Welfare criteria from choice: An axiomatic analysis

Games and Economic Behavior 99 (2016) 56–70

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Welfare criteria from choice: An axiomatic analysis ✩ Sean Horan, Yves Sprumont ∗ Département de Sciences Économiques and CIREQ, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal QC, H3C 3J7, Canada

a r t i c l e

i n f o

Article history: Received 9 January 2016 Available online 25 July 2016 JEL classification: D01 Keywords: Choice-based welfare analysis Bounded rationality

a b s t r a c t We propose an axiomatic approach to the problem of deriving a (linear) welfare ordering from a choice function. Admissibility requires that the ordering assigned to a rational choice function is the one that rationalizes it. Neutrality states that the solution covaries with permutations of the alternatives. Persistence stipulates that the ordering assigned to two choice functions is also assigned to every choice function in between. We prove that these properties characterize the sequential solution: the best alternative is the alternative chosen from the universal set; the second best is the one chosen when the best alternative is removed; and so on. We also discuss some alternative axioms and solutions. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The point of choice-based welfare analysis is to use information about choice behavior to draw inferences about welfare. Given an agent’s behavioral type, which captures all observations about her choices from subsets of a universal set X , the problem is to determine a (potentially incomplete) welfare ranking of the alternatives in X . When an agent’s behavior is fully rational, the standard answer is to adopt a revealed preference approach: the welfare ranking is simply the preference that is revealed to be maximized by the agent’s choices. (To simplify the subsequent exposition, we sometimes equate a “behavioral type” with an “agent.”) In this paper, we extend choice-based welfare analysis to the general setting where agents may fail to be fully rational. In this setting, there is no real consensus about how choice behavior relates to preference. Instead, there is a patchwork of conflicting “bounded rationality” theories, none of which accommodates the full range of possible individual behavior. This makes it difficult to single out one way to assign welfare relations to agents. In moving to the general setting, another issue is that the domain of behavioral types expands significantly (beyond the set of “rational” agent types) while the range of welfare relations remains fixed. This dramatically increases the number of possible ways to assign welfare relations to agents; and it forces the same welfare relation to be assigned to a potentially wide variety of agents who exhibit different behavior. To confront these issues directly, we consider the class of functions, called solutions, that assign a welfare relation to each behavioral type. Our approach is to impose axioms on such solutions. Of particular interest are “relational” axioms that

✩ The authors acknowledge financial support from the FQRSC. They also thank two referees and Miguel Ballester, Salvador Barberà, Tilman Börgers, Walter Bossert, Felix Brandt, Chris Chambers, Federico Echenique, Lars Ehlers, Jean-François Laslier, Bart Lipman, Paola Manzini, Marco Mariotti, Yusufcan Masatlioglu, Erkut Ozbay and Collin Raymond for helpful comments and discussions about the project. Last but not least, they also thank audiences at Mont Tremblant (Quebec Political Economy Workshop), Aix-Marseille, Michigan, Western (CETC Conference), Rochester, NYU (BRIC Conference), and UAB (IDGP Workshop) for their feedback. Corresponding author. E-mail addresses: [email protected] (S. Horan), [email protected] (Y. Sprumont).

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http://dx.doi.org/10.1016/j.geb.2016.07.005 0899-8256/© 2016 Elsevier Inc. All rights reserved.

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formulate restrictions on the welfare relations assigned to different behavioral types. Such axioms impose coherence on welfare judgments across agents. Not only does this axiomatic approach provide a principled basis to evaluate and compare solutions but it also helps to hone in on a solution. By insisting on a specific form of coherence, one can narrow the range of potential solutions tremendously. We believe that this leads to sound policy. Since our ultimate goal is to develop individual welfare measures that can be aggregated to evaluate social welfare, it is essential to make coherent welfare judgments across agents: this ensures that the resulting social welfare judgments are meaningful. Our axiomatic approach to welfare is quite flexible. In principle, the notion of a solution can be tailored to fit the type of input choice data observed and the kind of output relation required for policy making. For the sake of convenience, the current paper focuses on the simplest setting. For a solution in this canonical setting, the domain consists of all possible choice functions on X and the range consists of all possible (linear) orderings on X . In Section 5.3, we discuss how to extend our approach to a variety of non-canonical settings. We consider three natural axioms in the canonical setting: admissibility, neutrality and persistence. Admissibility requires that the ordering assigned to a rational choice function must be the one that rationalizes it. In turn, neutrality states that the solution covaries with respect to permutations of the alternatives. Finally, persistence stipulates that if the same ordering is assigned to two choice functions, then it is assigned to any choice function in between – that is, any choice function which, from each set, selects one of the alternatives chosen by the other two. Our main result shows that these three axioms uniquely determine a solution that is straightforward to compute from choice behavior. According to this sequential solution, the best alternative is the one chosen from the universal set X ; the second best alternative is the one chosen when the best alternative is removed from X ; and so on. In our view, the result illustrates the power of the axiomatic approach. First, it shows that a few natural properties can uniquely determine a simple solution even though the scope of possibilities is quite formidable.1 In addition, it shows that axioms can combine in unexpected ways. Indeed, the solution that we characterize is inherently sequential even though none of the axioms has this feature. What is more, it completes the welfare relation proposed by Bernheim and Rangel (2009) even though none of our axioms is clearly related to their approach. In accordance with the axiomatic method, we feel that the merits of a solution should be judged on the basis of its axiomatic foundations. Having said this, we are cautious about interpreting our result as conclusive support for the sequential solution. While there are compelling reasons to insist on each of our three axioms, there are also reasons (discussed at greater length in Section 2) to take issue with each. With this in mind, we are inclined to view our work as the first step towards a compelling theory of welfare based on the axiomatic approach. In Sections 4 and 6, we briefly touch on some of the most important issues that we feel still need to be resolved. Before turning to the related literature, we point out that the relevance of our characterization extends beyond welfare analysis. The sequential solution enjoys a certain “folk” status in the literature, having been used in a variety of different contexts (see Marschak, 1955; Arrow and Raynaud 1986, Ch. 7 or Moulin 1988, Exercise 11.9, for example). Clearly, its prevalence owes much to its simplicity as a method for extracting an ordering from choice data. Our result provides an independent normative justification of this solution. Whether this is ultimately compelling will, of course, depend on the specific purpose intended for the derived ordering. Related literature. The problem of choice-based welfare evaluation for boundedly rational agents has attracted considerable attention in the recent literature. By far the most popular suggestion is to extend the approach used in the standard setting – by defining revealed preference criteria appropriate for agents in the general setting. For proponents of this approach, the debate centers around which notion of revealed preference is best suited to the task (see Manzini and Mariotti, 2014 for a recent survey). Some, like Rubinstein and Salant (2012), advocate a model-specific approach where the revealed preference criteria are derived from a specific model of bounded rationality. Others, like Bernheim and Rangel (2009), favor a model-free approach. Specifically, they propose the following welfare criterion: an agent is better off with alternative x than alternative y if the agent never chooses y when x is available.2 While certainly intuitive, this Pareto-like criterion does not rely on an explicit model of behavior. Conceptually, our axiomatic approach is quite distinct from this revealed preference approach. Instead of treating each agent in isolation, it views the set of agents as a whole – with the explicit goal of making coherent welfare judgments across agents. Among the few other papers in the literature that do not follow a revealed preference approach, two are worth mentioning. In the first paper, Nishimura (2014) axiomatizes a function, called the transitive core, that assigns a reflexive and transitive (but potentially incomplete) welfare relation to each complete (but potentially cyclic) binary relation on X . The fundamental difference from our work is that Nishimura does not work with choice data directly. Instead, he considers a binary relation that can be defined (or derived) from choice data. This situates his work much closer to the vast literature on extracting orderings from tournaments (see Bouyssou, 2004 for a survey). In the second paper, Apesteguia and Ballester (2015) axiomatize an inconsistency index which, for each behavioral type, measures the “swaps difference” from the closest orderings on X . Implicitly, their approach defines a multivalued solution:

1 2

For | X | = n in the canonical setting, there are exactly n! K (n) solutions where K (n) := Independently, Green and Hojman (2009) propose the same welfare criterion.

n

k=1 k

n k

.

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to each behavioral type, it assigns the set of closest orderings. As they point out, these orderings may be interpreted as welfare rankings. The key difference is that Apesteguia and Ballester do not characterize their multivalued solution. In Section 5.2, we briefly discuss the issue of axiomatizing single-valued selections from it. 2. Definitions and axioms Let X := {1, ..., n} denote a finite universal set of alternatives such that n ≥ 2; and let X := { A ∈ 2 X : | A | ≥ 2} denote the collection of non-singleton subsets of X . Similarly, for every A ∈ X , let A := { B ∈ 2 A : | B | ≥ 2} denote the collection of non-singleton subsets of A. A choice function on A is a function C : A → A such that C ( B ) ∈ B for every B ∈ A. In words, a choice function on A selects a single alternative from every subset of A that contains more than one alternative. Let C ( A ) denote the set of choice functions on A. Let R( A ) denote the set of complete, asymmetric and transitive binary relations on A, which we call (linear) orderings.3 For an ordering R ∈ R( A ), we use interchangeably (x, y ) ∈ R and xR y. When convenient, we also denote R by listing the alternatives in decreasing order of preference. The natural ordering R 1 := {(x, y ) | 1 ≤ x < y ≤ n} on X , for instance, can also be written as R 1 = 1, ..., n. For the canonical setting that we consider throughout most of the paper, the object of interest is a function that assigns an ordering to every choice function. Formally, a (canonical) solution on A is a function f : C ( A ) → R( A ). Let F ( A ) denote the collection of solutions on A. The solution that we describe in the Introduction is recursive. For all choice functions C ∈ C ( A ), let A 1C := A; and let





A kC := A kC−1 \ C ( A kC−1 ) for k = 2, ..., | A |. Using these definitions, the sequential solution

ϕ A ∈ F ( A ) is given by

ϕ A (C ) := C ( A 1C ), ..., C ( A |CA | ) for all C ∈ C ( A ). By convention, we let C ({x}) := x for all x ∈ A so that the choice C ( A |CA | ) is well-defined. Our characterization of the sequential solution relies on three axioms. To formalize the first axiom, let max R ∈ C ( A ) denote the rational choice function induced by R ∈ R( A ). For each set B ∈ A, max R ( B ) is defined to be the best alternative in B according to R. A solution f ∈ F ( A ) is admissible if

f (max) = R for all R ∈ R( A ). R

To formalize the second axiom, let ( A ) denote the set of permutations (or bijections) on A. For all π ∈ ( A ), R ∈ R( A ) and C ∈ C ( A ), define the ordering π R ∈ R( A ) by π R := {(π (x), π ( y )) : (x, y ) ∈ R }; and define the choice function π C ∈ C ( A ) by π C ( B ) := π (C (π −1 ( B ))) for all B ∈ A. Then, a solution f ∈ F ( A ) is neutral if

f (π C ) = π f (C ) for all C ∈ C ( A ) and all π ∈ ( A ). Finally, to formalize the third axiom, define a choice function C ∈ C ( A ) to be between C 1 ∈ C ( A ) and C 2 ∈ C ( A ) if C ( B ) ∈ {C 1 ( B ), C 2 ( B )} for all B ∈ A. To denote this relationship, we write C ∈ [C 1 , C 2 ] when C is between C 1 and C 2 . Then, a solution f ∈ F ( A ) is persistent if

f (C 1 ) = f (C ) = f (C 2 ) for all C , C 1 , C 2 ∈ C ( A ) such that f (C 1 ) = f (C 2 ) and C ∈ [C 1 , C 2 ] . Each of these axioms has a natural interpretation in terms of welfare analysis. Admissibility is a very basic requirement of non-paternalism: welfare judgments should respect choice behavior that is rational. In turn, neutrality stipulates that welfare judgments should not depend on the names of the alternatives, a requirement that is desirable when the nature of the alternatives is left unspecified. Finally, persistence imposes a specific kind of coherence on the similarity judgments involved in welfare analysis. The idea is that two choice functions are less similar to one another than either is to any choice function between the two. So, if two choice functions are similar enough for welfare analysis, then the same should be true for any choice function in between. Having said this, there may be reasons to take issue with each of our axioms. According to some theories of choice, agents behave rationally, but the rationale that underlies their choice does not coincide with their true preferences. In such contexts, admissibility is not compelling. For other theories, specific alternatives play a special role in choice behavior – that of a status quo or reference point, for instance. Theories of this kind may undermine neutrality. Regarding persistence, one might argue that our definition of betweenness is too permissive. To illustrate, consider the case of three alternatives where X = {1, 2, 3}. Each choice function on X may be regarded as a vector (C ({1, 2, 3}), C ({1, 2}), C ({1, 3}), C ({2, 3})) in the set {1, 2, 3} × {1, 2} × {1, 3} × {2, 3}. Following this Cartesian product notation, let C 1 = (1, 1, 1, 2) and C 2 = (3, 1, 3, 3) represent the rational choice functions induced by the orderings 1, 2, 3 and 3, 1, 2. According to our definition, the choice function C = (1, 1, 3, 2) is between C 1 and C 2 . Since C exhibits a cyclic pattern of choice on two-element 3 A binary relation R on A is: complete if for all x, y ∈ A, x = y, (x, y ) ∈ R or ( y , x) ∈ R; asymmetric if for all x, y ∈ A, (x, y ) ∈ R ⇒ ( y , x) ∈ / R; and, transitive if for all x, y , z ∈ A, [(x, y ) ∈ R and ( y , z) ∈ R] ⇒ (x, z) ∈ R. While convenient, our use of the word “ordering” to designate a binary relation with these three properties is somewhat non-standard.

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sets, one might object that it reflects behavior which is fundamentally different from C 1 and C 2 ; and, thus, should not be viewed as lying between these rational choice functions. To amplify, observe that the rational choice function C 1 is itself between two cyclic choice functions, namely C and C  = (1, 2, 1, 3). The point is that our definition of betweenness cannot preserve “global” properties of choice functions (that relate to choice behavior across feasible subsets). Since it is defined through independent subset-by-subset comparisons, it can only hope to preserve “local” properties of choice functions. When considering alternatives to persistence, one should bear in mind that, on average, the same welfare ordering must be assigned to a huge number of choice functions.4 With | X | = 5 alternatives, for instance, each of the 120 welfare orderings must represent a group of 2,579,890,176 choice functions. To be compelling, an alternative axiom should group choice functions in a way that is at least as coherent than persistence.5 3. The result Theorem. A solution f ∈ F ( X ) is admissible, neutral and persistent if and only if f = ϕ X . It is straightforward to show that the sequential solution is admissible, neutral and persistent. Since it is quite involved to show that it is the only solution with these three features, we postpone the proof until Section 6. Below, we simply outline the basic intuition for uniqueness; and establish the independence of the axioms through some examples. (i) Uniqueness. To highlight the arguments used to prove uniqueness, it is instructive to consider the special case of three alternatives where X = {1, 2, 3}. Fix a solution f that is admissible, neutral and persistent. Think again of a choice function C ∈ C ( X ) as a vector in {1, 2, 3} × {1, 2} × {1, 3} × {2, 3}. By neutrality, it is enough to show that the set of choice functions to which f assigns the natural ordering R 1 = 1, 2, 3 coincides with the set of choice functions to which the sequential solution assigns the natural ordering. The key insight is that the former defines a Cartesian product: for each set of alternatives A ∈ X , there exists a nonempty subset of alternatives ( A ) ⊆ A such that

f −1 ( R 1 ) = ({1, 2, 3}) × ({1, 2}) × ({1, 3}) × ({2, 3}). This separability property is precisely the formal content of the persistence axiom. Since admissibility requires that the rational choice function generated by R 1 belongs to f −1 ( R 1 ), it then follows that 1 ∈ ({1, 2, 3}) ∩ ({1, 2}) ∩ ({1, 3}) and 2 ∈ ({2, 3}). Since admissibility also requires that the rational choice function generated by the ordering 1, 3, 2 cannot belong to f −1 ( R 1 ), it is in fact the case that ({2, 3}) = {2}. The rest of the argument exploits neutrality. Since there are 3 × 23 = 24 choice functions and 3! = 6 orderings on {1, 2, 3}, exactly 24/6 = 4 choice functions must be assigned the natural ordering R 1 . Combined with the observations in the last two paragraphs, it follows that

|({1, 2, 3})| × |({1, 2})| × |({1, 3})| = 4. Since this implies that |({1, 2, 3})| is either 1 or 2, it is enough to rule out the latter. To do so, consider the sub-class of choice functions where some alternative is chosen from both two-element sets to which it belongs. Since there are 32 × 2 = 18 such choice functions, neutrality requires that 18/6 = 3 must be assigned the natural ordering R 1 . If |({1, 2, 3})| = 2 however, the set f −1 ( R 1 ) must contain either 2 or 4 choice functions from this sub-class. Therefore |({1, 2, 3})| = 1; and, thus, |({1, 2})| = |({1, 3})| = 2. We conclude that

f −1 ( R 1 ) = {1} × {1, 2} × {1, 3} × {2} . In other words, f −1 ( R 1 ) is the set of choice functions to which the sequential solution assigns R 1 . (ii) Independence. The following examples establish that the three axioms are independent. Example 1. Consider the solution that assigns the inverse of

ϕ X (C ) to each C ∈ C ( X ).

This “anti-sequential” solution is neutral, persistent, but not admissible. Example 2. Consider the tournament (i.e., the complete and asymmetric binary relation) T C on X defined by (x, y ) ∈ T C if C ({x, y }) = x. Based on this relation, one can define a variety of solutions that depend only on pairwise choices. When n = 3, for instance, consider the following: For | X | = n, a neutral solution must assign the same welfare ordering to K (n)/n! choice functions. Related to this observation, we point out that persistence cannot account for the perceived “insensitivity” of the sequential solution. Every neutral solution must be constant (or insensitive) on the same number of groups of equal size. While some solutions would seem to depend on more choice data, they cannot do so on average. 4 5

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⎧ ⎪ ⎨1, 2, 3 if 1T C 2T C 3T C 1, τ (C ) := 1, 3, 2 if 1T C 3T C 2T C 1, ⎪ ⎩ TC

otherwise.

This solution advocates the use of the tournament T C provided that it is acyclic. Otherwise, it breaks the cycle in T C in favor of the alternative that comes first in the natural ordering. Since it provides an inherent advantage to alternative 1, this solution is not neutral. However, it is admissible and persistent. To see this, simply re-write τ using the Cartesian product notation:

⎧ ⎪ 1, 2, 3 ⎪ ⎪ ⎪ ⎪ 1, 3, 2 ⎪ ⎪ ⎪ ⎨2 , 1 , 3 τ (C ) = ⎪ 2, 3, 1 ⎪ ⎪ ⎪ ⎪ ⎪ 3, 1, 2 ⎪ ⎪ ⎩ 3, 2, 1

if

C ∈ {1, 2, 3} × {1} × {1, 3} × {2},

if

C ∈ {1, 2, 3} × {1, 2} × {1} × {2},

if

C ∈ {1, 2, 3} × {2} × {1} × {2},

if

C ∈ {1, 2, 3} × {2} × {3} × {2},

if

C ∈ {1, 2, 3} × {1} × {3} × {3},

if

C ∈ {1, 2, 3} × {2} × {3} × {3}.

σ (C ) on X defined by  |{ A ∈ X : C ( A ) = x}| > |{ A ∈ X : C ( A ) = y }| ; or (x, y ) ∈ σ (C ) ⇐⇒ |{ A ∈ X : C ( A ) = x}| = |{ A ∈ X : C ( A ) = y }| and C ({x, y }) = x.

Example 3. Consider the tournament

In words: alternative x beats alternative y if x is chosen more frequently than y or both alternatives are chosen equally frequently and x is chosen from the pair {x, y }. When n = 3, σ defines a solution because the Cartesian product notation:

⎧ ⎪ 1, 2, 3 ⎪ ⎪ ⎪ ⎪ 1, 3, 2 ⎪ ⎪ ⎪ ⎨2, 1, 3 σ (C ) = ⎪ 2, 3, 1 ⎪ ⎪ ⎪ ⎪ ⎪ 3, 1, 2 ⎪ ⎪ ⎩ 3, 2, 1

σ (C ) is transitive for all C ∈ C ( X ). To see this, it is helpful to re-write σ using

if

C ∈ {(1, 1, 1, 2), (2, 1, 1, 2), (1, 1, 3, 2), (3, 1, 1, 2)},

if

C ∈ {(1, 1, 1, 3), (3, 1, 1, 3), (1, 2, 1, 3), (2, 1, 1, 3)},

if

C ∈ {(2, 2, 1, 2), (1, 2, 1, 2), (2, 2, 1, 3), (3, 2, 1, 2)},

if

C ∈ {(2, 2, 3, 2), (3, 2, 3, 2), (2, 1, 3, 2), (1, 2, 3, 2)},

if

C ∈ {(3, 1, 3, 3), (1, 1, 3, 3), (3, 1, 3, 2), (2, 1, 3, 3)},

if

C ∈ {(3, 2, 3, 3), (2, 2, 3, 3), (3, 2, 1, 3), (1, 2, 3, 3)}.

When written this way, it is easy to check that σ is admissible and neutral; but it is not persistent. Examples 2 and 3 define solutions on a universal set of size n = 3. They can be extended to solutions on a universal set X of size n ≥ 4 by leveraging the sequential solution. Given a collection F3 := { f A ∈ F ( A ) : A ∈ X such that | A | = 3} of “base” solutions defined on the subsets of cardinality 3 of X , the sequential extension of F3 is the solution ϕ X ⊗ F3 ∈ F ( X ) that, on the “top” n − 3 alternatives, coincides with the sequential solution ϕ X ∈ F ( X ) and, on the “tail” of 3 alternatives, coincides with the relevant base solutions in F3 . Formally,



(x, y ) ∈ ϕ X ⊗ F3 (C ) ⇐⇒

x ∈ X \ X nC−2 x, y ∈ X nC−2

and (x, y ) ∈ ϕ X (C ); or and (x, y ) ∈ f X C (C | X C ), n −2

n −2

where X nC−2 denotes the “tail” of 3 alternatives according to C (as per the definition in Section 2 above) and C | X C denotes n −2 the restriction of C to (the subsets of) X nC−2 . A key feature of this construction is the separability between the “top” and the “tail” of the extension. By virtue of this feature, the extension generally inherits the properties that the base solutions share with the sequential solution. Indeed, it is easy to check that the sequential extension of τ (i.e., ϕ X ⊗ F3 where F3 contains the τ solutions on the three-alternative subsets of X ) is admissible and persistent. Similarly, the sequential extension of σ is admissible and neutral. 4. The solution in action In many theories of behavior that fall short of preference maximization, choice is nonetheless based on some underlying (potentially incomplete) preference relation(s). In our view, the underlying preferences associated with a particular theory should be used to evaluate welfare if that theory can be identified as the correct model of behavior. For this reason, it is important to investigate how the ordering produced by the sequential solution relates to the underlying preference relations

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associated with various theories of bounded rationality. Since the sequential solution is not based on a particular model of behavior, one should not always expect it to retrieve these preferences faithfully. Ultimately, one should judge the merits of the sequential solution on whether it performs adequately well for a reasonable number of compelling models.6 While this question is well beyond the scope of the current paper, it may be instructive to briefly discuss how well the sequential solution performs for a small selection of models. Let us begin with theories of behavior where the agent has a unique true preference ordering but her choices are “distorted” by a behavioral bias. One such model, due to Sen (1993, p. 501), contemplates an agent who maximizes her preference subject to following a social norm against greed. For each set, this causes her to forgo the most preferred alternative available. Example 4. Given A ∈ X and an ordering R ∈ R( A ), let S R ∈ C ( A ) denote the choice function where, from each set B ∈ A, the agent chooses the second best option according to R:

S R ( B ) := max( B \ {max( B )}). R

R

Another way to drive a wedge between true preference and choice behavior is the status quo bias. Although the literature suggests a variety of ways to model this phenomenon, we focus here on a version proposed by Apesteguia and Ballester (2013, p. 92). In their model, the agent chooses rationally if the status quo is unavailable or the menu contains an alternative unaffected by the status quo bias. Otherwise, she picks the status quo option. Example 5. Given A ∈ X and an ordering R ∈ R( A ), a status quo (or default) option d ∈ A and a set of alternatives U ⊆ {a ∈ A : aRd} unaffected by the presence of d, let A B R ,d,U ∈ C ( A ) denote the choice function where, from each set B ∈ A, the agent chooses as follows:

 A B R ,d,U ( B ) :=

max R ( B )

if d ∈ / B or U ∩ B = ∅,

d

otherwise.

In Example 4, if A = X and the agent’s true preference R coincides with the natural ordering R 1 ∈ R( X ), the sequential solution leads to the following ordering on X :

ϕ X ( S R 1 ) = 2, ..., n, 1.

(1)

For Example 5, again suppose that A = X and R = R 1 ∈ R( X ). Where u := min R (U ) denotes the least preferred alternative in U , it is easy to see that the sequential solution leads to the following ordering on X :

ϕ X ( A B R 1 ,d,U ) = 1, ..., u , d, u + 1, ..., d − 1, d + 1, ..., n.

(2)

In both cases, the sequential solution leaves only one alternative (in bold) out of place: in Example 4, it (significantly) underestimates the value of the most preferred option 1; and, in Example 5, it overestimates the value of the status quo d. We are skeptical that one could do much better without a specially tailored solution. In Example 4, since alternative 1 is never chosen from any set, there seems to be no basis to rank it above any other alternative. In Example 5, the issue is similar if less extreme: alternatives u + 1, ..., d − 1 are never chosen when d is available. It goes without saying that there are examples where the sequential solution perform less well. Pushing Example 4 to the extreme, suppose that the agent always chooses her least preferred alternative. For this choice procedure, the sequential solution yields the inverse of the agent’s true preference. Likewise, one could define a variant of Example 5 where the agent chooses: max R ( B ) if d ∈ / B; max R (U ∩ B ) if d ∈ B and U ∩ B = ∅; and d otherwise. For this model of “extreme” status quo bias (also due to Apesteguia and Ballester, 2013), the sequential solution only ranks the alternatives in U before d (rather than every alternative that is preferred to u). Next, we discuss some examples where behavior is based on multiple binary relations. In our view, such models are intrinsically more difficult to handle: while it seems quite reasonable for the welfare criterion to be related to the binary relations that underlie choice, it is unclear how (or whether) it should aggregate these binary relations. Our first example is a choice procedure due to Kalai et al. (2002, p. 2482) where the agent has two orderings. She maximizes the first ordering unless that leads to an alternative which is ranked too low with respect to the second; and, in that case, she maximizes the second ordering. Example 6. Given A ∈ X and R 1 , R 2 ∈ R( A ) and a cutoff k ∈ A, let K R S R 1 , R 2 ,k ∈ C ( A ) denote the choice function where, from each set B ∈ A, the agent chooses as follows:

6 Conversely, the proportion of choice functions ruled out by a theory of bounded rationality should be relevant for evaluating the usefulness of that theory for welfare analysis.

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 K R S R 1 , R 2 ,k ( B ) :=

max R 1 ( B )

if (k, max R 1 ( B )) ∈ / R2,

max R 2 ( B )

otherwise.

In this example, the “top” of the sequential solution ϕ X ( K R S R 1 , R 2 ,k ) reflects R 1 while the “tail” reflects R 2 . The boundary between the two parts of the solution depends on the location of the cutoff alternative k.7 When it ranks high for both R 1 and R 2 , the sequential solution closely approximates R 2 ; and, when k ranks low, the sequential solution closely approximates R 1 . Our second example is Manzini and Mariotti’s (2007) well-known model of choice, where the agent applies two binary relations sequentially: Example 7. Given A ∈ X and two (possibly incomplete) asymmetric relations R 1 , R 2 on A, let M M R 1 , R 2 ∈ C ( A ) denote the choice function where, from each set B ∈ A, the agent chooses as follows:

M M R 1 , R 2 ( B ) := max(max B ), R2

R1

where max R i B denotes the set of maximal elements of R i in B. It is easy to show that the sequential solution ϕ X ( M M R 1 , R 2 ) completes the first rationale R 1 . (In fact, this is a direct consequence of our theorem and Remark 1 of Dutta and Horan, 2015.) In our view, the examples discussed above highlight that the sequential solution performs reasonably well for some models of choice. Even if one sets aside the interpretive issues that arise when multiple relations are involved in the choice process, a more rigorous assessment would require one to measure the gap between the sequential solution and the rationale(s) underlying the agent’s choice behavior. As we discuss in Section 5.2 below, several measures might be adapted for this purpose. To complicate matters, it is not at all clear that this is the best way to judge the merits of a solution relative to a model of behavior. One could equally focus on how much it improves upon what the agent chooses for herself. Indeed, an attractive feature of the sequential solution in Examples 4 and 5 is that it always does better than the agent herself. Put more formally, (C ( B ), max B ϕ X (C )) ∈ / R for all B ∈ X when C = S R or C = A B R ,d,U . 5. Discussion We conclude with a discussion of some directions for future research. 5.1. Alternative axioms Besides the three axioms that we study, a variety of additional axioms merit consideration. As a point of departure, we propose some natural variations on admissibility and persistence; and briefly describe their relationship to our result. (i) Admissibility. Given a choice function C ∈ C ( A ), define the binary relation R C on A by (x, y ) ∈ R C if and only if C ( B ) = y for all B ∈ A such that x, y ∈ B. This is the (unambiguous choice) welfare relation due to Bernheim and Rangel (2009). Call a solution f ∈ F ( A ) consistent if

R C ⊆ f (C ) for all C ∈ C ( A ). In words, a solution is consistent if it completes the Bernheim–Rangel welfare relation. Obviously, consistency implies admissibility; but it is a much stronger condition. Interestingly, the sequential solution is consistent. To see this, fix distinct x, y ∈ X , C ∈ C ( X ), and note that

( y , x) ∈ ϕ X (C )

=⇒

∃k ∈ {1, ..., n} such that x, y ∈ XkC and C ( XkC ) = y

=⇒

(x, y ) ∈ / RC .

Since ϕ X (C ) ∈ F ( X ) is complete, it then follows that (x, y ) ∈ R C =⇒ (x, y ) ∈ ϕ X (C ). Combined with our result, this observation immediately gives the following corollary. Corollary. (i) There exists a unique solution which is consistent, neutral and persistent. (ii) What is more, every admissible, neutral and persistent solution is consistent.

7 Formally, the boundary between the “top” and the “tail” is k := max R 1 {x ∈ X : kR 2 x}. On K := {x ∈ X : xR 1 k }, the sequential solution coincides with R 1 ; and, on X \ K , it coincides with R 2 .

ϕ X ( K R S R 1 , R 2 ,k )

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63

To elaborate on the significance of this result, notice that Bernheim and Rangel’s approach to welfare analysis is punctual: R C does not impose any form of coherence between the welfare relations assigned to different choice functions. Statement (i) shows that simple relational axioms (namely neutrality and persistence) can be used to complete the Bernheim–Rangel welfare relation in a unique way. In principle, the same kind of completion exercise (based on relational axioms) could be carried out for other incomplete welfare criteria suggested in the literature. In turn, statement (ii) offers a justification of the Bernheim–Rangel relation from a relational perspective: it shows that a strong punctual axiom (consistency) follows from the combination of two mild conditions (admissibility and neutrality) and a powerful relational axiom (persistence). (ii) Persistence. As discussed in Section 2, persistence relies on a notion of betweenness that is quite permissive. The following axiom naturally limits the potential implications of betweenness. Call a solution f ∈ F ( A ) convex if



C ∈ C 1 , max

f (C 1 )

implies f (C ) = f (C 1 ) for all C , C 1 ∈ C ( A ).

Intuitively, the ordering f (C 1 ) assigned to C 1 is also assigned to any choice function between C 1 and the rational choice function generated by f (C 1 ).8 Since persistence and admissibility imply convexity, the sequential solution is consistent, neutral and convex. However, the sequential extension of σ (described in Section 3 above) also satisfies these three properties. These observations beg the following question: which other solutions are consistent, neutral and convex? 5.2. Alternative solutions In our view, alternatives to the sequential solution also merit consideration. As a point of departure, we describe two general approaches for defining solutions and briefly discuss their connection to the broader literature discussed in the Introduction. (i) Tournament-based solutions. Examples 2 and 3 illustrate one possible approach. Instead of using the choice data to define a solution directly, one could define a solution indirectly by using a tournament relation derived from the data. This approach makes it possible to leverage the vast literature on extracting orderings from tournaments. At the same time, it raises a difficult question: how should one go about constructing a tournament from individual choice data? Clearly, there are many reasonable constructions besides those described in Examples 2 and 3 above. (ii) Difference-based solutions. Another natural approach consists in assigning to every choice function C an ordering whose maximization produces a choice function that closely approximates C . This approach can be implemented with a difference function d that assigns to each pair (C , R ) ∈ C ( A ) × R( A ) a measure d(C , R ) of the gap between C and (the rational choice function based on) R. Call d-based any solution f ∈ F ( A ) such that

f (C ) ∈ arg min d(C , R ) for all C ∈ C ( A ). R ∈R( A )

Among the wide range of plausible difference functions, the simplest is probably the Hamming function dh , which counts the number of disagreements between a choice function C and the rational choice function generated by an ordering R:

    dh (C , R ) :=  B ∈ A : C ( B ) = max( B )  .9 R

Interestingly, dh -based (or Hamming-based) solutions are compatible with the convexity axiom proposed in Section 5.1. To see this, fix an ordering  on R( A ) to serve as a tie-breaker: to each choice function C , assign the ordering R ∈ R( A ) that maximizes  among the subset of orderings minimizing the Hamming difference with C . It is easy to check that this defines a convex (but non-neutral) Hamming-based solution. In our view, it would be interesting to describe the class of all such solutions; and determine whether it contains any neutral solutions. A more sophisticated proposal is the swaps difference d s studied by Apesteguia and Ballester (2015).10 For each pair (C , R ), this function counts the alternatives in each set B ∈ A that are R-preferred to C ( B ):

d s (C , R ) :=



|{x ∈ B : xRC ( B )}| .

B ∈A

Apesteguia and Ballester characterize the swaps index I s , which measures the degree of irrationality of a choice function C by the number I s (C ) := min R ∈R( A ) d s (C , R ). 8

Can and Storcken’s (2013) update monotonicity is a similar condition in the preference aggregation context. Klamler (2007) axiomatizes the related Hamming metric on the space of choice functions. 10 The deterministic choice data that we consider is more specialized than Apesteguia and Ballester’s stochastic choice data. For choice functions, their definition of the swaps difference function reduces to the definition given. 9

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More related to our own work is the task of axiomatizing the d s -based (or swaps-based) solutions. A natural starting point is the following analog of the convexity axiom. Say that C ∈ C ( A ) is swaps between C 1 ∈ C ( A ) and R ∈ R( A ) if C ( B ) RC 1 ( B ) or C ( B ) = C 1 ( B ) for all B ∈ A. Denote this relationship by C ∈ [C 1 , R ]s . Then, a solution f ∈ F ( A ) is swaps convex if

C ∈ [C 1 , f (C 1 )]s implies f (C ) = f (C 1 ) for all C , C 1 ∈ C ( A ). In other words, the ordering f (C 1 ) assigned to C 1 is also assigned to any choice function C that either leaves the choices made by C 1 unchanged or improves them with respect to f (C 1 ). It is straightforward to see that the sequential solution satisfies swaps convexity. Ideally, one would hope that swaps-based solutions also satisfy this natural property. Strikingly: Proposition. No swaps-based solution f ∈ F ( X ) is swaps convex for n ≥ 4. The proof is postponed to Section 7. For n = 3, the unique swaps-based solution is σ (as defined in Example 2 above); and it is clearly swaps convex. For each n ≥ 4, there are multiple swaps-based solutions: for every choice function C with multiple closest orderings, there is some freedom about how to sub-select the ordering assigned to C . In spite of this freedom, the proposition shows that one can never achieve swaps convexity. The issue is that “small” changes in choice behavior can lead to “big” changes in the set of closest orderings. This suggests a potential difficulty in providing axiomatic foundations for swaps-based solutions. In our view, the problem of measuring the degree of inconsistency of a choice function and the problem of deriving from it a welfare ordering, while formally related, are quite distinct. A good solution to the former does not automatically produce a good solution to the latter. 5.3. Non-canonical settings The canonical setting that we consider is not always the “right” setting for welfare analysis. One issue is that the input behavioral data may not be describable by choice functions. Another issue is that the output welfare relations required for policy making may have less structure than linear orderings. In what follows, we briefly discuss three natural extensions of our approach. (i) Incomplete data. Our approach relies on observing an agent’s choice from each subset of alternatives. When some of  → A that selects one her choices are unobservable, the agent’s behavior is described by an incomplete choice function  C :A  ⊂ A. A natural way to translate the sequential solution to this setting alternative  C ( B ) from each set B in a sub-domain A is as follows: assign to  C the intersection of the orderings ϕ A (C ) assigned by the sequential solution to the choice functions  to A (i.e. the C ∈ C ( A ) such that C |  =  C that extend  C from A C ). A We emphasize that this extension procedure is not specific to the sequential solution. In principle, the same approach can be used to extend any canonical solution to incomplete choice data. One drawback is that it does not necessarily yield a linear welfare ordering. In general, incompleteness in the input data translates into incompleteness in the output welfare relation.11 While unfortunate, this is a direct consequence of a desirable feature for solutions in this setting: less choice data should lead to weaker inferences about welfare. (ii) Multiple choices. Another departure from the canonical setting arises when the agent chooses multiple alternatives from some sets. In that case, her behavior is described by a choice correspondence C : A → 2 A that selects a (nonempty) subset C( B ) ⊆ B for each B ∈ A. There are two distinct ways to extend the sequential solution to this setting. The first follows from interpreting a choice correspondence as a union of choice functions. Then, it is natural to assign to C the intersection of the orderings ϕ A (C ) assigned by the sequential solution to the choice functions C that are selections from C, i.e. the choice functions C ∈ C ( A ) such that C ( B ) ∈ C( B ) for all B ∈ A. As in the case of missing choice data, the resulting welfare relation may be incomplete. The difference is that this results from indeterminacy (rather than incompleteness) in the input data. Another way to extend the sequential solution to this setting follows from interpreting multivalued choices as indifference classes of alternatives, and treating such classes as single alternatives. Then, it is natural to generalize the sequential solution by assigning to C the weak ordering (i.e., the complete, transitive, but not necessarily asymmetric relation)  A (C) computed as follows: the highest indifference class of  A (C) is C( A ); the second highest indifference class is C( X \ C( A )); and so on. Formally,  A defines what we call a weak solution: it assigns to each choice correspondence a weak ordering on A. Our axioms have natural reformulations for weak solutions. Admissibility and neutrality translate directly to this setting. The only real subtlety relates to the appropriate formulation of betweenness. In our view, a choice correspondence C on A is between two other choice correspondences C1 and C2 on A if

11

To some extent, this problem can be mitigated by taking the transitive closure.

S. Horan, Y. Sprumont / Games and Economic Behavior 99 (2016) 56–70

65

C1 ( B ) ∩ C2 ( B ) ⊆ C( B ) ⊆ C1 ( B ) ∪ C2 ( B ) for each B ∈ A.12 With this definition in place, our original formulation of persistence then translates verbatim to weak solutions. For a universal set X of n = 3 alternatives, it can be shown that  X is the unique weak solution that satisfies the natural reformulation of our three axioms (proof available on request). We conjecture that this characterization extends to n ≥ 4 alternatives. Not only would that provide a natural generalization of our theorem, but it would show that the requirement for a solution to be linear may be less crucial to our theorem than it would appear. In the setting of weak solutions, the fact that a linear ordering is assigned to each (single-valued) choice function does not need to be assumed. Instead, it follows directly from the axioms. (iii) Coarse welfare relations. The goal of extracting a linear ordering from every choice function is quite ambitious; and, in many cases, it may be unnecessary for effective policy making. For this reason, it may be worth studying incomplete solutions that assign a quasi-transitive (i.e. asymmetric and transitive but potentially incomplete) welfare relation to each choice function.13 The Bernheim–Rangel welfare relation provides a prime example of such incompleteness. However, this welfare relation is empty for a significant number of choice functions on n ≥ 4 alternatives. This observation suggests that it may be quite difficult to identify compelling solutions that are incomplete for some choice functions and nonetheless non-empty for every choice function. Echoing our observations about linearity (in (ii) above), the requirement for a solution to be complete may be less restrictive than it would appear. 6. Proof of theorem: uniqueness Fix an admissible, neutral and persistent rule f ∈ F ( X ). We claim that f = ϕ X . The proof is by induction on n, the size of X . The claim is trivially true if n = 2. For the induction step, suppose n ≥ 3 and suppose that, for all x ∈ X , the only admissible, neutral and persistent solution on X \ {x} is ϕ X \{x} . Recall that R 1 = 1 1 1, ..., n denotes the natural ordering on X . Because f is neutral, it is sufficient to show that f −1 ( R 1 ) = ϕ − X ( R ). For any R ∈ R( X ) and C ∈ C ( X ), let R | X \{1} ∈ R( X \ {1}) denote the restriction of the ordering R to X \ {1} and let C | X \{1} ∈ C ( X \ {1}) denote the restriction of the choice function C to (the subsets of) X \ {1}. Finally, define

f −1 ( R )| X \{1} := {C ∈ C ( X \ {1}) : ∃C  ∈ f −1 ( R ) such that C = C  | X \{1} }. 1 1 Step 1. We show that f −1 ( R 1 )| X \{1} = ϕ − X ( R )| X \{1} .

For any C ∈ C ( X \ {1}), first define the choice function C 1 ∈ C ( X ) by



1

C ( A ) := For any C ∈ (C | X \{1} )1 ∈

1

if 1 ∈ A ,

C ( A)

otherwise.

  f −1 ( R 1 ), observe that max R 1 ∈ f −1 ( R 1 ) by admissibility. Since (C | X \{1} )1 ∈ C , max R 1 , persistence then implies f −1 ( R 1 ). In other words:

C ∈ f −1 ( R 1 ) ⇒ (C | X \{1} )1 ∈ f −1 ( R 1 ).

(3)

Next, define

  R1 ( X ) := R ∈ R( X ) : max( X ) = 1 and R

1

C ( X ) := {C ∈ C ( X ) : C ( A ) = 1 for all A ∈ X such that 1 ∈ A } . Observe that

C ∈ C 1 ( X ) ⇒ f (C ) ∈ R1 ( X ).

(4)

If f (C ) ∈ / R ( X ), consider the ordering R obtained from  f (C ) by pushing alternative 1 to the first rank without altering the relative ranks of the other alternatives. Since max R ∈ max f (C ) , C and f (max f (C ) ) = f (C ), we obtain f (max R ) = f (C ) = R, contradicting admissibility. Finally, define the solution f 1 ∈ F ( X \ {1}) by f 1 (C ) := f (C 1 ) | X \{1} for all C ∈ C ( X \ {1}). It is straightforward to check that f 1 is an admissible, neutral and persistent solution on X \ {1}. By the induction hypothesis, 1

f 1 = ϕ X \{1} .

(5)

12 Not only does this definition collapse to the usual one for choice functions, but it also has a certain pedigree in the literature (see Albayrak and Aleskerov, 2000 and Klamler, 2007 for prior work relying on this definition). 13 We point out that the extensions described in (i) and (ii) are also quasi-transitive.

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To complete Step 1, notice that:

C ∈ f −1 ( R 1 )| X \{1}

⇔

∃C  ∈ f −1 ( R 1 ) such that C = C  | X \{1}

⇔

C 1 ∈ f −1 ( R 1 ) [by implication (3)]

⇔

f (C 1 ) = R 1

⇔

f (C 1 )| X \{1} = R 1 | X \{1} [by implication (4)]

⇔

f 1 (C ) = R 1 | X \{1} [by definition of f 1 ]

⇔

ϕ X \{1} (C ) = R 1 | X \{1} [by identity (5)]

⇔

1 1 C ∈ ϕ− X \{1} ( R | X \{1} )

⇔

1 1 C ∈ ϕ− X ( R )| X \{1} [by definition of ϕ X \{1} and ϕ X ].



Because f is  persistent, f −1 ( R 1 ) is a Cartesian product set. For each A ∈ X , there exists a nonempty set ( A ) ⊆ A such that f −1 ( R 1 ) = A ∈X ( A ). Moreover, max R 1 ∈ f −1 ( R 1 ) by admissibility. Hence,

max( A ) ∈ ( A ) for all A ∈ X .

(6)

R1

Denoting the cardinality of the set ( A ) by

   −1 1   γ ( A ).  f ( R ) =

γ ( A ), we have (7)

A ∈X

From Step 1, ({x, ..., n}) = {x} for each x ∈ {2, ..., n} and ( A ) = A for every other set A ∈ X that does not contain 1. To 1 1 prove that f −1 ( R 1 ) = ϕ − X ( R ), it remains to be shown that ( X ) = {1} and ( A ) = A for every set A ∈ X \ { X } such that 1 ∈ A. Note: For ease of notation from now on, we drop any reference to X unless this causes confusion. Thus, we write R instead of R( X ), C instead of C ( X ),  instead of ( X ) and ϕ instead of ϕ X . Step 2. We show that

γ ( A ) = n − 1 for every set A such that | A | = n − 1 and A = {2, ..., n}.

Let us call a set D ⊆ C symmetric if, for all C ∈ D and that, for every symmetric set D ⊆ C ,

π ∈ , we have π C ∈ D . Because f is neutral, it is easy to see

  |D | | D |  −1 1  = .  f ( R ) ∩ D = |R | n!

(8)

It is straightforward to compute14 that n −1

|C |  n−1 kk . = n!

(9)

k =2

Since C is a symmetric set, (7) and (8) imply

 A ∈X

γ ( A) =

|C | . n!

(10)

For x ∈ X , define Cx, n−1 := {C ∈ C : C ( A ) = x if | A | = n − 1 and x ∈ A }; and let Cn−1 := ∪x∈ X Cx, n−1 . The symmetric set Cn−1 contains all the choice functions on X where some alternative x ∈ X is selected from every set of size n − 1 that contains it. It is easy to compute that n −2

 n |Cn−1 | k k −1 . =n× n!

(11)

k =2

Since R 1 ranks alternative 1 first, (6) implies that 1 ∈ ( A ) for all A such that | A | = n − 1 and x ∈ A. Therefore alternative 1 may be chosen from every set of size n − 1 which contains it. In other words, C1, n−1 ⊆ f −1 ( R 1 ). Suppose f −1 ( R 1 ) ∩ Cn−1 = f −1 ( R 1 ) ∩ C1, n−1 so that 1 is the only such alternative. Since γ ({2, ..., n}) = 1, it then follows that

14

An easy way is to check that |ϕ −1 ( R 1 )| =

n−1 k=2

k

n k

−1

and note that |ϕ −1 ( R 1 )| = |C |/n! because

ϕ is neutral.

S. Horan, Y. Sprumont / Games and Economic Behavior 99 (2016) 56–70

n −2    −1 1  γ ( A ).  f ( R ) ∩ Cn−1  = γ ( X ) × 1 ×

67

(12)

| A |=2

Denote the last factor by G n−2 . Since Cn−1 is a symmetric set, (8) and (12) imply

γ ( X ) × G n −2 =

|Cn−1 | . n!

Dividing (10) by this equation and simplifying using (9) and (11) gives



γ ( A) =

| A |=n−1

|C | (n − 1)n−1 = . |Cn−1 | n

Denote the term on the left side of this expression by G n−1 . Since n and n − 1 are co-prime, we conclude that G n−1 is not an integer, which is a contradiction. So, it must be that some alternative other than 1 may be chosen from every set of size n − 1 to which it belongs. Since ({2, ..., n}) = {2}, this other alternative must be 2. In other words, f −1 ( R 1 ) ∩ Cn−1 = f −1 ( R 1 ) ∩ (C1, n−1 ∪ C2, n−1 ). Since there are γ ({1, 3, ..., n}) ways to guarantee that 2 is chosen from every set of size n − 1 that contains it,

   −1 1   f ( R ) ∩ Cn−1  = γ ( X ) × (1 + γ ({1, 3, ..., n})) × G n−2 .

(13)

Since Cn−1 is a symmetric set, (8) and (13) imply

γ ( X ) × (1 + γ ({1, 3, ..., n})) × G n−2 =

|Cn−1 | . n!

Dividing (10) by this equation and using (9) and (11) gives

G n −1 1 + γ ({1, 3, ..., n})

=

|C | (n − 1)n−1 (n − 1)n−1 = × [1 + γ ({1, 3, ..., n})]. or G n−1 = |Cn−1 | n n

Since G n−1 is an integer and n and n − 1 are co-prime, it must be that n = 1 + γ ({1, 3, ..., n}) or, equivalently, γ ({1, 3, ..., n}) = n − 1. Plugging this back into the above formula establishes that G n−1 = (n − 1)n−1 . Since γ ({2, ..., n}) = 1, we conclude that γ ( A ) = n − 1 for every set A of size n − 1 other than {2, ..., n}. This completes Step 2.  Note: If n = 3, Steps 1 and 2 imply that ({1, 2}) = {1, 2}, ({1, 3}) = {1, 3} and ({2, 3}) = {2}. From (10), it then follows that γ ({1, 2, 3}) = 1. Hence, ({1, 2, 3}) = {1} by (6). This means that f −1 ( R 1 ) = ϕ −1 ( R 1 ). So, f is the sequential solution. From now on, we assume that n ≥ 4. Step 3. We show that

γ ( X ) = 1 or γ ( X ) = n.

Using Step 2, we can rewrite (10) as

γ ( X ) × (n − 1)n−1 × G n−2 =

|C | . n!

(14)

Define C nX−1 := {C ∈ C : C ( A ) = C ( X ) if | A | = n − 1}. This is the symmetric set of choice functions where the alternative selected from X is never chosen from any set of size n − 1. It is straightforward to compute that n −2

 n |C nX−1 | k k −1 . = (n − 2)n−1 × n!

(15)

k =2

On the other hand,

   −1 1   f ( R ) ∩ C nX−1  = [(n − 2)n−1 + (γ ∗ ( X ) − 1)(n − 1)(n − 2)n−2 ] × G n−2  γ ( X ) − 1 if 2 ∈ ( X ), ∗ where γ ( X ) := γ (X) otherwise.

This is because there are: (i) (n − 2)n−1 ways of not choosing 1 from any set of size n − 1; (ii) no ways of not choosing 2 from any set of size n − 1 (because ({2, ..., n}) = {2}); and, (iii) (n − 1)(n − 2)n−2 ways of not choosing any other alternative from any set of size n − 1.

(16)

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Since C nX−1 is a symmetric set, (8) and (16) imply

[(n − 2)n−1 + (γ ∗ ( X ) − 1)(n − 1)(n − 2)n−2 ] × G n−2 =

|C nX−1 | . n!

Dividing (14) by this equation and simplifying using (9) and (15) gives

(n

− 2)n−1

|C | (n − 1)n−1 γ ( X ) × (n − 1)n−1 = n −1 = . ∗ n − 2 + (γ ( X ) − 1)(n − 1)(n − 2) |C X | (n − 2)n−1

Further simplifying this expression gives (γ ∗ ( X ) − 1)(n − 1) = (γ ( X ) − 1)(n − 2). Since n − 1 and n − 2 are co-prime: (i) γ ∗ ( X ) − 1 = γ ( X ) − 1 = 0; or (ii) γ ∗ ( X ) − 1 = n − 2 and γ ( X ) − 1 = n − 1. In case (i), γ ( X ) = 1; and, in case (ii), γ ( X ) = n. This completes Step 3.  Step 4. We show that

γ ( X ) = 1.

∗ := {C ∈ C : ∃ R ∈ R such that C ( A ) = max ( A ) if | A | = k}. This is the symmetric set of For any k ∈ {2, ..., n}, define C− R k choice functions that are rational except possibly on sets of size k. It is straightforward to compute that

   ∗  C−k  n!

n

=k k .

(17)

γ ( X ) = n. Let R 2 := 2, 1, 3, ..., n. Since max R 1 ∈ f −1 ( R 1 ) and max R 2 ∈/ f −1 ( R 1 ), there   exists some A ∈ X suchthat 1 ∈ ( A ) and 2 ∈  A \ ( A ). Let kˆ := | A |. From Step 2 and γ ( X ) = n, kˆ ∈ {2, ..., n − 2}. To simplify the notation, let G k := | A |=k γ ( A ). By way of contradiction, suppose

n −1 k

for all k ∈ {2, ..., n − 1} \ {kˆ } when γ ( X ) = n. n  ˆ Fix k ∈ {2, ..., n − 1} \ {k}. By Step 2, G n−1 = (n − 1)n−1 = (n − 1) n−1 −1 . This proves the claim for k = n − 1. It also proves the claim for n = 4 since in that case {2, ..., n − 1} \ {kˆ } = {2, 3} \ {2} = {3} = {n − 1}. Next, assume n ≥ 5 and k = n − 1. We

Substep 4.1. We claim that G k = k

claim that

   −1 1 ∗   f ( R ) ∩ C− k  ≤ kG k .

(18)

∗ . By definition of C ∗ , there exists an ordering R ∈ R such that To see why, consider a choice function C ∈ f −1 ( R 1 ) ∩ C− −k k C ( A ) = max R ( A ) whenever | A | = k. By Step 1,  ({i , ..., n}) = {i } for i = 2, ..., n − 1. So, C ({i , ..., n}) = i for i = 2, ..., (n − k), (n − k + 2), ..., (n − 1). Therefore

2 R ... R (n − k) R (n − k + 2) R ... R n

and

(n − k) R (n − k + 1).

(19)

Since 1 ∈ ( A ) and 2 ∈  A \ ( A ), it must be that

1R2.

(20)

Exactly k different orderings R on X satisfy (19) and (20): these are obtained from R by pushing the alternative n − k + 1 to any rank lower than or equal to n − k + 1. This proves  (18).  1







n



∗ is a symmetric set, f −1 ( R 1 ) ∩ C ∗ ∗ k −1 . But, Since C−  −k  = C−k  /n!. Using (17) and (18), it then follows that G k ≥ k k n n  since γ ({n − k + 1, ..., n}) = 1 and | A |=k | A | = k k , we also know that G k ≤ k k −1 . Combining these two inequalities gives

Gk = k

n k −1

. This completes Substep 4.1.

Substep 4.2. To complete the proof of Step 4, we derive a contradiction from Given the assumption that G n := γ ( X ) = n, Step 1 and Substep 4.1 imply

γ ( X ) = n.

   n  −1 1  k k −1 .  f ( R ) = n × G kˆ ×

(21)

k =kˆ ,n

Since C is a symmetric set, (8), (9) and (21) then imply

G kˆ =

kˆ

n −1 kˆ

n

(22)

. ˆ

ˆ

ˆ

For each x ∈ X , define Cx−k := {C ∈ C : C ( A ) = x if | A | = kˆ } and let C −k = ∪x∈ X Cx−k . This is the symmetric set of choice

ˆ It is straightforward to compute that functions where some alternative is never chosen except possibly from sets of size k.

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69

⎤ ⎤ ⎡ ⎡ ˆ −1 n −1   k ˆ n−1 n n−1 n−1 n −1 |C −k | − 1 − − 1 1 k k (k − 1) k−1 ⎦ × kˆ kˆ ×⎣ k k (k − 1) k−1 ⎦ . = (n − 1) × ⎣ n! k =2

k=kˆ +1

This simplifies to ˆ

|C −k | = (n − 1) × n!

n−1

k =2 k

kˆ

n−1 −1 k

n−1 kˆ

−1

n−1

(k − 1)

k −1

n−1

(kˆ − 1)

× kˆ

n kˆ

−1

(23)

.

kˆ −1

Since, by Step 1, ({x, ..., n}) = {x} for each x = 1, alternatives 1 and (n − kˆ + 1) are the only two alternatives that can be

ˆ That is, f −1 ( R 1 ) ∩ C −kˆ = f −1 ( R 1 ) ∩ (C −kˆ ∪ C −kˆ never chosen from any set of size other than k. 1 ˆ

n−k+1

). Therefore,

⎤ ⎤ ⎡ ⎡ ˆ −1 n −1   k   n−1 n−1 n−1 n −1 ˆ  −1 1  k k −1 (k − 1) k−1 ⎦ × G kˆ × ⎣ k k −1 (k − 1) k−1 ⎦  f ( R ) ∩ C −k  = (n − 1) × ⎣ k=kˆ +1

⎡ + (n − 1) × ⎣

n −1

k

n−1 k

k=kˆ +1

n−1

(k − 1)

k −1

k =2

⎤

⎡

−1 ⎦

× G kˆ × ⎣

ˆ −1 k n−1

k

k

−1

n−1

(k − 1)

k −1

⎤ ⎦.

k =2

Given (22) and (23), this simplifies to

    |C −kˆ | 1 + n−ˆ 1  −1 1 k −kˆ  . ×  f (R ) ∩ C  = n! n

(24)

ˆ

Since C −k is a symmetric set, (8) and (24) establish that (kˆ − 1) × (n − 1) = 0. Since n ≥ 4 and kˆ ∈ {2, ..., n − 2}, this is a contradiction. This completes Substep 4.2 and, hence, Step 4.  Steps 1 and 4 establish that γ ({x, ..., n}) = 1 for each x ∈ X . It then follows from (6) and (10) that ( X ) = {1} and ( A ) = A for every set A ∈ X \ { X } such that 1 ∈ A. Together with Step 1, this implies that ({x, ..., n}) = {x} for each x ∈ X and ( A ) = A for every other set A ∈ X . In turn, this establishes that f −1 ( R 1 ) = ϕ −1 ( R 1 ), which completes the proof. 7. Proof of proposition Fix a swaps-based solution f ∈ F ( X ). Step 1. First, we establish the claim for n = 4. Recall the Cartesian product notation C := (C ( X ), C ({1, 2, 3}), ..., C ({2, 4}), C ({3, 4})) introduced in Section 2; and, consider C := (2, 3, 4, 1, 2, 1, 1, 1, 2, 2, 3) ∈ C ( X ). To show that f violates swaps convexity, we identify a C  ∈ C ( X ) such that (i) C  ∈ [C , f (C )]s and (ii) f (C  ) = f (C ). Let R 2 := 2, 1, 3, 4. By direct computation, d s (C , R 1 ) = d s (C , R 2 ) = 5. Similarly, it is straightforward (but somewhat involved) to show that d s (C , R ) ≥ 6 for all R ∈ R( X ) \ { R 1 , R 2 }. So, f (C ) ∈ { R 1 , R 2 }, which gives two cases to consider. If f (C ) = R 1 , then C 1 := (2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 3) ∈ C ( X ) satisfies requirements (i) and (ii). In particular, (i) C 1 ∈ [C , f (C )]s = [C , R 1 ]s because C 1 improves the choices of C from {1, 2, 3} and {1, 2, 4}; and (ii) f (C 1 ) = R 2 = R 1 = f (C ) since d s (C 1 , R 2 ) = 1 and d s (C 1 , R 1 ) = 3. If f (C ) = R 2 , then C 2 := (2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3) ∈ C ( X ) satisfies requirements (i) and (ii). In particular, (i) C 2 ∈ [C , f (C )]s = [C , R 2 ]s because C 2 improves the choices of C from {1, 2, 3} and {1, 2, 4}; and (ii) f (C 2 ) = R 1 = R 2 = f (C ) since d s (C 2 , R 1 ) = 1 and d s (C 2 , R 2 ) = 3. Step 2. Next, we establish the claim for n > 4 by extending the construction in Step 1. Let B := {n − 3, n − 2, n − 1, n} denote the “tail” consisting of the last four alternatives in X according to the natural ordering R 1 ; and let π −1 ∈ ( X ) denote the permutation on X such that



π

−1

(x) :=

x+4−n

if x ∈ B ,

x+4

otherwise.

This permutation shifts every alternative in X down four spots according to R 1 , mapping the tail B of R 1 to the “top” {1, 2, 3, 4} of R 1 . Conversely, the permutation π shifts every alternative up four spots, mapping the top to the tail. Consider the choice function  C ∈ C ( X ) given by

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π C ( A) if A ⊆ B ,  C ( A ) := max R 1 ( A ) otherwise. Within the tail,  C behaves symmetrically to the choice function C from Step 1 (since  C | B = π C ). Everywhere else, it is rationalized by R 1 . To establish that f violates swaps convexity, we identify a choice function C  ∈ C ( X ) such that (i) C  ∈ [ C , f ( C )]s and (ii) f (C  ) = f ( C ). Let R 2 := 1, ..., n − 4, n − 2, n − 3, n − 1, n. Then, f ( C ) ∈ { R 1 , R 2 } by the same reasoning as Step 1, which again gives two cases. If f ( C ) = R i (for i = 1, 2), then consider the choice function  C i ∈ C ( X ) given by

 π C i ( A) if A ⊆ B ,  C i ( A ) := max R 1 ( A ) otherwise.

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