aa r X i v : . [ ec on . T H ] O c t A Model of Choice with Minimal Compromise ∗ Mario V´azquez Corte † This version: July 2016
Abstract
I formulate and characterize the following two-stage choice behavior. The deci-sion maker is endowed with two preferences. She shortlists all maximal alternativesaccording to the first preference. If the first preference is decisive, in the sense thatit shortlists a unique alternative, then that alternative is the choice. If multiplealternatives are shortlisted, then, in a second stage, the second preference vetoesits minimal alternative in the shortlist, and the remaining members of the shortlistform the choice set. Only the final choice set is observable. I assume that the firstpreference is a weak order and the second is a linear order. Hence the shortlist isfully rationalizable but one of its members can drop out in the second stage, leadingto bounded rational behavior. Given the asymmetric roles played by the underlyingbinary relations, the consequent behavior exhibits a minimal compromise betweentwo preferences. To our knowledge it is the first Choice function that satisfies Sen’s β axiom of choice, but not α . J.E.L. codes:
D0.
Keywords:
Bounded Rationality, Multiple Preferences, Two-Stage Choice, Short-listing, Altruism. ∗ This work draws from my work under the supervision of Levent ¨Ulk¨u, who erroneously appeared asa coauthor in a previous draft. † Department of Economics, ITAM Introduction
The standard model of rational choice centers around a decision maker (DM) who max-imizes a given preference in every menu. Experimental evidence and field data containrobust deviations from this model. The accumulation of such evidence has created aninterest in developing new models of bounded rationality which rely on a richer set ofpsychological variables.A particularly prominent idea which has been explored in this literature is that theDM might use multiple preferences in making choices. In the presence of multiple pref-erences, any conflict which may arise between preferences need to be resolved beforemaking choices. Recent work has studied various ways of resolving such conflicts, mainlyby explicitly attributing different roles to different preferences. In Manzini and Mariotti(2007) and Bajraj and ¨Ulk¨u (2015), for example, the DM uses one preference to identifya shortlist of viable alternatives and a second preference to choose from the shortlist .In this work I will study a model which, similarly, features a compromise between twopreferences. In my model, the DM will choose to maximize a preference, with the provisothat, in case multiple alternatives are maximal, a second preference will be able to veto analternative. To be precise, the model works as follows. The DM is endowed with a weakorder (a utility) and a linear order (a utility where no distinct alternatives are indifferent.)The DM first shortlists all best alternatives in the weak order. If a unique alternative isshortlisted, it is chosen. Otherwise, in a second stage, she eliminates from the shortlistthe worst alternative according to the linear order. The remaining alternatives form thechoice set.In view of the very asymmetric roles played by the two underlying preferences, thismodel features an idea of a minimal compromise. Note that the linear order plays no roleif the weak order is decisive in the first round. Only if the weak order shortlists exactlytwo candidates, does the linear order make the choice in the second round. If more thantwo are shortlisted, the linear order can only veto one of them. Hence the departure fromthe maximization of the first stage preference as a result of a conflict with the second Of course, conflicts between criteria need not be resolved and the DM may choose any alternativewhich is the best according to some preference. The resulting behavior is characterized by the famousPath Independence axiom. See, for instance, Moulin (1985). This is further discussed in Horan (2016), and Garc´ıa-Sanz and Alcatud (2015). Both analyze thetwo-stage procedure inspired by Manzini and Mariotti (2007). α axiom, however itnecessarily satisfies various other rationality axioms such as γ , β and No Binary Cycles(NBC). My main result is a characterization of this model using five novel conditions.My model is largely inspired by Manzini and Mariotti (2007). They study a two-stagechoice procedure which depends on two asymmetric binary relations. In the first stage,the DM forms a shortlist consisting of all maximal alternatives according to the first bi-nary relation. In the second stage she chooses from the shortlist using the second binaryrelation. They show that this two-stage procedure explains cyclical behavior whereby x is chosen over y , y over z and z over x . The main difference between the present workand Manzini and Mariotti (2007) is that I study a choice correspondence, while they char-acterize a choice function, which is a restrictive form of a choice correspondence whichselects a unique alternative in every menu. Furthermore, in my model the role of thesecond binary relation is different. Instead of choosing its best-preferred alternative fromthe shortlist, it vetoes the choice of its least preferred shortlisted alternative. I shouldnote that, as does the related two-stage model of Bajraj and ¨Ulk¨u (2015), my model failsto account for cyclical behavior. Instead, my model can explain violations of α , wherebyan alternative drops out of the choice set in a smaller menu which contains it. The Choice with Minimal Compromise can be interpreted as a two different agents choos-ing over a menu. Imagine you are in a restaurant with your significant other and decide toorder pizza. You read the menu and enumerate your favorite options: Hawaiian, supreme,veggie and 4 cheeses. Your significant other then says he doesn’t want to eat supreme.So you compromise and decide to choose a pizza from the remaining three alternatives.Formally, the firs agent is represented by the preference relation R shortlist her bestalternatives. Then asks the second agent, to take out his least favorite option from theshortlist encoded by the preference relation L . then, she proceeds to choose any itemfrom the remaining shortlist. 3 Model
I consider a standard choice environment. Let X be a finite set of alternatives. A binaryrelation R on X is (1) complete if for all x, y ∈ X , either xRy or yRx , (2) transitive if forall x, y, z ∈ X , if xRy and yRz , then xRz , and (3) antisymmetric if for all x, y ∈ X , if xRy and yRx , then x = y . A weak order is a complete and transitive binary relation. A linear order is an antisymmetric weak order. We will typically refer to weak orders by R and linear orders by L . For any weak order R , I will denote by I and P the symmetricand asymmetric parts of R , respectively: xIy ⇔ xRy and yRx ; and xP y ⇔ xRy and ¬ ( yRx ).A menu is any nonempty subset of X . 2 X = { A ⊆ X : A = ∅ } denotes the set ofmenus. If R is a weak order, let max( A, R ) denote the set of maximal alternatives in A according to R , in other words, max( A, R ) = { x ∈ A : xRy for all y ∈ A } . If L isa linear order, then max( A, L ) is a singleton, as is min(
A, L ) = { x ∈ A : yLx for all y ∈ A } . In this case, I will refer to the unique alternative in max( A, L ) (resp. min(
A, L ))by max(
A, L ) (resp. min(
A, L )) as well.A choice correspondence is a map c : 2 X → X satisfying c ( A ) ⊆ A for every menu A .If c ( A ) = { x } for some menu A and some alternative x ∈ A , I will say that c is decisive at A . A choice correspondence c is rational if there exists a weak order R such that c ( A ) = max( A, R ) for every menu A . A choice correspondence c satisfies the weak axiomof revealed preference (WARP) if for every x, y, A and B , if x, y ∈ A ∩ B , x ∈ c ( A ) and y ∈ c ( B ), then x ∈ c ( B ) as well. It is well known that WARP is a necessary and sufficientcondition for the rationality of choice correspondences. (See for instance Moulin, 1985.)I can now define the class of choice correspondences of interest in this work. Definition 1
A choice correspondence c admits a minimal compromise representation ifthere exist a weak order R and a linear order L such that for every menu Ac ( A ) = ( max( A, R ) if max(
A, R ) is a singleton,max(
A, R ) \ min(max( A, R ) , L ) otherwise.If this is the case, I will call c an MC choice correspondence.Hence a choice correspondence c with a minimal compromise representation operates4n two stages. In the first stage maximal alternatives according to R are shortlisted. Inthe second stage the choice is made from the shortlist using L as follows. If the shortlistcontains only one alternative, then it is the choice, and L has no role to play. If the shortlistcontains multiple alternatives, however, the alternative which is the worst according to L is eliminated. All remaining alternatives form the choice set. Hence the compromisebetween R and L is minimal, in the sense that L can veto only one alternative from theshortlist, if the shortlist contains two or more alternatives.MC choice correspondences can usefully explain failures of WARP. Consider the fol-lowing two axioms: α : If x ∈ c ( A ) and x ∈ B ⊂ A , then x ∈ c ( B ). β : If x, y ∈ A ⊂ B, x, y ∈ c ( A ) , and y ∈ c ( B ) , then x ∈ c ( B ) as well. Fact 1: (Moulin, 1985) A choice correspondence satisfies WARP if and only if it satisfies α and β .My first result indicates that MC choice correspondences can fail α , but they have tosatisfy β . Lemma 1
Let c admit a minimal compromise representation. Then c satisfies β . How-ever c may fail α . Proof.
Let c admit a minimal compromise representation. Fix x, y ∈ A, B ⊆ X . Suppose x, y ∈ c ( A ), A ⊂ B and y ∈ c ( B ) . I need to show that x ∈ c ( B ) . If x = y the conditionis automatically satisfied since y ∈ c ( B ) . Suppose x = y. Since y ∈ c ( B ), y ∈ max( B, R )and therefore yRb for all b ∈ B. The fact that x ∈ c ( A ) implies that x ∈ max( A, R ),so xRy since y belongs to the set A. Now transitivity of R gives xRb for all b ∈ B, and therefore x ∈ max( B, R ) . Notice that max(
B, R ) can not be a singleton since x and y are different, therefore c ( B ) = max( B, R ) \ min(max( B, R ) , L ) . I will now show that x = min(max( B, R ) , L ) . By hypothesis x, y ∈ c ( A ) so max( A, R ) is not a singleton, recall x = y, which implies c ( A ) = max( A, R ) \ min(max( A, R ) , L ). Hence there exists some z distinct from x and y , such that z = min(max( A, R ) , L ). Hence xLz. Since z ∈ max( A, R ), zRy . Since y belongs to the set A, then, by transitivity of R, zRb for all b ∈ B which5hows that z ∈ max( B, R ). Hence x can not be the L -worst alternative in the shortlistmax( B, R ).The following example shows that c may fail α . Let c admit a minimal compromiserepresentation. Additionally, let X = { x, y, z } , with R ranking all three alternativesindifferent and xLyLz. The consequent MC choices are given by the following table: A { x, y } { x, z } { y, z } { x, y, z } c ( A ) { x } { x } { y } { x, y } Note y is chosen from the menu { x, y, z } , but not from the menu { x, y } where it belongs,leading to a failure of α. This happens because the vetoed alternative is different betweenthe two menus: y is the L -worst alternative in the shortlist { x, y } = max( { x, y } , R ) , but z is the L -worst alternative in the shortlist { x, y, z } = max( { x, y, z } , R ) . There is another sense in which the deviation of MC choice correspondences from fullrationality comes in the form of α failures. Consider the following two axioms: γ : If x ∈ c ( A ) ∩ c ( B ), then x ∈ c ( A ∪ B ).No binary cycles (NBC): If x ∈ c ( { x, y } ) and y ∈ c ( { y, z } ), then x ∈ c ( { x, z } ). Fact 2 : (Moulin, 1985) A choice correspondence satisfies WARP if and only if it satisfies α, γ and NBC.My next result indicates that MC choice correspondences satisfy γ and NBC. Hencein view of the particular characterization given in Fact 2, MC choice correspondences failto be rational only because they may fail α . Lemma 2
Let c admit a minimal compromise representation. Then c satisfies γ andNBC. Proof.
Suppose c admits a minimal compromise representation. I will first show that c satisfies γ . Let x ∈ c ( A ) ∩ c ( B ). Then it must be the case that x ∈ max( A, R ) and x ∈ max( B, R ) , which means x ∈ max( A ∪ B, R ) . There are two cases: x is the only R -maximal element in both menus or there exists an other R -maximal element, distinctfrom x, in at least one menu. In the first case, { x } = max( A ∪ B, R ) and c ( A ∪ B ) =6ax( A ∪ B, R ) = { x } and x is chosen in A ∪ B as desired. In the second case, there exists y = x in, say, menu A such that y ∈ max( A, R ). Since yIx , y ∈ max( A ∪ B, R ) as well.Since x, y ∈ max( A, R ) it must be the case that c ( A ) = max( A, R ) \ min(max( A, R ) , L ) . Notice that there exists z ∈ max( A, R ) such that xLz, since x ∈ c ( A ) and therefore x = min(max( A, R ) , L ) . This last statement means zRx hence z ∈ max( A ∪ B, R ) and x = min(max( A ∪ B, R ) , L ) and x ∈ c ( A ∪ B ), as desired.To show that c satisfies NBC, let x ∈ c ( { x, y } ) and y ∈ c ( { y, z } ) . I have to show that x ∈ c ( { x, z } ) . Since c has a minimal compromise representation it must be the case that x ∈ max( { x, y } , R ) and y ∈ max( { y, z } , R ) , so xRy and yRz. Then, by transitivity of
R,xRz which means x ∈ max( { x, z } , R ) . If ¬ ( zRx ) then c ( { x, z } ) = max( { x, z } , R ) = { x } . If zRx . then zRy and yRx since R is transitive. Consequently max( { x, y } , R ) = { x, y } , max( { y, z } , R ) = { y, z } and max( { x, z } , R ) = { x, z } , hence L vetoes an alternative in allthree doubleton menus. Recall x ∈ c ( { x, y } ) and y ∈ c ( { y, z } ) , then xLy and yLz , whichimplies xLz by transitivity of L . Hence { x } = c ( { x, z } ) once again. This finishes theproof. Suppose that c admits a minimal compromise representation with the underlying prefer-ences R and L . How does the resulting behavior reveal R and L ? The following exampleshows that there may be more than one way to rationalize observed choices by this model. Example 1
Consider the following choice correspondence: A { x, y } { x, z } { y, z } { x, y, z } c ( A ) { y } { x } { y } { y } Note that this choice correspondence is decisive in every menu. Furthermore it is ratio-nalizable in the standard sense. One way to see that this behavior admits a minimalcompromise representation is to let L be any linear order and take R as follows: yP xP z ,where P is the strict part of R . Alternatively, if I take xIyP z for R , and zLyLx , theresulting choices are identical. N Yet, through violations of α , a MC choice correspondence can quite tractably revealat least parts of the underlying weak order R and the linear order L . First, note that any7hosen alternative must be at least as good as any other feasible alternative in R . Hence xRy if x ∈ c ( A ) and y ∈ A in some menu A . This mirrors the revelation of the underlyingpreference in a standard rationality framework. However in my model, if x ∈ c ( A ) and y ∈ A \ c ( A ), this does not mean that x is strictly preferred to y . This is because x and y are perhaps indifferent, but y has been vetoed by the linear order L . However, any such y can easily be detected as their removal would impact behavior. Indeed, if y has beenvetoed, then its removal from A may lead to the veto of the next alternative in L , meaning c ( A \ y ) = c ( A ) even though y c ( A ). If this happens, then I can also conclude that yRa for all a ∈ A . Furthermore I can also conclude that xLy for all x ∈ c ( A ). Howeverthis revelation of L is not complete. Suppose that R shortlists only two alternatives x and y but y is vetoed by L . In this case the removal of y has no impact on behavior as c ( A ) = { x } since y is vetoed, but c ( A \ y ) = { x } as well, as only x has been shortlisted,stripping L of its veto power.The characterization exercise of the next section contains my main result, Theorem 1,where these revelations play the key role. In it, I define the binary relation R as follows: xRy iff y belongs to a menu where the removal of x affects behavior. I also define a binaryrelation L as follows: xLy iff c is decisive in the menu { x, y } in favor of x . I show that,under certain conditions which I will specify, R is complete and transitive, i.e., a weakorder. I also show that, again under conditions, L can be completed to a linear order.Furthermore, any c satisfying the conditions I will identify behaves identically to a MCchoice correspondence defined by R and L . In this section I will characterize the class of MC choice correspondences. Let me beginwith some notation which will help in the statement of two of my conditions. For anychoice correspondence c and any menu A , let r c ( A ) = { x ∈ A : c ( A \ x ) = c ( A ) } . In words, r c ( A ) collects all members of A whose removal impacts behavior. Note r c isa choice correspondence itself: r c ( A ) ⊆ A and, since c ( A ) ⊆ r c ( A ), r c ( A ) is nonempty.Clearly, the removal of any chosen element will impact behavior. However the removal of8lternatives which are not chosen may also impact behavior. The following observationindicates that this never happens if c is rational. Lemma 3
If a choice correspondence c is rational, then r c ( A ) = c ( A ) for every menu A . Proof.
Take a choice correspondence c . Suppose c is rational and let R be the weakorder which c maximizes, i.e., c ( A ) = max( A, R ) for all A . I will show that r c = c . Bydefinition, c ( A ) ⊆ r c ( A ). Take any x ∈ A \ c ( A ). If a ∈ max( A, R ), then a = x , a ∈ A \ x and a ∈ max( A \ x, R ) as well. Hence c ( A ) ⊆ c ( A \ x ). If a max( A, R ) and a = x , onthe other hand, then there exists a ′ = x such that a ′ P a and a max( A \ x, R ), giving c ( A \ x ) ⊆ c ( A ). I conclude that a r c ( A ) and c ( A ) = r c ( A ).The next example show that the statement in the preceding Lemma can not be re-versed, i.e., r c = c is not sufficient for the rationality of c . Example 2
Consider the following choice correspondence on X = { x, y, z } . A { x, y } { x, z } { y, z } { x, y, z } c ( A ) { x, y } { z } { y } { x, y } Note r c = c but c is not rational as it fails α : x ∈ c ( { x, y, z } ) but x c ( { x, z } ). N I am now ready to state the characterizing conditions for MC choice correspondences.
Condition 1. If x, y ∈ A ⊂ B, x ∈ c ( A ) , and y ∈ c ( B ) , then x ∈ c ( B ) . This condition says that if an alternative y is chosen in a menu, and x is chosen in asmaller menu where y is present, then x must also be chosen in the larger menu. Thiscondition strengthens β by weakening its ”if-part” as it does not insist that y should bechosen in the smaller menu A for the conclusion to follow. The following example showsthat the strengthening is strict, as there are choice correspondences which satisfy β butfail Condition 1. 9 xample 3 The following choice correspondence satisfies β but fails Condition 1. A { x, y } { x, z } { y, z } { x, y, z } c ( A ) { x } { x } { y } { y } Note β holds vacuously as c is decisive in every menu. However Condition 1 fails: x, y ∈{ x, y } ⊂ { x, y, z } , x ∈ c ( { x, y } ) , and y ∈ c ( { x, y, z } ) , but x c ( { x, y, z } ) . N The next condition says that Condition 1 should hold for the map r c associated withthe choice correspondence c . Condition 2.
For any c , r c satisfies Condition 1: If x, y ∈ A ⊂ B, x ∈ r c ( A ) , and y ∈ r c ( B ) , then x ∈ r c ( B ) . Hence if the removal of y changes behavior in menu B , and the removal of x changesbehavior in a smaller menu A where y belongs, then the removal of x should changebehavior in menu B as well.The next condition says that c should not choose every feasible alternative, except ofcourse in singletons. Condition 3.
For any nonsingleton menu A , there exists x ∈ A such that x c ( A ).Note that Condition 3 implies, in particular, that c should be decisive in doubletonmenus. Furthermore, I have the following result. Lemma 4 If c satisfies Conditions 1 and 3, then it must satisfy NBC as well. Proof.
Suppose c satisfies Conditions 1 and 3, x ∈ c ( { x, y } ) and y ∈ c ( { y, z } ) but { z } = c ( { x, z } ). Note, by Condition 3, this means { x } = c ( { x, y } ) and { y } = c ( { x, y } ).Consider the menu { x, y, z } . Since all alternatives in { x, y, z } have been chosen in asmaller menu, Condition 1 implies that if one of them belongs to c ( { x, y, z } ), then all doso. This violates Condition 3.Next consider the following condition. 10 ondition 4. If x ∈ A \ c ( A ) and x ∈ c ( A ∪ { y } ) , then y / ∈ c ( A ∪ { y } ) . Imagine adding a new alternative y to a menu A . Condition 4 says that this can notlead to the inclusion of both y and a previously unchosen alternative x in the choice set.If x jumps in the choice set as a result of the inclusion of y , then y should not belong tothe choice set.Finally, my last condition is as follows. Condition 5.
For all A , for all nonsingleton B ⊆ r c ( A ) there exists some x c ( B ) suchthat B = c ( B ) ∪ { x } .Imagine choice in a menu of alternatives all of which impact choice in a larger menu.Condition 5 says that all but one of these alternatives must belong to the choice set.I am now ready to state the main result. Theorem 1
A choice correspondence admits a minimal compromise representation if andonly if it satisfies Conditions 1-5.
Proof.
To begin, suppose that c admits a minimal compromise representation and let R and L be the underlying weak and linear orders.To see that c satisfies Condition 1, fix x, y ∈ A ⊂ B such that x ∈ c ( A ) and y ∈ c ( B ).I need to show that x ∈ c ( B ) . There is nothing to show if x = y , so suppose x = y . Bydefinition x ∈ max( A, R ) and y ∈ max( B, R ), hence x ∈ max( B, R ) and y ∈ max( A, R )as well. Since max(
A, R ) is not a singleton, there exists some z ∈ max( A, R ) \ c ( A ) suchthat xLz . Note z ∈ max( B, R ) as well, so x = min( { max( B, R ) } , L ), giving x ∈ c ( B ).Next take x, y ∈ A ⊂ B such that x ∈ r c ( A ) and y ∈ r c ( B ). To establish Condition 2,I need to show that x ∈ r c ( B ) . There is nothing to show if x = y or if x ∈ c ( B ). Suppose x = y and x c ( B ). Note that minimal compromise representation implies r c ( S ) ⊆ max( S, R ) for any menu S . Hence x ∈ max( A, R ), y ∈ max( B, R ) and consequentlymax(
A, R ) ⊆ max( B, R ) and x ∈ max( B, R ) as well. This means x is vetoed by L in B . Ineed to show that c ( B ) contains at least two alternatives so that x ∈ r c ( B ). Suppose x ∈ c ( A ). Note y ∈ max( A, R ) as well, so max(
A, R ) contains at least two alternatives. Hencean alternative must be vetoed in A , say some a = x . Then a ∈ max( A, R ) ⊆ max( B, R )and xLa , hence x cannot vetoed in B , a contradiction. If x c ( A ), on the other hand,then x is vetoed in A . However x ∈ r c ( A ), meaning c ( A ) contains at least two distinct11lternatives a , a ∈ max( A, R ) ⊆ max( B, R ) such that a i Lx . Then a , a ∈ c ( B ), and theremoval of x from B will result in a change in behavior, as I needed to show. I concludethat x ∈ r c ( B ).To show that c satisfies Condition 3, take a menu A which is not a singleton andsuppose c ( A ) = A . Since c ( A ) ⊂ max( A, R ) whenever c ( A ) contains multiple alternatives, A = c ( A ) ⊂ max( A, R ) , and impossibility.Next, to show c satisfies Condition 4, take x, y and A such that x ∈ A \ c ( A ) and x ∈ c ( A ∪ { y } ) . I have to show that y / ∈ c ( A ∪ { y } ) . By definition of r c , x, y ∈ r c ( A ∪ { y } ),which means x, y ∈ max( A ∪ { y } , R ) as well. Hence x ∈ max( A, R ) but for all a ∈ c ( A ), aLx . Now take any a ∈ max( A ∪ { y } , R ) \{ y } . Note such a ∈ max( A, R ) as well. If yLa , then yLx also and consequently x is the L -worst alternative in max( A ∪ { y } , R ),meaning x c ( A ∪ { y } ), a contradiction. Hence aLy for all a ∈ max( A ∪ { y } , R ) \{ y } and y c ( A ∪ { y } ).Finally, to see that c satisfies Condition 5 take nonsingleton menus A and B such that c ( A ) = c ( A \ x ) for all x ∈ B . Then B ⊆ max( A, R ). Consequently B = max( B, R ). Let x = min( B, L ) so that c ( B ) = B \{ x } , and Condition 5 follows.In the reverse direction, suppose c satisfies Conditions 1-5. Define xRy iff there existssome menu A such that x ∈ r c ( A ) and y ∈ A . Also define xLy iff { x } = c ( { x, y } ). I willnow show that R is a weak order and L is a linear order.Completeness of R follows from the definition of a choice correspondence. Take any x, y ∈ X . If x = y , then xRx since { x } = c ( { x } ). Otherwise since c ( { x, y } ) = ∅ , xRy or yRx (or both). To see that R is transitive, suppose xRy and yRz . Then there exist menus A and B such that x ∈ r c ( A ) and y ∈ r c ( B ), y ∈ A and z ∈ B. Take any w ∈ r c ( A ∪ B ).If w ∈ A , then x ∈ r c ( A ∪ B ) by Condition 2. If w ∈ B , then y ∈ r c ( A ∪ B ) and therefore x ∈ r c ( A ∪ B ), again, by Condition 2. Since z ∈ A ∪ B , xRz as desired. This proves that R is a weak order.To see that L is complete, take any x, y ∈ X . If x = y , then xLx as { x } = c ( { x } ).Otherwise Condition 3 implies { x } = c ( { x, y } ) or { y } = c ( { x, y } ). Hence xLy or yLx and L is complete. Furthermore I can not have xLy and yLx for distinct x and y , hence L isasymmetric. The transitivity of L follows from Lemma 4 and Condition 3 as follows. If xLy , yLz , then Lemma 4 implies x ∈ c ( { x, z } ) and Condition 3 implies { x } = c ( { x, z } ).Hence xLz and L is transitive. This proves that L is a linear order.Now let c R,L be the MC choice correspondence defined by R and L . I will show that c =12 R,L . First suppose that x ∈ c ( A ). I need to show that x ∈ c R,L ( A ). Since c ( A ) ⊆ r c ( A ), x ∈ r c ( A ). By definition of R , then, x ∈ max( A, R ). There are two cases to consider.If max(
A, R ) = { x } , then max( A, R ) = { x } = c R,L ( A ) by definition and x ∈ c R,L ( A ) asdesired. Suppose now that max( A, R ) contains multiple alternatives. I need to show that x is not the L -worst alternative in max( A, R ). In other words, I need to find an alternative a ∈ max( A, R ) \{ x } such that c ( { a, x } ) = { x } . Let y ∈ max( A, R ) \ x . By Condition 3, c ( { x, y } ) is a singleton. If c ( { x, y } ) = { x } , then xLy and I am done. Suppose that c ( { x, y } ) = { y } . Since x ∈ c ( A ) and { x, y } ⊂ A , as I keep adding alternatives to menu { x, y } to reach menu A , x must jump in the choice set at some point. In other words, theremust exist a menu D and an alternative z ∈ A \ D such that { x, y } ⊆ D ⊆ A , x c ( D )and x ∈ c ( D ∪ { z } ). (If no such D and z exist, then x c ( A ).) Note that z ∈ r c ( D ∪ { z } )and therefore zRx . Hence z ∈ max( A, R ) as well. However z c ( D ∪ { z } ) by Condition4. Now consider the menu { x, z } . If { z } = c ( { x, z } ), then Condition 1 dictates that z ∈ c ( D ∪ { z } ), a contradiction. Then, by Condition 3, c ( { x, z } ) = { x } and xLz . Thisproves that x is not the L -worst in max( A, R ). I conclude that x ∈ c R,L ( A ).To finish, take x ∈ c R,L ( A ). I need to show that x ∈ c ( A ). By definition x ∈ max( A, R ). If max(
A, R ) = { x } , there exists no y ∈ A \{ x } such that yRx . This impliesthat c ( A ) can only contain x . Since c ( A ) is nonempty, it has to contain x , as desired. Nowsuppose max( A, R ) = { x } , i.e., that there exists some z = x such that { x, z } ⊆ max( A, R ).Since x ∈ c R,L ( A ) and max( A, R ) contains multiple alternatives, x is not the L -worstalternative in max( A, R ). Hence I can take z such that xLz , i.e., c ( { x, z } ) = { x } .Towards a contradiction suppose x c ( A ) and pick y ∈ c ( A ). If x ∈ c ( { x, y } ), thenby Condition 1, x ∈ c ( A ) as well, a contradiction. By Condition 3, then { y } = c ( { x, y } ).Since zRx and zRy , there exist menus B x and B y such that x ∈ B x , y ∈ B y and z ∈ r c ( B x ) ∩ r c ( B y ). By Condition 2, then z ∈ r c ( B x ∪ B y ). Since c ( { x, z } ) = { x } , x ∈ r c ( { x, z } ) and Condition 2 implies x ∈ r c ( B x ∪ B y ). Similarly, since c ( { x, y } ) = { y } , y ∈ r c ( B x ∪ B y ).Now I will use Condition 5. Consider the menu { x, y, z } ⊆ r c ( B x ∪ B y ). Conditions3 and 5 imply that c ( { x, y, z } ) contains exactly two alternatives. If c ( { x, y, z } ) = { y, z } ,then Condition 1 fails since c ( { x, z } ) = { x } . Similarly if c ( { x, y, z } ) = { x, z } , as { y } = c ( { x, y } ). Hence c ( { x, y, z } ) = { x, y } . Now Condition 1 implies x ∈ c ( A ), as y ∈ c ( A ).This contradiction finishes the proof. 13 Conclusion