Abstract Various notions of risk aversion can be distinguished for the class of rank-dependent expected utility (RDEU) preferences.

Transcription

1 Abstract Various notions of risk aversion can be distinguished for the class of rank-dependent expected utility (RDEU) preferences. We provide the rst complete characterization of the RDEU orderings that are risk-averse in the sense of Jewitt (1989). We also extend Chew, Karni and Safra s (1987) important characterization of strong risk aversion (Rothschild and Stiglitz, 197) by relaxing strict monotonicity and di erentiability assumptions, and allowing for discontinuities in the probability transformation function. The important special case of maximin choice falls within this relaxed RDEU class. It is shown that any strongly risk-averse RDEU order is a convex combination of maximin and another RDEU order with concave utility and continuous, concave probability transformation. Our proof of the result on strong risk aversion is also simpler (as well as more general) than that of Chew, Karni and Safra (1987). 1

2 Risk Aversion in RDEU Matthew J. Ryan February 26 1 Introduction The rank-dependent expected utility (RDEU) model is the most widely used, and arguably the most empirically successful, generalization of expected utility (EU). 1 Given a cumulative distribution function F on [; 1], the RDEU model evaluates F according to the functional: U (F ) = u d (h F ) (1) In addition to the utility function u, this expression involves a non-decreasing transformation function h : [; 1]! [; 1], satisfying h () = and h (1) = 1. The EU special case is obtained if h is linear. Risk attitude in the RDEU model resides jointly in u and h. In particular, aversion to risk can be separated from diminishing marginal utility of wealth (Chateauneuf and Cohen, 1994; Wakker, 1994). Moreover, unlike EU, RDEU preferences allow us to discriminate amongst di erent notions of risk aversion (Cohen, 1995). Chew, Karni and Safra (1987, Corollary 2) were the rst to demonstrate that risk aversion in the sense of Rothschild and Stiglitz (197) also known Special thanks to Suren Basov, Simon Grant, Warren Moors, John Quiggin and Arkadii Slinko for valuable comments on earlier drafts. My thanks also to seminar audiences at the Universities of Auckland and Melbourne. The usual disclaimer applies. 1 RDEU began life as anticipated utility (Quiggin, 1982). Yaari (1987) introduced an important special case which he called the dual theory. References to the Quiggin-Yaari functional under the rank-dependent nomenclature seem to have begun with Chew, Karni and Safra (1987). The RDEU (sometimes, RDU) terminology is now rmly established. 2

3 as strong risk aversion imposes straightforward and independent restrictions on u and h: each function must be concave. Chew, Karni and Safra s result is an important benchmark in RDEU theory. Unfortunately, their proof requires strong assumptions on u and h, including strict monotonicity and di erentiability, and employs an argument based on Gâteaux derivatives not a standard part of most economists toolkit. In Section 3, we extend Chew, Karni and Safra s result by showing that it su ces to assume weak monotonicity of u and h, and continuity of u. 2 We also employ a more straightforward method of proof. Transformation functions that are dis-continuous or non-strictly increasing are more than just an esoteric curiosity. They include all-but-one members of the NEO-additive class of RDEU preferences (Chateauneuf, Eichberger and Grant, 24). This is a tractable, two-parameter generalization of EU that o ers a more realistic description of human decision-making. The added realism comes from the fact that all NEO-additive transformations exhibit the inverse-s shape consistently observed in experimental studies of risky choice (Wakker, 21). The NEO-additive transformation functions are those that are linear on (; 1). Functions within this class may be discontinuous at zero or one, and may be constant on (; 1). The indicator function for (; 1], for example, is a NEO-additive transformation and corresponds to maximin choice. Under our weaker assumptions on u and h, we show that any strongly risk-averse RDEU functional is a convex combination of maximin and another RDEU functional with concave and continuous u and h. In other words, maximin is the canonical example of a strongly risk-averse RDEU ordering with a discontinuous transformation function. More recently, Chateauneuf, Cohen and Meilijson (25) 3 have provided a characterization of monotone risk aversion (Quiggin, 1991) for RDEU preferences. This is a less demanding notion than strong risk aversion, and imposes a joint restriction on u and h: non-concavity of the utility function may be compensated through su ciently pronounced pessimism embodied in h. We adapt the ideas of Chateauneuf, Cohen and Meilijson (25) in Section 4 to provide a characterization of another important risk aversion concept Jewitt s (1989) notion of aversion to location-independent risk. A random 2 Note that U cannot be continuous with respect to the weak convergence topology unless h is continuous. 3 This paper rst appeared in the Cahiers d Ecomath in

4 variable X 2 is said to exhibit greater location-independent risk than X 1 if the di erence E [X 1 j X 1 x 1 (p)] E [X 2 j X 2 x 2 (p)] is non-increasing in p 2 (; 1), where x i (p) denotes the (1p) th percentile of X i. This notion was originally motivated by expected utility considerations, but has proved useful beyond the EU context. For example, Jewitt s is the weakest notion of risk aversion consistent with Arrow s famous theorem on the optimality of deductible insurance: see Landsberger and Meilijson, (1994b, p.664), Vergnaud (1997) and Chateauneuf, Cohen and Vergnaud (21). Ours is the rst characterization of the RDEU preferences that are risk averse in this sense. The following section presents some preliminary ideas. The RDEU model is described rst. We then recall four notions of risk aversion, and show that no two are equivalent within the RDEU class. For each notion of risk aversion, we also describe a class of elementary transformations of distribution functions that add risk of the appropriate sort. These elementary transformations are essential to the proofs that follow. Section 3 proves our result on the characterization of strong risk aversion. Results for Jewitt s locationindependent notion are in Section 4. Section 5 o ers concluding remarks. Less informative proofs are relegated to an Appendix. 2 Preliminaries 2.1 Rank-Dependent Expected Utility (RDEU) Let D denote the set of all cumulative distribution functions (CDF s) on the unit interval [; 1]. Capital letters, such as F and G, will be used to denote elements of D, and the same letters with an overbar F, G, and so forth to denote the corresponding decumulative distribution functions (DDF s). If c 2 [; 1] then c 2 D is the indicator function for [c; 1]. For any F 2 D we use (F ) to denote the mean of F. We also de ne, for any F 2 D, the following inverses : inf fx j F (x) > pg if p < 1 F 1 (p) = sup fx j F (x) < pg if p = 1 inf x j F (x) < p if p > F 1 (p) = F 1 (1 p) = sup x j F (x) > p if p = 4

5 It is useful to observe that if F 1 is interpreted as a random variable on [; 1], endowed with the usual Borel -algebra and Lebesgue measure, then its CDF is F. An RDEU preference ordering % on D is such that: F % G, u d (h F ) u d (h G) (2) for some utility function u : [; 1]! [; 1] and CDF transformation function h : [; 1]! [; 1]. Throughout the paper, we shall maintain the assumptions that u and h are non-decreasing and normalized: u () = h () = and u (1) = h (1) = 1. We also assume that u is continuous (and hence surjective) to ensure that the Riemann-Stieltjes integrals in (2) exist (Wheeden and Zygmund, 1977, Theorem 2.24). Let us pause to consider an important special case of (2) in which h is the indicator function for (; 1]. Then: F % G, u F 1 () u G 1 () (3) In other words, each distribution is assigned a value equal to the in mum of utility over outcomes in its support. If u is strictly increasing, this is maximin choice. For our purposes, it is convenient to re-express the functional (1) in an equivalent form. To do so, we rst de ne the following dual to h: 4 h (z) = 1 h (1 z). Thus, h : [; 1]! [; 1] is also non-decreasing and normalized. We observe, using integration by parts (ibid., Theorem 2.21), that u d (h F ) = h F du. For example, if F is the CDF of a discrete random variable with values then u d (h F ) = where x =. x 1 < x 2 < < x n nx h F (x i 1 ) [u (x i ) u (x i 1 )] i=1 4 Chateauneuf, Cohen and Meilijson (25) call h the probability perception function. 5

6 2.2 Notions of risk aversion Recall the notion of risk aversion proposed by Rothschild and Stiglitz (197), also known as strong risk aversion (Cohen, 1995). We say that F second order stochastically dominates (SOSD) G if Z x F G dz for every x 2 [; 1]. De nition 1 Preferences % on D exhibit strong risk aversion if F % G whenever F and G have the same means and F SOSD G. This is the strongest of the extant risk aversion concepts. It is also immediate from (3) and the de nition of SOSD that: Lemma 2 If h is the indicator function for (; 1], then the RDEU ordering (2) exhibits strong risk aversion. We next recall four alternative notions of risk aversion. Say that F is less location-independent risky than G (F LLIR G) if the di erence E F 1 jf 1 F 1 (p) E G 1 j G 1 G 1 (p) is non-increasing in p 2 (; 1). This is equivalent to F 1 (p) Z F 1 (p) F (z) dz G 1 (p) Z G 1 (p) G (z) dz (4) for all p 2 (; 1). 5 Chateauneuf, Cohen and Meilijson (24, 2.3.4) show that the LLIR order on equal-mean distributions is important in optimal search problems, such as a buyer who pays a xed search cost per seller sampled while searching for the best price. Assuming that the buyer follows 5 I have expressed all de nitions in terms of the DDF s for comparison with Figure 1. But Jewitt s notion is easier to parse in its equivalent CDF version (Jewitt, 1989, Theorem 3): Z F 1 (p) F (z) dz Z G 1 (p) G (z) dz. 6

7 an optimal stopping rule, the expected total cost (expected price plus search cost) and the expected search cost are both monotone with respect to LLIR: if sellers prices are independent random draws from G then the expected total cost (respectively, search cost) is lower (respectively, higher) than if prices are drawn from F, where F LLIR G. Jewitt (1989) introduced the following: De nition 3 Preferences % on D exhibit Jewitt risk aversion if F % G whenever F and G have the same means and F LLIR G. We de ne F to be less Bickel-Lehmann dispersed than G (F LBLD G) if G 1 F 1 is non-increasing on (; 1). The following notion is due to Quiggin (1991) and Landsberger and Meilijson (1994a): De nition 4 Preferences % on D are monotone risk averse if F % G whenever F and G have the same means and F LBLD G. The nal two concepts are based on the idea that one may increase risk by splitting an atom while preserving the mean. They do not de ne increases in risk starting from any atomless distribution. First is the well-known concept of weak risk aversion, which requires that any distribution G is weakly dispreferred to the distribution that places unit mass on the mean of G. De nition 5 Preferences % on D exhibit weak risk aversion if (G) % G. Finally, we have the less familiar conditional certainty equivalent risk aversion. De nition 6 Preferences % on D exhibit conditional certainty equivalent risk aversion if pf + (1 p) (G) % pf + (1 p) G for any F; G 2 D and any p 2 [; 1]. To understand the di erences between these concepts, it is useful to consider the discrete CDF s F and G whose associated DDF s are depicted in Figure 1. The bold lines describe G and the lighter lines F. The di erence in means Z Z x df x dg is equal to the area of region A less that of region B. As indicated in Figure 1, we assume (a a) ( ) = b b (5) so that both distributions share the same mean. 7

8 1 a a A b b B α α β β 1 ( α α )( a a ) = ( β β )( b b ) Figure 1: DDF s F and G (bold) with equal means Notice that F has more weight concentrated near the middle of the distribution than G: moving from the random variable F 1 to the random variable G 1, weight is shifted from to and from to. When should we regard F 1 as less risky than G 1, and hence require that F % G for any risk-averse decision-maker? This depends on which notion of risk aversion we adopt. For example, it is clear that F SOSD G in Figure 1. We say that G is an elementary mean-preserving spread (MPS) of F. 6 More precisely, let us de ne M = a; a; b; b; ; ; ; < < 1; b < b a < a 1 and (5). Then G is an elementary MPS of F if F; G 2 D are nite step functions and there is some a; a; b; b; ; ; ; 2 M with F, G constant on [; ) and 6 Compare Rothschild and Stiglitz (197, p.229) and Dasgupta, Sen and Starrett (1973, Lemma 2(iii)). Hong and Hui (1995, p.413) de ne this term slightly di erently, requiring that all probabilities are rational and =. 8

9 ;, and F (x) G (x) = 8 < : a a if x 2 [; ) b b if x 2 ; otherwise It is straightforward to observe that F SOSD G whenever G is an elementary MPS of F. Thus, a necessary condition for strong risk aversion is aversion to elementary MPS s: that is, F % G whenever G is an elementary MPS of F. In fact, this condition is also su cient see Theorem 1. Returning to Figure 1, F LLIR G i a = 1, since otherwise condition (4) is violated for p 2 (a; a). Say that G is an elementary Jewitt mean-preserving spread (JMPS) of F if G is an elementary MPS of F and (6) holds for some a; a; b; b; ; ; ; 2 M with a = 1. The concept of an elementary JMPS strengthens that of an elementary MPS by requiring that the worst possible outcome become even worse. A necessary condition for Jewitt risk aversion is aversion to elementary JMPS s: F % G whenever G is an elementary JMPS of F. Once again, this condition turns out to be su cient as well see Theorem 2. For F LBLD G in Figure 1 we need both a = 1 and b =. Say that G is an elementary mean-preserving out-stretch (MPOS) of F if G is an elementary JMPS of F and (6) holds for some a; a; b; b; ; ; ; 2 M with a = 1 and b =. 7 In this case, not only must the worst outcome become worse, but the best outcome must also become better. Therefore, a necessary condition for monotone risk aversion is aversion to elementary MPOS s: F % G whenever G is an elementary MPOS of F. For F in Figure 1 is the indicator for [; (G)) i G is an elementary MPOS of F and =. It follows that a necessary condition for weak risk aversion is that F % G whenever G is an elementary MPOS of F and (6) holds for some a; a; b; b; ; ; ; 2 M with a = 1, b = and =. Finally, a necessary condition for conditional certainty equivalent risk aversion is that F % G whenever G is an elementary MPS of F and (6) holds for some a; a; b; b; ; ; ; 2 M with =. The discussion of Figure 1 suggests a clear ordering amongst the rst four risk aversion concepts. This impression is con rmed by the following Proposition, whose proof may be found in the Appendix: 8 7 Compare Chateauneuf, Cohen and Meilijson (25, p.662). 8 The content of Proposition 7 is well known. However, proofs of its various parts are scattered through the literature, so a proof is included here for completeness and ease of (6) 9

10 Proposition 7 The following implications hold: % exhibit strong risk aversion + % exhibit Jewitt risk aversion + % exhibit monotone risk aversion + % exhibit weak risk aversion However, none of the converse implications holds within the RDEU preference class. Only by admitting non-linear transformation functions can we distinguish these various notions of risk aversion. Within the class of expected utility preferences, all four notions are equivalent. This follows, for example, from Proposition 7 and Rothschild and Stiglitz (197, Theorem 2). Since weak risk aversion is clearly implied by conditional certainty equivalent risk aversion, weak risk aversion is the weakest of the ve notions. We shall later show that conditional certainty equivalent risk aversion is identical to strong risk aversion for RDEU preferences (Corollary 13). 9 Finally, let us observe that F % G in Figure 1 if and only if h F du which is equivalent to u () u () h (a) h (a) a a h G du, u u! given the equal-mean condition (5). We therefore obtain: h b! h (b) b b (7) Proposition 8 The RDEU preferences (2) exhibit (i) aversion to elementary MPS s i (7) for all a; a; b; b; ; ; ; 2 M; reference. 9 See also Theorem 2 in Hong and Hui (1995), which implies this result for the case of u and h continuous and strictly increasing. 1

11 (ii) aversion to elementary JMPS s i (7) for all with a = 1; a; a; b; b; ; ; ; 2 M (iii) aversion to elementary MPOS s i (7) for all a; a; b; b; ; ; ; 2 M with a = 1 and b = ; and (iv) weak risk aversion only if (7) for all a = 1, b = and =. a; a; b; b; ; ; ; 2 M with (v) conditional certainty equivalent risk aversion only if inequality (7) holds for all a; a; b; b; ; ; ; 2 M with =. Propositions 7 and 8 are fundamental to the arguments that follow. We rst use Proposition 8(iv) to establish the following useful fact: Lemma 9 Unless h is the indicator on (; 1], a necessary condition for the preferences % in (2) to exhibit weak risk aversion is that there exists some ^x 2 (; 1] with u strictly increasing on [; ^x] and u (^x) = 1. Proof. See the Appendix. Lemma 9 and Proposition 7 show that risk aversion of any sort imposes strict monotonicity of utility up to its maximum value, unless preferences are of the maximin variety (3). 3 Strongly risk-averse RDEU preferences The main result of this section is that the strongly risk averse RDEU orderings are precisely those with u and h concave, or h the indicator for (; 1] (Corollary 12). To clarify the underlying logic, we argue in two steps. First, we prove the result under a slight strengthening of our assumptions on u, which supposes the existence of some x 2 (; 1) at which u is di erentiable with u (x) >. In general, continuity, monotonicity and surjectivity do not su ce for this recall the Cantor-Lebesgue function. Next, we show that strong risk aversion implies a strengthening Lemma 9: the derivative of u must be strictly positive at every point of di erentiability in (; ^x). The main result then follows as a corollary. 11

12 Theorem 1 Suppose there exists an x 2 (; 1) at which u is di erentiable with u (x) >. Then the preferences % in (2) exhibit strong risk aversion i (i) h is concave, and (ii) either u is concave or h is the indicator for (; 1]. This result shows that strong risk aversion imposes a good deal of continuity and monotonicity on the transformation function. Concavity of h excludes discontinuities at any x >, and further implies that h is strictly increasing up to its maximum value. Theorem 1 also reveals that maximin choice is fundamental to strong risk aversion. Since h can be discontinuous only at zero, we may write: ( where and u d (h F ) = u (F 1 ()) if h (+) = 1 h (+) u (F 1 ()) + [1 h (+)] R 1 u d ~h F if h (+) < 1 ~h (z) = h (+) = lim z# h (z) ( h(z) h(+) 1 h(+) if z 2 (; 1] if z = Note that ~ h is continuous. Thus, any strongly risk-averse RDEU ordering is a convex combination of maximin and a strongly risk-averse RDEU ordering with a continuous (and concave) probability transformation function. Proof of Theorem 1. We prove Theorem 1 in two steps: (I) we rst show that % exhibit aversion to elementary MPS s i (i) and (ii) hold; and then (II) we show that aversion to elementary MPS s su ces for strong risk aversion. Aversion to elementary MPS s i (i) and (ii). Su ciency is obvious from Proposition 8. For the necessity, let x 2 (; 1) be such that u (x) exists and u (x) >. Consider sequences f n g 1 n=1 n no and 1 in [; 1] such that n " x, n # x and n=1 (a a) (x n ) = b b n x 12. (8)

13 for each n. 1 Then u (x) u ( n ) h (a) x n a h (a) u n n 1 u (x) A h b! h (b) x b b for each n. Taking limits: h (a) a h (a) a h b h (b) b b for any a < a b < b. But this just says that h is convex, and hence h is concave. Therefore, aversion to elementary MPS s implies (i). We may apply a similar argument to show u concave if there is some z 2 (; 1) at which h (z) exists and h (z) >. But concavity of h ensures that this is so, unless h is the indicator for (; 1]. See this as follows. Concavity entails that h is strictly increasing up to its maximum value, and is Lipschitz continuous on any closed subinterval of (; 1) Wheeden and Zygmund (1977, Theorem 7.43). Therefore, either h (x) = 1 for all x >, or there exists a subinterval [x; y] (; 1) with h (x) < h (y) and some z 2 (x; y) at which h is di erentiable and h (z) > (ibid., Theorem 7.43). Aversion to elementary MPS s implies strong risk aversion. If h is the indicator for (; 1], then preferences exhibit strong risk aversion by Lemma 2. Suppose, therefore, that h is not the indicator for (; 1]. Rothschild and Stiglitz (197, Lemmas 1 and 2) show that F and G are the L 1 -limits of sequences ff n g 1 n=1 and fgn g 1 n=1 of CDF s with nite ranges, such that G n is obtained from F n by a ( nite) sequence of elementary MPS s. However, potential discontinuity of h at zero dictates the need for extra care when inferring F % G, since standard arguments based on the continuity of integrals no longer apply That is: n + (1 ) n = x for each n, where (a a) = (a a) + b b. 11 Since maximin preferences are not continuous with repect to the weak convergence topology, we cannot apply the technique in Hong and Hui (1995, 4) either, as their Lemma 1 requires continuous preferences. 13

14 Recall the decomposition (8). Since ~ h is continuous: 12 ~h F n du! and ~h G n du! ~h F du. ~h G du. It is also obvious from Rothschild and Stiglitz s construction of ff n g 1 n=1 and fg n g 1 n=1 that (F n ) 1 ()! F 1 () and (G n ) 1 ()! G 1 (). Thence, using (8) and the continuity of u: h F n du! h F du and h G n du! h G du. Since Z h F n 1 du for each n, the desired result follows. h G n du Lemma 11 Let ^x be as de ned in Lemma 9. Unless h is the indicator for (; 1], a necessary condition for the preferences % in (2) to exhibit strong risk aversion is that there not exist any z 2 (; ^x) at which u is di erentiable and u (z) =. Proof. See the Appendix. Since u is monotone, it is di erentiable almost everywhere (Wheeden and Zygmund, 1977, Corollary 7.23). Lemma 11 implies that its derivative is strictly positive at every point of di erentiability in (; ^x). We may now immediately deduce the main result of this section: 12 Recall that u is strictly increasing on [; ^x] with u (^x) = 1. We may therefore treat du as a measure on the Borel subsets of [; 1] that is absolutely continuous with respect to the uniform measure on the Borel subsets of [; ^x]. 14

15 Corollary 12 The preferences % in (2) exhibit strong risk aversion i (i) h is concave, and (ii) either u is concave or h is the indicator for (; 1]. We also obtain: Corollary 13 The preferences % in (2) exhibit strong risk aversion i they exhibit conditional certainty equivalent risk aversion. Proof. The only if part is obvious. For the if part, note that, in the proving the equivalence of aversion to elementary MPS s and (i) and (ii) in Theorem 1, we used = = x. Therefore, the same proof shows that the necessary condition for conditional certainty equivalent risk aversion (Proposition 8(v)) implies (i) and (ii). Hence conditional certainty equivalent risk aversion implies strong risk aversion. So far as we are aware, Chew, Karni and Safra (1987, Corollary 2) is the rst characterization of strong risk aversion in the RDEU context. Their result assumes u and h are strictly increasing and di erentiable. This ensures the Gâteaux di erentiability of the RDEU functional; a fact which Chew, Karni and Safra exploit in their proof. Our proof makes no appeal to such exotica. Indeed, if we had assumed di erentiability of u and h, our proof would become very straightforward. The only technical complexities arise from the need to accommodate potential discontinuities in h and potential lack of absolute continuity of u. But apart from an appeal to results on Lipschitz continuity at one point, even these di culties are overcome entirely by consideration of the fundamental inequality (7). After completing a rst draft of our proof, we learned of three related results. Grant and Kajii (1994, Proposition 2) characterize strong risk aversion for their AUSI-EU model, in which h (z) = z with >. The published version of their paper Grant and Kajii (1998) makes mention of this result, but excludes the proof. Schmidt and Zank (22, Theorem 1) characterize strong risk aversion for cumulative prospect theory (a generalization of RDEU), assuming strictly increasing utility and probability transformation functions. Also, Schmidt and Zank restrict attention to random variables with nitely many outcomes. Nevertheless, their arguments share many common features with ours. Finally, Hong and Hui (1995) state a result similar to Theorem 1, but under the stronger assumptions that both u and h are continuous and strictly 15

16 increasing. Their proof, too, has many similarities to ours. However, the proof of their result is actually incomplete (ibid., Appendix 2). They implicitly assume that u and h (denoted v and g respectively in their paper) each have a strictly positive derivative somewhere on the interior of their domains. However, as the Cantor-Lebesgue function illustrates, this need not be so, even for continuous, strictly increasing and surjective functions. 13 Our Lemma 11 and the third paragraph in the proof of Theorem 1 provide the necessary auxiliary arguments. We may use Corollary 12 to obtain the following characterization of the SOSD relation between equal-mean distributions. Corollary 14 Suppose that F and G have equal means. Then F SOSD G i u d (h F ) u d (h G) for any concave, non-decreasing and continuous u : [; 1]! R and any concave, non-decreasing h : [; 1]! [; 1] with h () = and h (1) = 1. Proof. The only if part follows directly from Corollary 12. In particular, any non-constant utility function can be normalized by a positive a ne transformation; and the result is trivial for constant utility functions. Conversely, suppose Z x F dz > Z x G dz for some x 2 (; 1). Taking h to be the identity function and u (z) = z x if z < x if z x we nd that G F since u df = Z x F dz < Z x G dz = u dg. The following analogues of Corollary 14 are well-known. 13 An early draft of the present paper fell into the same error. We are most grateful to Suren Basov for having pointed it out. 16

17 Proposition 15 If F and G have equal means, then F SOSD G i u df u dg for any concave, non-decreasing and continuous function u : [; 1]! R. Proof. The only if direction is provided by Corollary 14. The converse follows by the same argument as in the proof of Corollary 14. Proposition 16 If F and G have equal means, then F SOSD G i z d (h F ) z d (h G) for any concave, non-decreasing function h : [; 1]! [; 1] that satis es h () = and h (1) = 1. Proof. The only if direction again follows from Corollary 14. For the converse, if for some x 2 (; 1), set h (z) = Z x 8 < : F dz > Z x G dz if z 1 x z x (1 x) x if z > 1 x and observe that G F (with u the identity function). Proposition 15 is familiar to economists from the work of Rothschild and Stiglitz (197). Proposition 16 is a less well-known result of Hardy, Littlewood and Polya (1929, Theorem 1). 14 It is used by Yaari (1987, Theorem 2) to show that RDEU preferences exhibit strong risk aversion when u is linear and h is concave, non-decreasing In fact, Hardy, Littlewood and Polya s result makes no reference to h being normalized which is clearly redundant but does make the further assumption that h is continuous. The latter may also be dropped. 15 Yaari also makes the assumption that h is continuous, since his axioms ensure that such exists. 17

18 4 RDEU and Jewitt risk aversion Chateauneuf, Cohen and Meilijson (25) characterize monotone risk aversion for u and h strictly increasing. Adapting aspects of their argument, we here provide an analogous characterization of the RDEU orders that are risk averse in the sense of Jewitt (1989). However, our proof does not require strict monotonicity of u or h. We begin with some de nitions. Let and ^P h = inf <z <z<1 Q^x u = sup < < ^x 1 h (z) 1 z u h (z) h (z ) z z u! u () u (). When ^x = 1, Chateauneuf, Cohen and Meilijson (25) refer to Q^x u as an index of greediness for u. 16 It measures the extent of non-concavity of the utility function. Note that Q^x u 1 when u is strictly increasing on [; ^x], with equality precisely when u is concave. The quantity ^P h is a variant on 1 P h = inf <z<1 h (z) 1 z h (z) which Chateauneuf, Cohen and Meilijson (25) describe as an index of pessimism for h. Consider, for example, a two-outcome lottery whose better outcome occurs with probability z. Then 1 h (z) h (z) > 1 z z implies a pessimistic transformation of the odds ratio: the relative weighting of the bad outcome is increased. For example, if h is the indicator for f1g, corresponding to maximin preferences, then P h = 1 so h is maximally pessimistic. If we call 1 h (z) h (z) P h (z) = (9) 1 z z 16 This index is denoted G u in their paper. We have avoided this notation because of potential confusion with a CDF. 18 z

19 the degree of pessimism at z, then monotone risk aversion requires that the degree of pessimism over all z 2 (; 1) is bounded below by Q^x u 1 (Chateauneuf, Cohen and Meilijson, 25, Theorem 1). Suppose, for example, that P h < 1, which implies P h (z) < 1 for some z 2 (; 1). Then the RDEU preferences (2) cannot be monotone risk averse (or even weakly risk averse), 17 and it is easy to see why by considering Figure 1. Let a = 1, b = a = z, b = and = = x. In this case, the random variable G 1 shifts weight from x and distributes it in a mean-preserving fashion over and. If u were linear, then P h (z) < 1 would imply a strict preference for G over F. But even if u is not linear, provided we can nd some x 2 (; 1) at which it is di erentiable with u (x) >, then choosing and close enough to x delivers the same conclusion. Now consider ^P h. It is obvious that ^P h P h. In particular, we may have ^P h < 1 < P h. 18 We show below that aversion to location-independent risk requires ^P h Q^x u, which is a more demanding requirement than that for monotone risk aversion (as we should expect). For example, if h is linear in a neighbourhood of 1, then ^P h 1 so it is necessary for u to be concave in order for preferences to be Jewitt risk averse. This is not the case for monotone risk aversion see Chateauneuf, Cohen and Meilijson (25, Example 2). More generally, we can see that ^P h < 1 is inconsistent with Jewitt risk aversion as follows. Suppose 1 h (z) h (z) h (z ) < 1 z z z for some z; z with < z < z < 1. Setting a = 1, b = a = z, b = z and = = x in Figure 1, we infer a strict preference for G over F when u is linear, and hence a violation of Jewitt risk aversion. Since Lemma 19 ensures the existence of x 2 (; 1) at which u is di erentiable with u (x) >, then we may again choose and close enough to x to arrive at the same conclusion. The following lemmata will be useful for proving the main results of this section (Theorems 2 and 21). 17 This fact is also demonstrated by Chateauneuf and Cohen (1994). 18 For example, if then P h = 3 2 while ^P h =. 8 < h (z) = : if z < z 1 2 if z

20 Lemma 17 [Chateauneuf, Cohen and Meilijson (25, Lemma 1)] If u is strictly increasing on [; ^x], then u u! u () u () Q^x u = sup < < ^x for any >. ( )( )= Lemma 18 If h (x) =x is non-increasing on (; 1], then 1 ^P h = inf b < b a < 1 h (a) 1 a h b h (b) b b Lemma 19 Unless h is the indicator on (; 1], a necessary condition for the preferences % in (2) to exhibit Jewitt risk aversion is that there not exist any z 2 (; ^x) at which u is di erentiable and u (z) =. We these preliminaries in hand, we may proceed to the characterization of Jewitt risk aversion. As an intermediate step, we rst demonstrate the following theorem, which is of independent interest. Theorem 2 The preferences % in (2) exhibit Jewitt risk aversion if and only if they exhibit aversion to elementary JMPS s. Proof. The only if part is Proposition 8(ii), so we consider the if direction here. The argument is more complex than the corresponding part of the proof of Theorem 1 for two reasons. First, suppose F n and G n are nite step functions with equal means, such that F n LLIR G n. Because of the need to anchor a = 1, it will not in general be possible to transform F n into G n by a sequence of elementary JMPS s. Second, even if we can express F and G as the limits of nite step functions, because of potential discontinuities in h we need to modify the convergence argument. In particular, h ~ in (8) may no longer be continuous. Suppose that F LLIR G. Landsberger and Meilijson (1994, Theorem 1) guarantees the existence of a sequence ff m g 1 m= such that: F = F, ff m (x)g 1 m= converges to G (x) at every point x at which the latter is continuous, and F m is a mean-preserving left stretch of F m 1 (ibid., De nition 1) 2!.

21 for every m >. For our purposes, the essential content of the last statement is that, for each m >, F m has the same mean as F m 1 and Fm 1 crosses Fm 1 1 at most once in (; 1) (ibid., Lemma 1). We rst show that aversion to elementary JMPS s implies F m 1 % F m for each m >. To do so, we adapt arguments from Chateauneuf, Cohen and Meilijson (25). Fix some m >. It is easy to see that we may obtain F m 1 and F m (respectively) as the L 1 -limits of sequences Fm n 1 1 and ff n n=1 mg 1 n=1 of CDF s with nite ranges such that, for each n, Fm n 1 has the same mean as Fm n and Fm 1 n 1 crosses (F n m ) 1 at most once in (; 1). We now show that aversion to elementary JMPS s implies Fm n 1 % Fm n for each n. Fix n 1. We may assume, without loss of generality, that Fm n 1 and Fm n are non-identical. Because Fm n 1 and Fm n have nite ranges, we may write with s > 1, sx i=1 Z h F n 1 m 1 du [h (p i ) h (p i 1 )] u x m 1 i h F n m du = = p < p 1 < < p s 1 < p s = 1 u (x m i ) (1) and x m 1 i = F n 1 m 1 (pi ) and x m i = F n 1 m (pi ). Furthermore, since Fm n 1 and Fm n have equal means and cross only once in (; 1), there also exists k 2 f1; 2; :::; s 1g such that x m 1 i x m i x m 1 j x m j and u x m 1 i u (x m k ) u x m 1 j whenever i k < j. We may therefore choose < < such that u () u () u x m 1 i u (x m i ) for all j > k and u u u x m 1 j u x m j for all i k. 21

22 Hence, from (1) we have Z h F n 1 m 1 du h F n m du Let [1 h (p k )] [u () u ()] h (p k ) u u (11) ( ) =. In virtue of Lemma 17 it is without loss of generality to suppose that = p k (1 p k ). Therefore, aversion to elementary JMPS s implies [1 h (p k )] [u () u ()] h (p k ) u u, which, together with (11), gives the desired result: Fm n n1 % Fm. n o Next, consider the sequences of random variables 19 Fm n and n=1 (F n m ) 1 1. Since F n 1 d n=1 m 1! Fm 1 1 and (Fm) n 1! d Fm 1 (where! d denotes convergence in distribution) and u is continuous, it follows that u F n j 1 d! u F 1 j for each j 2 fm 1; mg (Rao, 1973, p.124). Therefore (Denneberg, 1994, Proposition 8.9): Z Z Z Z u d h Fj n = u Fj n 1 dh! u F 1 j dh = u d (h F j ) for j 2 fm 1; mg, and hence F m 1 % F m. To recap: we have shown that aversion to elementary JMPS s implies F % F m for each m >. It remains only to show that Z Z u d (h F m )! u d (h G). 19 These should all be considered as random variables on [; 1] endowed with the Borel -algebra and Lebesgue measure. 22

23 But this follows by the same sort of argument as we deployed in the preceding paragraph. This completes the proof. We can now prove: Theorem 21 The preferences % in (2) exhibit Jewitt risk aversion only if h (x) x is non-increasing for x 2 (; 1]. A necessary and su cient condition is that EITHER (i) h is the indicator on (; 1]; OR OTHERWISE (ii) there exists ^x 2 (; 1] with u strictly increasing on [; ^x] and u (^x) = 1, and ^P h Q^x u. Corollary 22 When u is concave, the preferences % in (2) exhibit Jewitt risk aversion if and only if h (x) x is non-increasing on (; 1]. If h is the identity, then u concave is both necessary and su cient. Proof of Theorem 21. Given Theorem 2, it su ces to show that the preferences exhibit aversion to elementary JMPS s i EITHER (i) OR OTH- ERWISE (ii). Given Lemma 19, we may argue as in the proof of Theorem 1 to deduce the necessity of 1 h (a) 1 a h b h (b) b b whenever b < b a < 1. This is equivalent to 1 h (x) 1 x (12) non-decreasing in x on [; 1), which is in turn equivalent to h (x) x nonincreasing on (; 1]. Using Lemmas 2 and 9 we further deduce the necessary condition that u be strictly increasing up to its maximum value, unless h is the indicator for (; 1]. Let us therefore exclude the trivial case (i), and assume that u is strictly increasing up to its maximum value, which is reached at ^x. It remains to demonstrate the necessity and su ciency of ^P h Q^x u. 23

24 Observe that ^P h is well-de ned (since (12) is non-increasing on [; 1), so h (x) < 1 when x < 1) and nite (since h is not the indicator for f1g). Note also that ^P h Q^x u entails ^P h 1, and hence that h (x) x is non-increasing on (; 1]. Preferences therefore exhibit aversion to elementary JMPS s i for any a; b; b with b < b a < 1, sup < < 1 ( )( ) =(b b)(1 a) u u! u () u () 1 h (a) 1 a h (b) h (b) b b. Since u is constant on [^x; 1] this is equivalent to sup < < ^x ( )( ) =(b b)(1 a) u u! u () u () 1 h (a) 1 a h (b) h (b) b b. By Lemma 17 it follows that aversion to JMPS obtains i 1 Q^x u inf b < b a < 1 h (a) 1 a h b h (b) b b!, which is equivalent to ^P h Q^x u by Lemma 18. Corollary 22 follows straightforwardly, so we omit its proof. Indeed, Chateauneuf, Cohen and Meilijson (24, Theorem 4) already implies most of Corollary 22. We now easily obtain the following characterization of the LLIR order on equal-mean distributions: Corollary 23 Given F and G with equal means, F LLIR G i u d (h F ) u d (h G) for any concave, non-decreasing and continuous u : [; 1]! R and any non-decreasing h : [; 1]! [; 1] that satis es h () = and h (x) =x nonincreasing on (; 1]. 24

25 Proof. Theorem 21 gives the only if part. For the converse, suppose Z F 1 (p) F (z) dz > Z G 1 (p) G (z) dz for some p 2 (; 1). We shall separate the analysis into two cases. Case I: F 1 (p) G 1 (p). In this case: Z 1 p F 1 dz = F 1 (p) p < G 1 (p) = 1 p Z p Z 1 F p 1 p G 1 dz If we let u be the identity function and de ne 1 (p) Z G 1 (p) F dx G dx h (z) = z=p if z 2 [; p] 1 if z > p then h (z) =z is non-increasing on (; 1], F 1 dh = 1 p Z p F 1 dz and so G F as required. G 1 dh = 1 p Z p G 1 dz, Case II: F 1 (p) > G 1 (p). By right-continuity it follows that there is some " 2 (; 1 p) such that F 1 (z) > G 1 (z) for all z 2 [p; p + "]. If G 1 () > F 1 () then G F according to maximin preferences (which are Jewitt risk-averse), so let us further 25

26 assume G 1 () F 1 (). We may now choose u to be the identity and 8 if z = >< " + z if z 2 (; p) h (z) = p + " if z 2 [p; p + "] >: z if z > p + " Observe that h (z) =z is non-increasing on (; 1]. Furthermore, since F and G have equal means: F 1 dh = < = F 1 dz G 1 dz G 1 dh Z p+" p Z p+" p F 1 dz + "F 1 () G 1 dz + "G 1 () so G F. Chateauneuf, Cohen and Meilijson (24, Theorem 4(iii)) demonstrate that, for equal-mean distributions F and G, F LLIR G if all RDEU maximizers with linear u and h (z) =z non-increasing on (; 1] weakly prefer F to G. In this connection, note that the if part of our Corollary 23 is proved using linear utility functions. 5 Concluding remarks Figure 1 and the associated inequality (7) prove to be simple but powerful tools for understanding various risk aversion concepts within the RDEU framework. As we have seen, a simple argument based on (7) can be used to characterize the strongly risk averse RDEU orders, even under very weak maintained assumptions about u and h. This characterization reveals the prominent role of maximin behavior in strong aversion to risk. It is also possible to fully characterize risk aversion in the sense of Jewitt (1989) as the requirement that (7) hold whenever a = 1. This simple test implies the necessary and, given u increasing up to its maximum, su cient 26

27 condition that ^P h Q^x u. Corollary 22 describes important special cases in which this condition may be signi cantly simpli ed. As corollaries to our main results, one also obtains characterizations of the underlying more risky than orders SOSD and LLIR on D. These may be of independent interest. Appendix: Proofs Proof of Proposition 7. For this proof, it is convenient to work with the various de nitions re-expressed in terms of CDF s, rather than DDF s. If F = (G), then F 1 is constant on (; 1) and hence G 1 F 1 is non-decreasing on (; 1). It follows immediately that monotone risk aversion implies weak risk aversion. Jewitt (1989, pp.68-69) shows that Z F 1 (p) F (z) dz Z G 1 (p) G (z) dz is equivalent to G 1 (p) F 1 (p) 1 p Z p G 1 (!) F 1 (!) d! (13) If G 1 F 1 is non-decreasing on (; 1), then (13) holds for all p 2 (; 1). Therefore, Jewitt risk aversion implies monotone risk aversion. Finally, suppose (13) holds for all p 2 (; 1). If F and G have the same mean, then This fact and (13) gives Z p G 1 (!) F 1 (!) d! =. G 1 (!) F 1 (!) d! for all p 2 [; 1] with equality for p = 1. It follows that F SOSD G (Roell, 1987, p.151; Vergnaud, 1997, Proposition 1). Therefore, strong risk aversion implies Jewitt risk aversion. 27

28 Next, consider the converses. To see that Jewitt need not imply strong risk aversion, choose u linear and any non-concave h with h (x) x non-increasing on [; 1] (compare Theorem 1 in Section 3 and Corollary 22 in Section 4). Likewise, to see that monotone risk aversion does not imply Jewitt risk aversion, choose u linear and h satisfying h (x) x but not such that h (x) x is non-increasing on [; 1] (compare Chateauneuf, Cohen and Meilijson (24, Theorem 3(i)) and Corollary 22 in Section 4 below). 2 Finally, Chateauneuf and Cohen (1994, Example 3 with h = f 1 ) reveals that weak risk aversion does not imply monotone risk aversion. In particular, recalling Chateauneuf, Cohen and Meilijson (25, Theorem 1), we observe that Q 1 u = 1 while P f is nite for this example. Proof of Lemma 9. Suppose that h is not the indicator on (; 1] and there exist ; 2 [; 1) with < and u () = u () < 1. We deduce a contradiction follows. Since u is continuous and monotone, it is without loss of generality to assume = sup fz 2 [; 1] j u (z) = u ()g (14) Set =, a = 1 and b = in Figure 1. Then weak risk aversion requires that F % G (Proposition 8(iv)). If we can nd 2 (; 1] and a = b 2 (; 1) such that h b > and 1 b ( ) = b, = 1 b b (15) we are done, since (7) then implies u = u (), which contradicts (14). But h b > for any b su ciently close to 1 recall that h is not the indicator on (; 1] so we can always satisfy (15) with 2 (; 1]. 2 A suitable example may be found in Quiggin (1991, pp.243-4). 28

29 Proof of Lemma 11. Suppose h is not the indicator on (; 1] and there exists z 2 (; ^x) with u (z) =. We shall deduce a contradiction. Let p 2 (; 1). We show that h has a left-hand derivative equal to zero at p. To do so, consider Figure 1 and x a = 1, a = b = p, = = z and = 1. Now choose a sequence fb n g 1 n=1 such that bn < p for each n and We need to show that lim n!1 lim n!1 bn = p. Let f n g 1 n=1 be a sequence with n < z and for each n, 21 and h (p) h (b n ) p b n = (16) (1 p) (z n ) = (p b n ) (1 z) lim n!1 n = z. Since u (z) = and u is strictly increasing on (; ^x) we have and lim n!1 u (z) u ( n ) z n = u (1) u (z) 1 z >. Therefore, strong risk aversion and (7) give (16) as required. Now let p 2 [; 1). Using a similar argument, we may show that h has a right-hand derivative equal to zero at p. In this case, we start by n xing a = 1, b = p, = = z, = 1; and choose sequences fa n g 1 n=1 and b no 1 with p < a n = b n 1 for each n and lim n!1 an = p. The rest of the argument follows, mutatis mutandis, as before. 21 This is always possible, if necessary by rst eliminating nitely many initial members of the sequence fb n g 1 n=1. 29 n=1

30 We have therefore established that h has a derivative equal to zero at every p 2 (; 1), and a right-hand derivative equal to zero at p =. Given the normalization of h, it follows that h is the indicator function for f1g. But this implies that h is the indicator for (; 1] a contradiction. Proof of Lemma 18. The arguments used in the proof of Chateauneuf, Cohen and Meilijson (25, Proposition 2(vi)) are easily adpated to the present case. Proof of Lemma 19. Observe that throughout the proof of Lemma 11 we x a = 1. Therefore, we can use the same argument to prove the present Lemma, substituting Jewitt risk aversion for strong risk aversion where the latter appears. References [1] Chateauneuf, A. and M. Cohen (1994). Risk Seeking with Diminishing Marginal Utility in a Non-Expected Utility Model, Journal of Risk and Uncertainty 9, [2] Chateauneuf, A., M. Cohen and I. Meilijson (24). Four Notions of Mean-Preserving Increase in Risk, Risk Attitudes and Applications to the Rank-Dependent Expected Utility Model, Journal of Mathematical Economics 4, [3] Chateauneuf, A., M. Cohen and I. Meilijson (25). More Pessimism than Greediness: A Characterization of Monotone Risk Aversion in the Rank-Dependent Expected Utility Model, Economic Theory 25, [4] Chateauneuf, A., M. Cohen and J.-C. Vergnaud (21). Left-Monotone Reduction of Risk, Deductible and Call, mimeo. [5] Chateauneuf, A., J. Eichberger and S. Grant (24). Choice Under Uncertainty with the Best and Worst in Mind: Neo-Additive Capacities, mimeo. 3

31 [6] Chew, S.H., E. Karni and Z. Safra (1987). Risk Aversion in the Theory of Utility with Rank-Dependent Probabilities, Journal of Economic Theory 42, [7] Cohen, M. (1995). Risk-Aversion Concepts in Expected- and Non- Expected-Utility Models, Geneva Papers in Risk and Insurance Theory 2, [8] Dasgupta, P., A. Sen and D. Starrett (1973). Notes on the Measurement of Inequality, Journal of Economic Theory 6, [9] Denneberg, D. (1994). Non-Additive Measure and Integral, Dordrecht: Kluwer. [1] Grant, S. and A. Kajii (1994). AUSI Expected Utility: An Anticipated Utility Theory of Relative Disappointment Aversion, CORE Discussion Paper [11] Grant, S. and A. Kajii (1998). AUSI Expected Utility: An Anticipated Utility Theory of Relative Disappointment Aversion, Journal of Economic Behavior and Organization 37, [12] Hardy, G.H., J.E. Littlewood and G. Polya (1929). Some Simple Inequalities Satis ed by Convex Functions, Messenger of Mathematics 58, [13] Hong, C.S. and M.M. Hui (1995). A Schur Concave Characterization of Risk Aversion for Non-Expected Utility Preferences, Journal of Economic Theory 67, [14] Jewitt, I. (1989). Choosing Between Risky Prospects: The Characterization of Comparative Static Results and Location Independent Risk, Management Science 35, 6 7. [15] Landsberger, M. and I. Meilijson (1994a). Co-Monotone Allocations, Bickel-Lehmann Dispersion and the Arrow-Pratt Measure of Risk Aversion, Annals of Operations Research 52, [16] Landsberger, M. and I. Meilijson (1994b). The Generating Process and an Extension of Jewitt s Location Independent Risk Concept, Management Science 4,

32 [17] Quiggin, J. (1991). Increasing Risk: Another De nition, in A. Chikán (ed.) Progress in Decision, Utility and Risk Theory, Dordrecht: Kluwer. [18] Rao, C.R. (1974). Linear Statistical Inference and its Applications (2nd edition), New York: John Wiley and Sons. [19] Roell, A. (1987). Risk Aversion in Quiggin and Yaari s Rank-Order Model of Choice Under Uncertainty, Economic Journal 97, [2] Rothschild, M. and J.E. Stiglitz (197). Increasing Risk: I. A De nition, Journal of Economic Theory 2, [21] Schmidt, U. and H. Zank (22). Risk Aversion in Cumulative Prospect Theory, mimeo, University of Manchester. [22] Vergnaud, J.-C. (1997). Analysis of Risk in a Non-Expected Utility Framework and Application to the Optimality of the Deductible, Finance 18, [23] Wakker, P.P. (1994). Separating Marginal Utility and Risk Aversion, Theory and Decision 36, [24] Wakker, P.P. (21). Testing and Characterizing Properties of Nonadditive Measures Through Violation of the Sure-Thing Principle, Econometrica 69, [25] Wheeden, R.L. and A. Zygmund (1977). Measure and Integral: An Introduction to Real Analysis, New York: M. Dekker. [26] Yaari, M. (1987). The Dual Theory of Choice Under Risk, Econometrica 55,