The Epistemic Value of a Menu and Subjective States Asen Kochovy September 20, 2010

Abstract A major theme in behavioral economics focuses on experimental evidence that individuals learn from the choice problems they face and consequently violate the consistency requirements of revealed preference theory. Despite the experimental evidence, the testable implications of such contextual inference remain unclear. In particular, it is an open question if learning from the choice problem imposes any restrictions on observed behavior. Motivated by the Sha…r and Tversky [17] experiments, the paper models contextual inference in the framework of preference for ‡exibility introduced by Kreps [10] and extended by Dekel, Lipman, and Rustichini [4]. Within this framework, the paper proposes a relaxation of the weak axiom which formalizes the identi…cation strategy in Sha…r and Tversky [17]. A subjective state space that may depend on the subset of actions faced by the individual is uniquely identi…ed from behavior, and ‘local’preferences are partially recovered.

This work has bene…ted from the helpful comments of Paulo Barelli, Eddie Dekel, and Bart Lipman. I am especially grateful to my advisor Larry Epstein. y Department of Economics, University of Rochester, Rochester, NY 14627. E-mail: [email protected].

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1 1.1

Introduction Objectives

Luce and Rai¤a’s Dinner [11, p.288] provides a prominent example of an individual who learns from the choice problem she faces. A customer orders chicken when the restaurant o¤ers chicken and steak, but chooses steak when frog legs are added to the menu. The customer is not irrational: the presence of an exquisite dish (frog legs) changes her perception of the quality of food served at the restaurant. When quality is perceived to be high, the customer orders steak; otherwise she chooses the safer option (chicken). In this example, ‘irrelevant’alternatives matter as they provide information about the uncertainty an individual considers relevant to her choice. Adding alternatives may also suggest contingencies the individual did not conceive of prior to their inclusion. Sen [15, 16] describes the following thought experiment. Jack comes back to town and asks his friend Jill to join him for co¤ee. Facing the alternative of staying at home, Jill gladly accepts the invitation. But when Jack o¤ers co¤ee versus a shot of cocaine at his hotel room, the host of possibilities spurred by the o¤er of cocaine sway Jill to stay at home. As in the previous example, Sen attributes the violation of the weak axiom to the informational content of the choice problem or its epistemic value, and argues that a fully rational individual would often infer the possible states of the world by examining what choices are being o¤ered to her. What are the testable implications of such contextual inference? Can a theory of revealed preference distinguish inference from arbitrary behavior? In the abstract choice setting, Sen [15, 16] gives a negative answer and argues that there is an “inescapable need to go beyond the internal features of a choice function to understand its cogency and consistency”. This paper provides an a¢ rmative answer by retaining the revealed preference approach while specializing the abstract choice setting. The abstract setting is restrictive for two reasons. One challenge, as Sen points out, arises when each choice problem induces a di¤erent perception of uncertainty. In that case, behavior exhibits no evident consistency across problems making the distinction between arbitrary behavior and rational inference impossible. The second di¢ culty precedes the question of consistency. In the abstract setting, the act of choice does not by itself reveal if the individual faces uncertainty at the time of choice. A customer who selects 2

chicken over steak might be perfectly con…dent in her enjoyment of the dish or completely unsure of the quality of food being served. This problem is …rst addressed by Kreps’ work [10] on preference for ‡exibility. Kreps’ seminal idea is that an individual’s desire to defer the choice of action reveals her perception of uncertainty. To model choice deferral, he postulates a preference ordering over menus or subsets of actions. By choosing a menu, the individual e¤ectively defers her choice of action until uncertainty resolves at a later (unmodeled) stage. This paper characterizes rational inference in the richer framework of preference for ‡exibility. To illustrate the main argument, imagine Jill’s choice problem when Jack does not o¤er cocaine. Suppose Jill had the possibility to defer choice. As Jill sees no reason to stay at home, the added ‡exibility would be of no value to her. Conversely, Jill’s indi¤erence to the added ‡exibility reveals that the actions home and co¤ee alone induce no uncertainty in her mind. In the formal framework, having the option to defer choice leads to a new and distinct choice problem which nonetheless retains the same basic actions (home and co¤ee) and therefore the same epistemic content as Jill’s original problem without deferral. Thus, a subcollection of epistemically equivalent problems can be constructed by varying the possibility to defer choice while keeping the subset of basic actions …xed. The main question then becomes: Is the collection of epistemically equivalent problems rich enough to reveal a unique preference ordering and pin down Jill’s perception of uncertainty? Standard uniqueness results in the theory of revealed preference require that behavior is coherent across all possible choice problems. It is therefore unsurprising that a proper subset of choice problems may not identify a unique preference ordering. The paper characterizes the extent to which a modeler who observes choice deferral may recover a local preference within each collection of epistemically equivalent problems. It then proves that the individual’s conceptualization of what might happen can be pinned down uniquely in the form of an endogenous state space. Since local preferences may not be fully recovered, the state space is de…ned directly in terms of observable behavior, and independently of preference. The construction of epistemically equivalent problems is closely motivated by the experimental work of Sha…r and Tversky [17] who employ choice deferral in order to test how di¤erent subsets of actions may in‡uence individuals’ perception of uncertainty. By adopting Kreps’framework for modeling choice deferral, this paper proposes a relaxation of the weak axiom that is consistent 3

with the experimental evidence and formalizes the identi…cation strategy in Sha…r and Tversky [17]. One of the main …ndings in Sha…r and Tversky [17] is that the inclusion of new alternatives often increases the likelihood of choice deferral. The suggested interpretation is that new alternatives lead subjects to conceive of new contingencies. In the present model, such behavior implies that subjective state spaces are nested: each state induced by a given subset of actions corresponds to an event, that is, a potentially incomplete resolution of uncertainty in the state space induced by larger choice problems. This result parallels recent epistemic models of unawareness. In particular, the syntactic construction of Heifetz, Meier, and Schipper [8] identi…es an individual’s awareness with a subset of a given language. The set of sublanguages in turn induces a lattice of nested state spaces. This paper provides a behavioral analogue of their construction, where each subset of actions faced by the individual shapes her awareness of relevant contingencies.

1.2

Formal Framework

To model choice deferral, the paper adopts the Dekel, Lipman, and Rustichini [4] framework of preference for ‡exibility, which extends Kreps’ work by modeling choice between sets of lotteries. Thus let Z = fz1 ; :::zK g denote a …nite set of actions and let Z be the set of lotteries over Z endowed with the Euclidean metric. A generic lottery is denoted by 2 Z. For an arbitrary metric space X, de…ne K(X) as the set of nonempty closed subsets endowed with the Hausdor¤ metric. Thus x 2 K( Z) is a menu or a generic subset of actions and constitutes an object of choice, whereas A 2 K(K( Z)) is a collection of menus or a choice problem. The primitive of the model is a choice correspondence c : K(K( Z)) K( Z) such that: ? 6= c(A)

A for all choice problems A 2 K(K( Z)).

When the individual chooses a menu x 2 c(A), she e¤ectively defers her choice of action (lottery) 2 x until after the uncertainty she perceives is resolved. This timeline is sketched below: ____ choose x2A

____ state realized

____ choose 2x

4

payo¤

The individual is forward-looking and her ex ante choice c(A) re‡ects, and under suitable conditions, reveals her perception of the relevant contingencies.

1.3

The Sha…r-Tversky Experiments

In the Sha…r and Tversky [17] experiments, subjects can choose either of two actions z; z 0 or they can defer choice. The alternative to defer choice is described as the option to "wait until [subjects] can learn more" about the actions z and z 0 . The possibility of obtaining information is therefore explicit in the experimental design. This is consistent with the interpretation of Kreps’framework, in which the option to defer choice is formally modeled by the menu fz; z 0 g. Sha…r and Tversky [17] …nd that the inclusion of a new alternative z 00 may increase the likelihood of deferring choice or, equivalently, of choosing the menu fz; z 0 g. The interpretation is that the new alternative z 00 leads subjects to conceive of new contingencies and so to defer their decision until after the uncertainty is resolved. If one takes z 00 to be cocaine, the behavior observed by Sha…r and Tversky [17] can be illustrated in terms of Sen’s example: c{{home}, {co¤ee}, {cocaine}} = {{home}}, (1.1) c{{home}, {co¤ee}, {cocaine}, {home, co¤ee}} = {{home, co¤ee}} The …rst problem corresponds to Jill’s original decision after the o¤er of cocaine. In the second, Jill faces the same actions, but has, in addition, the ‡exibility to defer her decision between home and co¤ee. In the interim, she might seek the advice of a friend and try to learn more about Jack. Thus Jill’s desire to defer choice reveals her uncertainty between the states ‘good Jack’, in which case she would accept the invitation for co¤ee, and ‘bad Jack’ when she would stay at home. Compare (1.1) with Jill’s behavior in the absence of cocaine. As Sen argues, the possibility of ‘bad Jack’is induced by the o¤er of cocaine. In its absence, Jill foresees no contingencies when staying at home is preferable. Consequently, she is indi¤erent to having the ‡exibility of postponing her decision: c{{home}, {co¤ee}} = {{co¤ee}}, (1.2) c{{home}, {co¤ee}, {home, co¤ee}} = {{home, co¤ee}, {co¤ee}} 5

Examples (1.1) and (1.2) illustrate the identi…cation strategy in the Sha…r and Tversky [17] experiments. By keeping the subset of basic actions …xed and simply adding the possibility to defer choice, one generates a subcollection of problems having the same informational content. Within any such collection, a fully rational individual would satisfy the weak axiom of revealed preference. De…nition 1 in this paper provides a formal construction of epistemically equivalent problems. Then Theorem 4 shows that the modeler can recover a preference ordering within each such collection. This local preference is unique in a strong albeit limited sense. To see why uniqueness may fail in general, suppose that in (1.1), Jack o¤ered co¤ee, cocaine and going to the movies: c{{home}, {co¤ee}, {movies}, {cocaine}} = {{home}}

(1.3)

Since the o¤er of cocaine sways Jill to stay at home, the modeler can never observe Jill’s ranking between having co¤ee and going to the movies conditional on the o¤er of cocaine. In the presence of cocaine, neither of the alternatives is ever chosen. Theorem 4 shows that this example is in fact tight: the ranking of alternatives that are chosen in at least one problem of the collection is unique. Despite the partial recoverability of local preferences, the paper shows that Jill’s perception of uncertainty or the states of the world she foresees can be fully identi…ed from behavior. To see why this is the case, imagine the following example: 1 co¤ee + c{{home}, { 10 9 c{{home}, { 10 co¤ee +

9 cocaine}, 10 1 cocaine}, 10

{co¤ee}} = {home}, {co¤ee}} = {co¤ee}

(1.4)

Jill can stay at home, o¤er co¤ee at her place or go out with Jack. Jack’s prior dates have resulted either in taking cocaine or having co¤ee. Two scenarios with di¤erent dating histories are depicted above. In both, the presence of cocaine induces the states ‘good Jack, bad Jack’, but Jill’s beliefs are likely to be correlated with the relative frequency of cocaine: She o¤ers co¤ee when cocaine is the exception, but stays at home when cocaine is the rule. In other words, the two choice problems are epistemically di¤erent yet they induce the same perception of relevant contingencies. Based on the joint data of such collections, Theorem 1 recovers the state space uniquely even when local preferences change depending on the relative frequency of 6

cocaine. This identi…cation requires that the state space be de…ned directly in terms of behavior and independently of local preferences. The intuitive appeal of example (1.4) depends crucially on the interpretation of the objective lottery as the relevant history of prior dates. In many economic settings, objective lotteries arise when the decision maker randomizes on her own or simply conceptualizes the set of possible lotteries. In such settings, objective lotteries have no epistemic content beyond the deterministic actions in their support. Under this assumption, a fully rational individual would satisfy the weak axiom across all problems with the same support. Then Theorem 5 recovers a local preference which is unique in the class of all continuous preferences. The restriction to continuous preferences is essential since continuity is used to ‘approximate’and pin down Jill’s ranking across alternatives that are never chosen, such as co¤ee and going to the movies in example (1.3).

2 2.1

Local State Spaces A Model of Rational Inference

This section describes the formal model of rational inference. First, the collections of epistemically equivalent problems are constructed based on the identi…cation strategy in the Sha…r and Tversky [17] experiments. Behavior is assumed to be coherent within each such collection. Second, the notion of a local state space is de…ned. As suggested by example (1.4), the de…nition is stated directly in terms of choice behavior and independently of local preferences. This separation facilitates the identi…cation of the local state space which is more general than, and logically precedes, the notion of local preference. Say that a choice problem B provides more ‡exibility than A, and write B B A, if B = A [ f[m i=1 xi g and xi 2 A for all i = 1; :::; m. If there exists a …nite sequence B1 ; :::; Bn such that B = Bn and Bn B Bn 1 :::B1 B A, write B Bn A and de…ne the collection A := fB : B Bn A; n 2 Ng. Choice problems in A retain the same basic actions as problem A but provide more ‡exibility to defer choice. Thus the collection A consists of epistemically equivalent problems and the following de…nition requires that behavior is coherent within A .

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De…nition 1 The choice correspondence c is weakly locally rationalizable if it is rationalizable within every collection of epistemically equivalent problems A for A 2 K(K( Z)). Weak local rationalizability ensures that violations of the weak axiom arise only in response to changes in the underlying actions or the probability distribution over actions. This requirement captures a notion of local coherency and underlies the experiments conducted by Sha…r and Tversky [17]. For any choice problem A, de…ne the support of a choice problem A, or simply supp(A), to be the subset of deterministic actions which receive strictly positive probability for at least one lottery in some menu x 2 A. For any nonempty, nonsingleton subset of actions Z 0 Z, let Z 0 denote the collection of all choice problems supported by the subset Z 0 . Choice problems in Z 0 retain the same deterministic actions in their support and, as example (1.4) suggests, they induce the same perception of uncertainty. To de…ne the local state space induced by Z 0 , let PZ 0 be the set of all nondegenerate 0 von Neumann-Morgenstern (vNM) preferences over Z 0 , and let SZ 0 RjZ j be a corresponding collection of vNM indices such that the mapping P : SZ 0 ! PZ 0 is one-to-one and onto. A local state space is a subset PZ 0 of PZ 0 . Each state or vNM preference in PZ 0 characterizes the individual’s ex post behavior, or her choice from the menu, conditional on some resolution of her subjective uncertainty. The next de…nition summarizes the behavioral implications of a local state space. Notationally, it is convenient to state the de…nition in terms of a subset of vNM indices SZ 0 SZ 0 . The de…nition does not depend on the particular normalization and both SZ 0 and its image PZ 0 := P (SZ 0 ) will be used interchangeably to mean a local state space. De…nition 2 A nonempty set of vNM indices SZ 0 SZ 0 is a local state space for the choice correspondence c on Z 0 if, for every problem A 2 Z 0 and menus x; y 2 A, the following conditions are ful…lled: (i) Su¢ ciency: If max

2x

s = max

s for every s 2 SZ 0 , then:

2y

x 2 c(A) if and only if y 2 c(A). (ii) Dominance: If max 2x s strict inequality for some s, then:

max

2y

s for every s 2 SZ 0 , with

x 2 A implies y 2 Anc(A). 8

Su¢ ciency requires that the state space SZ 0 be large enough to encompass all contingencies foreseen by the individual. Thus, any two menus that achieve the same ex post payo¤s are revealed indi¤erent. In contrast, Dominance requires that the state space SZ 0 be small in the sense that it includes only states that are behaviorally relevant. Together, Su¢ ciency and Dominance determine the individual’s subjective perception of uncertainty as revealed by her behavior. To illustrate the de…nition, recall Jill’s desire to defer her choice between home and co¤ee as in (1.1). By Su¢ ciency, S{home, co¤ee, cocaine} must include at least two payo¤ functions: one for the state when co¤ee is an optimal choice and another when home is preferred. In contrast, Dominance and the indi¤erence between {co¤ee} and {home, co¤ee} in example (1.2) require that S{home, co¤ee} excludes any state in which staying at home is optimal. Prior to the o¤er of cocaine, such contingencies are simply not relevant for Jill’s behavior. Jointly, weak local rationalizability and the existence of local state spaces provide a framework for modeling rational inference.

2.2

Axioms

In the theory of choice, the weak axiom of revealed preference (WARP) is the paradigmatic requirement of coherent or rational behavior. The axiom, see Arrow [3], takes the following form: WARP: For all choice problems A and B such that A

B:

c(B) \ A 6= ? implies c(A) = c(B) \ A: The weak axiom requires that an alternative y cannot be revealed strictly preferred to x if x has been revealed preferred to y in another choice problem. The following axiom preserves the substantive content of the weak axiom, but applies it only to some choice problems. To illustrate, consider a problem A and any two alternatives x; y 2 A. Adding the union of x and y to the choice problem A keeps the set of basic actions …xed while enabling the individual to defer choice between x and y. Consequently, the individual is contemplating the same set of actions but is less constrained to make the choice ex ante, that is, prior to the resolution of any contingencies she foresees. As no new information is obtained, the individual’s behavior is coherent: 9

Weak Consistency: For any choice problems A and menus x; y 2 A, let B := A [ fx [ yg. Then: c(B) \ A 6= ? implies c(A) = c(B) \ A: Weak Consistency ensures that violations of the weak axiom arise only in response to changes in the underlying lotteries or actions. One such response is illustrated in example (1.4). There Jill’s perception of uncertainty is determined by the support of the choice problem, but her beliefs are revised depending on the probability of cocaine. To characterize Jill’s local state space, it is necessary to isolate behavior that depends only on Jill’s conceptualization of what might happen, but not on her beliefs. For this purpose, consider any menus x and y such that y x and observe that Jill reveals a strict preference for x if and only if she perceives a contingency that makes the ‡exibility provided by x valuable. This argument plays a key role in the next axiom. To state the axiom, de…ne the mixture x + (1 )x0 of two menus 0 x; x 2 K( Z) for some real number 2 [0; 1] as follows: x + (1

)x0 := f

+ (1

)

0

:

2 x;

0

2 x0 g.

Observe that x y implies x + (1 )x0 y + (1 )x0 for all menus 0 x . Suppose then that x y is revealed strictly preferred to y in some choice problem A. This choice reveals that the individual perceives a contingency in which some action 2 xny is strictly more desirable than any 2 y. But if that same contingency obtains, an individual who satis…es vNM independence would …nd the lottery + (1 ) 0 strictly preferable to 0 0 + (1 ) for any 2 y and . Ex ante, a forward-looking individual should therefore prefer the ‡exibility provided by x+(1 )x0 over y +(1 )x0 in any choice problem that induces the same perception of uncertainty. Intuitively, these are choice problems having the same support: Weak Independence: For all choice problems A; B with supp(A) = supp(B), all menus x; y; x0 ; x00 with x y and ; 0 2 (0; 1]: [ x + (1

)x0 2 c(A) and [ 0 x + (1

y + (1 0

)x0 2 Anc(A)]

)x00 2 c(B) )

0

y + (1

implies 0

)x00 2 Bnc(B)].

The next axiom ensures that ‡exibility is always weakly preferable. 10

Monotonicity: For any choice problem A and menus x; y 2 A such that x y: x 2 c(A) implies y 2 c(A): The …nal axioms impose a weak continuity requirement and a standard nontriviality assumption. Weak continuity ensures that perturbing the menus slightly without changing the subset of actions in their support does not drastically alter the individual’s preference for ‡exibility. Weak Continuity: For any sequence An := fxn ; yn g converging to A := fx; yg such that 8n supp(An ) =supp(A) and xn yn : yn 2 c(An ) 8n implies y 2 c(A): Nontriviality: For any nonsingleton, nonempty subset Z 0 of Z, there exists a choice problem A such that supp(A) = Z 0 and c(A) 6= A. The next theorem shows that the above axioms fully characterize the model of rational inference. Theorem 1 The choice correspondence c satis…es Weak Consistency, Weak Independence, Monotonicity, Weak Continuity and Nontriviality if and only if c is weakly locally rationalizable and has a local state space PZ 0 on Z 0 for every nonempty, nonsingleton subset Z 0 of Z . Moreover, if PZ0 0 is another 0 local state space for c on Z 0 , then cl(PZ 0 ) = cl(PZ 0 ).1 Remark 1 If PZ 0 is a local state space then so is its closure cl(PZ 0 ). In view of the uniqueness result in Theorem 1, refer to a local state space as the closed set of preferences satisfying De…nition 2. For every subset of actions Z 0 , the local state space PZ 0 is derived from choice behavior restricted to the collection of problems Z 0 . This derivation assumes weak local rationalizability. The property is strictly weaker than the existence of a complete and transitive ordering that rationalizes behavior on the entire collection Z 0 . Accordingly, Theorem 1 provides a choicetheoretic characterization of a subjective state space that is, to a large extent, independent of the hypothesis of utility maximization. 1

cl(PZ 0 ) denotes the closure of PZ 0 in an appropriate topology on PZ 0 . Appendix 1 provides details.

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Weakening rationalizability requires a de…nition of a state space that is strictly stronger than the one posited by Dekel, Lipman, and Rustichini [4]. In their most general formulation, they de…ne a state space to be the smallest set of vNM preferences for which Su¢ ciency (cf. De…nition 2) obtains. In the absence of rationalizability, the latter de…nition imposes few restrictions on observed behavior and poses questions about the interpretation of certain choices. To elaborate, consider Sen’s example and focus on the collection of choice problems whose support is {home, co¤ee}. Under the Dekel, Lipman, and Rustichini [4] de…nition of a state space, it is possible that an individual is indi¤erent between the menus {home, co¤ee} and {co¤ee} in one problem of the collection, but that she exhibits a strict preference in another. In so far as both problems have the same support, such reversals cannot be attributed to the epistemic content of a new action. In turn, it is no longer possible to isolate the e¤ect of o¤ering {cocaine} on the individual’s perception of uncertainty.

3

Small Worlds

The o¤er of cocaine leads Jill to think of new contingencies. They re…ne her understanding of the world and of the possible implications her actions might have. The next axiom requires that larger supports induce a richer perception of uncertainty and thereby a greater demand for ‡exibility: Nestedness: For all choice problems A; B such that supp(A) all menus x; y; x0 with x y and 2 (0; 1]: [x 2 c(A) and y 2 Anc(A)] [ x + (1

supp(B);

implies )x0 2 c(B) ) y + (1

)x0 2 Bnc(B)].

The formulation of Nestedness and its interpretation are analogous to Weak Independence. The rest of this section addresses its implications in terms of how state spaces re‡ect an individual’s increased awareness of uncertainty. In his seminal work on subjective probability, Savage [13, p. 83] alludes to the practical necessity that the relevant state space depend on the isolated decision problem, and at best, approximate an idealized universal state space. Consequently, he postulates a ‘small world’that is a partition of the universal 12

state space or ‘the grand world’. Each small state constitutes an event in the grand world or an incomplete resolution of uncertainty. This formulation is substantiated in the recent work of Heifetz, Meier, and Schipper [8] who construct a canonical model of knowledge and awareness. In their syntactic framework, the individual’s awareness is identi…ed with a subset of a universal language. The set of sublanguages in turn induces a lattice of nested state spaces - each state described by some language corresponds to an event in the state space of a richer language. The next theorem provides a behavioral analogue of their construction where the support of each choice problem shapes the individual’s awareness of relevant contingencies. To illustrate, take Jill’s initial preference for co¤ee over staying at home. Following the o¤er of cocaine, this preference is still a contingency that may eventuate but need no longer exhaust, or resolve, all uncertainty. More formally, let Z 0 ; Z 00 be nonempty, nonsingleton subsets of 00 actions such that Z 0 Z 00 and let tZZ 0 : RZ 00 ! RZ 0 denote the restriction of each preference pro…le in RZ 00 to the set of lotteries Z 0 . The theorem shows that every state % in the subjective state space PZ 0 corresponds to a 00 nonempty event, (tZZ 0 ) 1 (%) \ PZ 00 , in the state space PZ 00 induced by the richer support Z 00 . Theorem 2 Let c be a nontrivial choice correspondence having a local state space PZ 0 for every nonempty, nonsingleton subset Z 0 of Z. The choice corre00 spondence c satis…es Nestedness if and only if PZ 0 tZZ 0 (PZ 00 ) for all subsets Z 0 , Z 00 s.t. Z 00 Z 0 .

4

Recovery of Local Preferences

This section investigates the uniqueness of local preferences. A central result in revealed preference theory is the following theorem due to Arrow [3]: Theorem 3 (Arrow [3]) If a collection includes all choice problems of up to three alternatives, then the choice correspondence c satis…es WARP if and only if it is rationalizable. The preference relation % that rationalizes c is unique. It is evident from examples (1.1) and (1.2) that collections of epistemically equivalent problems in general fail Arrow’s condition. In (1.1), the modeler tries to infer Jill’s local preference over the actions {home}, {co¤ee} 13

and {cocaine}, yet the binary problem {{home}, {co¤ee}} is epistemically di¤erent. Its inclusion to the collection of problems in (1.1) leads to a violation of the weak axiom. It is important to recognize that the failure of Arrow’s condition jeopardizes both the existence and uniqueness of a rationalizing preference relation. The problem of existence is fully resolved by Richter [12]. He identi…es a requirement stronger than the weak axiom whereby a choice correspondence c is rationalizable on any collection . At the same time, the generality of Richter’s nonconstructive approach provides no answer to the problem of uniqueness. Intuitively, uniqueness depends on the speci…c structure and richness of the collection . The next theorem identi…es a broad class of collections which includes the collections of epistemically equivalent problems A and admits both a constructive approach to preference and a suitably de…ned uniqueness property. To state the theorem, de…ne c( ) := [A2 c(A) to be the range of the choice correspondence c, that is, the set of alternatives chosen from some problem in the collection . Theorem 4 If is closed under …nite unions, then the choice correspondence c satis…es WARP if and only if it is rationalizable. If two preference relations % and %0 rationalize c on , then % and %0 agree on the range of chosen alternatives c( ). Theorem 4 shows that within each collection of epistemically equivalent problems the modeler can recover a local preference which is unique when restricted to the set of chosen alternatives c( ). Notice that if all binary problems are observable as in Arrow’s theorem, then the range c( ) includes the whole set of feasible alternatives (except for possibly one) and behavior is uniquely rationalizable. In general, the range c( ) is a proper subset of all alternatives within . In example (1.3), the actions {co¤ee} and {movies} are never chosen by Jill and their ranking conditional on the o¤er of cocaine cannot be inferred from behavior. Thus the partial recoverability in Theorem 4 is in general binding for a model of rational inference.

4.1

A Special Case: Only Supports Matter

The rest of this section studies the special case when objective lotteries have no epistemic content beyond the deterministic actions in their support. While 14

violated in example (1.4), this assumption is satis…ed in many economic settings in which the decision maker randomizes on her own or simply conceptualizes the set of possible lotteries. In experiments, the modeler can also utilize an extraneous randomization device which has no epistemic content. In these cases, the question becomes: Can we perturb the probabilities of unchosen actions and …nd ‘nearby’ lotteries whose ranking is observable? Put di¤erently, can we approximate the ranking of unchosen actions by extending continuously the unique ranking on c( )? Before answering the question, it is useful formalize the stronger restrictions on behavior. The following axioms require that behavior is coherent and also continuous within each collection of problems having the same support. These requirements capture the fact that only supports have informational content. Consistency: For all choice problems A and B such that supp(A) =supp(B) and A B, c(B) \ A 6= ? implies c(A) = c(B) \ A. Continuity: For any sequence of choice problems An converging to A such that 8n supp(An ) =supp(A) and yn 2 c(An ): yn ! y implies y 2 c(A). The topological closure of a preference ordering does not, in general, preserve transitivity. In Fig. 1 below, there is a continuous ordering in the interior of a simplex. Two of its indi¤erence curves, however, intersect in the limit. Consequently, any extension of the ranking to the entire simplex must violate either transitivity or continuity. It is possible to cast the problem of this section in the context of Fig. 1. To do so, recall that, for any subset Z 0 , Z 0 denotes the collection of problems supported by Z 0 , and interpret the set of chosen actions, c( Z 0 ), as the interior of a simplex. The boundary of the simplex represents actions that are never chosen and whose ranking is therefore unobservable. If behavior satis…es Consistency, Theorem 4 delivers a unique ranking on c( Z 0 ). When Continuity, Monotonicity and Weak Independence are further assumed, the next theorem proves that the tradeo¤ in Fig. 1 does not arise: the unique ranking on c( Z 0 ) admits a transitive and continuous extension to all alternatives within Z 0 . Moreover, the ranking is unique in the class of all continuous preferences that rationalize behavior. Notice that this result is still weaker 15

Figure 1: The closure of a preference ordering may not preserve transitivity

than Arrow [3]. Other preference relations may rationalize behavior but by the next theorem they are necessarily discontinuous. Theorem 5 A choice correspondence c satis…es Consistency, Weak Independence, Monotonicity, Continuity and Nontriviality if and only if for every nonempty, nonsingleton subset Z 0 of Z there exists a continuous local preference %Z 0 that rationalizes c on Z 0 , and a local state space SZ 0 . The local preference is unique in the class of all continuous, complete and transitive relations that rationalize c on Z 0 . Remark 2 The unique continuous preference %Z 0 inherits the local state space SZ 0 . That is, SZ 0 is the subjective state space of %Z 0 in the sense of Dekel, Lipman, and Rustichini [4]. The content of Theorem 5 is to prove the existence of a unique continuous preference %Z 0 that rationalizes c on Z 0 .

5

Related Literature

Motivated by Luce and Rai¤a’s dinner, Kalai, Rubinstein, and Spiegler [9] explore the rationalization of a choice function by multiple preferences. To describe their approach, consider an abstract set of actions K and a collection of nonempty choice problems. A choice function c is locally rationalizable if there exists a partition of such that c is rationalizable within each cell 16

in . It is evident that any choice function can be locally rationalized by taking to be the discrete partition, that is, by assigning a possibly di¤erent preference to each choice problem in . This rationalization is extreme in that it fails to account for the consistency of behavior within any subset of choice problems. Kalai, Rubinstein, and Spiegler [9] focus on the minimal number of orderings su¢ cient to explain behavior, or equivalently, on a partition such that (i) the choice function c is rationalizable within each of its cells, and (ii) there is no strictly coarser partition having the same property. Such ‘minimal’representations exist for any choice function, but neither the partition nor the implied local rankings are in general unique. As the authors show, moreover, many ‘simple’choice procedures do not admit a small number of orderings. Kalai, Rubinstein, and Spiegler [9, p. 2487] conclude that the appeal of a certain collection of local preferences “does not emanate only from its small number of orderings, but also from the simplicity of describing in which cases each one of them is applied”and that “more research is needed to de…ne and investigate ‘structured’forms of rationalization.” Sen [15, 16] argues that there is no such ‘structure’, or no way of restricting a priori the choice problems for which the weak axiom should hold. The necessary structure in this paper is provided by the framework of preference for ‡exibility. Within that domain, the collection of choice problems is partitioned into classes of epistemically equivalent problems. The partition admits a clear and intuitive interpretation. Namely, adding the option to defer choice does not alter the informational content of a choice problem. The same rationale is utilized in the Sha…r and Tversky [17] experiments on contextual inference. The domain permits an analysis of uniqueness that is not possible in an abstract setting. The signi…cance of the domain is evident from Theorems 4 and 5 in which the stated uniqueness results depend critically on the speci…c properties of the collection of choice problems. Identifying these properties and the implied degree of uniqueness is the main analytical question addressed in the paper. In recent papers, Green and Hojman [7] and Ambrus and Rozen [2] use multi-selves or mutli-criteria models in order to accommodate violations of the weak axiom documented in experimental work. Drawing on work in social choice, a rationalization in these papers consists of a family of preference orderings, or multiple selves, together with an aggregation rule. In the language of this paper, multiple selves correspond to possible states of the world. Both Green and Hojman [7] and Ambrus and Rozen [2], however, operate in 17

an abstract choice setting and admit all possible choice functions. The cost of such generality is that models are not identi…ed. In contrast, this paper shows that models of contextual inference are both testable and identi…able in the richer framework of preference for ‡exibility.

6 6.1 6.1.1

Appendix Preliminary Results Support Functions

D denotes the collection of nonempty, nonsingleton subsets of Z. For any Z 0 2 D, endow K( Z 0 ) with the Hausdor¤ metric: h(x; y) := maxfsup inf d( ; 0 2x

2y

0

); sup inf d( ; 0 2y

0

)g;

2y

0

where d( ; ) denotes the Euclidean distance in RjZ j . Let X K( Z 0 )P denote theP set of closed and convex subsets of Z 0 and 0j 2 jZ SZ 0 := fs 2R : i si = 1g. Identify each x 2 X with its i si = 0; support function x : SZ 0 !R de…ned by: x (s)

= max 2x

s:

Note that C := f x : x 2 X g is a convex subset of C(SZ 0 ), the space of continuous functions on SZ 0 . Endow C(SZ 0 ) with the sup-norm jjf jj := supfjf (s)j : s 2 SZ 0 g and the pointwise order: f g if and only if f (s) g(s) 8s 2 SZ 0 . For any f; g 2 C(SZ 0 ), de…ne the function f _ g 2 C(SZ 0 ) by (f _ g)(s) = maxff (s); g(s)g 8s 2 SZ 0 . The following lemmas summarize results obtained in [4, 5]. Lemma 1 The mapping x 7 ! x is an isometry from X to C such that: (i) x+(1 )y = x + (1 ) y; (ii) x y if and only if x y; (iii) x _ y p = co(x[y); (iv) 8c 2 [0; 2K2 ) there exists a menu x 2 X such that x c. In particular, the function that is identically zero on SZ 0 lies in C i.e. 0 2 C C(SZ 0 ). Lemma 2 The set H := f ( of C(SZ 0 ).

0

):

18

0; ;

0

2 Cg is a vector sublattice

6.1.2

The Set of vNM Preferences

For any subset of actions Z 0 2 D, let RZ 0 denote the set of vNM preferences over Z 0 endowed with the quotient topology, i.e., the strongest topology 0 such that the mapping P : RjZ j ! RZ 0 is continuous. The quotient topology on RZ 0 is used by Dhillon and Mertens [6] and as they remark a subbase 0 is given by the collection ff%: g : ; 0 2 Z 0 g. One feature of the quotient topology is that it is not Hausdor¤: RZ 0 is the only open neighborhood of the degenerate preference %0 (complete indi¤erence). On the other hand, PZ 0 := RZ 0 nf%0 g with its relative topology is homeomorphic to SZ 0 and therefore a compact, Hausdor¤ space. This delivers the following lemma. 00

Lemma 3 Let Z 0 Z 00 and tZZ 0 : RZ 00 ! RZ 0 denote the projection that maps a preference in RZ 00 into its induced preference in RZ 0 , Then: 00 (i) tZZ 0 is continuous; 00 (ii) If F PZ 00 is relatively closed in PZ 00 , then tZZ 0 (F ) \ PZ 0 is relatively closed in PZ 0 . Proof: (i) Take a basic set U = f%2 RZ 0 : construction: 00

Z 00 Z0

(tZZ 0 ) 1 (U ) = f%2 RZ 00 :

k

k 0 k

0 k

for k = 1; :::; ng. By

for k = 1; :::; ng

Thus (t ) 1 (U ) is basic in the quotient topology on RZ 00 proving the conti00 nuity of tZZ 0 . 00 To prove (ii), let f%n g be a sequence in tZZ 0 (F )\PZ 0 converging to %2 PZ 0 0 in the relative topology. Take any sequence f%n g such that: %0n 2 (tZZ 0 ) 1 (%n ) \ F 8n. 00

0

Since F PZ 00 is relatively closed and PZ 00 is compact, f%n g has a subse00 00 quence converging to %0 2 F . By continuity, %= tZZ 0 (%0 ) 2 tZZ 0 (F ) \ PZ 0 . As in Dekel, Lipman, and Rustichini [4], the closure of a local state space P is de…ned in terms of the relative topology on PZ 0 . Z0

6.2

Proof of Theorem 1

Necessity of the axioms follows from the properties of support functions obtained in Lemma 1. To prove su¢ ciency, …x some Z 0 2 D and de…ne: Z0 Z0

: = fA 2 K(K( Z)) : supp(A) = Z 0 g, and : = fA 2 K(K( Z)) : supp(x) = Z 0 8x 2 Ag. 19

De…ne the possibly incomplete relation % on K supp(x) = Z 0 g as follows: x % y if and only if 9A 2

Z0

:= fx 2 K( Z) :

s.t. x [ y 2 A and x 2 c(A):

(6.1)

Let and denote the asymmetric and symmetric parts of %, respectively. The following lemmas summarize some useful properties of %. Lemma 4 If the choice correspondence c satis…es Weak Independence and Monotonicity then: (i) x y implies x % y; (ii) x % y if and only if x x [ y; (iii) % satis…es Independence, i.e., for all x0 and 2 (0; 1]: x % y if and only if x + (1

)x0 % y + (1

)x0 ;

(6.2)

(iv) x % y implies x [ x0 % y [ x0 ; (v) % is transitive. Proof: The …rst two properties follow easily from the de…nition of %. To see (iii), take x % y and note that: x + (1

)x0 [ y + (1

)x0 =

x [ y + (1 )x0 x + (1 )x0 :

The indi¤erence follows from Weak Independence and the fact that x by (ii). Applying property (ii) again gives x + (1 )x0 % y + (1 The proof of the ‘if’part is analogous. To see (iv), let: 1 1 1 1 1 1 A = f x + x [ y [ x0 ; x + x [ x0 ; x [ y + x [ x0 g 2 2 2 2 2 2

x[y )x0 .

(6.3)

By Monotonicity, 21 x [ y + 12 x [ x0 2 c(A). Then Weak Independence implies that: 1 1 x + x [ x0 2 c(A): (6.4) 2 2 Applying Monotonicity again: 1 1 x + x [ y [ x0 2 c(A). 2 2 20

(6.5)

Weak Independence and equations (6.4), (6.5) imply: x [ x0 x [ y [ x0 . To see (v), let x % y % x0 and A = fx; x [ y; x [ x0 ; x [ y [ x0 g. By Monotonicity, x [ y [ x0 2 c(A): But then y % x0 and (iv) imply that y [ x y [ x0 [ x. By the de…nition of %, x [ y is chosen in some problem A0 2 Z 0 such that y [ x0 [ x 2 A0 . By Weak Independence, x [ y 2 c(A). Similarly, Weak Independence and x % y then imply that x 2 c(A). By Monotonicity, x [ x0 2 c(A): Conclude that x x [ x0 and so x % x0 . ) = Z 0 and for every x; y 2 K( Z 0 ), extend

Fix some such that supp( 0 % to K( Z ): x % y if and only if

1 1 x+ f 2 2

1 1 g% y+ f 2 2

g.

By Weak Independence the extension is well-de…ned. Furthermore, the extended % is a preorder that satis…es Independence (6.2). By Weak Continuity, % is also closed in the product topology on K( Z 0 ) K( Z 0 ) and so a continuous a¢ ne preorder. The construction of % is inspired by Kreps’notion of a dominance relation whose main feature is given by property (ii) in Lemma 4: x % y if and only if x

x [ y.

The dominance relation % summarizes the part of individual’s behavior that pertains only to her perception of contingencies but not to her beliefs. Lemmas (5-7) stated next establish this formally. The lemmas show that there exists a bijection between the set of continuous a¢ ne preorders on K( Z 0 ) that satisfy property (ii) above and the set K(SZ 0 ) of possible state spaces. The proof proceeds by identifying the preorder % with a cone D in 0 the linear space H = f ( ) : 0; ; 0 2 Cg. Property (ii) then implies that the dual cone of D equals the set of probability measures on some closed subset of SZ 0 . First, use [5, Lemma S6] to show that for any menu x, x conv(x). By Lemma 1, % de…nes a continuous a¢ ne preorder on C = f x : x 2 X g. De…ne the cone of % as: D := f (

0

):

0;

21

%

0

8 ;

0

2 Cg:

Let L denote a generic linear functional on H and de…ne the dual cone of D as: D := fL : L( ) L( 0 ) whenever % 0 g. The next two lemmas prove that D is a nonempty set of linear functionals representing %. Lemma 5 The linear span of D, denoted by span(D), equals H. D has nonempty algebraic interior in H. Proof: By monotonicity, the positive orthant H+ := ff 2 H : f 0g of H is a subset of D and so span(H+ ) span(D). By Lemma 2, H = H+ H+ which implies that span(H+ ) = H and so span(D) = H. To prove the second assertion, it su¢ ces to show that H+ has nonempty topological interior in H. Fix any f 2 H+ such that M := minff (s) : s 2 SZ 0 g > 0 and note that the open ball ff 0 2 H : jjf 0 f jj < M2 g is contained in H+ . The next result is due to Shapley and Baucells [18]. Lemma 6 Let C be a mixture space embedded in a linear space H such that 0 2 C and span(C) = H and let % be a mixture-continuous a¢ ne preorder whose cone D has nonempty algebraic interior in H. Then D is nonempty and for all ; 0 2 C; % 0 if and only if L( ) L( 0 ) 8L 2 D . By Monotonicity, each L 2 D is a positive linear functional. Lemma 5 shows that H+ = ff 2 H : f 0g has nonempty topological interior and so each L is a continuous linear functional by a theorem in Schaefer [14, p. 225]. Since H is a linear subspace of C(SZ 0 ) and each L 2 D is a continuous, linear functional on H, [1, Lemma 6.13] implies that L can be extended to a continuous linear functional on C(SZ 0 ). By Riesz Representation Theorem [1, Lemma 13.15], for each L 2 D there exists a countably additive …nite R Borel measure L on SZ 0 such that L(f ) = L (f ) := S 0 f d L 8f 2 C(SZ 0 ). Z Since each L 2 D is positive, L can be normalized to a Borel probability measure in SZ 0 . Letting := f L : L 2 D g, it follows that: =f 2

( 0 ) whenever

SZ 0 : ( )

22

%

0

g.

And for every ; %

0

2 C: 0

if and only if ( )

( 0 ) for all

2

(6.6)

:

De…ne the support of each measure 2 as the closed set supp( ) satisfying: (i) (supp( )c ) = 0, and (ii) If G is open and G \ supp( ) 6= ?, then (G \ supp( )) > 0. By [1, Lemma 2.5], supp( ) exists and is unique. Lemma 7 The set SZ 0 := [

2

supp( ) is closed in SZ 0 and

=

SZ 0 .

Proof: Take a sequence fsn g in SZ 0 converging to s. It is enough to show that s , the Dirac measure on s, belongs to . Fix some % 0 . By Lemma 1, _ 0 2 C and by the construction of %, % 0 if and only if _ 0. 0 By equation (6.6), it must be the case that (sn ) = _ (sn ) for all n. The continuity of support functions implies that (s) = _ 0 (s). Conclude that 0 _ 0 (s) (s) = s ( 0 ) and so s 2 . s ( ) = (s) = To prove the second assertion, take 2 SZ 0 . As argued above, % 0 if and only if _ 0 and so (s) = _ 0 (s) for all s 2 SZ 0 . Conclude 0 that ( ) = ( _ ) ( 0 ) and so 2 : The following lemma shows that SZ 0 is a local state space for c on

Z0 .

Lemma 8 For every choice problem A 2 Z 0 and x; y 2 A: (i) x jSZ 0 = y jSZ 0 implies [x 2 c(A) , y 2 c(A)]; (ii) x jSZ 0 y jSZ 0 implies [x 2 A ) y 2 Anc(A)]; Proof: Abusing notation, write x for x jSZ 0 . To see (i), take a set A such that x 2 c(A); y 2 A and x = y . Let B = A[fx[yg. By Monotonicity and Weak Consistency, fx[yg 2 c(B). Furthermore, y = x[y implies y x[y and by the de…nition of %, y belongs to c(B). By Weak Consistency, we must have y 2 c(A). To prove (ii), suppose by way of contradiction that y 2 c(A). As before, x [ y 2 c(B). But then x = x[y implies that x 2 c(B). By Weak Consistency, y belongs to c(B). This implies y x [ y. On the other hand, x [ y y, a contradiction. x y and equation (6.6) imply x The next two lemmas establish the uniqueness of SZ 0 . The …rst adopts an argument in Dekel, Lipman, and Rustichini [4, Lemma 6]. 23

Lemma 9 For every s 2 SZ 0 and every open neighborhood O of s, there exist 0 0 0 0 0 0 x ; y 2 C such that x (s ) = y (s ) 8s 2 SZ 0 nO and x (s ) y (s ) 8s 2 O and x (s) > y (s). Proof: Fix a state s 2 SZ 0 . Take any two concentric closed balls x; x0 such that x ( x0 f : supp( ) = Z 0 g and let ; 0 be the unique points in x; x0 respectively such that x (s ) = s and x0 (s ) = 0 s . Clearly we have x (s ) < x0 (s ). Let f n g be a sequence of real numbers converging to 1 and 0 de…ne n := n +(1 and xn := conv(x [ f n g) n ) . Note that n ! x 8n. De…ne D(x; xn ) := cl(fs : xn (s) > x (s)g). Note that s 2 \1 i=1 D(x; xn ). Proceed to show that fs g = \1 D(x; x ). To see this, take any s0 6= s . n i=1 For such s0 , we have s0 < x (s0 ) by construction. Since n ! , for n 0 0 su¢ ciently large n s < x (s ) or s0 2 = D(x; xn ). It follows that fs g = \1 D(x; x ). But then fs g equals the Hausdor¤ limit of fD(x; xn )g and n i=1 so D(x; xn ) must be a subset of any neighborhood of s for n su¢ ciently large. Lemma 10 If S SZ 0 .

SZ 0 is another local state space for c on

Z0 ,

then cl(S) =

Proof: Suppose by way of contradiction that there exists s 2 cl(S)nSZ 0 . The case s 2 SZ 0 ncl(S) is analogous. Since SZ 0 is closed there exists an open neighborhood O of s that is disjoint from SZ 0 . Further, s 2 cl(S) implies that O \ S 6= ?. Take s0 2 O \ S and x , y as constructed in Lemma 9. But then for any choice problem A 3 x; y: x jSZ 0

=

x jS

y jSZ 0

implies [x 2 c(A) , y 2 c(A)]

and y jS implies [x 2 A ) y 2 Anc(A)]

The two implications above are incompatible for A = fx; yg 2

Z0 .

It remains to prove that c is rationalizable on A for every A 2 K(K( Z)). By Theorem 4, it is enough to show that A is closed under unions and that c satis…es WARP on A . Lemma 11 For all B 2

A,

c(B) \ A 6= ? implies c(B) \ A = c(A). 24

Proof: De…ne B B0 A if B = A [ fx [ yg and x; y 2 A. If there exists a …nite sequence B1 ; :::; Bn such that B = Bn and Bn B0 Bn 1 :::B1 B0 A, write B B0n A. First, we prove by induction that B B0n A for some n and c(B) \ A 6= ? implies c(B) \ A = c(A). For n = 1, we have B B0 A and the argument follows from Weak Consistency. Fix some n 2 N and suppose the result holds for all B such that B B0n A. Take some B B0n+1 A and a sequence B = Bn+1 B0 Bn ::: B0 B1 B0 A. Then c(Bn+1 ) \ A 6= ? and A Bn imply c(Bn+1 ) \ Bn 6= ?. By Weak Consistency, c(Bn ) = c(Bn+1 ) \ Bn and so c(Bn ) \ A = c(Bn+1 ) \ A 6= ?. But then c(Bn+1 ) \ A = c(Bn ) \ A = c(A), where the last equality follows from the induction hypothesis. Next take B B A such that c(B) \ A 6= ?. That is, B = A [ f[m i=1 xi g and xi 2 A 8i. We are going to show that c(B) \ A = c(A). The proof of the lemma then follows from an inductive argument identical to the one above. 1 0 0m 1 De…ne B 0 := B [ fx1 [ x2 g::: [ f[m A i=1 xi g. By construction, B B 0 0m 2 0 0m 1 0 and B B B. Since B B A, we must have either c(B ) \ A 6= ? 0 0 or [m i=1 xi 2 c(B ) (by Monotonicity). Conclude that c(B ) \ B 6= ? and so B 0 B0m 2 B implies that c(B 0 ) \ B = c(B). But then c(B 0 ) \ A = c(B) \ A 6= ?, which implies c(A) = c(B 0 ) \ A since B 0 B0m 1 A. Combining the last two equalities, we have c(A) = c(B 0 ) \ A = c(B) \ A as desired. To see that c is rationalizable on A , take some B; B 0 2 A such that B 0 B and c(B 0 ) \ B 6= ?. Since B 0 nB B 0 nA and the latter is a …nite set, B 0 nB is …nite. Conclude that B 0 2 B . By Lemma 11, c(B 0 ) \ B = c(B). To see that A is closed under unions, take some B; B 0 2 A . Since (B [ B 0 )nB (B [ B 0 )nA = (BnA) [ (B 0 nA), (B [ B 0 )nB is …nite. Conclude that B [ B 0 2 B A as desired.

6.3

Proof of Theorem 2

For each Z 0 2 D, let %Z 0 be the preorder constructed above and let P (SZ 0 ) be the local state space for c on Z 0 . Proceed to show that for all Z 0 Z 00 : P (SZ 0 )

00

tZZ 0 (P (SZ 00 )) if c satis…es Nestedness.

It su¢ ces to show that P (SZ 0 ) diction, suppose:

00

tZZ 0 (P (SZ 00 )) \ PZ 0 . By way of contra-

9 %2 P (SZ 0 )n(tZZ 0 (P (SZ 00 )) \ PZ 0 ) 00

25

Since P : SZ 0 ! PZ 0 is a homeomorphism, we have: s := P

1

S := P

1

(%) 2 (SZ 0 nS)

where 00

SZ 0 ,

00

((tZZ 0 (P (SZ 00 )) \ PZ 0 )).

By Lemma 3, tZZ 0 (P (SZ 00 )) \ PZ 0 is relatively closed in PZ 0 and so S is closed in SZ 0 . Apply Lemma 9 to …nd menus x y 2 K( Z 0 ) such that: x jSZ 0

y jSZ 0

and

x jS

=

y jS .

(6.7)

Since SZ 0 is a local state space, Dominance implies x Z 0 y. Furthermore, if F denotes the …eld on SZ 00 generated by projZ 0 ( ), then s 7 ! max 2y s is an F-measurable function, whenever y 2 K( Z 0 ). Then x jS = y jS implies that: x jSZ 00 = y jSZ 00 . The su¢ ciency of SZ 00 then implies x ness. Necessity is easily established.

6.4

Z 00

y in contradiction of Nested-

Proof of Theorem 4

Let X be an arbitrary set and let be a collection of subsets of X that is closed under unions. De…ne x % y if x; y 2 c( ) := [A2 c(A) and there exists a problem A 2 such that x; y 2 A and x 2 c(A). Proceed to show that the binary relation % on c( ) is complete and transitive. To prove completeness, take x; y 2 c( ) and let Ax and Ay be the sets such that x 2 c(Ax ) and y 2 c(Ay ). Without loss of generality suppose that c(Ax [ Ay ) \ Ax 6= ?. Apply WARP to conclude that x 2 c(Ax [ Ay ) and so x % y. To prove transitivity, take x % y % x0 and let Axy be the set such that x 2 c(Axy ) and y 2 Axy . De…ne Ayx0 analogously. If c(Axy [Ayx0 )\Axy 6= ?, then WARP implies that x 2 c(Axy [ Ayx0 ) and so x % x0 . If c(Axy [ Ayx0 ) \ Ayx0 6= ?, then WARP implies that y 2 c(Axy [ Ayx0 ). But y 2 Axy \ c(Axy [ Ayx0 ) and WARP imply that x 2 c(Axy [ Ayx0 ) as desired. Extend % to X as follows: if x; y 2 Xnc( ) set x y; if x 2 c( ) and y 2 Xnc( ) set x y. The extended % is easily seen to be complete and transitive on X.

26

To prove that % rationalizes c, de…ne c (A) := fx 2 A : x % y 8y 2 Ag for any A 2 . Fix some A and note that by de…nition of %, c(A) c (A). Next take x 2 c (A) and y 2 c(A). Since x % y, there exists a set Axy such that x 2 c(Axy ) and y 2 Axy . If c(Axy [ A) \ Axy 6= ?, then WARP implies that x 2 c(Axy [ A). It follows that c(Axy [ A) \ A 6= ?. Applying WARP again, conclude that x 2 c(Axy [ A) \ A = c(A). If c(Axy [ A) \ A 6= ?, then WARP implies that y 2 c(Axy [ A). It follows that c(Axy [ A) \ Axy 6= ?. Conclude that x 2 c(Axy [ A) \ A = c(A). To prove uniqueness, suppose % and %0 both rationalize c. Take x; y 2 c( ), that is x 2 c(Ax ) and y 2 c(Ay ) for some Ax ; Ay 2 , and without loss of generality suppose x % y. But then x 2 c(Ax [ Ay ) and so x %0 y.

6.5

Proof of Theorem 5

Fix Z 0 2 D and, abusing notation, write for the menu Z 0 . By Theorem 3, there exists a unique preference relation %0 on K = fx 2 K( Z 0 ) : supp(x) = Z 0 g that rationalizes c on the collection Z 0 . Let %:= cl(%0 ) denote the closure of %0 in the product topology on K( Z 0 ) K( Z 0 ). By de…nition, % is a consistent extension of %0 , that is, %0 % and 0 . The following steps establish that % is complete and transitive and rationalizes c on Z 0 . (i) % is complete. To see this, note that K is dense in K( Z 0 ) and so, for any (x; y) 2 K( Z 0 ) K( Z 0 ), there exists a sequence (xn ; yn ) 2 K K converging to (x; y). The completeness of %0 implies that xn %0 yn in…nitely often or yn %0 xn in…nitely often. The de…nition of % implies that x % y or y % x, as desired. (ii) 8y 2 K( Z 0 ), the sets fx : x yg and fx : y xg are open. By completeness, fx : x yg is the complement of fx : y % xg in K( Z 0 ). The latter is closed by construction. (iii) x y implies x % y. The monotonicity of %0 implies that + (1 0 )x % + (1 )y 8 . Taking ! 0 gives x % y. (iv) x y implies y . By way of contradiction, suppose that there exist sequences yn ! y and yn0 ! such that yn %0 yn0 8n. Since x , 0 0 0 0 there exists a sequence xn yn such that xn ! x. But then yn % yn % xn implies y % x contradicting x y. (v) y implies cf ; yg = f g. By way of contradiction, suppose cf ; yg = f ; yg. By Weak Independence, cf ; + (1 )y; yg = 27

0 f ; + (1 )y; yg 8 2 (0; 1]. By Consistency, + (1 )y 8 . Taking ! 0 implies y, establishing a contradiction. (vi) y implies + (1 )y y 8 . By Weak Independence and step (v), cf + (1 )y; yg = f + (1 )yg. By Continuity, there exists no sequence (xn ; yn ) ! ( + (1 )y; y) such that xn %0 yn , for otherwise, y 2 cf + (1 )y; yg. (vii) For any x y, there exists x 2 K such that x x y. Since +(1 )y ! y for ! 0 and fx0 : x x0 g is open by (ii), x +(1 )y for small enough. By (vi), x + (1 )y y for some . (viii) x x and x 2 K if and only if cfx; x g = fxg. Take x x. By (vii), there exists y 2 K such that x y x . For any sequence xn in K converging to x and n large enough, we have cfxn ; y ; x g = fxn g. By Continuity, x 2 cfx; y ; x g. By Consistency, x 2 = cfx; y ; x g. Applying Consistency again gives cfx; x g = fxg. The ‘if’part follows from Continuity. (ix) % is transitive. Take any x; y; y such that x % y % y 0 . That is, there exist sequences (xn ; yn ) ! (x; y) and (b yn ; yn0 ) ! (y; y 0 ) such that xn %0 yn and ybn %0 yn0 8n. By way of contradiction, suppose y 0 x. By step (vii), there exists x 2 K such that y 0 x x. For all n large enough, step (ii) implies that: ybn %0 yn0 0 x 0 xn %0 yn .

But then ybn 2 cfb yn ; yn0 ; x g and x 2 cfx ; xn ; yn g. By Continuity, 0 y 2 cfy; y ; x g and fx g = cfx ; x; yg. But y 0 x and step (viii) imply 0 0 0 cfy ; x g = fy g. By Consistency, x 2 = cfy; y ; x g. But then y 2 cfy; y 0 ; x g and x 2 cfx ; x; yg violate Consistency. (x) % rationalizes c on Z 0 . Take any choice problem A 2 Z 0 , x 2 c(A). Suppose by way of contradiction that 9y 2 A such that y x. By (vii), 9x 2 K such that x x and y x x. By step (viii) and Consistency, x 2 = c(y; x ; x). By Monotonicity, x 2 = c(y; x ; x). But then Consistency implies that x 2 = c(A) which is a contradiction. Suppose next that x % y 8y 2 A. We have to show that x 2 c(A). Take any x0 2 c(A). Since Z 0 is …nite, there exists a …nite subset A0 of A such that x; x0 2 A0 and supp(A0 ) = Z 0 . By Consistency, x0 2 c(A0 ) and so it remains to show that x 2 c(A0 ). Let y = [y2A0 y. By Monotonicity, y + (1 )x % x. But x % x0 and the transitivity of % imply that y + (1 )x % x0 . By (viii), y + (1 )x 2 cf y + (1 )x; x0 g and by Consistency, y + (1 )x 2 cff y + (1 )xg [ A0 g 8 . Letting ! 0 0 implies that x 2 c(A ) as desired. 28

b is another continuous preference To see the uniqueness of %, suppose % that rationalizes c on Z 0 . Take x % y and a sequence (xn ; yn ) 2 K K 0 b converging to (x; y) such that xn % yn 8n. The latter implies that xn %yn b x%y. b Conversely, take x y. By step (vii), 8n and by the continuity of %, there exists x 2 K such that x x y, implying x b x b y as desired. To complete the proof of the theorem, let SZ 0 be the local state space for c on Z 0 . We have to show that SZ 0 is su¢ cient for %, that is, for all x; y 2 K( Z 0 ), x jSZ 0 = y jSZ 0 implies x y. Suppose by way of contradiction, that x y. Then, step (vii) implies that x [ y x y for some x 2 K . Since % rationalizes c on Z 0 , it follows that cfx [ y; x ; yg = fx [ yg. But x jSZ 0 = y jSZ 0 implies x[y jSZ 0 = y jSZ 0 , contradicting the su¢ ciency of SZ 0 for c on Z 0 . Since SZ 0 satis…es Dominance for c on Z 0 , SZ 0 is the smallest closed set su¢ cient for %. By [4, Theorem 1], % has a weak expected utility representation (SZ 0 ; uZ 0 ).

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