 -p 1 )) = (x 2 -x 1 )(1 -p 1 -f (1 -p 1 )) ( 32
)
is equal to the difference

1 0 Ψ H (H -1 (1 -q))dν(q) - 1 0 Ψ H ( H-1 (1 -q))dν(q) = 1-p 1 1-p 1 -p 2
Ψ H (H -1 (1 -q))dν(q) +

The two are equal if and only if ν(1 -p 1 , 1] = 1 -f (1 -p 1 )/(1 -p 1 ), or ν[0, 1 -p 1 ) = f (1 -p 1 )/(1 -p 1 ). Again, this follows from the definition of the probability measure ν.

This finished the proof of Lemma 5. Returning now to the proof of Theorem 5, it follows from (27) that

V I,f (X) -V I,f (Y ) = E[X] -E[Y ] + 1 0
[Ψ G (G -1 (1 -q)) -Ψ F (F -1 (1 -q))]dν(q) . (34) Hence, if the expression in square brackets in [START_REF] Ross | Some stronger measures of risk aversion in the small and the large with applications[END_REF] is non-negative, then V I,f (X) ≥ V I,f (Y ) as soon as E[X] ≥ E[Y ]. But, as pointed out earlier, non-negativity of the integrand Ψ G (G -1 (1 -q)) -Ψ F (F -1 (1 -q)) for all q is one of the equivalent definitions of rightmonotone increase in risk. In other words, decision makers with probability transformation function star-shaped at 0 are averse to (mean-reducing or mean-preserving) left-monotone increase in risk. This proves statement (ii) of Theorems 4 and 5.

To prove Theorem 5 (iii), suppose that G does not represent a right-monotone increase in risk with respect to F . We have to display a transformation function f supported from above by the identity function and star-shaped at 0, under which the Yaari functional shows strict preference for G. Under the current assumption, the clue to the construction of a transformation function under which G is strictly preferred is provided by expression [START_REF] Ross | Some stronger measures of risk aversion in the small and the large with applications[END_REF]. The set N of the previous case (see [START_REF] Landsberger | Comonotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion[END_REF]) is replaced by the set M ⊆ (0, 1) where Ψ G (G -1 (1 -q)) -Ψ F (F -1 (1 -q)) is strictly negative, a non-empty open set. In analogy to the previous case, let the measure ν in (34) be defined by x and I A is the indicator function of the set A.

1 1-p 1 Ψ 1 1-p 1 -p 2 Ψ

 1112 H (H -1 (1 -q))dν(q) -H ( H-1 (1 -q))dν(q) = Ψ(x 2 )ν(1 -p 1 -p 2 , 1 -p 1 ] + Ψ(x 1 )ν(1 -p 1 , 1] -Ψ(x 2 )ν(1 -p 1 -p 2 , 1] = (Ψ(x 1 ) -Ψ(x 2 ))ν(1 -p 1 , 1] = (x 2 -x 1 )(1 -p 1 )ν(1 -p 1 , 1] .

  ν(A) = λI A (0) + (1 -λ) Leb(A ∩ M ) Leb(M ) ,for Borel subsets A of the unit interval, where λ ∈ [0, 1) is a desired value of ν({0}) = lim x↓0 f (x)

Proof of Theorems 2, 3, 4 and

The four statements (i) of Theorems 2, 3, 4 and 5 are corollaries of Lemma 4, since the four canonical examples are special cases of the corresponding stochastic orders.