Yusufcan Masatlioglu - Rational Choice With Status Qup Bias

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Rational Choice with Status Quo Bias¤

Yusufcan Masatliogluy EfeA. Okz

J anuary 11, 2003

A bstract

M otivated by the empirical …ndings concerning the importance of one’s current situation on her

choice behavior, the main objective of this paper is to propose a rational choice theory that

allows for the presence of a status quo bias, and that incorporates the standard choice theory

as a special case. We follow a revealed preference approach, and obtain two nested models of

rational choice that allow phenomena like the status quo bias and the endowment e¤ect, and

that are applicable in any choice situation to which the standard (static) choice model applies.

J EL Classi…cation: D11, D81.

K eywords: Revealed Preferences, Incomplete Preferences, Status Quo Bias, Endowment E¤ect,

Expected Utility, Overpricing, P reference Reversal Phenomenon.

¤Various conversations with J ean-P ierre Benôit, Faruk Gul, and J acob Sagi have helped to the development of

this paper to a signi…cant degree; the second author gratefuly acknowledges his intellectual dept to them. We are

also grateful to the support of the C. V. Starr Center for Applied E conomics at New Y ork University.yDepartment of E conomics, New Y ork U niversity. E -mail: [email protected]: Department of Economics, New York U niversity, 269 M ercer Street, New York, NY 10003.

E-mail: [email protected].

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1 I ntroduction

In thelast two decades, a sizable amount ofexperimental data on theindividual choicebehavior have

been obtained, and a number of startling regularities have been noted.1 Among such regularities

is the observation that, relative to other alternatives, a current choice or a default option is oftenevaluated markedly positively by the individuals. T his phenomenon is termed the status quo bias

(Samuelson and Zeckhauser (1988)), and is documented not only by experimental studies but also

by empirical work in the case of actual markets.2 Motivated by these…ndings, the main objective

of this paper is to propose a rational choicetheory that allows for thepresence of a statusquo bias,

and that incorporates thestandard choicetheory as a special case. Our approach is axiomatic, and

yields an individual choice model that is general enough to be applicablein any situation that the

standard (static) choice model is applicable.

We think of a choice problem in this paper either as a feasible set S of alternatives, or as a

feasible set S and a point x in S; which is interpreted as thedefault option of theindividual.3 As

usual, a choice correspondence is then de…ned as assigning to any given choice problem a subset

of thefeasible set of theproblem. We introduce to the model a status quo bias by requiring that

if an alternative is chosen when it is not a status quo, it should be chosen uniquely when it is

itself a status quo, other things being equal. I n addition to this property, we consider four other

rationality requirements in the …rst part of the paper. Two of theseare straightforward re‡ections

of the standard axioms of revealed preference theory, and the other two are new properties that

link the choice behavior of thedecision maker across problems with and without a status quo.

These …ve axioms characterize a decision-making model which is quite reminiscent of some

earlier suggestions present in the literature. According to this model, the agent has a complete

preference relation, and in the absence of a status quo choice, she solves her choice problems

by maximizing this relation. T his is, of course, nothing but the choice model of the classical

revealed preference theory. When, however, there is a status quo x in the problem, then the

agent’s preferences become incomplete. In a sense (to be formalized below), the agent evaluates

1See Camerer (1995) and Rabin (1998) for careful surveys.2Samuelson and Zeckhauser (1988) identi…ed this e¤ect experimentally by an extensive study concerning portfolio

choices. M ost of the experimental studies that …nd gaps between buying and selling prices provide support for the

status quo bias as we understand the term here; see, for instance, K netsch (1989) and K ahneman, K netsch and

T haler (1990, 1991). Hartman, Doane and Woo (1991), J ohnson et al. (1993), and Madrian and Shea (2001) report

substantial amounts of status quo bias in the …eld setti ngs (where individual choices concern reliability levels of

residential electrical services, car insurance, and participation in 401(k) plans, respectively).3 T hisl atter model of choice problems was alsoconsidered by Zhou (1997) and Bossert and Sprumont (2001); more

on the relation between the present work and thesepapers shortly.

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the alternatives in terms of several decision cri teria (i.e. several complete preference relations)

and unless there are some alternatives that dominate the status quo choice x with respect to this

incomplete preference relation (that is, in terms of all criteria), shesticks with x. Put di¤erently,

having an initial entitlement all ows the agent to get “confused/indecisive,” when comparing the

other alternatives with her current holdings, and she always resolves this situation in favor of

her status quo; hence the status quo bias: If no such confusion arises (because some alternatives

are unambiguously superior to the status quo), then the initial position of the agent becomes

irrelevant, and the agent settles her decision making problemby maximizing her original complete

preferencerelation(that aggregates her decision criteria). Figure1providesa geometric illustration

of this choice procedure whichis, as weshall discuss below, closely linked to the choice procedures

suggested by Simon (1955) and Bewley (1986).

By strengthening the axioms that connect how the problemswith and without status quo are

solved, one can provide sharper characterizationsof thenature of the“indecisiveness” of the agent

in the presence of a status quo. Our second characterization theorem is a case in point. This result

provides a model which is apparently suitable for capturing the famous endowment e¤ect. In this

model, again, the agent solves the standard problems by maximizing a utility function u(¢); but

when there is a status quo x (which, in view of the strengthened set of axioms, is best interpreted

as an alternative that the agent is endowed with), she gives up x if, and only if, another choice

item provides her higher utility than u(x) plus a “utility pump” of ' (x) > 0 whichmay perhapsbe

thought of as a psychological switching cost. Put di¤erently, if the “value” of an object x is some

number u(x) when the object is not owned, its value is u(x) + ' (x); when it is owned; hence the

endowment e¤ect.4 What is more, this decision-making model is easily extended to the risky choice

situations (via positing the independence axiom on choice correspondences).

A few remarks on the relationship of the present work to the literature are in order. The

motivation of the papers by Zhou (1997) and Bossert and Sprumont (2001) are very close to this

paper, and there are some similarities between the models that we consider. In particular, both of

thesepapers adopt therevealed preferenceapproach and consider thechoiceproblems with a status

quo as we de…nethem here. A major di¤erence between thesepapers and the present work is that

we allow here problems without a status quo as well in our domain of choice problems. Not onlythat this allows us produce a theory that admits the standard rational choice model as a special

case, but it also givesriseto a rich setup in which onecan consider properties regarding how choices

4While this representation has certain attracti ve features for applications, the issue of “uniqueness” of u and '

naturally arises at this junction. Su¢ ces it to say that our theory is close to being ordinal (in the riskless case), but

given t he nonstandard nature of the representations that we propose, this issue will have to be carefully addressed.

We will do so in t he body of the paper.

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are made in problems with and without status quo in a consistent manner. The implications of

this are surprisingly far reaching.

In particular, our choice model is in stark contrast with that of Zhou (1997) who works with

choice functions. While the only axiom considered by Zhou (menu-independence) is ill-de…ned

for choice correspondences in general, even the single-valued choice correspondences of the form

characterized here (such as the Simon-Bewley choice procedures) need not satisfy this axiom. On

the other hand, in the case of abstract choice problems (the case we consider here), Bossert and

Sprumont (2001) characterize thosechoice correspondences that choosein a problem a set only if

every element of this set is superior to thestatusquo (according to a partial order on the universal

set of alternatives). While quite interesting, this approach gives only little information about how

to model thestatusquo bias in concrete situations. Certainly all choiceprocedureswe consider here

are in linewith the model of Bossert and Sprumont (2001), but thelatter model allows for rather

coarsechoicebehavior such as “choose everything that is better than the statusquo.” By contrast,

our objective is to provide rational choice models that have relatively sharp representations that

may be used in applications in a straightforward way. To reiterate, we do this by studying in

conjunction the choice problems with and without status quo, and examining certain consistency

properties that tie such problems together.

Our work is also related to the literature on reference dependent preferences. In particular,

T versky and Kahneman (1991) suggest a deterministic utility theory over a …nite-dimensional

commodity spacein which analternativeispreferredto another alternativeaccordingto a preference

relation that depends on a reference state, which can be interpreted in our context as the status

quo point. Yet, Tversky and Kahneman assume a particular choice behavior generated by such

referencedependent preferences (that is, impose a model that parallels thechoicebehavior that we

derive axiomatically here), and do not discussthe structure of representation for such preferences.

In the case of risky prospects, however, this situation is remedied by Sagi (2001) who provides

an axiomatic foundation for the reference dependent decision model of T versky and K ahneman

(1991).5

A major di¤erence of the present work fromthese studies is that wetakeherethechoices as the

startingpoint, and derive the(referencedependent) preferences thereof, as opposed to following theopposite direction. While, under certain assumptions, the two approaches are dual to each other,

theappeal and strength of rationality axioms di¤er across thesemodels. A second major di¤erence

is our insistence of developing a model that allows for reference independence (the absence of a

5We view this as a much needed remedy. For, noti ons like loss aversion and diminishing sensitivity are cardinal

in nature, and are thus arguably ill-founded in the context of riskless choice.

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status quo), and exploiting this in order to link thestandard theory to reference-based choice. As

noted earlier, this approach yields a decision theory which is somewhat more comprehensive than

the one in which agents maximize a given (reference dependent or otherwise) preference relation.

(See Theorem 1and the discussion that follows.) Wealso note that this formulation contrasts with

(and should be viewed complementary to) the axiomatization provided by Sagi (2001) in that the

analysis here is nonlinear (henceordinal) for the most part, as in the classical revealed preference

theory. As such it is arguably more appropriate for models of riskless choice, even though, as we

shall show in Section 3.3, it is not di¢ cult to extend the proposed choice theory to risky choice

situations through theclassical independence axiom.

The rest of the paper is organized as follows. T he next section begins with introducing a set

of basic axioms that seem unexceptionable for a rational choice theory that allows for a status

quo bias. Choice correspondences that satisfy these properties are thus called basic, and are char-

acterized in Section 2.2 under the hypothesis that the alternative space is …nite. I n Section 2.3,

in a setup that all ows for in…nitely many alternatives, we characterize a subclass of basic choice

correspondences that appear particularly suitable for modeling the endowment e¤ect. In Section

3 we consider a few applications of the resulting individual choice model. First weshow that this

model predicts a discrepancy between buying and selling prices of commoditiesin concert with the

related experimental …ndings. We next point to the implications of the model for general equi-

librium theory; in particular, we show that the present model entails that the endowment e¤ect

may not only cause a considerable decline in the predicted volume of trade (which was already

noted in the literature), but it may also cause considerable problems with regard to the existence

of equilibrium. Finally, Section 3.3 shows how to linearizethe present axiomatic model in order to

cover risky choice situations, and Section 3.4 appliesthe resultingmodel to o¤er a new explanation

for the (in)famous preference reversal phenomenon. Section 4concludes with a few remarks about

future research. All proofs are contained in Section 5.

2 A n A xiomatic M odel of Status Q uo B ias

2.1 B asic A xioms

We consider an arbitrary compact metric spaceX ; and interpret each element of X as a potential

choice alternative(or prize).6 Theset X is thus viewed as the grand alternativespace. For reasons

that will become clear shortly, wedesignate thesymbol 3 to denotean object that does not belong

6 T hroughout this paper we adopt the innocuous convention of assuming that every metric space is nonempty.

cl(¢), int(¢); and N"(¢) (with " > 0) denote the closure, interior, and "-neighborhood operators, respectively.

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to X : Also let X denotetheset of all nonempty closed subsets of X . By a choice problem in this

paper, wemean a list (S;x) whereS 2 X and either x 2 S or x =3 : Theset of all choice problems

is denoted as C(X ):

If x 2 S 2 X ; then the choice problem (S;x) is referred to as a choice problem with a

status quo. T he interpretation is that the individual is confronted with theproblem of choosing

an alternative from the feasible set S while either she is currently endowed with the alternative

x or her default option is x: Viewed this way, choosing an alternative y 2 Snf xg means that the

subject individual gives up her status quo x and switches to y: We denote by Csq(X ) the set of all

choice problems with a status quo.

On theother hand, many real-life choicesituationsdonot have a natural statusquo alternative.

Within the formalism of this paper, thechoice problems of the form (S;3 ) model such situations.

Formally, then; wede…ne a choice problem without a status quo as the list (S;3) for any set

S in X : (While theuse of the symbol 3 is clearly redundant here, it will prove quite convenient in

the foregoing analysis.)

J ust to …x ideas, and help illustratesome of what follows, we shall carry with us a concrete(yet

hypothetical) example which is rather close to home. We have in mind the choice problem of an

economist, call her Prof. ¾, who is currently employed at Cornell and is pondering over two new

job o¤ers, one from NY U and one from UCSD. I n the terminology of thispaper, then, the choice

problem of P rof. ¾is one with a statusquo, where the status quo point is to stay at Cornell, and

the feasible set is thejobs she might have in all three of the schools. If we change thescenario a

little bit, and instead say that Prof. ¾is about to graduate from UPenn, and after a successful job

market experience, has now three o¤ers from Cornell, NY U and UCSD, then it would make sense

to model her choice problem instead as one without a status quo.

By a choice correspondence in the present setup, wemean a map c : C(X ) ! X such that

c(S;x) µ S for all (S;x) 2 C(X ):

(Notice that a choice correspondence must be nonempty-valued by de…nition.) We shall next

consider some basic properties for choice correspondences. The …rst two of these properties are

straightforward re‡ections of the classical theory of revealed preference. As in thestandard theory,they allow one to identify when a “choice” can be viewed asan outcome of a utility maximization

exercise. While in the standard theory such axioms regulatethe alteration of choices in response to

the alteration of feasible sets of alternatives, our properties condition things here also with regard

to the changes of statusquo points across choice problems.

Property ®. For any (S;x) 2 C(X ); if y 2 c(S;x) and x;y 2 T µ S; then y 2 c( T; x):

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Property ¯ . For any (S;x) 2 C(X ); if z; y 2 c(S;x), S µ T; and z 2 c( T; x); then y 2 c( T;x):

Thereis littleneed to motivatetheseproperties; they arenoneother than theobvious extensions

of the classical axioms of revealed preference theory. (See, for instance, Kreps (1988, pp. 11-15)

for an expository account.) Su¢ ces it to say that these properties are jointly equivalent to thestatement that, for any x 2 X [ f 3g; thecorrespondencec(¢; x) : X ! X satis…es Houthakker’ss

weak axiom of revealed preference, or Arrow’s choice axiom (Arrow (1959)).

Axiom D. (Dominance) For any (S;x) 2 C(X ); if f yg = c(S;x) for some S µ T; and

y 2 c( T;3); then y 2 c( T; x):

Recall that if f yg = c( T;3 ); we understand that y is the most preferred alternative in the

alternative set T in the absence of a status quo. So the only reason why y would not be chosen

from ( T;x) is because x may defeat y when it is endowed with the additional strength of being

thestatus quo. But if f yg = c(S;x) for some S µ T; then it is clear that this cannot be the case,

because y is then revealed to be (strictly) preferred to x even when x is designated in the choice

situation as the status quo point. T hus, it seemsthat y dominates everything feasible in the choice

problem( T; x) alongwith thestatusquo point x, and hence, so AxiomD posits, it should bechosen

from ( T; x) as well.

To illustrate, suppose that Prof. ¾, who is currently employed at Cornell, would accept a job

o¤er from NY U. We somehow also know that, upon getting her degree (and thus not having a

status quo), she would have taken the NY U o¤er over the o¤ers of Cornell and UCSD. Assuming

that her tastes have not changed through time, what would one expect Prof. ¾to do, when she

gets the NY U and UCSD o¤ers simultaneously while she is at Cornell? We contend that most

people would not be surprised to see her at NY U the following year, for our knowledge about her

preferences indicates that NY U dominates the o¤ers of both Cornell andUCSD, andthis regardless

of her being currently employed at Cornell. Axiom D is based on precisely this sort of a reasoning.

Axiom SQI. (Status-quo I r relevance) For any (S;x) 2 Csq(X ); if y 2 c(S;x) and there

does not exist any nonempty T µ S with T 6= f xg and x 2 c( T;x); then y 2 c(S;3).

To understand the intuitive appeal of this property, assume that x is never chosen from any

subset of S despite the fact that it is the status quo. Thus x cannot be thought of as playing a

signi…cant role in the choice situation (S;x); it is completely irrelevant for the problem at hand.

T his means that thereis practically no di¤erence between the choice problemwithout a status quo

(S;3 ) and the choice problem (S;x) in theeyes of thedecision maker. So, if y is chosen from the

latter problem, it should also be chosen from theformer. T his is thegist of Axiom SQI .

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To illustrate more concretely, suppose again that Prof. ¾, who is currently tenured at Cornell

and who has two o¤ers, onefrom NY U and one from UCSD, has indicated that she would take the

NY U o¤er. Moreover, it is somehow known that both of the o¤ers dominate staying at Cornell.

What would oneexpect Prof. ¾to do when confronted with theo¤ersof Cornell, NY U and UCSD,

upon getting her degree (and thus not having a status quo)? T he information at hand indicates

that shedeems beingemployed or not being employed at Cornell as irrelevant; theproblemis really

to choose between the o¤ers of NY U and UCSD. But we already know that she likes the former

at least as much as thelatter, so it appears rational that she take theNY U o¤er; this is precisely

what Axiom SQI posits.

The…nal property that we will consider here for choice correspondences is central to the devel-

opment of this paper. It speci…es a distinctive rolefor the status quo point in choiceproblems with

a status quo.

Axiom SQB. (Status-quo Bias) For any (S;x) 2 C(X ); if y 2 c(S;x) then f yg= c(S;y):

This axiom is based on the idea that if the decision maker reveals that y is no worse than any

other alternativein a feasibleset S, including the status quo point x (if thereis such a point), then,

when y is itself the status quo, its position can only be stronger relative to thealternatives in S:

T he axiom posits that in this case y must be the only choice from the alternative set S; thereby

requiring a choice correspondence to exhibit some bias towards the status quo. I f it is somehow

revealed elsewhere that the status quo is at least as desirable as all other feasible alternatives,

then “why move?”; SQB requires in this case the individual to keep the status quo point. This

property seems quite appealing for a rational choice theory whose primary objective is to model

the phenomenon of statusquo bias.

2.2 B asic Choice C orrespondences

Each of the…rst four propertiesconsideredabovecorrespondsto aparticularly appealingrationality

trait in the case of choice problems that may possess a status quo point. The…fth property, on the

other hand, introduces to the model a form of inertia towards the status quo, but it does this in

a very conservative manner. Consequently, the choice correspondences that satisfy all …veof these

properties appear to befocal for a rational choice theory that would allow for statusquo bias. This

prompts the following de…nition.

De…nition. Let X be a compact metric space. Wesay that a choice correspondence c on C(X )

is basic if it satis…es properties ®and ¯; and Axioms D, SQI and SQB.

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The determination of the basic structure of basic choice correspondences is the primary aim

of this paper. In this section weshall providea complete characterization to this e¤ect under the

assumption of …nitenessof the prize space. Towards this theorem, we will …rst state a lemma that

gives the general outline of the sought characterization.. Let us …rst introduce the order-theoretic

nomenclature we adopt in this paper. A binary relation on a nonempty set is called a preorder if

it is re‡exive and transitive. An antisymmetric preorder is called a partial order, and a complete

partial order is called a linear order. If < is some preorder on X ; we say that <¤ is a completion of

< if <¤ is a complete preorder with » µ »¤ and Â µ Â¤, where Â is the strict (asymmetric) part

of < and » is the weak (symmetric) part of <; and similarly for »¤ and Â¤. For any nonempty

subset S of X; by M (S;<) wemean the set of all maximal elements in S with respect to<; that

is, M (S;<) := f x 2 S : y Â x for no y 2 Sg where Â denotes the strict part of <. Finally,

for any x 2 X ; by UÂ(x); we denote the strict upper contour set of x with respect to<; that is,

UÂ(x) := f y 2 X : y Â xg:We are now ready to exhibit what sort of structure onewould expect a basic choice correspon-

dence to possessin general.

Lemma 1. Let X be a nonempty set. I f the choice correspondence c on C(X ) is basic, then

there exists a partial order <and a completion <¤ of this partial order such that c(¢;3 ) = M (¢;<¤)

and

c(S;x) =

8<

:

f xg; if x 2 M (S;<)

M (S \ UÂ (x);<¤); otherwise

for all (S;x) 2 Csq(X ):

Unfortunately, absent any continuity requirements on the choice correspondence, one cannot

hope to prove the converse of this fact since one cannot then guarantee in general the existence

of a maximal element in a feasible set. However, when X is …nite, then we can not only escape

this technical problem, but also get a multi-dimensional utility representation for the partial order

found in Lemma 1. Before we discuss the choice structure found in this lemma, therefore, we make

note of the ensuing situation in this case. As a …nal bit of notation, we note that, for any positive

integer n and any S 2 X ; we denotetheupper contour set of any x 2 X with respect to a function

u : X ! Rn as Uu(S;x); that is,

Uu(S;x) := f y 2 S : u(y) >u(x)g:

The following characterization theorem identi…es all basic choice correspondences in the case

where the grand alternativespaceX is …nite.

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T heorem 1. Let X be a nonempty …nite set. A choice correspondence c on C(X ) is basic

if, and only if, there exists a positive integer n; an injective function u : X ! Rn and a strictly

increasing map f :u(X ) ! R such that

c(S;3

) = argmaxx2S f (u

(x)) for all S 2 X ; (1)

and

c(S;x) =

8><

>:

f xg; if Uu(S;x) = ;

argmaxy2Uu(S;x)

f (u(y)); otherwise(2)


Theorem1shows that abasicchoicecorrespondencemodelsa surprisingly well-structured choice

behavior, at least when X is …nite. An agent whosechoicebehavior ischaracterizedby such a choice

correspondence evaluates all alternatives by means of a vector-valued utility function u. Wemayinterpret this as theevaluation of thealternatives on the basis of various distinct criteria; in this

interpretation, the ith component of u can be thought of as representing the agent’s (complete)

ranking of the alternatives with respect to the ith criterion:

If the subject agent is dealingwith a choice problem without a statusquo, then breakingdown

her preferencesin this way is not essential. For, in this case, shehas a particular way of aggregating

all criteria (that is thecomponents of u) by meansof a map f : A moment’s re‡ection showsthat this

entails that the standard maximizing choice paradigm is but a special case of the present setting.

When, however, there is a status quo x in the problem, the multidimensional way in whichthe agent makes a …rst pass at evaluating the alternatives is detrimental. First of all, the agent

compares the status quo point x with all other feasible alternatives with respect to all criteria that

shedeems relevant.7 If noneof the alternatives weakly dominates x in each criterion, and strictly

in at least onecriterion, then the agent sticks with her statusquo, thereby depictinga pronounced

status quo bias. If at least onealternative passes this test, then thedecision maker decides to leave

her status quo, and considers choosing an alternative among all thosethat “beats” her status quo

in each criterion. T he…nal choice among such alternatives is madeon the basis of maximizing her

aggregateutility, where to aggregate thevarious criteria she uses the same aggregator f (hence thesame trade-o¤s between thesecriteria) that she uses in choice problems without a status quo.

Consider the now familiar exampleof Prof. ¾who has job o¤ersfromCornell, NY U and UCSD,

and whose choice correspondence is basic. I f she does not havea status quo job (say, because sheis

7I t is worth noting here that there are some axiomatic studies about individual preferences that admit such a

multi-utility representati on. See, for instance, Shapley and Baucells (1998), Dubra, Maccheroni and O k (2001), and

Ok (2002).

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fresh out of thegraduate school), then her choiceproblemis completely standard. Shedistinguishes

between theo¤ers on the basis of a number of criteria that shedeems relevant, for instance, salary,

location, preferences of the spouse and reputation of the school (or some linear combinations of

these criteria). But, given that she does not have a status quo at present, there is no benchmark

for her to compare these dimensions against, and hence she somehow aggregates the potential of

each o¤er across therelevant criteria, thereby assigning an aggregate utility to each o¤er, and then

chooses the one that yields her the maximum aggregate utility. If, on the other hand, Prof. ¾

is already tenured at Cornell, so her Cornell o¤er is really her status quo, then she compares the

promises of NY U and UCSD with those of Cornell with respect to every criteria, and unless at

least one of her outsideo¤ers dominates Cornell in every dimension, she stays at Cornell. If only

NY U passes this test, then she moves to NY U. If, …nally, both NY U and UCSD o¤ers dominate

Cornell with respect to each of her criteria, then theproblembecomes choosing between NY U and

UCSD, and Prof. ¾settles this problem as if her choice set consists only of these two alternatives

and there is no statusquo.

It is interesting that the behavioral choice procedures stipulated by both Simon (1955) and

Bewley (1986) are closely related to basic choice correspondences (at least when X is …nite). Si-

mon (1955) suggests a choice procedure in which an agent tries “to implement a number of values

that do not have a common denominator - e.g., he compares two jobs in terms of salary, climate,

pleasantness of work, etc.,” and then searches for the set of feasible alternatives which is “satis-

factory” in terms of all these values. Simon is a bit imprecise about what “satisfactory” means

in this context, but it is clear that he has in mind some sort of a dominance (in terms of all val-

ues/ criteria) over what is guaranteed to theagent, presumably at his statusquo. In the same spirit

the inertia assumption of Bewley (1987, p. 1) “asserts that in some circumstances onecan de…nea

status quo, which the decision maker abandons in favor of an alternative only if doing so leads to

an improvement.”

Theformalization of this procedure, which we shall refer to as the Simon-Bewley choice proce-

dure, is straightforward in the present setup. If u denotes themultidimensional evaluation criteria,

given a choice problemwith a statusquo (S;x); the agent’s aspiration levels in each of these criteria

is given by thevector u(x): So the Simon-Bewley procedure chooses the status quo x if no y 2 Sdominates x in all these criteria (i.e., if u(y) > u(x) for no y 2 S), and choose all dominating

alternatives if there is such an alternative y; that is, choose the set Uu(S;x) whenever this set is

nonempty.

To illustrate, let X := f x;y;z;wg; and assumethat the decision maker uses exactly two criteria

(so u(X ) ½R2), and we haveu(x) := (0;0); u(y) := (1;4); u(z) := (2;1); u(w) := (¡ 1; 4): Then

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theSimon-Bewley procedure chooses precisely the set f y;zg in the case of the problem (X ;x): By

contrast, a basic choice correspondence can be more re…ned than this. It certainly agrees with

the Simon-Bewley procedure, in that it chooses a subset of f y;zg; but the …nal decision requires

comparingy and z asif thereis nostatusquoin theproblem. (After all, x did its job; it eliminatedw

from consideration, and in turn, it is itself eliminated by both y and z.) I f theindividual aggregates

thecomponents of u bymeansof a function f : u(X ) ! R in theabsence of a status quo point, then

the agent’s unique choice from (S;x) is y whenever f (u(y)) > f (u(z)): This situation is depicted

in Figure 2, where f is represented by the indi¤erence curves drawn in the criteria space.8

Note that a basic choice correspondence makes use of a status quo points in two ways: (i)

to eliminate those alternatives that do not (Pareto) dominate the status quo; and (ii) to act as

the unique choice if all alternatives are eliminated by the test of (i). To make this point clear,

let X := f x;y;zg; and consider the basic choice correspondence de…ned as in Theorem 1 with

u(x) := (2; 2);u(y) :=( 3;3); u(z) := (1; 10); and with f :u(X ) ! R de…ned by f (a) := a1a2: This

correspondencechooses z from the problem(X;3) while it chooses y from the problem (X ; x); see

Figure3. Theupshot is that, a basic choice correspondence maintains that, a status quo point may

alter one’s choices even if it will itself not be chosen! If Prof. ¾hassuch a choicecorrespondence,

then she may choose to go to UCSD over NY U when she is not employed anywhere (a problem

without a status quo), but may as well choose instead NY U over UCSD when sheis employed at

Cornell.

This observation reveals that status quo-dependent choices is not really irrational, at least

insofar as one would agree that there is a good deal of rationality contained in a basic choice

correspondence. In an intuitivesense too there is no need to view the choice behavior of Prof. ¾in

this exampleas irrational. For instance, suppose that UCSD is indeed her best choice (with other

alternatives being Cornell and NY U) when shedoes not have a status quo job. Now consider the

case where sheis employed at Cornell, and an o¤er from UCSD camethrough. If UCSD does not

dominate Cornell in all criteria that Prof. ¾deems relevant, and she indeed possesses a status quo

bias the way modeled by a basic choice correspondence, then it makes good sense that she will

stay at Cornell. If after she had turned down the UCSD o¤er, comes along an NY U o¤er, and if

the NYU o¤er dominates Cornell according to all her criteria, she will move to New York. Afterthe dust settles, then, there is hardly anything surprising about seeing Prof. ¾employed at NY U.

8I t is worth noting that a basic choice correspondencecollapses to (our formulation) of the Simon-Bewley choice

procedure in sequential decision problems where (S;x) is such that S = f x;yg with y standing for a new alternative

o¤ered to the agent against x: In this sense, and informally speaking, our characterization i n T heorem 1 can be

thought of as providing an axiomatic support for the inertia assumption of Bewley’s theory of Knightian uncertainty

which is often criticized for being ad hoc.

12





implications; in particular, it launches an unexpected attack on the famousCoase theorem. (More

on this in Section 3.2.)

While this sort of a phenomenon seems at …rst quite distinct fromthe choice behavior we have

characterized in Theorem 1 (where the status quo bias materializes through the multidimensional

way one ranksthe alternatives relative to the statusquo), perhaps somewhat surprisingly, it turns

out that it can be considered as a special case of such behavior. More precisely, every choice

correspondence that envisages theendowment e¤ect (in a way that is formalized below) is in fact a

basic choice correspondence. The main objective of this section is in fact to characterize precisely

thissubclassof basicchoicecorrespondencesbymeansof strengtheningtheset of axiomsweutilized

when proving Theorem 1.

The…rst property we need states simply that if two alternatives are chosen in the absence of a

status quo, then in the presence of a status quo that is distinct from thesealternatives, either they

should be chosen together or neither of them should be chosen.

Axiom SQI ¤. (Status-quo I ndependence) For any (S;x) 2 Csq(X ); if y;z 2 c(S;3)nf xg

and z 2 c(S;x), then y 2 c(S;x):

This assumption cuts one of the channels through which choices might depend on the status

quo. As discussed by means of two examples above(recall Figure 3), a basic choicecorrespondence

need not satisfy this assumption. But if one is really attempting to model the endowment e¤ect,

then SQI¤ is an apparently reasonable rationality requirement. If y and z are equally good when

an individual does not own an object x distinct from y and z), they should also be equally good

(so it is not the case that one is chosen but theother is not) when the individual owns x:

Thesecond property we will consider is a straightforward strengthening of the SQB axiom.

Axiom SQB¤. (Strong Status-quo Bias) For any (S;x) 2 Csq(X ); the followinghold:

(i) If y 2 c(S;x); then f yg= c(S;y);

(ii) If y 2 c(S;x)nf xg; then y 2 c(S;3 );

(iii) There exists an " > 0 such that x 2 c(cl(N"(x)); x):

While the statement (i) here is a restatement of SQB, (ii) says that if the decision maker

quali…es y no worse than any other alternativein a feasibleset S; including the status quo (if there

is such a point), then, when thereis a statusquo, its position should not deterioraterelativeto the

alternatives in S: While it need not be satis…ed by a basic choice correspondence, it is clear that

this property sits well with the intuitive understanding of the endowment e¤ect. Finally, (iii) is a

nontriviality requirement that says that thepower of being the status quo makes any alternative x

14



be thechoicefroma set that consists only of alternatives thenature of which are arbitrarily close

to x: T his requirement is trivially satis…ed when X is …nite (and is thus endowed with the discrete

metric).

Axioms SQI¤ and SQB¤ are enough to transform Theorem 1 into a characterization of choice

correspondences that are arguably suitable for the modeling of the endowment e¤ect. With the

help of the followingstandard continuity assumption, however, we can in fact state our new char-

acterization in a more general framework that allows for in…nitely many alternatives. Of course,

we need to introduce a metric on X for this purpose, and as usual, we adopt theHausdor¤ metric

towards this end.10

Axiom UHC. (Upper H emicontinuity) T he map c(¢;3 ) is upper hemicontinuous on X :11

Moreover, for each x 2 X and a sequence(Sm) in X ; if Sm ! S and x 2 c(Sm;x) for each m; then

x 2 c(S;x):

The following theorem characterizes those basic choice correspondences that satisfy the above

three properties. Due to some redundancies in the set of axioms we posited so far, we can state

this result by using only …ve properties.

T heorem 2. Let X be a compact metric space. A choice correspondence c on C(X ) satis…es

properties ®and ¯; and Axioms SQI ¤, SQB¤ and UHC if, and only if, there exist a continuous

map U : X ! R and a function ' : X ! R++ such that

c(S;3 ) = argmaxx2S

U(x) for all S 2 X ; (3)

and

c(S;x) =

8><

>:

f xg; if U(x) + ' (x) > U(y) for all y 2 S

argmaxy2S

U(y); otherwise(4)


The interpretation of the choice behavior identi…ed by this result is quite straightforward. In

the absenceof a status quo point, an agent with such a choice correspondence solves his/ her choice

problemsbymaximizinga utility function U (asin thestandard theory). But if thereis a statusquo

10 T he metric structure postulated here is not essential to the analysis. As far as T heorem 2¤ is concerned, X

can actually be taken as any Hausdor¤ topological space, provided that we topologizeX by means of the Vietoris

topology.11Given that c(¢;3 ) is compact-valued, this is equivalent to say that, for any sequences (Sm ) in X and (xm) in X ;

if Sm ! S and xm 2 c(Sm;3) for each m; then there exists a subsequence of (xm ) that converges to somepoint in

c(S;3 ).

15



x; then we interpret x as the object that the agent “owns.” While, materially speaking, the utility

of x for the agent is U(x); out of owning the object, the agent gets a “utility pump” ' (x) > 0

in the sense that to move away from his/ her status quo (that is, to exchange x for some other

alternative) she needs to be compensated by ' (x) in addition to U(x); that is, she must be “paid”

at least as much as U(x) + ' (x); hence the phenomenon of the endowment e¤ect. (An arguably

good way to interpret ' (x) is thus to view it as a psychological switching cost.) If no alternative

in his feasible set passes this test, the agent sticks with his/her status quo (i.e. endowment). I f,

on the other hand, some alternatives are materially more desirable than x despite theendowment

e¤ect, (i.e. U(y) ¸ U(x)+' (x) for some feasibley), then theagent choosesthe alternative with the

highest utility. This model seems to correspond well to the experimental observation that there is

in general a discrepancy between the willingness to buy and willingness to sell. If the agent deems

the worth of an alternative U(x) when (s)he does not possesses it, she values it at U(x) + ' (x)

when she owns it.

Remark 4. One way to see that T heorem 2 is a special case of Theorem 1 (when jX j < 1 ) is

to note that if a choicecorrespondencesatis…es theaxioms of T heorem2, then it can berepresented

by (<;<¤); where< is an interval order on X ; and<¤ is a completion of < (in thesenseof Remark

2), but not conversely.12 2

Remar k 5. It isworth noting that, when X is a connected metric space, then theknowledgeof

theutility function U determines thepsychological switching cost function ' essentially uniquely.

More precisely, (U;' ) and (U; Á) represent a given choice correspondence c on C(X ) asin T heorem

2 if, and only if, ' jX c= Á jXc

; where Xc := f x 2 X : x =2 c(S; x) for some S 2 X with x 2 Sg:13 2

Remar k 6. In thecasewhereX hasadditional structure, onecangivesharper characterizations

than theone given in Theorem 2. For instance, if X is a compact subset of a Euclidean spaceRd

(interpreted perhaps as a commodity space), and if we further postulate the natural rationality

12A binary relation R on a nonempty set A isan interval order if it is re‡exive, antisymmetric, and xRa and yR b

imply either xRbor yRa; for all x;y;a;b2 A : Real functi onal representati ons of interval orders are studied extensively

within order theory (cf. Chapter 7 of Fishburn (1985)). T hesetheorems, however, makeuse of algebraic separability

conditions which do not sit well with the revealed preference approach. T he proof of T heorem 2 is, in fact, not based

on a standard interval order representation theorem.13Proof . Suppose that there is an x 2 X c such that ' (x) > Á(x): T hen thereexists a y 2 X with f yg= c(f x; yg; x),

and hence U(y) > U(x) + ' (x) > U(x) + Á(x) > U(x): Since the continuous image of a connected set is connected,

we must have [U(x);U(y)] µ U (X ). T hus there exists a z 2 X nf xg such that U(x) + ' (x) > U(z) > U(x) + Á(x):

But then f xg = c(f x;zg;z) and f zg = c(f x;zg;x) contradicting the hypothesis that (U;' ) and (U;Á) represent the

same choicecorrespondence. Reversing the roles ' and Á in this argument completes the proof.

16



property that c(S;3) µ f y 2 S : x > y for no x 2 Sg; then we guarantee that U found in T heorem

2 is stri ctly increasing in addition to being continuous. (T his follows from the argument sketched

for the proof of Theorem 2 in Section 5, and T heorem 1 of Ok and Zhou (1999).) If we further

assumethat thedecision maker hasa preferencefor compromisesin convex choiceproblems without

a status quo, and formalize this by requiring that jc(S;3) j 6= 2 for any convex S in X ; then Uturns out to be a continuous, strictly increasing and quasiconcave function. (T his follows from the

previous observation and L emma 3 of Ok and Zhou (1999).)142

2.4 M onotonicity of the E ndowment E¤ect

T heorem 2 does not give any structure for the psychological switching cost function ' other than

its strict positivity. Consequently, we cannot answer at present the following interesting question:

Does the status quo bias increase or decrease with the value of the initial endowment? To make

things precise, let us consider the following situation: y 2 c(f x;yg;x) and x 2 c(f x;zg;3): In

words, the agent in question views y more valuable than x even if x is the status quo, while she

values x (weakly) higher than z (free of any status quo bias). What would this agent choose from

f y;zg if thestatusquo wasz?

According to the choice model of T heorem 2, the agent may choose either of the alternatives;

both f yg = c(f y;zg; z) and f zg = c(f y;zg; z) are consistent with the axioms we have considered

so far. However, the latter situation is possible only if the psychological switching cost fromz is

higher than that from x; even though x is deemed more valuable than z by the decision maker.

T herefore, if onewishes to model theendowment e¤ect as monotonically increasingin thevaluation

of the alternatives, then she would wish to see instead that f yg= c(f y;zg;z): While its empirical

plausibility is not self-evident, it is true that this sort of a requirement would give rise to a more

re…ned model of choice. In particular, as we shall show formally in Section 3.1, it would entail a

model which envisages that an agent would chargea higher pricefor thealternative that he values

more in theabsence of any endowment e¤ect.

This discussion prompts the following hypothesis:

Axiom M EE¤

. (M onotonicity of the E ndowment E ¤ect) For any (S;x) 2 Csq(X ); if f yg= c(S; x) and f xg= c( T;3 ) for some T 2 X ; then y 2 c(S [ f zg; z) for all z 2 T:

14I f we assume that the choice correspondences under considerati on are all de…ned only for convex problems,

however, substanti al changes in the theorems would be needed, for then the property ®looses much of its strength.

In this case, a less appealing theory would obtain, wherethe properties ®and ¯ are replaced with the strong axiom

of revealed preference as analyzed by Peters and Wakker (1991).

17



In words, AxiomMEE says simply that if an agent prefers to move away from the statusquo x

in favor of an alternative y; hewould also do so if her initial endowment was less valuable than x:

T he …nal result of this section shows the implication of this property for thechoice model we have

developed thus far.

T heorem 3. Let X be a compact and connected metric space. A choice correspondence c on

C(X ) satis…es properties ®and ¯; and Axioms SQI ¤, SQB¤, UHC, and MEE if, and only if, there

exist a continuous map U : X ! R and a function ' : X ! R++ such that

(i) (3) and (4) hold for all (S;x) 2 Csq(X );

(ii) U and U + ' are comonotonic.15

Theinterpretation of this result is identical to that of T heorem2, except that thechoice model

of T heorem 3 warrants in addition that an agent who …nds an alternative x more valuable than

another alternative y in the absence of status quo bias (i.e. U(x) ¸ U(y)), must be compensated

more generously to moveaway from her status quo, when her status quo is x as opposed to y (i.e.

U(x) + ' (x) ¸ U(y) + ' (y)): Once again, we note that we do not see a strong reason why this

is a normatively more compelling theory; the …nal arbiter of the usefulness of T heorem 3 is the

experimental testing of Axiom MEE.16

3 A pplications

3.1 T he Overpricing P henomenon

A major channel through which theendowment e¤ect was discovered in experimental environments

is thediscrepancy found between buying and sellingpricesof commodities bytheindividuals, which

is often referred to asthegap between willingness to pay and willingness to buy of a person. (See

Kahneman, K netsch and T haler (1991) and Camerer (1995, pp. 665-670) for detailed surveys.)

We will show below that the model of the endowment e¤ect derived in the previous section would

indeed predict this sort of a gap.

Since we wish to talk about buying and selling prices of commodities, we need to introduce

slightly more structure to the present setup. Consequently, we let Y stand for a compact metric

space, and in order to interpret this space astheset of all non-monetary choice alternatives (such

as physical goods and/or lotteries), we assume Y \ R = ; : On the other hand, take any M > 0

15 T hat is, U(x) ¸ U(y) implies U(x) + ' (x) ¸ U(y) + ' (y) for any x;y 2 X :16One test of the model is through the classical experiment of the preference reversal phenomenon (Grether and

Plott, 1979). As we shall show in Secti on 3.4, this test would refute the model of T heorem 3, but not of Theorem 2.

18



and let [0;M ] denote the set of all potential prices. T he outcome space of the model is then

obtained by putting [0; M ] and Y together. Lettingd denote the disjoint-union operation, then,

we let X := [0; M ]d Y; and make this set a compact metric space in a natural way.17 T he choice

correspondences in this context are thus de…ned on C(X ): A particularly interesting subclass of

thesecorrespondences is theone that consists of themonotonic ones, that is, thosec that satisfy

c(f a;bg;3 ) = f ag whenever M ¸ a > b¸ 0; (5)

and

M 2 c(f M ;yg; y) and y 2 c(f 0;yg; 0) for all y 2 Y: (6)

Property (5) is an obviously appealingcondition that requires that more money is preferred to less

in theabsence of a status quo. On theother hand, the…rst part of (6) says that there is a price for

every feasible commodity at which that commodity would be sold, and the second part says that

all goodsare “good” - any member of Y is better than holding $0.

Let us consider an individual whose choice correspondencec on C(X ) is monotonic. We de…ne

the map Sc : Y ! [0;M ] by

Sc(y) := inf f a 2 [0;M ] : c(f y;ag; y) 3 ag

and themap Bc : Y ! [0;M ] by

Bc(y) := supf a 2 [0;M ] : c(f y;ag;a) 3 yg:

(Observe that these functions are well-de…ned in view of (6).) In words, Sc(y) is the minimum

selling price (or willingness to sell) for thenonmonetary alternative y according to the individual

with thechoice correspondence c: Similarly, Bc(y) is interpreted as themaximum buying price

(or willingness to pay) for y in theeyes of this agent.

Theso-called price gap between buying and selling prices translates, therefore, into thecompar-

ison of the maps Bc and Sc: The following result thus establishes that the present model predicts

precisely the overpricing phenomenon found repeatedly in various experimental settings. It also

shows that, provided that theagent abidesby AxiomMEE, then her willingness to sell is monoton-ically decreasing in the valueof theobject.18

17Since [0; M ] \ Y = ; ; there is an obvious way of doing this. L et d stand for the metric of Y and let µ >

maxf M ;diam( Y )g: We de…ne D : X 2 ! R+ by D(y; y0) := d(y;y0); D(a; a0) := ja ¡ a0 j ; and D (y;a) := D (a;y) := µ

for all y; y02 Y and a; a02 [0; M ]: It is easy to check that D is a metric on X which makes X compact.18Here by the “value” of the object wemean the utility of the object as obtained in the absence of status quo bias

and/ or endowment e¤ect.

19



P roposition 1. ( The Overpricing Phenomenon) Let M > 0; Y a compact metric space and

X := [0;M ]d Y: I f c is a monotonic choice correspondence on C(X ) that satis…esproperties ®and

¯; and Axioms SQI ¤, SQB¤ and UHC, then

Sc(y) > Bc(y) for all y 2 Y:

Moreover, if c also satis…es Axiom MEE, then

x 2 c(f x;yg;3)) implies Sc(x) ¸ Sc(y):

Proof. By Theorem 2, there exist a function ' : X ! R++ and continuous U : X ! R

such that (3) and (4) hold for all (S;x) 2 Csq(X ): Consequently, Sc(y) = inf f a 2 [0; M ] : U(a) ¸

U(y) + ' (y)g for any …xed y: But (3) and monotonicity of c imply that U is strictly increasing and

U(M ) ¸ U(y) + ' (y) ¸ U(0): SinceU is continuous, therefore, we …nd U(Sc(y)) = U(y) + ' (y):

Now de…ne By := f a 2 [0;M ] : U(y) ¸ U(a) + ' (a)g and apply Theorem 2 to conclude that

Bc(y) = supBy: Since ' (a) ¸ 0; for any a 2 By we have

U(Sc(y)) = U(y) + ' (y) ¸ U(a) + ' (a) + ' (y) ¸ U(a) + ' (y):

T hus by continuity of U; we…nd

U(Sc(y)) ¸ supU(By) + ' (y) = U(supBy) + ' (y) > U(Bc(y)):

Given that U is strictly increasing, this impliesthat Sc(y) > B

c(y) as we sought.

To see the second claim, observe that x 2 c(f x;yg;3)) implies U(x) ¸ U(y); so since U and

U + ' are comonotonic by Theorem 3, we get

U(Sc(x)) = U(x) + ' (x) ¸ U(y) + ' (y) = U(Sc(y)):

T hus theclaim follows from themonotonicity of U: ¤

Whileour axiomatic model entails that the endowment e¤ect is strictly positive (albeit possibly

in…nitesimal) even for monetary outcomes, onemay wish to usethe choicemodel found in T heorem

2 coupled with the assumption that ' (a) = 0for all a 2 [0; M ] (the psychological switching cost of

money is nil) and ' (y) > 0for all y 2 Y: Wenotethat P roposition 1 appliesalsoto this marginally

more general model; in fact, the proof we have given for this result above does not use the strict

positivity of ' on [0; M ]:

20



3.2 C ompetitive E quilibrium with the Endowment E ¤ect

T his section discusses brie‡y the potential implications of the decision-making model obtained in

Section 2.3 for competitive markets. For brevity, we shall only provide here a simple example

in which we contrast the implications of the basic choice model with and without the endowment

e¤ect. Wetakea 2£ 2exchange economy where the initial endowment vector of agent 1is(0; 1) and

that of consumer 2 is (2;1): Consumer 2 is free from theendowment e¤ect; sheis interested only in

maximizing her utility function u(x2;y2) := x2y2; x2; y2 ¸ 0: T he choice behavior of consumer 1 is,

in turn, modeled in concert with T heorem 2, where X :=R2+; and U and ' are given respectively

as U(x1;y1) := x1 +y1 and ' ´ " with " being any positive real number.

If we depart fromthe model given in T heorem2 by ignoring the endowment e¤ect, and choose

the second commodity as the numeraire, oneeasily veri…es that the uniqueequilibrium priceof the

…rst good is 1; and the equilibrium allocation of consumer 1 is (1=2;1=2): By contrast, with the

endowment e¤ect modelled as in Theorem 2, theoptimization problem of the agent 1 becomes

Maximize x1+ y1 such that px1 +y1 · 1 and x1+ y1 ¸ 1+ ";

where p > 0 denotes the price of the …rst good, and with the understanding that the “choice” of

the individual is her endowment (0; 1) if the constraint set of the problem above is empty. Thus,

in this case thedemand functions of theagents are found as

(x1(p;");y1(p;")) =

8<

:

(0; 1); if 1=p < 1+"

(1=p;0); if 1=p ¸ 1+"

and (x2(p);y2(p)) =

µ

1+1

2p

; p+1

2

¶

:

It is readily checked that there is no equilibrium of this economy for any " 2 (0; 1]. For instance,

p = 1 can no longer serve as an equilibrium price in this case, because person 1 does not view

this price low enough to persuade her move away from her endowment by trade. If, on the other

hand, " ¸ 1; then p = 1=2 is the equilibrium price with the unique equilibrium allocation being

((0;1);(2; 1)): T hus, introducing the endowment e¤ect through our model changes the behavior of

equilibrium drastically. If the endowment e¤ect is small, then there is no equilibrium, and if the

endowment e¤ect is large, then the volume of tradeis nil.19

This simple example shows that amending the standard economic models by introducing the

endowment e¤ect might well havenovel implications. For instance, onestandard approach towards

19I n this example, if person 2 (and not person 1) exhibited the endowment e¤ect (as modeled in Theorem 2 with

' ´ " > 0), then we would …nd that the autarchy is the unique equili brium allocation for any " > 1=4; while the

equilibrium coincides with that of the standard one for 0 < " · 1=4: Finally, if both individuals have the same

(constant) psychological switching cost " > 0; then we would again …nd that there is no equilibrium for su¢ cientl y

small " > 0; and when t here is an equilibrium, the volume of trade vanishes.

21



the demonstration of the Coase theorem is introducing markets for the externalities present in the

economy (the so-called “missing markets” approach), and showing that any equilibrium in this

extended economy is Pareto optimal regardless of whichparty is given rights to sell the externality.

Yet theexampleabove shows that, depending on whois allocated the property rights, theextended

economy may not have an equilibrium when we introduce to the model even a su¢ ciently small

amount of endowment e¤ect, thereby pointing formally to yet another reason for the failure of the

Coasetheorem. Moreover, even when an equilibriumexists, thepredicted volume of tradewith the

endowment e¤ect may bedrastically lower than that envisaged without this e¤ect. This particular

implication is indeed veri…ed experimentally by K ahneman, Knetch and Thaler (1990).

3.3 T he E ndowment E ¤ect under R isk

In this section wecextend the results of Section 2.4to thecase of choicesets that consist of lotteries.

Our development parallels the classical von Neumann-Morgenstern expected utility theory. We

designate an arbitrary compact metric space Z (that contains at least two elements) as the set of

all (certain) prizes, and let C(Z) denote the Banach spaceof all continuous real maps on X (under

the sup-norm). By a lottery, we mean a Borel probability measure on Z; and denote the set of

all lotteries by P (Z): Of course, Z is naturally embedded in P (Z) by identifying any certain prize

a 2 Z with the (Dirac) probability measure that puts full mass on the set f ag: This identi…cation

allows us to write a 2 P (Z) with a slight abuse of notation.

For any p 2 P (Z) and any continuous real function f on Z; we denote the Lebesgue integral

RZ f dpbyEp(f ): Weendow P (Z) with some metric that induces thetopology of weak convergence,

so for any sequence(pn) in P (Z); pn ! p means that pn converges to p weakly, that is, Epn(f ) !

Ep(f ) for all f 2 C(Z):20 It is well-known that this renders P (Z) a compact metric space.

The following axiom is a straightforward re‡ection of the classical independence axiom of ex-

pected utility theory, but note that it is stated only in terms of the choice problems without a

status quo.

Axiom I. ( I ndependence) For any (S;q) 2 - (P (Z)); ¸ 2 (0;1); and r 2 P (Z);

p 2 c(S;3) impli es ¸p +(1¡ ¸)r 2 c(¸S + (1¡ ¸)r;3 );

where ¸S +(1¡ ¸)r = f ̧ q+ (1¡ ¸)r : q2 Sg:

It turns out that adding this property to the set of assumptionsemployed in Theorem 2 yields

20 T here are variousdistance functions that may beused for this purpose (e.g the P rohorov metric). For thepresent

purposes, however, it is inconsequential which of thesemetrics is chosen.

22



easily a characterizationof choicecorrespondences that exhibit theendowment e¤ect in risky choice

situationsas well.

T heorem 4. Let Z be a compact metric space. A choicecorrespondence c on C(P (Z)) satis…es

properties ®and ¯; and Axioms SQI¤

, SQB¤

, UHC, and I if, and only if, there exist a continuousfunction u : Z ! R and a function ' : P (Z) ! R++ such that

c(S;3 ) = argmaxp2S

Ep(u)

for any nonempty closed subset S of P (Z); and

c(S;q) =

8><

>:

f qg; if Eq(u) + ' (q) >Ep(u) for all p2 S

argmaxp2S

Ep(u); otherwise


The interpretation of this result is analogous to that of T heorem 2, so we do not discuss it

further here. We should note, however, that adding Axiom M to T heorem 4 would ensure in this

result the comonotonicity of themap V : p7! Ep(u) and V + ' : (Recall T heorem 3.)

3.4 T he P reference R eversal P henomenon as an E ndowment E¤ ect

Amongthemanyexperimental observations that refute thebasic premisesof expectedutility theory,

a particularly striking one is the so-called preference reversal phenomenon. First noted by Slovic

and Lichtenstein (1968), this phenomenon has caused a great deal of theoretical and experimental

debate among decision theorists, especially after the seminal contribution of Grether and Plott

(1979). The basic experiment behind this phenomenon asks thedecision maker to choose between

two lotteries, one o¤ering a high probability of winning a small amount of money, and the other

o¤ering a low probability of winning a large payo¤. For concreteness, let us couch the discussion

by means of a speci…c example: Let h stand for the lottery that pays $10 with probability 8/ 9 and

nothing otherwise, and let ` stand for the lottery that pays $85 with probability 1/ 9 and nothing

otherwise. When confronted with such a choice problem, most individuals were observed to preferh over ` in the experiments. The subjects were then asked to state the minimumprice at which

they would be willing to sell h and ` (had they owned these lotteries), and surprisingly, about half

of them were found to charge a strictly higher price for ` than for h; hence the term preference

reversal (PR) phenomenon.

23



While there have been a largenumber of experimental and theoretical studies concerning the

explanationsfor this phenomenon,21 to our knowledge no onehas suggested that this phenomenon

is but a particular instant of the endowment e¤ect. T his is quite surprising because the structure

of the PR phenomenon is very reminiscent of the overpricing phenomenon (Section 3.1) which is

often explained by this e¤ect. At any rate, it may be worthwhile to note that the present choice

model (as envisaged by Theorem 4) provides an immediate test of whether or not it is mainly the

endowment e¤ect that underlies thePR phenomenon.

Let Z := [0;z]for somez > 85; and assumethat thechoicecorrespondencec onC(P (Z)) satis…es

theaxiomsof Theorem4alongwiththefollowinginnocuousmonotonicity property: c(f ±a;±bg;3 ) =

f ±ag for all a;b 2 Z with a > b. J ust as in Section 3.1, we de…ne the minimum selling price

induced by c asthemap Sc : P (Z) ! Z given by

Sc(p) := inf f a 2 Z : c(f p; ±ag;p) 3 ±ag:

T his formulation recognizes the fact that a seller prices a lottery when she is in possession of it, a

potentially important aspect of thePR experiments. The data of the PR phenomenon is then the

following:

f hg = c(f h;`g;3) and Sc(h) < Sc( )̀:

In the languageof T heorem 4, these statements are tantamount to the followingtwo inequalities:

Eh(u) > E`(u) and E`(u) + ' ( )̀ > Eh(u) + ' (h); (7)

with u and ' as found in that theorem. Since one can easily choose u and ' in a way to satisfy

thesei nequalities, we …nd that theP R phenomenon is consistent with the choicemodel of T heorem

4. Perhaps more importantly, the inequalities of (7) provide us with an immediate experimental

test of the model (which will be undertaken in future research): Ask the subjects to choose from

f h;`g when they are endowed with :̀ On the basis of (7), our model predicts that a substantial

fraction of the agents with f hg = c(f h;`g;3) will in fact change their choices to ` when ` is the

status quo of the problem, i.e., f ̀ g = c(f h; g̀; )̀; for Eh(u) + ' (h) > E`(u) + ' ( )̀ > E`(u): If the

subjects keep choosing h from f h; g̀ even when endowed with ,̀ then this would, in turn, refute

themodel envisaged by Theorem 4.

21See, inter alia, Holt (1986), Karni and Safra (1987), Segal (1988), Cox and Epstein (1989), Tversky, Sl ovic and

K ahneman (1990), and Loomes, Starmer and Sugden (1991).

24



4 C onclusion

Wehave sketched a revealed preferencetheory in this paper that modi…esthestandard static choice

theory by introducing to the model the possibil ity that the decision maker may have an initial

reference point which can beinterpreted as a default option, current choiceand/ or an endowment. T his expands the classical setup, and leads to some intuitive representations of choice behavior. In

particular, the representations we provide here allow for phenomena like the status quo bias and

the endowment e¤ect, and notably, draw a connection between how problems with and without a

status quo are settled.

There are, of course, several directions that needs to be explored. For one thing, like other

related papers mentioned in Section 1, our analysis applies to only static choice problems. While

the status quo bias phenomenon is presumably more pressing in static problems, it is not known

if and how dynamic choice procedures would induce static choice behavior that would exhibit a

status quo bias. T his sort of an analysis would provide a deeper model in which the status quo

bias is endogenized. While the present work might provide useful in modeling such a phenomenon

in the stage problems of adynamic choice model, its exclusively static natureis of course a serious

shortcoming.22 Secondly, on the applied front, it will be interesting to see if and how the choice

models introduced here might a¤ect theconventional conclusions of thestandard search and buyer-

seller models, where a status quo bias and/ or the endowment e¤ect are likely to play important

roles.

5 P roofs

P roof of Lemma 1. Take any choice correspondence c that satis…es all …ve of the postulated

properties, and de…ne the binary relation< on X by

y< x if and only if y 2 c(f x;yg;x):

Sincec is nonempty-valued, < is re‡exive. B y SQB, y 2 c(f x;yg;x) and x 2 c(f x;yg;y) can hold

simultaneously if andonly if y = x: T hus< is antisymmetric. To see that < is also transitive, take

any y;x;z 2 X with y< x<z; that is,

y 2 c(f x;yg;x) and x 2 c(f x;zg;z):

We assume that y;x and z are distinct lotteries, otherwise the claim is trivial. T he…rst expression

above andSQB then jointly imply that x =2 c(f x;yg; x) so, by property ®, x =2 c(f x;y;zg; x): Then,

22 T he only paper in this regard that we know is Vega-Redondo (1995) who provides a dynamic decision model

(with learning) the limit behavior of which covers decisions that depend on the status quo.

25



by SQB, x =2 c(f x;y;zg; z): But property ®and SQB imply that z 2 c(f x;y;zg;z) is possible only

if f zg = c(f x;zg;z) which contradicts x < z: T hus we have y 2 c(f x;y;zg; z); and by property ®,

it follows that y 2 c(f y;zg;z); that is, y <z: Consequently, we conclude that < is a partial order

on X .

Claim 1. For any choiceproblem with a status quo (S;x) 2 Csq(X );

c(S;x) µ

8<

:

f xg; if UÂ(x) \ S = ;

UÂ (x); otherwise.

Proof. Take any (S;x) 2 Csq(X ); and assume that UÂ(x) \ S = ; : If x 6= y 2 c(S;x); then, by

property ®, y 2 c(f x;yg;x); which yields the contradiction y 2 UÂ(x) \ S: T hus, in this case, we

have c(S;x) = f xg: Assume next that UÂ(x) \ S 6= ; ; and pick any y 2 c(S;x): I f y = x; then, by

property ®and SQB, f xg = c(f x;zg;x) for all z 2 S; andthisyieldsUÂ (x)\ S = ; ; a contradiction.

So y 6= x: T hen, by property ®, y 2 c(f y;xg;x) so that y 2 UÂ(x): k

Claim 2. For any choiceproblem with a status quo (S;x) 2 Csq(X ); if UÂ(x) \ S 6= ; ; then

c(S;x) = c(UÂ (x) \ S;3 ):

Proof. Take any (S;x) 2 Csq(X ); and assume that UÂ(x) \ S 6= ; : L et y 2 c(S;x): By Claim 1,

y 2 UÂ(x) \ S; so by property ®; y 2 c((UÂ (x) \ S) [ f xg;x): But by Claim 1, x =2 c( T;x) for any

nonempty T µ (UÂ(x) \ S) [ f xg with T 6= f xg: T hus SQI gives y 2 c((UÂ (x) \ S) [ f xg;3 ); and

so by property ®; y 2 c(UÂ (x) \ S;3):

To prove the converse containment, let y 2 c(UÂ (x) \ S;3 ); and notice that if z 2 c((UÂ(x) \

S) [ f xg;3); then by property ®, we must havez 2 c(UÂ(x) \ S;3); and hence, by property ¯; we

obtain y 2 c((UÂ(x) \ S) [ f xg;3 ): Moreover, sincey 2 UÂ(x); wehavef yg= c(f x;yg;x) by SQB,

and therefore, wemay apply Axiom D to concludethat y 2 c((UÂ(x) \ S) [ f xg;x): Now take any

z 2 c(S;x); and apply property ®to get z 2 c((UÂ (x) \ S) [ f xg;x): It then follows fromproperty

¯ that y 2 c(S;x): k

Given that c satis…es the properties®and ¯; by a standard result of choice theory, there must

exist a completepreorder <¤ such that c(¢;3) = M (¢;<¤): Tocomplete the proof, then, it is enough

to show that Â is contained in Â¤. But for any distinct x;y 2 X with y 2 c(f x;yg;x); SQB implies

that x 2 c(f x;yg;3 ) cannot hold, so it follows that y Â¤ x:

P roof of T heorem 1. The “if” part of theclaimis easily veri…ed. To provethe“only if” part,

take any choice correspondence c that satis…es all …ve of the postulated properties, and consider

the binary relations< and<¤ found in L emma 1.

26



Claim. There exists a positive integer n and an injection u : X ! Rn such that

y< x if and only if u(y) ¸ u(x) for all x;y 2 X :

Proof. L et e(<) stand for the set of all linear orders such that Â µ R: Given that X is …nite, it

is obviousthat e(<) is a nonempty …nite set. Let us enumeratethis set as f R1;:::;Rng: It is readily

checked that < = T n

i=1Ri: Moreover, since X is …nite, there exists a map ui : X ! R such that

yRix i¤ ui(y) ¸ ui(x) for all x;y 2 X: Thus, de…ningu(x) := (u1(x);:::;un(x)); we …nd that y< x

i¤ u(y) ¸ u(x) for all x;y 2 X : Since< is antisymmetric, u must be an injection. k

Now observethat c(¢;3 ) : X ! X is a standard choicecorrespondencethat satis…es theclassical

properties ®and ¯: Given that X is …nite, it follows that there exists a map v : X ! R such that

c(S;3) = argmaxz2S

v(z) for all S 2 X : (8)

Consequently, by Lemma 1, we may concludethat

Uu(S;x) 6= ; implies c(S;x) = argmaxz2Uu (S;x)

v(z) (9)

for any (S;x) 2 Csq(X ): To complete the proof, we de…ne f : u(X ) ! R by f (a) := v(u¡ 1(a)):

Sinceu is injective, f is well-de…ned. Moreover, if u(y) = a > b = u(x) for some x;y 2 X; then

f yg= c(f x;yg;xg by Lemma 1 and theclaimproved above. But then x 2 c(f x;yg;3) cannot hold

by SQB, and hence (8) yields f (a) = v(y) > v(x) = f (b): Weconcludethat f is strictly increasing.

Finally, observethat, by Lemma 1 and the claim proved above, we have

Uu(S;x) = ; implies c(S;x) = f xg (10)

for any (S;x) 2 Csq(X ): Combining (10), (9), and (8), and noting that v = f ±u; completes the

proof.

P roof of T heorem 2. T he only nontrivial statement in the “if” part of the claim concerns

the veri…cation of UHC, but this is also easily established by using the maximum theorem. To

prove the “only if” part, take any choice correspondence c that satis…es all …ve of the postulated

properties, and de…ne the binary relation%on X by

y%x if and only if y 2 c(f x;yg;3):

It is easily veri…ed that % is complete preorder on X by using properties ®and ¯: We now verify

that this relation is continuous. To verify itsupper semicontinuity at anarbitrary point x 2 X , take

27



any sequence (ym) in X such that ym%x and ym ! y for some y 2 X : To derive a contradiction,

assume that x Â y; where Â is the asymmetric part of %. This means that y =2 c(f x;yg;3) = f xg:

Now denote themetric of X by d; let " := d(x;y) > 0; and de…ne

A :=

n

S 2 X : dH (S;f xg) ·

"

2

o

=

n

S 2 X : dH(S;c(f x;yg;3

)) ·

"

2

o

;

wheredH is theHausdor¤ metric on X inducedby d: Clearly, A is aclosed subset of X ; c(f x;yg;3 ) 2

int(A ), and f yg =2 A: But the Hausdor¤ metric dictates that f x;ymg ! f x;yg; so by UHC, there

must exist an M > 0such that c(f x;ymg;3 ) 2 int(A ) for all m ¸ M: T herefore, if x 2 c(f x;ymg;3)

for …nitely many m; then there exists an M0> 0 such that f ymg 2 int(A ) for all m ¸ M 0: Since A

is closed, wethen get f yg=limf ymg2 A; a contradiction. I f, on the other hand, x 2 c(f x;ymg;3)

for in…nitely many m; then there exists a subsequence (ymk) such that c(f x;ymk

g;3 ) = f x;ymkg

for all k: Thus in this case there exists a K > 0 with f x;ymkg 2 int(A ) for all k ¸ K : SinceA is

closed, this implies that f x;yg= limf x;ymk g2 A; but this is impossible since dH (f x;yg;f xg) = ":

T his proves that % is upper semicontinuous. Lower semicontinuity of % is veri…ed similarly.

Given that % is upper semicontinuous, and any S in X is compact, it follows that

f y 2 S : y%x for all x 2 Sg6= ; for all S 2 X :

Moreover, by using properties ®and ¯; one may easily verify that

c(S;3 ) = f y 2 S : y%x for all x 2 Sg for all S 2 X :

But, given that % is continuous, by the Debreu utility representation theorem, there exists a

continuous real function U on X (which is compact, hence separable) such that y %x i¤ U(y) ¸

U(x) for all x;y 2 X: Combining this fact with the previous observation, we obtain

c(S;3) = argmaxz2S

U(z) for all S 2 X : (11)

Now de…ne

I (x) := f y 2 X nf xg : y 2 c(f x;yg; x)g; x 2 X

and

Xc := f x 2 X : x =2 c(S;x) for some S 2 X with x 2 Sg:

Claim 1. I (x) is a nonempty compact subset of X for any x 2 Xc:

Proof. Fix any x 2 Xc: By de…nition, there exists an S 2 X with x 2 S and x =2 c(S;x): Let

z 2 c(S;x): T hen z 6= x; and z 2 c(f x;zg;xg by property ®; that is, z 2 I (x); establishing that

I (x) 6= ; :

28



Given that X is compact, it is then enough to show that I (x) is a closed set: To this end,

takeany sequence (ym) in I (x) with ym ! y for somey 2 X : Clearly, dH (f x;ymg;f x;yg) ! 0 and

ym 2 c(f x;ymg;x) for each m; so by UHC, thereexistsa strictly increasingsequence(mk) of positive

integers such that (ymk) converges to some point in c(f x;yg;x): Since limymk

= y; we clearly have

y 2 c(f x;yg; x): Moreover, by SQB¤; there exists an " > 0 such that f xg = c(cl(N"(x))); so if

y = x; then there exists an integer K such that ymk2 N"(x) for all k ¸ K : But then, by property

®; f xg= c(f x;ymkg; xg which gives x = ymk

for each k ¸ K ; a contradiction. T hus, y 6= x; that is,

y 2 I (x): k

Given Claim 1 and thefact that U is continuous, we may de…ne ¸ : X c ! R by

¸(x) := minz2I (x)

U(z):

Claim 2. ¸(x) > U(x) for all x 2 Xc:

Proof. For any x 2 Xc and z 2 X nf xg; if U(x) ¸ U(z) holds, then x 2 c(f x;zg;3 ); so by SQB,

f xg= c(f x;zg;x); that is, z =2 I (x): T hus, U(z) > U(x) holds for all z 2 I (x); and hence theclaim.

k

Claim 3. For any (x;y) 2 Xc £ X; U (y) ¸ ¸(x) implies that y 2 I (x):

Proof. By hypothesis, U(y) ¸ U(z) for some z 2 I (x): By Claim 2, therefore, y 2 c(f x;y;zg;3):

Now suppose that y =2 c(f x;y;zg;x): Since z 6= x; SQB and property ® imply that f zg =

c(f x;y;zg;x); but this contradicts SQI¤: T hus y 2 c(f x;y;zg; x); and hence y 2 c(f x;yg;x) byproperty ®: But, by Claim 2, U(y) ¸ ¸(x) > U(x); so y 6= x: T hus y 2 I (x). k

Claim 4. For any x 2 Xc and (S;x) 2 Csq(X ); if U(y) ¸ ¸(x) for some y 2 S; then

y 2 c(S;x) if and only if U(y) = maxU(S):

Proof. Take any (S;x) 2 Csq(X ) with x 2 Xc and

Y := f y 2 S : U(y) ¸ ¸(x)g6= ; :

Suppose…rst that x 2 c(S;x): Takeany y 2 Y; and note that y 6= x by Claim 2. Then by property

®and SQB, y =2 c(f x;yg;x); that is, y =2 I (x); which contradicts Claim 3. Thus x =2 c(S;x); so

y 2 c(S; x) implies that y 6= x: T hen, by SQB¤; y 2 c(S;x) holds only if y 2 c(S;3); that is,

U(y) = maxU(S) by (11). Conversely, if y 2 argmaxz2S U(z); then y 2 c(S;3): But if z 2 c(S;x);

thepreviousargument yield that z 6= x; so z 2 c(S;3 ) by SQB¤: Then, by SQI¤; weget y 2 c(S;x):

k

29



Claim 5. For any (S;x) 2 Csq(X ); if ¸(x) > U(y) for all y 2 S; then c(S;x) = f xg:

Proof. By de…nition of ¸; ¸(x) > U(y) implies that y =2 I (x); so f xg= c(f x;yg; x) for all y 2 S:

T hus, by property ®; wehave c(S;x) = f xg: k

To complete the proof, we de…ne ' : X ! R++ as

' (x) :=

8<

:

¸(x) ¡ U(x); if x 2 Xc

maxU(X ) ¡ minU(X ) + 1; otherwise.

Take any (S;x) 2 Csq(X ); and suppose that U(x) + ' (x) > U(y) for all y 2 S: If x 2 Xc;

then ¸(x) > U(y) for all y 2 S; so c(S;x) = f xg by Claim 5. If x =2 Xc; then x 2 c(S;x) by

de…nition of Xc; so by SQB, we have c(S;x) = f xg: Now suppose that U(x) + ' (x) · U(y) for

some y 2 S: If x =2 Xc; then maxU(X ) ¡ minU(X ) + 1 · U(y) ¡ U(x) for somey 2 X ; which is

impossible. Thus x 2 Xc; and in this case, we have¸(x) · U(y) for somey 2 S; and there followsc(S;x) = argmaxz2S U(z) by Claim 4. T he proof of Theorem 2 is now complete.

P roof of T heorem 3. We only need to talk about the “only if” part. To this end, we de…ne

the set Xc and the maps I ; ¸; U and ' exactly as in the proof of T heorem 2, and note that, by

T heorem 2, we only need to establish the comonotonicity of U and U +' . Wewill usethefollowing

claim for this purpose.

Claim 1. For any x;y 2 Xc; U(x) ¸ U(y) implies ¸(x) ¸ ¸(y):

Proof. Takeany x;y 2 Xc with U(x) ¸ U(y); and to derivea contradiction, assume¸(y) > ¸(x):

Since I (y) and I (x) are nonempty compact sets (by Claim 1 of the proof of T heorem 2) and U is

continuous, there exists a (zx; zy) 2 I (x) £ I (y) such that U(zx) = ¸(x) and U(zy) = ¸(y): Given

that X is connected, wemay then use the intermediate value theorem to …nd a z 2 X such that

¸(y) > U(z) > ¸(x): Now, by Claim3 of theproof of T heorem 2, z 2 I (x); that is, z 2 c(f x;zg;x);

so we have x =2 c(f x;y;zg;x) by Property ®and SQB. On the other hand, (4) and U(x) ¸ U(y)

imply that y =2 c(f x;y;zg;x); so we must have f zg = c(f x;y;zg;x): But again by (4) we have

x 2 c(f x;yg;3); so by Axiom MEE we get z 2 c(f x;y;zg;y): By property ®; this means that

z 2 c(f y;zg; y); that is, z 2 I (y): But this is impossible, for U(z) < ¸(y) = minz2I (y) U(z): k

Now take any x;y 2 X with U(x) ¸ U(y): Consider …rst the case where x =2 Xc: In this case, if

y 2 Xc; then

U(x) + ' (x) = maxU(X ) + (U(x) ¡ minU(X ) ) +1

> maxU(X )

30



¸ ¸(y)

= U(y) + ' (y);

and if y =2 Xc; then

U(x) + ' (x) = U(x) + maxU(X ) ¡ minU(X ) + 1

¸ U(y) +maxU(X ) ¡ minU(X ) + 1

= U(y) + ' (y);

as we sought. Now let x 2 Xc: This implies that x =2 c(S;x) for some S 2 X with x 2 S;

so by properties ®and ¯; we have f zg = c(f x;zg;x) for some z 2 X: Moreover, U(x) ¸ U(y)

implies x 2 c(f x;yg;3 ); so it follows from Axiom MEE that z 2 c(f x;y;zg;y): T hen by SQB,

y =2 c(f x;y;zg; y); that is, y 2 X c: By Claim 1, therefore,

U(x) + ' (x) = ¸(x) ¸ ¸(y) = U(y) + ' (y);

and we are done.

P roof of T heorem 4. (Sketch) Weproceed exactly as in theproof of Theorem 2 (by setting

X := P (Z)). To prove the “only if” part, then, we takeany choice correspondencec that satis…es

all six of the postulated properties, and de…ne the complete and continuous preorder %on P (Z)

by

p%q if and only if p2 c(f p;qg;3):

Using Axiom I , it is readily veri…ed that % satis…es the classical independence axiom so that by

the von Neumann-Morgenstern expected utility theorem (see K reps (1988)), there must exist a

function u 2 C (Z) such that p%q i¤ Ep(u) ¸ Eq(u) for all p; q2 P (Z): But the properties®and

¯ imply

c(S;3) = f p2 S : p%q for all q2 Sg

so that c(S;3) = argmaxp2S Ep(u) for all nonempty closed subsets S of P (Z): Therest of the proof

is identical to that of Theorem2 with p7!Ep(u) playing therole of p7! U(p):

31



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