Maxmin expected utility through statewise combinations

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1 Eonomis Letters 66 (2000) loate/ eonbase Maxmin expeted utility through statewise ombinations Ramon Casadesus-Masanell, Peter Klibanoff, Emre Ozdenoren* Department of Managerial Eonomis and Deision Sienes, Kellogg Graduate Shool of Management, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, USA Reeived 14 May 1999; aepted 22 July 1999 Abstrat We provide an axiomati foundation for a maxmin expeted utility over a set of priors (MMEU) deision rule in an environment where the elements of hoie are Savage ats. The key axioms are stated using statewise ombinations as in Gul [Gul, F., Savage s theorem with a finite number of states. Journal of Eonomi Theory 57, Elsevier Siene S.A. All rights reserved. Keywords: Unertainty; Ambiguity; Ellsberg paradox; Multiple priors; Utility JEL lassifiation: D81 1. Introdution This paper provides an axiomati foundation for a maxmin expeted utility over a set of priors (MMEU) deision rule in an environment where the elements of hoie are Savage (1954) ats. This haraterization omplements the original axiomatization of MMEU developed in a lottery-ats (or Ansombe and Aumann, 1963) framework by Gilboa and Shmeidler (1989). MMEU preferenes are of interest primarily beause they provide a natural and tratable way of modeling deision makers who display an aversion to unertainty or ambiguity. In Casadesus-Masanell et al. (1999), we haraterized an MMEU rule over Savage ats using axioms stated in terms of standard sequenes, a measurement theory onstrution (see Krantz et al., 1971). In ontrast, the key axioms here involve statewise ombinations of ats (as defined below). Statewise ombinations have the advantage that they look muh like the onvex ombinations used in Ansombe Aumann and von Neumann Morgenstern style theories. Thus, as in Gul (1992), they allow one to transfer muh of the intuition from these settings to a setting with Savage ats. The disadvantage of this approah is that it is less *Corresponding author. Tel.: ; fax: address: peterk@nwu.edu (E. Ozdenoren) / 00/ $ see front matter 2000 Elsevier Siene S.A. All rights reserved. PII: S (99)

2 50 R. Casadesus-Masanell et al. / Eonomis Letters 66 (2000) general than the one in Casadesus-Masanell et al. (1999): the theory here will imply the existene of a binary partition of the state spae over whih subjetive expeted utility holds. The remainder of the paper presents a set of axioms and a theorem proving the equivalene between these axioms and an MMEU rule. The novel axioms are weakenings of Gul (1992, Assumption 2) at-independene ondition. Results in Nakamura (1990) and Gilboa and Shmeidler (1989) are useful for the proofs. 2. Notation and framework V is the set of states. A state in V is represented by v. S is an algebra of subsets of V. Events are elements of S. X 5fm,M g, R, m, M is the set of prizes or outomes. A (Savage) at f is a funtion f :V X. A simple at is an at with only finitely many distint values. A simple at is S-measurable if hv [ Vu f(v) [ Wj [ S for all W # X. F is the set of all S-measurable ats defined as the losure in the supnorm of all S-measurable simple ats. A set G # F is losed if it is losed in the supnorm. A onstant at f is one for whih fsvd5 x for all v [ V, for some x [ X; we denote this onstant at by x* or simply x when no onfusion would result. F * is the subset of F onsisting of all onstant ats. For any event [ S and x, y [ X, x y denotes f [ F suh that fsvd5 x for v [ and fsvd5 y for v [ ; suh ats are referred to as -measurable. The event V 2 is denoted. For f,g,h [ F and [ S, ifh(v) f(v) g(v) for all v [ V then h is a statewise ombination of f and g over the event. 3 is the set of all finitely additive probability measures P:S f0,1 g. Finally, K is a binary relation on F. Note that this environment is similar to that in Savage (1954) with the differene that we impose more struture on the prize set (X). The important aspet of this struture is that X is onneted and separable. This will allow the non-singleton set of states in our theory to be of any size, finite or infinite. 3. Axioms Axiom 1. (Weak order) K is omplete and transitive. Definition 3.1. An event is ordered non-null if there exist x, y and z in X with xayaz suh that xz yz. An event is ordered non-universal if there exist x, y and z in X with zayax suh that zx zy. Note that sine we impose restritions on the ordering of x, y and z, our definitions of non-null and non-universal (borrowed from Nakamura, 1990) are weaker than the orresponding notions in Savage (1954). See Casadesus-Masanell et al. (1999) for an explanation of why these are the appropriate notions. Axiom 2. (Struture) (a) x. y x* s y *. (b) There exists an event A [ S suh that A and A are ordered non-null and ordered non-universal. Part (a) is purely a simplifying assumption. The event A identified here is used in Axioms 5, 6 and 7 below.

3 R. Casadesus-Masanell et al. / Eonomis Letters 66 (2000) Axiom 3. (Continuity) For all f [ F, the sets Mf5 sd hg [ FgKf u jand Wf5 sd hg [ FfKg u jare losed. Axiom 4. (Monotoniity) (a) For all f, g [ F, iff v Kg v, for all v [ V then f Kg. (b)if [ S is s d s d ordered non-null and zkx s y, then xzs yz.if [ S is ordered non-universal and x s ykz, then zxs zy. Observe that part (a) is weak monotoniity, and part (b) is strit monotoniity on ordered non-null and non-universal events. Axiom 5. (A-at-independene) Let x 1, x 2, y 1, y 2, z1 and z 2[ X and let f 5 x1ax 2, g 5 y1a y2 and h 5 z1az 2. If f 9, g9 [ F are suh that, for either 5 A or 5 A, f 9sv d hsvd fsvd and g9sv d hsvd gsvd for all v [ V then, f Kg f 9Kg9. This axiom imposes at-independene (as introdued in Gul, 1992) only for A-measurable ats and the events A and A. In words, given A-measurable ats f, g and h and given the event 5 A or 5 A,iff9is a statewise ombination of h and f over the event and g9 is a statewise ombination of h and g over the event, then preferene between f and g is the same as between f 9 and g9. As disussed in Gul (1992), at-independene is analogous to the independene axiom in the theory of expeted utility over lotteries. These first five axioms guarantee the existene of an expeted utility representation for A- measurable ats. That is, there exists a stritly inreasing and ontinuous funtion u:x R and r [ s0,1d suh that if x, y, v, w [ X then xykv w rusxd1s1 2 rduy$ s d rusvd1s1 2 rduw. s d A A Moreover, u is unique up to positive affine transformations and r is unique. How do preferenes extend from A-measurable ats to all ats? If at-independene is required to hold for all ats and non-null events, expeted utility preferenes result (see Gul, 1992; Chew and Karni, 1994). This at-independene is too strong for MMEU as it is inompatible, for example, with the Ellsberg Paradox (Ellsberg, 1961). We now develop the appropriate weakening of at-independene. Definition 3.2. (Ghirardato et al., 1998) Two ats f and g are affinely related if there exist a $ 0 and b [ R suh that either u( f(v)) 5 au( g(v)) 1 b for all v [ V or u( g(v)) 5 au( f(v)) 1 b for all v [ V. The key is the use of statewise ombinations over the event A to form what will turn out to be sets of affinely related ats. This requires the following definitions: Definition 3.3. A set S, F ontains all statewise ombinations over the event A if f 1, f 2[ S, and for all v [ V, fsv d f svd f svd implies f [ S. 1 A 2 f f f Definition 3.4. Fix f [ F. We define S 05hj f < F *. Let S $ S 0 be the smallest losed set ontaining all statewise ombinations over the event A. f It an be shown that given an at f, the set S may be onstruted by the following iterative method: at eah step i 5 1,2,3,... we produe a set

4 52 R. Casadesus-Masanell et al. / Eonomis Letters 66 (2000) h f i i i21 i21 i21 i21 f i s d 1 s da 2 s d 1 2 i21 f ` f f i51 i 1 S 5 f [ F:f v f v f v, for all v [ V where f, f [ S. Finally, S is the losure of < S # F. Observe that S onsists of statewise ombinations over the event A of either f or a onstant at with either f or a onstant at. y the expeted utility representation for A-measurable ats, we end up with either a onstant utility at or a positive affine transformation of the state-by-state utility of f. Either way the resulting ats are affinely related to f. f f 2 1 To form S, we take any two ats in S and ombine them statewise over A. Sine both of these ats are affinely related to the at f, the resulting at will also be affinely related to f. This argument plus f ontinuity shows that S onsists only of ats that are affinely related to f. In fat, if f is not a onstant f at, the set S ontains all ats that are affinely related to f. l Axiom 6. (S-at-independene) For any f, g, h [ F suh that there exist l, k [ hf,gj for whih f, h [S k and g, h [S, if f 9, g9 [ F are suh that, for either 5 A or 5 A, f 9sv d hsvd fsvd and g9sv d hsvd gsvd for all v [ V then, f Kg f 9Kg9. f Using the representation for A-measurable ats and the definition of S, it an be shown that the axiom applies only if: (1) h is a onstant at; or (2) h is not a onstant at, but f, g and h are pairwise affinely related. It is useful to ompare S-at-independene to the ertainty-independene (C-independene) axiom of Gilboa and Shmeidler (1989): (C-independene) For any ats f and g, any onstant at h, and any a [ (0,1), if f 9, g9 are suh that, f 95af 1 (1 2 a)h and g95ag1 (1 2 a)h then, f Kg f 9Kg9. Note that the onvex ombination operation is defined statewise and is well-defined sine ats in their setting are funtions from states to probability distributions over prizes. C-independene relaxes the independene axiom of Ansombe and Aumann (1963) so that it is only required to hold when the third at, h, is a onstant at. S-at-independene and C-independene are quite similar in form, with two salient differenes. First, statewise ombinations over A or A replae onvex ombinations. Seond, as pointed out in possibility (2) above, S-at-independene applies to some h whih are not onstant ats. In fat, the first differene leads to the seond one. In an Ansombe Aumann framework, all probabilities in the unit interval are available. Consequently, to express the fat that preferene is preserved by homogeneous transformations (of utility), one need only onsider onvex ombinations of the at in question and a onstant at. In ontrast, the only probabilities that are available through statewise ombinations over A are those of A and A. For example, if the revealed probability of A happens to be 1/3 and we want to show that multiplying the utility of a pair of ats by 1/2 preserves the preferene ordering between them, we annot onstrut the 1/ 2-ats through statewise ombinations over A without taking ombinations of two non-onstant ats. This is why we annot restrit h to be a onstant at in the S-at-independene axiom. The final axiom, at-unertainty aversion, restrits the way that at-independene an be violated. It requires that the deision-maker weakly likes to smooth utilities aross states of the world, sine this leaves her less exposed to any unertainty or ambiguity about the probability of various states. Speifially it modifies Gilboa and Shmeidler (1989) unertainty aversion axiom by replaing onvex ombinations with statewise ombinations over A. Axiom 7. (At-unertainty aversion) For all f,g, f 9 [ F, iff g and f 9sv d fsvda gsvd for all v [ V then, f 9Kf. j

5 R. Casadesus-Masanell et al. / Eonomis Letters 66 (2000) See Casadesus-Masanell et al. (1999) for a disussion relating this type of unertainty aversion axiom to reent alternatives suggested by Epstein (1999) and Ghirardato and Marinai (1998). 4. A representation theorem Theorem 4.1. Let K be a binary relation on F. Then K satisfies Axioms 1 7 if and only if there exists a ontinuous and stritly inreasing funtion u:x R, and a non-empty, ompat and onvex set # of finitely additive probability measures on S suh that f g F P[# P[# G f Kg min Eu + fdp $ min Eu + gdp for all f and g [ F. Furthermore, there exists an event A [ S and a r [ (0,1) suh that P [ # implies P(A) 5 r. Moreover, u is unique up to positive affine transformations and the set # is unique. Proof. We sketh suffiieny; neessity is omitted. Axioms 1 5 imply versions of the appropriate axioms of Nakamura (1990) using events A and A. Apply Nakamura s Theorem 1 to yield an expeted utility representation for A-measurable ats. The ontinuity axiom and struture axiom part (a) guarantee that u is ontinuous and stritly inreasing. Let r be the probability of A. Let K 5 ux. s d We normalize u suh that K 522,2 f g. We now onstrut a funtional J:F R that represents preferenes. For any onstant at f 5 x [ X we define Jf5 sd ux. s d For general ats f [ F, let Jf5 sd u, s d where is the ertainty equivalent of f. Continuity ensures that exists. Let be the spae of bounded (in the supnorm), S-measurable, real valued funtions on V. For g [ R, we denote by g * the element of that assigns g to every v. Let K s d be the subset of funtions in with values in K. Observe that for f [ F, u + f [ K, s d and for d [ K s d there exists f [ F suh that u + f 5 d. Define the funtional I:sKd R by Iu+ s fd5 Jf. s d Sine J represents preferenes, it is lear that I does as well. It an be shown that an extension of I to satisfies the following properties: (i) Is1* d5 1; (ii) (I is monotoni) For all a, b [, a $ b implies Ia s d$ Ib; s d (iii) (I is homogeneous of degree 1) For all b [, a $ 0, Isabd5 aisb; d (iv) (I is C-independent) For all b [, g [ R, Ib1 s g * d5 Ib s d1 Isg * d; and, (v) (I is superadditive) For all a, b [, Ia1 s b d$ Ia s d1 Ib. s d Properties (i) and (ii) on (K) are immediate. The proof of (iii) on (K) uses S-at-independene heavily. Then extend I to all of by homogeneity. This preserves homogeneity and monotoniity. Now (iv) and (v) an be shown for the extension of I by arguments similar to those in Gilboa and Shmeidler (1989). Properties (i) (v) imply the existene of the minimum expetation over a set of measures representation. This is a fundamental lemma variations of whih have been proved by, for example, Gilboa and Shmeidler (1989), Chateauneuf (1991) and Marinai (1997). h

6 54 R. Casadesus-Masanell et al. / Eonomis Letters 66 (2000) Aknowledgements We thank Eddie Dekel, Paolo Ghirardato and Massimo Marinai for helpful disussions. Referenes Ansombe, F.J., Aumann, R.J., A definition of subjetive probability. The Annals of Mathematial Statistis 34, Casadesus-Masanell, R., Klibanoff, P., Ozdenoren, E., Maxmin expeted utility over Savage ats with a set of priors, mimeo, Northwestern University. Chateauneuf, A., On the use of apaities in modeling unertainty aversion and risk aversion. Journal of Mathematial Eonomis. 20, Chew, S.H., Karni, E., Choquet expeted utility with a finite state spae: ommutativity and at-independene. Journal of Eonomi Theory 62, Ellsberg, D., Risk, ambiguity, and the Savage axioms. Quarterly Journal of Eonomis 75, Epstein, L., A definition of unertainty aversion. The Review of Eonomi Studies (in press). Ghirardato, P., Klibanoff, P., Marinai, M., Additivity with multiple priors. Journal of Mathematial Eonomis 30, Ghirardato, P., Marinai, M., Ambiguity made preise: a omparative foundation, Soial Siene Working Paper 1026, California Institute of Tehnology. Gilboa, I., Shmeidler, D., Maxmin expeted utility with non-unique prior. Journal of Mathematial Eonomis 18, Gul, F., Savage s theorem with a finite number of states. Journal of Eonomi Theory 57, Krantz, D.H., Lue, R.D., Suppes, P., Tversky, A., In: Foundations of Measurement, Vol. I, Aademi Press, New York. Marinai, M., A simple proof of a basi result for multiple priors, mimeo, Univ. Toronto. Nakamura, Y., Subjetive expeted utility with non-additive probabilities on finite state spaes. Journal of Eonomi Theory 51, Savage, L.J., In: The Foundations of Statistis, Wiley, New York.