revealed incomplete preferences under status-quo bias

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Mathematical Social Sciences 53 (2007) 274–283
www.elsevier.

Revealed incomplete preferences under status-quo bias☆

İpek Gürsel TapkıSabanci University, Faculty of Arts and Social Sciences, Orhanli, Tuzla, Istanbul, 34956 Turkey

Received 25 April 2006; received in revised form 12 October 2006; accepted 11 December 2006Available online 7 February 2007

Abstract

We propose a theory of revealed preferences that allows both the status-quo bias and indecisivenessbetween any two alternatives.© 2007 Elsevier B.V. All rights reserved.

Keywords: Revealed preferences; Incomplete preferences; Status-quo bias

JEL classification: D11

1. Introduction

Recently, (standard) revealed preference theory has been criticized for not being able toaddress two phenomena: (i) incomplete preferences and (ii) a status-quo bias. The aim of this noteis to propose a theory of revealed preferences that allows both of these features.

Quoting Aumann (1962), “… Of all the axioms of utility theory, the completeness axiom isperhaps the most questionable”. Aumann argues that in daily life there are many situations inwhich an individual has incomplete preferences and thereby exhibits indecisiveness. Hisarguments are supported by experimental studies. Danan and Ziegelmeyer (2004) experimentallytest the descriptive validity of the completeness axiom and they show that a significant number ofsubjects (around two-thirds) violate completeness. In an experiment, Brady and Ansolabehere(1989) find that approximately 20 percent of their subjects have incomplete preferences over the

☆I am very much indebted to Özgür Kıbrıs for his invaluable comments and guidance. I also thank Efe Ok, YusufcanMasatlioglu, Mehmet Barlo, Remzi Sanver, Haluk Ergin, and Emre Ozdenoren for helpful comments.E-mail address: [email protected].

0165-4896/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.mathsocsci.2006.12.003

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http://dx.doi.org/10.1016/j.mathsocsci.2006.12.003

275˙I.G. Tapkı / Mathematical Social Sciences 53 (2007) 274–283

candidates in the 1976 and 1984 Democratic Presidential primaries. Similar results are obtainedby Eliaz and Ok (2006) who show that completeness of the revealed preferences is closely relatedto a Property β (Sen, 1971) of choice and that this property is violated by a significant number ofsubjects. (Therefore, they weaken Property β to represent incomplete preferences.)

The second feature, status-quo bias, refers to an agent whose choice behaviour is affected bythe existence of an alternative he holds at the time of choice (called the status-quo). Thisphenomenon has been repeatedly demonstrated in experiments (see Kahneman et al., 1991 forsurveys). Particularly, Samuelson and Zeckhauser (1988) report on several decision-makingexperiments where a significant number of their subjects exhibit a status-quo bias. Masatliogluand Ok (2005) model the status-quo bias as an agent having an incomplete preference relation thathe uses to compare the status-quo to another alternative (and whenever indecisive, to choose thestatus-quo).

There is no experimental study that demonstrates both incomplete preferences and a status-quobias. However, we believe that there are choice situations in which a decision maker can exhibitboth features. As an example, consider a professor who has job offers. He may be evaluating theseoffers with respect to several criteria and thus, may be indecisive between some of them. Inaddition, his current job (if it exists) may bias his choices. Models that exhibit both features arealready used in political theory. For example, Ashworth (2005) considers voters whosepreferences are incomplete. He also assumes that there is a status-quo action that the voters takeunless some alternative dominates it.

Motivated by these observations, we propose a theory that encompasses both Masatlioglu andOk (2005) and Eliaz and Ok (2006). In our model, a choice problem is (i) a feasible set S ofalternatives and (ii) a status-quo point x in S (allowed to be null when there is no status-quo). Ourmain result is that if an agent's choice behavior satisfies a set of basic properties, then it isrationalizable1 by a pair of incomplete preference relations (one “more incomplete” than theother): when there is a status-quo, the agent first compares the nonstatus-quo alternatives to thestatus-quo by using the more incomplete preference relation. He chooses the status-quo if noalternative is strictly preferred to it. However, if there are some alternatives that are strictlypreferred to the status-quo, then among them the agent chooses alternatives that maximize thesecond (less incomplete) preference relation. 2,3

Existence of two distinct preference relations is essential in capturing certaincharacteristics of the choice behaviour that we observe. We show that agents whosechoice behaviour can be rationalized by a single (however incomplete) preference relationsatisfy a property that significantly limits the implications of status-quo bias (see Corollaries1 and 2).

Our model is rich enough to make a distinction between an agent being indecisive orindifferent between two alternatives. There is an observational distinction between these twocases (e.g. see Eliaz and Ok, 2006). In both of them, the agent's choices switch between the twoalternatives in repetitions of the same choice problem. However, an agent being indecisivebetween two alternatives also implies that in terms of comparison to some third alternatives, these

1An agent's choice behaviour is rationalizable if there exists a preference relation such that for any choice problem, itsmaximizers coincide with the agent's choices.2This process is similar to Masatlioglu and Ok (2005). However, they require one of the preference relations to becomplete. As a result, the agent in their model is never indecisive between two non-statusquo alternatives.3Note that, this is different than simply maximizing the second preference relation on the whole set.

276 ˙I.G. Tapkı / Mathematical Social Sciences 53 (2007) 274–283

two alternatives differ. This feature of indecisiveness leads to certain “inconsistencies” in thechoice behavior (which do not exist in the case of indifference). 4

In addition to Masatlioglu and Ok (2005) and Eliaz and Ok (2006), our model is similar toZhou (1997) and Bossert and Sprumont (2003). However, these authors do not considerincomplete preferences and they only analyze problems with a status-quo (thus unlike them, wecan also discuss properties that link the choice behaviour in problems with and without a status-quo). Our model is also similar to Tversky and Kahneman (1991) and Sagi (2006) who analyzecases where an agent's preferences are dependent on a reference state (which, in our case, is astatus-quo alternative). However, these authors focus on properties of preferences (rather thanchoices as we do).

2. Properties of a choice correspondence

Let X be a nonempty metric space of alternatives and X be the set of all nonempty closedsubsets of X. A choice problem is a pair (S, x) where SaX and x∈S or x=⋄.5 The set of all choiceproblems is CðX Þ. If x∈ S, then (S, x) is a choice problem with a status-quo and we denote the setof such choice problems by CsqðX Þ. If x=⋄ , then (S,⋄) is a choice problem without a status-quo.A choice correspondence is a map c : CðX ÞYX such that for all ðS; xÞaCðX Þ, c(S, x)⊆S.

A binary relation ⪰ on a nonempty set X is called a preorder if it is reflexive (x⪰x for allx∈X ) and transitive (x⪰y and y⪰ z imply x⪰ z for all x, y, z∈X ). An antisymmetric (x⪰yand y⪰x imply x=y for all x, y∈X) preorder is called a partial order and a complete (x⪰y ory⪰x for all x, y∈X ) partial order is called a linear order. Let ⪰ be any binary relation on X.Let x, y∈X. Then, x≻y if and only if x⪰y and yqx and x∼y if and only if x⪰y and y⪰x. Let⪰ and ⪰′ be two binary relations on X and x, y∈X. Then, ⪰′ is an extension of ⪰ if and only ifx≻y implies x≻′y and x∼y implies x∼′y.

Let x∈X and SaX . Following Masatlioglu and Ok (2005), we let UdðS; xÞ ¼ fyaSjydxg bethe strict upper contour set of x in S with respect to ⪰ and MðS;vÞ ¼ fxaSjUdðS; xÞ ¼ tg bethe set of all maximal elements in S with respect to ⪰. For any positive integer n and any functionu : XYℝn;UuðS; xÞ ¼ fyaSjuðyÞNuðxÞg is the upper contour set of x in S with respect to u6 andMðS; uÞ ¼ fyaSjUuðS; yÞ ¼ tg is the set of all maximal elements in S with respect to u.

Now, we define some properties. The first two are borrowed from Masatlioglu and Ok (2005).Property α is a straightforward extension of the “standard Property α” in the revealed preferencetheory.

Property α: For any ðS; xÞ; ðT ; xÞaCðX Þ if y∈T⊆S and y∈c(S, x), then y∈c(T, x).For the second property, suppose y is not worse than any other alternative in a feasible set S,

including the status-quo alternative x (if there is one). Then, status-quo bias requires that when ybecomes the status-quo, it will be revealed strictly preferred to every other alternative in S. (For adetailed discussion, see Masatlioglu and Ok, 2005).

Status-quo Bias: For any ðS; xÞaCðX Þ, if y∈c(S, x), then c(S, y)={ y}.

4As an example, consider the following voter. He has two favorite parties, A and B. If he has to vote between one of thesetwo, he could vote for either. First, let him face the problem of voting among A, X, and Y. Suppose that he votes for A.Now, in this choice problem, replace A with B. Being indifferent between A and B means that he chooses B in thisproblem. However, being indecisive between A and B refers to the case in which he chooses an alternative different thanB. For further discussion, please see Eliaz and Ok (2006).5⋄ denotes a null alternative and is used to represent cases when there is no status-quo.6For vectors in ℝn, the inequalities are defined as follows: x≥y if and only if xi≥yi for all i=1,…, n and xNy if and onlyif x≥y and x≠y.


Now, we introduce a new property which is a weakening of the counterpart of Sen's (1971)Property β for choice problems with status-quo (see Masatlioglu and Ok, 2005 for a strongerformulation). To see the main difference between Properties β and β′, take any alternative y froma feasible set of alternatives S and suppose there is a chosen alternative z in S such that thefollowing condition holds: there is a subset T of S containing both y and z such that both arechosen in T. Property β then says that y must also be chosen from S. Our weaker Property β′ onthe other hand requires the above condition to hold for every chosen z in S for y also to be chosen.

Property β′: For any ðS; xÞaCðX Þ and y∈S, if for all z∈c(S, x), there exists T⊆S such thatx, y, z∈T7 and y, z∈c(T, x), then y∈c(S, x).

Properties β′ and α are together equivalent to a “revealed non-inferiority” property (introducedby Eliaz and Ok, 2006) which is weaker than the weak axiom of revealed preferences.

The following three properties relate the choice behavior of a decision maker across problemswith and without a status-quo. The first two of them are borrowed form Masatlioglu and Ok(2005). (For a detailed discussion of these properties, see their paper.)

Dominance: For any ðT ; xÞaCðX Þ, if c(T, x)={y} for some T⊆S, and y∈c(S,⋄), then y∈c(S, x).SQ-irrelevance: For any ðS; xÞaCsqðX Þ, if y∈c(S, x) and x∉c(T, x) for any nonempty T⊆S

with T≠{x}, then y∈c(S, ⋄).For the third property, take any alternative x from a set S. Suppose that x is never chosen from

a subset T≠{x} of S despite the fact that it is the status-quo. Thus, x does not play a significantrole in the choice problem (S, x). In such cases, strong SQ-irrelevance requires that dropping outthe status-quo alternative does not affect the agent's choices.

Strong SQ-irrelevance: For all ðS; xÞaCsqðX Þ, if x∉c(T, x) for any nonempty T⊆S suchthat T≠{x}, then c(S, ⋄)=c(S, x).

Strong SQ-irrelavence is weaker than the combination of Masatlioglu and Ok (2005)'s“dominance” and “SQ-irrelevance”. It implies “status-quo irrelevance”, but not “dominance” asnoted in the following example: let X={x, y, z} and

cðfx; y; zg;>Þ ¼ fyg; cðfx; yg;>Þ ¼ fyg; cðfx; zg;>Þ ¼ fzg; cðfy; zg;>Þ ¼ fyg:

cðfx; y; zg; xÞ ¼ fx; zg; cðfx; yg; xÞ ¼ fyg; cðfx; zg; xÞ ¼ fxg:

However, together with Property α and β′, strong SQ-irrelevance implies both properties.

Lemma 1. (i) If a choice correspondence c satisfies SQ-irrelevance and dominance, then itsatisfies strong SQ-irrelevance.

(ii) If a choice correspondence c satisfies Property α, Property β′, and strongSQ-irrelevance, then it satisfies SQ-irrelevance and dominance.

Proof. (i) Let c satisfy SQ-irrelevance and dominance. Let ðS; xÞaCsqðX Þ. Suppose x∉c(T, x)for any nonempty T⊆S such that T≠{x}. Then, for any z∈S \{x}, c({x, z}, x)={z}.Let y∈S. First, let y∈c(S, ⋄). Since c({x, y}, x)={y}, by dominance, y∈c(S, x).Second, let y∈ c(S, x). Then, by SQ-irrelevance, y∈ c(S, ⋄). Therefore, c(S, x)=c(S, ⋄) and c satisfies strong SQ-irrelevance.

(ii) Let c satisfy the given properties. To show that c satisfies SQ-irrelevance, letðS; xÞaCsqðX Þ and y∈S. Suppose that y∈c(S, x) and x∉c(T, x) for any non-empty T⊆S such that T≠{x}. By strong SQ-irrelevance, c(S, x)=c(S, ⋄). Thus,

7If x=⋄, then consider only y, z∈T.


y∈ c(S,⋄). To show that c satisfies dominance, let y∈ c(S,⋄) and suppose thereis T⊆S such that ðT ; xÞaCsqðX Þ and c(T, x)={y}. Suppose for a contradiction thaty∉ c(S, x). Then by Property β′, there is z∈ c(S, x) such that there is no T′⊆Swith x, y, z∈T′ and y, z∈ c(T′, x). Note that z≠ x, because otherwise by Propertyα, x∈ c(T, x). Now, consider the problem ({x, y, z}, x). By Property α, z∈ c({x, y,z}, x). Then, y∉ c({x, y, z}, x). Also, x∉ c({x, y, z}, x), because otherwise,by Property α, x∈ c({x, y}, x) and this implies by Property β′ that x∈ c(T, x).Thus, c({x, y, z}, x)={z}. By Property α, z∈ c({x, z}, x). This implies byProperty β′ that c({x, z}, x) ={z}, because otherwise x∈ c({x, y, z}, x). Thus,x∉ c(T′, x) for any T′⊆{x, y, z} with T ′≠{x}. Then, by strong SQ-irrelevance,c({x, y, z}, ⋄)=c({x, y, z}, x). Thus, c({x, y, z}, ⋄)={z}. But y∈c(S, ⋄) impliesby Property α that y∈c({x, y, z}, ⋄), a contradiction. □

Thus, the class of choice correspondences that satisfy Property α, Property β′, and strongSQ-irrelevance is the same as the class of choice correspondences that satisfy Property α,Property β′, dominance, and SQ-irrelevance. For simplicity, we use strong SQ-irrelevanceinstead of dominance and SQ-irrelevance.

3. Results

The following lemma discusses the implications of the properties introduced in Section 2.

Lemma 2. If the choice correspondence c on CðX Þ satisfies Propertyα, Property β′, status-quobias, and strong SQ-irrelevance, then there is a partial order ⪰ and a preorder ⪰′ such that ⪰′is an extension of ⪰ and

cðS;>Þ ¼ MðS;⪰ VÞ for all SaX ;

and

cðS; xÞ ¼ fxg if UdðS; xÞ ¼ t;MðUdðS; xÞ;⪰ VÞ otherwise

�

for all ðS; xÞaCsqðX Þ.Proof. Assume that c satisfies the given properties. For any SaX, x∈S and y∉S, let Sy,−x=(S⋃{y}) \ {x}. Let

PðcÞ :¼ fðx; yÞaX � X : x p y and cðfx; yg;>Þ ¼ fx; ygg;and let IðcÞ be the set of pairs of alternatives (x, y)∈X×X such that there is a finite set SaX withx∈S and y∉S and at least one of the following is true:

i) x∈c(S, ⋄) but y∉c(Sy,−x, ⋄),ii) x∉c(S, ⋄) but y∈c(Sy,−x, ⋄),iii) c(S, ⋄) \ {x}≠c(Sy,−x, ⋄) \ {y}.

Now, consider the binary relations ⪰, ≻′, and ∼′ defined on X by

x⪰y if and only if x∈c({x, y}, y),x≻′y if and only if c({x, y}, ⋄)={x} and x≠y,x∼′y if and only if ðx; yÞaPðcÞ⧹ IðcÞ or x=y.


Note that, ∼′ is symmetric. To see this, take any ðx; yÞaPðcÞ⧹ IðcÞ. Note that ðy; xÞaPðcÞ.Then we have to show that ðy; xÞgIðcÞ. Take any finite TaX with y∈T and x∉T. Let S=Tx,−y.Since ðx; yÞgIðcÞ, we have x∈c(S, ⋄) if and only if y∈c(Sy,−x, ⋄). That is y∈c(T, ⋄) if andonly if x∈c(Tx,−y, ⋄). Moreover, c(Tx,−y, ⋄) \ {x}=c(S, ⋄) \ {x}=c(Sy,−x, ⋄) \ {y}=c(T, ⋄) \ {y}.Then ðy; xÞgIðcÞ.

Also, note that ≻′ is asymmetric and disjoint from ∼′. Then, define⪰′:=≻′ [ ∼′. Thus,⪰′ isa binary relation on X with symmetric and asymmetric parts ∼′ and ≻′.

To show that ⪰′ is an extension of⪰, first let x, y∈X be such that x∼y. Then, x∈c({x, y}, y)and y∈c({x, y}, x). By status-quo bias, x∈c({x, y}, y) implies c({x, y}, x)={x}. Thus, x=yand by definition of ∼′, x∼′y. Now, suppose x≻y. Then, x∈c({x, y}, y) and y∉c({x, y}, x). Bystatus-quo bias, c({x, y}, x)={x} and c({x, y}, y)={x}. Thus, x≠y and by strong SQ-irrelevance,c({x, y}, ⋄)={x}. Thus x≻′y.

Now, we want to prove that for all SaX ; cðS;>Þ ¼ MðS;⪰ VÞ. First, let x∈S be such thatx∈c(S, ⋄). Suppose for a contradiction that xgMðS;v VÞ. Then, there is y∈S such that y≻′x. Then, c({x, y}, ⋄)={y}. On the other hand, since x∈c(S, ⋄), Property α implies x∈c({x,y}, ⋄). Thus, x=y contradicting y≻′x and so cðS;>ÞpMðS;⪰ VÞ.

Second, let xaMðS;⪰ VÞ and suppose for a contradiction that x∉c(S,⋄). Then, by Property β′,there is y∈S \{x} such that y∈c(S, ⋄) and for all T⊆S with x, y∈T, x∉c(T, ⋄). Then, c({x, y},⋄)={y}. Thus, y≻′x, contradicting xaMðS;⪰ VÞ and soMðS;⪰ VÞpcðS;>Þ.Claim 1. For any ðS; xÞaCsqðX Þ,

cðS; xÞp fxg if UdðS; xÞ ¼ t;UdðS; xÞ otherwise:

�

Proof of Claim 1. AssumeUdðS; xÞ ¼ t. Let y∈S \{x} and for a contradiction, suppose y∈c(S, x).By Property α and status-quo bias, c({x, y}, x)={y}. Thus, y≻x contradicting UdðS; xÞ ¼ t.Therefore, c(S, x)={x}.

Now, let UdðS; xÞ p t and first suppose x∈c(S, x). Thus, by Property α and status-quo bias,for all z∈S, c({x, z}, x)={x}. Then, there is no z∈S such that z≻x, contradicting UdðS; xÞ p t.Thus, x∉c(S, x). Then, let y∈S \ {x} be such that y∈c(S, x). By Property α and status-quo bias,c({x, y}, x)={y}. Thus, y≻x and so ya UdðS; xÞ. □

Claim 2. For any ðS; xÞaCsqðX Þ, if UdðS; xÞ p t; cðS; xÞ ¼ cðUdðS; xÞ;>Þ.Proof of Claim 2. We first show that cðS; xÞpcðUdðS; xÞ;>Þ. Let y∈c(S, x). By Claim 1,yaUdðS; xÞ. Thus, by Property α, y∈c(S, x) implies yacðUdðS; xÞ [ fxg; xÞ. Also by Claim 1,for any nonempty TpUdðS; xÞ; cðT [ fxg; xÞpUdðT [ fxg; xÞ. Thus, x∉c(T[{x}, x). Then,by strong SQ-irrelevance, yacðUdðS; xÞ [ fxg;>Þ. Then, by Property α, yacðUdðS; xÞ;>Þ.Thus, cðS; xÞpcðUdðS; xÞ;>Þ.

Now, we want to show cðUdðS; xÞ;>ÞpcðS; xÞ. Let yacðUdðS; xÞ;>Þ. Since (i) ⪰′ is anextension of⪰, and (ii) cðUdðS; xÞ [ fxg;>Þ ¼ MðUdðS; xÞ [ fxg;v VÞ, we have xgcðUdðS; xÞ[fxg;>Þ. Then, byProperty α, for any zacðUdðS; xÞ [ fxg;>Þ, we have y; zacðUdðS; xÞ;>Þ. Thus,by Property β′, yacðUdðS; xÞ [ fxg;>Þ. On the other hand, by Claim 1, for any TpUdðS; xÞ withT≠t, x∉c(T[{x}, x). Then, by strong SQ-irrelevance, yacðUdðS; xÞ [ fxg; xÞ. Since UdðS; xÞ pt, by Claim 1, cðS; xÞpUdðS; xÞ. Thus, x∉c(S, x). Then, byProperty α, for any z∈c(S, x), we havey; zacðUdðS; xÞ [ fxg; xÞ. Thus, by Property β′, y∈c(S, x). Thus, cðUdðS; xÞ;>ÞpcðS; xÞ and socðS; xÞ ¼ cðUdðS; xÞ;>Þ. □


Thus, by cðS;>Þ ¼ MðS;⪰ VÞ and by Claims 1 and 2, we prove that for all ðS; xÞaCsqðX Þ,

cðS; xÞ ¼ fxg if UdðS; xÞ ¼ t;MðUdðS; xÞ;⪰ VÞ otherwise:

�

The proofs of⪰ being a partial order and⪰′ being a preorder are identical to Masatlioglu andOk (2005, page 22) and Eliaz and Ok (2006, page 82), respectively. □

Note that, the agent in our model can be both indecisive and indifferent between two non-status-quo alternatives.

The following theorem shows that whenever X is finite, a choice correspondence, csatisfies our properties if and only if it is “rationalizable” by a pair of vector-valued utilityfunctions (one aggregating the other). Vector-valued utility functions exist also inMasatlioglu and Ok (2005) who interpret them as an evaluation of the alternatives on thebasis of various distinct criteria. The ith component of the vector-valued utility functionrepresents the agent's ranking of the alternatives with respect to the ith criterion. While inMasatlioglu and Ok (2005) the agent uses a real-valued function to aggregate these criteria(so that he has complete preferences on the alternatives), the agent in our model cannotalways do so.

Theorem. Let X be finite. A choice correspondence c on CðX Þ satisfies Property α, Property β′,strong SQ-irrelevance, and status-quo bias if and only if there are positive integers n, m suchthat n≥m, an injective function u : XYℝn, and a strictly increasing map f : uðX ÞYℝm suchthat for all SaX ,

cðS;>Þ ¼ MðS; f ðuÞÞ

and

cðS; xÞ ¼ fxg if UuðS; xÞ ¼ t;MðUuðS; xÞ; f ðuÞÞ otherwise

�

for all ðS; xÞaCsqðX Þ.Proof. It is straightforward to show that the described choice correspondence satisfies thegiven properties. Conversely, let c be a choice correspondence on CðX Þ. Assume that itsatisfies the given properties. Consider the partial order ⪰ and the preorder ⪰′ constructedin the Lemma.

Claim 1. There is a positive integer n and an injective function u : XYℝn such that for all x,y∈X,

y⪰x if and only if uðyÞzuðxÞ:

Proof of Claim 1. Let L(⪰) stand for the set of all linear orders⪰⁎ such that⪰⁎ is an extensionof ⪰. Since X is finite, L(⪰) is a nonempty and finite set. Then, enumerate L(⪰)= (⪰i)i=1

n andnote that ⪰=∩i=1

n ⪰i. Since for each i=1, …, n, ⪰i is a linear order on a finite set X, there exists afunction ui : XYℝ such that

x⪰i y if and only if uiðxÞzuiðyÞ:


Let u=(u1, …, un). Then, for all x, y∈X,

x⪰y if and only if uðxÞzuðyÞ:Since ⪰ is antisymmetric, u must be injective. □

Claim 2. There is a positive integer m with m≤n and a function u V : XYℝm such that for all x,y∈X,

y⪰ V x if and only if u VðyÞzu VðxÞ:

Proof of Claim 2. We can show the existence ofm and u V: XYℝm by using the same argument asin Claim 1. Since ⪰′ is an extension of ⪰, L(⪰)=(⪰i)i=1

n , and L(⪰′)=(⪰i′)i=1m , we have m≤n. □

To complete the proof, we define f : uðX ÞYℝm by f (u(x)) :=u′(x). Since u is injective, f iswell-defined. Moreover, if u(x)Nu(y) for some x, y∈X, by Claim 1, x≻y. Then, by the lemmaand Claim 2, x≻′y and f (u(x))=u′(x)Nu′(y)= f (u(y)). Thus, f is strictly increasing.

Thus, by Claims 1 and 2 and by the Lemma,

cðS;>Þ ¼ MðS; f ðu VÞÞ

and

cðS; xÞ ¼ fxg if UuðS; xÞ ¼ t;MðUuðS; xÞ; f ðu VÞÞ otherwise:

□�

Note that if the agent's choice behaviour satisfies our properties, then similar to Masatliogluand Ok (2005) the status-quo alternative affects the agent's choice in the following ways: (i) iteliminates the alternatives that do not give higher utility in all evaluation criteria, (ii) it becomesthe unique choice if all alternatives are eliminated, (iii) it affects the agent's choices even if it isnot chosen itself (please see Masatlioglu and Ok, 2005 for an example).

4. Independence of unique choice from the status-quo

In this section, we analyze the conditions under which the two preference relations can bereplaced with a single one. There are agents whose choice behaviour can satisfy all of ourproperties and yet cannot be rationalized by a single preference relation. In fact, if a choicebehaviour can be rationalized by a single incomplete preference relation, it then has to satisfy aproperty that we call independence of unique choice from the status-quo. To understand thisproperty , suppose x is the unique choice when there is no status-quo in the problem. Now,consider the effect of a non-status-quo alternative, y being the status-quo. Independence of uniquechoice from the status-quo then requires that x should be also chosen from the latter problem. Thatis if x is revealed to be superior to any alternative in the feasible set, making y the status-quo doesnot cause it to be revealed superior to x.

Independence of unique choice from the status-quo: For all SaX and x, y∈S such thatx≠y, if c(S, ⋄)={x}, then x∈c(S, y).

This property restricts the power of the status-quo bias significantly. Since x is chosenuniquely, it is strictly preferred to y when there is no status-quo. Then, independence of uniquechoice from the status-quo requires that y being a status-quo alternative does not create a“too”strong status-quo bias towards itself, i.e. y cannot be revealed strictly preferred to x. This


contradicts with one of the well-known experimental observations, “the preference reversalphenomenon as an endowment effect” (Slovic and Lichtenstein, 1968).

Unfortunately, it turns out that independence of unique choice from the status-quo is bothnecessary and sufficient for a choice behaviour to be rationalized by a single incomplete pre-ference relation.

Corollary 1 (to Lemma 2). If the choice correspondence c on CðX Þ satisfies Property α,Property β′, strong SQ-irrelevance, status-quo bias, and independence of unique choice fromthe status-quo, then there is a partial order ⪰ such that

cðS;>Þ ¼ MðS;⪰Þ for all SaX

and

cðS; xÞ ¼ fxg if UdðS; xÞ ¼ t;MðUdðS; xÞ;⪰Þ otherwise

�

for all ðS; xÞaCsqðX Þ.Proof. Suppose that the choice correspondence c satisfies the given properties. Then, by Lemma2, there is a partial order ⪰ and a preorder ⪰′ such that ⪰′ is an extension of ⪰ and

cðS;>Þ ¼ MðS;⪰ VÞ for all SaX ;

and

cðS; xÞ ¼ fxg if UdðS; xÞ ¼ t;MðUdðS; xÞ;⪰ VÞ otherwise

�

for all ðS; xÞaCsqðX Þ.Let ⪰ and ⪰′ be defined as in the proof of Lemma 2. It is sufficient to show that for any

SaX ;MðS;⪰ VÞ ¼ MðS;⪰Þ. For this, first let x∈S be such that xaMðS;⪰ VÞ and suppose for acontradiction that xgMðS;⪰Þ. Then, there is y∈S such that y≻x. Since⪰′ is an extension of⪰,y≻′x, contradicting xaMðS;⪰ VÞ. Second, let xaMðS;⪰Þ and suppose for a contradiction thatxgMðS;⪰ VÞ. Then, there is y∈S such that y≻′x. Thus, c({x, y}, ⋄)={y}. Then, by indepen-dence of unique choice from the status-quo, y∈c({x, y}, x) and by status-quo bias, c({x, y}, x)={y}. Thus, y≻x, contradicting xaMðS;⪰Þ. Thus, we have the desired conclusion. □

The implications of the independence of unique choice from the status-quo on the repre-sentation of the revealed preferences in Theorem are as follows:

Corollary 2 (to the Theorem). Let X be a nonempty finite set. A choice correspondence c onCðX Þ satisfies Property α, Property β′, strong SQ-irrelevance, status-quo bias, and indepen-dence of unique choice from the status-quo if and only if there is a positive integer n and afunction u : XYℝn such that for all SaX ,

cðS;>Þ ¼ MðS; uÞand

cðS; xÞ ¼ fxg if UuðS; xÞ ¼ t;MðUuðS; xÞ; uÞ otherwise

�

for all ðS; xÞaCsqðX Þ. □


Proof. It is straightforward to show that the choice correspondence satisfies the given properties.Conversely, let c satisfy the given properties. Consider the partial order ⪰ constructed in theLemma.

Claim. There exist a positive integer n and an injective function u : XYℝn such that for all x,y∈X,

y⪰x if and only if uðyÞzuðxÞ:

Proof of Claim. The proof is the same as the proof of Claim 1 in Theorem.Thus, by the Claim and Corollary 1,

cðS;>Þ ¼ MðS; uÞ

and

cðS; xÞ ¼ fxg if UuðS; xÞ ¼ t;MðUuðS; xÞ; uÞ otherwise

�

for all ðS; xÞaCsqðX Þ. □

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revealed incomplete preferences under status-quo bias

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