Completing the State Space with Subjective States 1

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1 Journal of Economic Theory 105, (2002) doi: jeth Completing the State Space with Subjective States 1 Emre Ozdenoren Department of Economics, University of Michigan, Ann Arbor, Michigan emreoumich.edu Received May 25, 2000; final version received February 8, 2001; published online November 7, 2001 We define an opportunity act as a mapping from an exogenously given objective state space to a set of lotteries over prizes, and consider preferences over opportunity acts. We allow the preferences to be possibly uncertainty averse. Our main theorem provides an axiomatization of the maxmin expected utility model. In the theorem we construct subjective states to complete the objective state space. As in E. Dekel et al. (Econometrica, in press), we obtain a unique subjective state space. We also allow for preference for flexibility in some of the subjective states and commitment in others. Journal of Economic Literature Classification Number: D Elsevier Science (USA) Key Words: subjective states; unforeseen contingencies; preference for flexibility; uncertainty aversion; ambiguity. 1. INTRODUCTION In a seminal paper, Kreps [9] provides a characterization of preference for flexibility. As a primitive he takes preferences over opportunity sets, which are nonempty subsets of prizes, and derives a set of subjective states. Dekel et al. [5], on the other hand, take as primitive preferences over sets of lotteries over prizes. Using the richness of this choice set, they derive a unique subjective state space. 2, 3 In many decision problems, however, there is a natural objective state space that we expect the decision maker to use. Indeed in Kreps's [10] interpretation, objective states correspond to contingencies that are 1 I thank an associate editor, Nabil Al-Najjar, Ramon Casadesus-Masanell, Ehud Kalai, and especially Peter Klibanoff for helpful comments. All errors are mine. 2 For an unforeseen contingencies interpretation of the model, and a discussion on the importance of uniqueness of the state space see Dekel et al. [5]. 3 In Dekel et al. [5] the decision maker may prefer flexibility in some subjective states and commitment in others. Gul and Pesendorfer [8], in a closely related framework, also provide a representation theorem where commitment is valued in some states Elsevier Science (USA) All rights reserved.

2 532 EMRE OZDENOREN foreseen by the decision maker. The decision maker is also aware that there may be unforeseen contingencies, and that her preferences may change in the future. Subjective states reflect this aspect of the problem. Therefore, it is useful to have a framework that takes the objective state space as part of the primitives. In contrast to Kreps [9] and Dekel et al. [5], our model explicitly specifies an objective state space. Having an objective state space also allows us to consider uncertainty averse preferences. Ellsberg [6] and many afterward demonstrated that uncertainty averse preferences seem common and are incompatible with the standard expected utility theory. 4 In this paper, we use the maxmin expected utility model, 5 which is a common way to model uncertainty averse preferences, and derive a set of subjective states within this framework. Formally, we define an opportunity act as a mapping from the set of objective states into the set of lotteries over prizes. We take as primitive preferences over opportunity acts, and we allow these preferences to be uncertainty averse. Our main theorem provides an axiomatization of the maxmin expected utility model. In the theorem we construct subjective states derived from preferences to complete the objective state space. As in Dekel et al. [5], we obtain additivity over subjective states and a unique subjective state space. We also allow for preference for flexibility in some of the subjective states and commitment in others NOTATION AND DEFINITIONS Let 0 be the nonempty set of states of nature and O be an algebra of subsets of 0. We call 0 as the objective states of nature, and we allow 0 to be of any cardinality. Let B=[b 1,..., b n ] be a finite set of prizes. Let 2B be the set of all (simple) probability distributions on B. LetX be the set of nonempty subsets of 2B. Elements of X are called opportunity sets. An opportunity act is a function from 0 into X. A simple opportunity act is an opportunity act which gives finitely many distinct opportunity sets. Denote by H* the constant opportunity acts. We sometimes denote the constant opportunity act that gives the opportunity set x for all # 0 by x*. 4 See, for example, Camerer and Weber [1] for a survey. 5 Gilboa and Schmeidler [7] first provided an axiomatization of maxmin expected utility model in an AnscombeAumann framework. In a Savage framework there are two axiomatizations by Casadesus-Masanell et al. [2,3]. 6 Nehring [11] looks at preferences over opportunity acts in a Savage framework. He does not consider uncertainty averse preferences, and his model allows for only preference for flexibility. For more on related literature see the survey by Dekel et al. [4].

3 SUBJECTIVE STATES 533 A simple opportunity act h is O-measurable if [ # 0 h( )#W]# O, where WX. Let H be the set of all O-measurable opportunity acts defined as the closure of the set of all O-measurable simple acts. 7 The interpretation of these acts is as follows. First the agent will learn which objective state has occurred. Then she will pick a lottery from the set h( )#X. Finally the objective uncertainty will resolve and the agent will receive a prize. For x, x$#x and for * # [0, 1], let *x+(1&*) x$ be the set of all ;"# 2B such that there exists ; # x and ;$#x$ with ;"=*;+(1&*) ;$. This is the natural way to think about taking the convex combination of two opportunity sets. That is, the convex combination of an element of the first set and an element of the second set are taken and these are put into the set of convex combinations, and this procedure is repeated for all possible pairs. The convex combination of two acts is obtained by taking the convex combination of opportunity sets for each state. Formally, for h and g in H and * # [0, 1], define *h+(1&*) g by (*h+(1&*) g)( )= *h( )+(1&*) g( ) for all # AXIOMS Suppose p is a binary relation on H. Next we consider axioms on this binary relation. The first four axioms are standard and are used throughout the analysis. Axiom 1 (Weak Order). p on H is a weak order. p on H induces a relation also denoted by p on X. Specifically, if x, y # X, xpy if and only if x*py*. 7 The distance between two opportunity sets is defined as follows. Let d denote any distance on 2B. For any pair x, x$#x, we define and d (:, x$)# inf d (:, ;) ; # x$ e(x, x$)#sup d (:, x$) : # x the excess of x over x$. Now, let the distance between x and x$ be defined as d hausdorff (x, x$)#max[e(x, x$), e(x$, x)]. Finally, we define the distance between any two opportunity acts h and h$ by d(h, h$)# sup d hausdorff (h( ), h$( )). # 0

4 534 EMRE OZDENOREN Axiom 2(Continuity). For any h # H, the sets L(h)=[h$#H hoh$] and U(h)=[h$#H h$oh] are both open. Axiom 3 (Monotonicity). For any f, g # H, if f( ) p g( ) for all # 0, thenf pg. The monotonicity axiom implicitly requires the preference relation to be state independent. That is the agent believes that her ex post preferences do not depend on which objective state occurs. The next axiom rules out preferences with complete indifference. Axiom 4 (Nondegeneracy). There exists h, g # H such that hog. Next, we introduce an independence condition for opportunity acts. Axiom 5 (Independence). For any f, g, h # H and * #[0,1], f pg if and only if *f +(1&*) hp*g+(1&*) h. To see how this axiom works, suppose that the decision maker is comparing *f +(1&*) h with *g+(1&*) h, and that an objective state has occurred. Then the decision maker is faced with *f( )+(1&*) h( ) and *g( )+(1&*) h( ). Suppose once she learns about her preferences, the decision maker chooses ; from f( ), ;$ from g( ) and ;" from h( ). Since she receives ;" with the same probability in both cases, her preferences should be determined only by how she feels about ; and ;$ which only depends on f( ) and g( ). Therefore her overall preference should depend only on how she feels overall about f and g. Axiom 6 (C-Independence). For any f, g # H, h # H* and * #[0,1], f pg if and only if *f+(1&*) hp*g+(1&*) h. Here, we are relaxing the previous axiom by requiring independence to hold only when h is a constant act. C-Independence was introduced in Gilboa and Schmeidler [7], and the idea is that it is easier for the decision maker to visualize the mixtures of f and g with constant acts, and therefore she is less likely to violate C-independence than independence. Axiom 7(Set-monotonicity). Suppose x, x$# H* then x$x$ O xpx$. Set-monotonicity says that, the agent faced with two constant acts prefers the act that gives her a bigger opportunity set. We say the decision maker has preference for flexibility if for some acts the preference is strict. This is the analogue of Kreps's [9] monotonicity axiom in our framework. Without this axiom the decision maker may prefer commitment in some subjective states and flexibility in other states.

5 SUBJECTIVE STATES 535 Axiom 8 (Uncertainty Aversion). For any f, g # H and * #[0, 1], ftg implies *f+(1&*) gpf. This is Gilboa and Schmeidler's uncertainty aversion axiom, and it roughly says that the decision maker likes hedging A REPRESENTATION THEOREM First we define additive maxmin expected utility (AMMEU) representation of p. In this representation, ex ante, the decision maker is a maxmin expected utility maximizer. She has subjective states determining her preferences in the implicitly modelled ex post stage which are in the expected utility form, and which enter the representation additively. Ex post for each subjective state the decision maker picks the best lottery from the set that is available to him, and her utility at any objective state is computed by adding her expected utility over all subjective states from each such lottery using some measure. The formal definition follows: Definition 4.1. An AMMEU representation of p, (C, S, +, U), is a set C of finitely additive probability measures on O, a set S, a finitely additive measure + with full support on S, and a continuous utility function U: 2B_S R such that V is continuous and represents p where V( f )=min P # C 0 _ S sup U(;, s) d+ & dp ; # f( ) and where U(., s) is an expected-utility function, i.e., U(;, s)= : b # B U(b, s) ;(b). AMMEU allows for uncertaintyuncertainty aversion about the objective states while simultaneously allowing for preference for flexibility or commitment. The complete state space is 0_S, and this is the sense in which state space is completed with subjective states. Now, we are ready for the main representation theorem: Theorem 4.2. p satisfies weak order, C-independence, continuity, monotonicity, uncertainty aversion and non-degeneracy if and only if p has an AMMEU representation, (C, S, +, U) and the set C is unique. If in addition to the previous axioms p also satisfies set-monotonicity then the measure + 8 See Casadesus-Masanell et al. [2, Section 4.2] for a detailed discussion of this axiom.

6 536 EMRE OZDENOREN is positive. If (C, S, +$, U$) is another AMMEU representation of p, then, there exists a>0, a$>0, and b such that +$=a$+ and U$=aU+b. See Section 7 for the proof of Theorem 4.2. It is clear from Theorem 4.2 that by assuming different versions of the independence axiom, we can obtain different representation theorems. In particular by replacing the C- independence axiom with the independence axiom, we would obtain a representation theorem where maxmin expected utility is replaced with expected utility. 9 See the remark at the end of Section 7 for the proof of this claim. So far we only stated uniqueness results for a fixed subjective state space. Next we look at the question of uniqueness of the subjective state space. 5. UNIQUENESS OF THE SUBJECTIVE STATE SPACE In this section we provide a uniqueness result for the subjective state space which follows from a similar result in Dekel et al. [5]. Given an AMMEU representation (C, S, +, U), define o s as, ;o s ;$ U(;, s)>u(;$, s). Let [ o s : s # S] be the subjective state space for (C, S, +, U). We need the following definition to state the result: Definition 5.1. Given a sequence [ o k ] of expected-utility preferences over 2B, we say that o is a limit of the sequence if it is a nontrivial expected utility preference such that ;o;$ O _K such that ;o k ;$, \kk. The closure of a set of expected utility preferences adds to the set all such limit points. We focus only on subjective state spaces with relevant states. A subjective state s is relevant if for any neighborhood of s, there are two constant acts among which the agent is not indifferent, even though the particular menus that these constant acts yield are evaluated identically at all subjective states that are not in this neighborhood Similarly if we assume comonotonic independence rather than C-independence, we would obtain Choquet expected utility. For a discussion of the comonotonic independence axiom and Choquet expected utility see Schmeidler [12]. 10 For a detailed discussion of the topology on the set of all expected-utility preferences see Dekel et al. [5].

7 SUBJECTIVE STATES 537 Definition 5.2. Suppose (C, S, +, U) and(c$, S$, +$, U$) are AMMEU representations of a preference relation p (where S and S$ contain only relevant states). We say that these representations have essentially equivalent subjective state spaces if the closures of [ o s : s # S] and [ o s : s # S$] are the same. Theorem 5.3. Suppose p has an AMMEU representation, then all AMMEU representations of p have essentially equivalent subjective state spaces. Proof. Follows from Theorem 1 part C of Dekel et al. [5]. K 6. CONCLUSION Dealing with unforeseen contingencies has been problematic in decision theory while it is widely agreed that it is an important aspect of most decision making problems. In this paper following Kreps [10] we tackle this problem by allowing for objective states (or foreseen states) as well as subjective states (or unforeseen states) in the representation. Formally we do this by generalizing the model by Dekel et al. [5] to opportunity acts. Our representation allows for different attitudes towards uncertainty about the objective states. This approach also pins down the subjective state space uniquely, which is a necessary condition for any fruitful application of such models. The representation theorem in this paper is general enough to allow for both preference for flexibility and commitment. 7. PROOFS We only prove sufficiency part of Theorem 4.2. Necessity is easy to show, and uniqueness follows from the uniqueness part of Lemma 7.1. Let Y be the set of closed and convex sets in X. Lemma 7.1. There exists a linear function v: Y R such that for all x, x$#y, xpx$ iff v(x)v(x$). For * #[0,1], v(*x+(1&*) x$)=*v(x)+(1&*) v(x$). v is unique up to affine transformations and continuous. Proof. Our axioms restricted to constant acts imply the axioms of Dekel et al. [5]. Therefore this lemma follows from their Proposition 2. K

8 538 EMRE OZDENOREN Lemma 7.2. For every x # X, xtconv(x) and xtcl(x). Proof. Similarly follows from lemmas 1 and 2 in Dekel et al. [5]. K By the previous lemma we can restrict our attention to Y. Lemma 7.3. Given a v: Y R from Lemma 7.1, there exists a unique J: H R such that: 1. f pg iff J( f )J(g), for all f, g # H. 2. for f =x*#h*, J( f )=v(x). Proof. On H*, J is uniquely defined by 2. We extend J to all acts as follows. Given f # H, by monotonicity there exits x, x # X such that x pf px. By continuity there exists a unique * # [0, 1] such that f t*x+ (1&*) x. Define J( f )=J(*x+(1&*) x ). By construction J satisfies 1, hence is also unique. By the nondegeneracy axiom there exists x 1, x 2 # X such that x 2 ox 1. Choose a specific v: X R such that, v(x 1 )=&2 and v(x 2 )=2. Denote by B the space of all bounded O measurable real valued functions on 0. Let B 0 be the space of functions in B which assume finitely many values. For # # R, let #*#B 0 be the constant function on 0 with the value #. Lemma 7.4. There exists a functional I: B 0 R such that: (i) For all f # H, I(v b f )=J( f ). (Hence I(1*)=1.) (ii) I is monotonic (i.e., for a, b # B 0, ab O I(a)I(b)). (iii) I is superadditive and homogenous of degree 1. (iv) I is C-independent (i.e., for any a # B 0 and # # R, I(a+#*)= I(a)+I(#*)). Proof. The proofs follow by using arguments that are similar to those in Gilboa and Schmeidler's proof of Lemma 3.3. K By a fundamental lemma from Gilboa and Schmeidler [7, Lemma 3.5], there exists a closed and convex set C of finitely additive probability measures on O such that for all b # B, I(b)=min bdp. (7.1) P # C

9 SUBJECTIVE STATES 539 Restricting our axioms to constant acts, from Dekel et al. [5, Theorem 4], we know that there is a finitely additive measure +, a set S and functions U: 2B_S R such that for all x # X, v(x)= sup U(;, s) d+, (7.2) S ; # x where each U(., s) is an expected utility function. If preferences also satisfy set-monotonicity + is a positive measure. The desired result follows by combining 7.1 and 7.2. We claimed in Section 4 that by replacing the C-independence axiom with the independence axiom, we obtain a representation where maxmin expected utility is replaced with expected utility. The proof of this claim follows by recognizing that independence axiom makes I additive, in this case there is a finitely additive probability measure P on O such that for all b # B, I(b)= bdp, and the result follows by combining this with 7.2. K REFERENCES 1. C. Camerer and M. Weber, Recent developments in modelling preference: Uncertainty and ambiguity, J. Risk Uncertainty 5 (1992), R. Casadesus-Masanell, P. Klibanoff, and E. Ozdenoren, Maxmin expected utility over Savage acts with a set of priors, J. Econ. Theory 92 (2000), R. Casadesus-Masanell, P. Klibanoff, and E. Ozdenoren, Maxmin expected utility through statewise combinations, Econ. Letters 66 (2000), E. Dekel, B. Lipman, and A. Rustichini, Recent developments in modeling unforeseen contingencies, Europ. Econ. Rev. 42 (1998), E. Dekel, B. Lipman, and A. Rustichini, Representing preferences with a unique subjective state space, Econometrica, in press. 6. D. Ellsberg, Risk, ambiguity, and the Savage axioms, Quart. J. Econ. 75 (1961), I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econ. 18 (1989), F. Gul and W. Pesendorfer, Temptation and self-control, Econometrica, in press. 9. D. Kreps, A representation theorem for `preference for flexibility', Econometrica 47 (1979), D. Kreps, Static choice and unforeseen contingencies, in ``Economic Analysis of Markets and Games'' (P. Dasgupta, D. Gale, O. Hart, and E. Maskin, Eds.), pp , MIT Press, Cambridge, K. Nehring, Preference for flexibility in a Savage framework, Econometrica 67 (1999), D. Schmeidler, Subjective probability and expected utility without additivity, Econometrica 57 (1989),