Inferring Rationales from Choice: Identification for Rational Shortlist Methods ∗ Rohan Dutta† and Sean Horan‡ September 2014

Abstract A wide variety of choice behavior inconsistent with preference maximization can be explained by Manzini and Mariotti’s Rational Shortlist Methods. Choices are made by sequentially applying a pair of asymmetric binary relations (rationales) to eliminate inferior alternatives. Manzini and Mariotti’s axiomatic treatment elegantly describes which behavior can be explained by this model. However, it leaves unanswered what can be inferred, from observed behavior, about the underlying rationales. Establishing this connection is fundamental not only for applied and empirical work but also for meaningful welfare analysis. Our results tightly characterize the surprisingly rich relationship between behavior and the underlying rationales. (JEL D01) Keywords: Revealed Preference; Identification; Two-stage Choice Procedures; Behavioral Industrial Organization; Comparative Advertising; Welfare.

∗

The paper began as two separate but closely related projects. Happily, Montreal brought the authors and their work together. We thank David Levine, Bart Lipman, John Nachbar, Paulo Natenzon, Larry Samuelson, and Pierre-Yves Yanni for their helpful comments on earlier drafts of the paper(s). We also thank the editor, Andrew Postlewaite, and three anonymous referees for suggestions that have improved the paper considerably. † CIREQ and Department of Economics, McGill University, Leacock 531, 855 Sherbrooke St. W, Montreal QC, Canada H3A 2T7; 514-398-3030 (ext. 00851); [email protected]. ‡ D´epartement de sciences ´economiques, Universit´e de Montr´eal, C-6018 Pavillon Lionel-Groulx, 3150 rue Jean-Brillant, Montr´eal QC, Canada H3T 1N8; 514-343-2404; [email protected].

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I. Introduction Manzini and Mariotti [2007] study a choice procedure, called Rational Shortlist Methods (RSM), where the decision-maker maximizes in stages using a pair (P1 , P2 ) of rationales (i.e. asymmetric but not necessarily transitive preferences). For any menu of alternatives, the decision-maker first eliminates any option which is dominated by another alternative according to P1 before selecting, from the remaining alternatives, the option which maximizes P2 . While they provide an axiomatization of the model, Manzini and Mariotti do not address the important issue of identification: to what extent and how does choice pin down the underlying rationales? The three results in our paper address these questions. Together, they show that identification in the RSM model is at once relatively straightforward and surprisingly rich. In the spirit of Samuelson’s [1938, 1950] revealed preference, we first define “revealed rationales” for choice behavior consistent with the RSM model. Unlike the standard model of preference maximization, there may be multiple ways to represent the same choice behavior in the RSM model. Proposition 1 shows that the revealed rationales capture the features common to all rationale pairs that can be used to represent behavior. Implicitly, the result provides a way to construct a “most cautious” and “least cautious” estimate for each rationale. As discussed at greater length in Section IV, these estimates have significant implications for welfare analysis based on the RSM model. In turn, Proposition 2 uses the revealed rationales to provide, for any choice behavior consistent with the RSM model, an exact characterization of all rationale pairs that can be used to represent the behavior. While Proposition 1 provides “partial” estimates of the rationales, Proposition 2 pins down the valid ways to “complete” these estimates into a representation. Finally, Proposition 3 provides a systematic way to modify a given rationale pair without affecting the induced choice behavior. Like the “uniqueness up to affine transformations” result for models with cardinal utility representations, this result characterizes the choice-invariant transformations of any representation without having to consider choice behavior directly. 2

Related Literature: Our work contributes to a growing literature concerned with identification in models of procedural decision-making (Au and Kawai [2011]; Lleras et al. [2011]; Masatlioglu et al. [2012]; Cherepanov et al. [2013]; Horan [2013]; and, de Clippel and Rozen [2014]). For the time being, we limit ourselves to three pointed remarks about the related literature, postponing a more detailed discussion until the Conclusion: (1) Like our work, each of the cited papers provides identification results for a particular model of two-stage choice. Yet, because they differ from the RSM model in terms of empirical scope, the identification results for these models bear no transparent relationship to our results.1 To illustrate this point, it is easiest to consider the issue of revealed preference (i.e. Proposition 1). In terms of empirical scope, the models of Lleras et al. [2011] and Masatlioglu et al. [2012] are the least related to the RSM model. Not only does the RSM model accommodate behavior inconsistent with these models, but these models accomodate behavior inconsistent with the RSM model. As a result, valid inferences about revealed preference in one setting may be invalid in the other. Intuitively, this makes it particularly difficult to go between Proposition 1 and the revealed preference definitions in these papers. For the specialized transitive RSM models of Au and Kawai [2011] or Horan [2013], the inferences about revealed preference in Proposition 1 remain valid. The problem is that the added structure in these models leads to much stronger inferences about revealed preference. To derive Proposition 1 from the revealed preference definitions in these papers, one faces the difficult task of isolating the inferences that depend only on the transitivity of the rationales. The opposite problem arises for Cherepanov et al. [2013], who consider a generalized version of the RSM model. Intuitively, our more restrictive setting should allow for more inferences about revealed preference. Surprisingly, this intuition is only half right: the second-stage revealed rationale turns out to be the same in the two models (see our Proposition 1 and their Proposition 2). 1

While this point also applies to de Clippel and Rozen [2014], there is an even more fundamental difference between our work and theirs. While they focus on the issue of identification with partial data, we are primarily concerned with complete data.

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One takeaway from these observations is that our work complements the prior literature—particularly the last three papers discussed. Only by considering all four papers together does one obtain a clearer picture of how revealed preference is sharpened by adding progressively more structure to the model. (2) Our three results address important questions of identification that are relevant for every model of procedural choice. While the cited papers all establish a revealed preference result like Proposition 1 and one (Horan [2013]) even characterizes the class of representations as in Proposition 2, none provides an invariance result like Proposition 3. Having said this, our work clearly owes a significant debt to each of these papers in terms of methodology.2 (3) Finally, it is worth mentioning that the binary relations defined in Proposition 1 were introduced, separately, in prior work by Rubinstein and Salant [2008] and Houy [2008].3 Relative to these papers, the key contribution of Proposition 1 is to establish that, taken together, these binary relations fully characterize revealed preference in the RSM model. Propositions 2 and 3 serve to amplify this point by showing that the revealed rationales “tell us almost everything we need to know” about identification in the RSM model.4 Examples: The kinds of identification results we provide are critical for understanding models of decision-making. They determine the extent to which meaning can be inferred from choice data and, conversely, the extent to which the parameters of the model are meaningful in terms of behavior (Dekel and Lipman [2011]). While central to the goal of decision theory, identification is equally relevant for empirical and applied work (Spiegler [2008]). To help illustrate this point (while, at the same time, highlighting some natural applications of the RSM model), consider the following examples: Example I (Multi-Criterial Choice) Frequently, a variety of product dimensions 2

Several papers (Lleras et al. [2011]; Masatlioglu et al. [2012]; Cherepanov et al. [2013]) were instrumental in developing the notion of revealed preference for models of procedural choice. Others (Au and Kawai [2011]; Cherepanov et al. [2013]; Horan [2013]) were key in emphasizing the role played by the range of possible representations. Finally, one paper (de Clippel and Rozen [2014]) tackled the subtleties raised by identification with limited data. 3 Neither paper does so for the purpose of addressing identification in the RSM model. 4 We are indebted to one of the referees for putting the point this way.

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are important to consumers. Evidence suggests that, instead of aggregating these criteria, consumers tend to evaluate the dimensions lexicographically (Tversky et al. [1988]; Dulleck et al. [2011]). After eliminating options which are inferior on the most important product dimension, the consumer selects the product which is most preferred on the second most important dimension. Given data consistent with this model, what can the analyst infer about the criteria used by consumers? Example II (Forced Choice) Sometimes, decision-makers are forced to pick a single alternative. Consumer surveys, for example, generally require respondents to choose one alternative from several different menus. When preference alone is not sufficient to discriminate among the feasible alternatives, the respondent must employ a “tie-breaking rule” to make her final choice. In that case, what can the analyst infer about the respondents’ “true” (first stage) preferences? Example III (Limited Consideration) In a variety of choice situations, consumers focus on a subset of the alternatives before selecting the most preferred alternative among those considered (see Wright and Barbour [1977]). If the consumer’s first stage consideration set is the result of a process where alternatives “compete” for the consumer’s attention, it is reasonable that: (i) an option that is considered on a given menu continues to be considered when some of the other feasible options are removed; and, (ii) an option considered on two different menus is also considered when the two menus are “merged” (into a single larger menu).5 What can the analyst infer about consumers’ “true” (second stage) preferences? Example IV (Advertising) A firm asks their advertising agency to conduct a market survey in order to determine why the sales of their new laundry detergent are lagging. The agency finds that many consumers are ignorant of the fact that the new product compares favorably with some of its competitors. Based on their findings, the agency recommends an aggressive comparative advertising strategy that targets these competitors. Should the firm follow this recommendation? Example V (Advertising, continued) Another producer of laundry detergent has an accurate picture of the rationales used by consumers (perhaps because it understands how advertising has made certain product comparisons more salient to consumers). Assuming the strategies of its competitors are fixed in the short run, can the firm cut advertising costs without affecting consumer choices? 5

If the options considered on the menu A are denoted by Γ(A), (i) is equivalent to Γ(A) ∩ B ⊆ Γ(B) for B ⊂ A, and (ii) is equivalent to Γ(A) ∩ Γ(B) ⊆ Γ(A ∪ B). In the choice theory literature, these properties are known as α and γ (Sen [1971]).

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Each example raises a question that is intimately related to the issue of identification in an application of the RSM model. After presenting our results in Section III, we revisit these examples in Section IV to show how our results address the questions they raise. In the course of the discussion, we also consider the broader implications for welfare analysis and policy making. In Section V, we conclude with a discussion of related work and extensions.

II. The RSM Model and an Example Before presenting our results in the next section, we first describe the RSM model more formally and illustrate the task of identification with an example. a. The RSM Model A rationale is an asymmetric binary relation on a finite domain X. A Rational Shortlist Method (RSM) is a choice function c(P1 ,P2 ) : 2X \ ∅ → X induced by a pair (P1 , P2 ) of rationales.6 Given a menu A ⊆ X, the choice induced by the pair (P1 , P2 ) is c(P1 ,P2 ) (A) ≡ max(max(A; P1 ); P2 ) where max(B; P ) = {a ∈ B : no b ∈ B s.t. bP a} denotes the set of maximal alternatives in B according to the binary relation P . A choice function c is said to be RSM-representable if there exists a pair of rationales (P1 , P2 ) such that c(A) = c(P1 ,P2 ) (A) for any menu of alternatives A ⊆ X. Manzini and Mariotti [2007] give an axiomatic characterization of the RSM model (which, for convenience, is restated in the Appendix).7 To establish sufficiency, they show that any choice function c which satisfies their axioms can be represented by a pair of rationales (P1∗ , P c ) where: - P1∗ is defined by aP1∗ b if c(A) 6= b for all A ⊇ {a, b}; and, - P c is the pairwise revealed preference defined by aP c b if c(a, b) = a. 6

As pointed out by one referee, not every rationale pair induces a choice function. The problem is that c(P1 ,P2 ) (A) is empty- or multi-valued for some A ⊆ X. Lemma 2 of the Appendix gives necessary and sufficient conditions for (P1 , P2 ) to induce a choice function. 7 Rubinstein and Salant [2008] provide an alternative axiomatization.

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In passing, they observe that (P1∗ , P c ) is not the only RSM-representation of behavior (see their Remark 1). A simple example will serve to illustrate this point and provide some intuition for the identification results in Section III. b. An Example Example 1 Let c be a choice function on X = {w, x, y, z} with P c given by: x

Pc

w

y

z

In addition, suppose that c(w, x, y) = w and c(x, y, z) = x.8 Note that, while the revealed preference P c contains cycles, these choices are consistent with the RSM model (see Remark 2 of the Appendix). To say something about identification, suppose c is represented by the pair (P1 , P2 ). In order to make inferences about (P1 , P2 ), we make the following observations about choice from two-element sets: (1) every revealed preference in P c must belong to some rationale (P c ⊆ P1 ∪ P2 ); and, (2) the first rationale must not contradict the revealed preference (P1 ⊆ P c ). The first observation formalizes the intuition that one of the rationales must determine the choice from every two-element set. In turn, the second captures the idea that the contents of the first rationale must affect choice from two-element sets. To appreciate the power of these observations, first consider the task of assigning the revealed preferences in {x, y, z} to the rationales P1 and P2 : 8

For {w, y, z} and {w, x, z}, P c is transitive. To be consistent with the RSM model, it must be that c(w, y, z) = max({w, y, z}; P c ) = y and c(w, x, z) = max({w, x, z}; P c ) = z.

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y

x

z

Figure 1: Revealed preferences P c on {x, y, z}

Notice that zP c x and c(x, y, z) = x together imply zP2 x. Otherwise, zP1 x by (1) so that x is eliminated in the first stage and, consequently, c(x, y, z) 6= x. Moreover, yP c z and c(x, y, z) = x together imply yP1 z. Otherwise, (2) ensures that z is not eliminated in the first stage and, again, c(x, y, z) 6= x. By similar reasoning about {w, x, y}, xP1 y and yP2 w. This exhausts what can be inferred from the limited data provided. By observing c(X), it is possible to say more. To be consistent with the RSM model, the only possibilities are (i) c(X) = w and (ii) c(X) = x (see Remark 2 of the Appendix). In case (i), the same type of reasoning given above establishes zP2 w. Otherwise, zP1 w by (1) so that w is eliminated from X and c(X) 6= w. In other words, the “reversal” between c(w, z) = z (i.e. zP c w) and c(X) = w reveals zP2 w. In this case, behavior does not determine how to assign wP c x. In case (ii), no such indeterminacy arises. Since c(X) = x “reverses” c(x, w) = w, wP2 x. In addition, zP1 w. Effectively, c(X) = x requires w to be eliminated from X in the first stage. Otherwise, w eliminates x in the second stage. Since c(w, x, y) = w precludes yP1 w, the only possibility is zP1 w. More succinctly, the “switch” between c(w, x, y) = w and c(X) = x reveals zP1 w. Table 1 summarizes the features of the rationales (P1 , P2 ) identified above:

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Choices \ Rationales

P1

P2

c(x, y, z) = x

y

z

z

x

c(w, x, y) = w

x

y

y

w

(i) c(X) = w

-

-

z

w

(ii) c(X) = x

z

w

w

x

Table 1: Inferences drawn from Example 1

For case (ii), Table 1 leads to a unique representation (P1 , P2 ) with P1 ≡ {(y, z), (x, y), (z, w)}

and

P2 ≡ {(z, x), (y, w), (w, x)}

For case (i), it suggests two representations: (P c \ P2 , P2 ) with rationales P c \ P2 ≡ {(y, z), (x, y), (w, x)}

and

P2 ≡ {(z, x), (y, w), (z, w)}

and (P1 , P c \ P1 ) with rationales P1 ≡ {(y, z), (x, y)}

and

P c \ P1 ≡ {(z, x), (y, w), (z, w), (w, x)}.

Intuitively, the reason for the multiplicity is the indeterminacy in wP c x. By observation (1), every pair in P c must belong to some rationale. However, the induced choices do not depend on whether wP c x is added to P1 or P2 .

III. Identification In this section, we present three results related to identification in the RSM model. First, we define revealed rationales for behavior consistent with the model. Next, we show how these revealed rationales may be used to describe the class of RSM-representations. Finally, we provide a systematic way to modify a given representation without changing the induced choice behavior. a. Revealed Rationales 9

Example 1 suggests that the RSM model is amenable to a simple revealed preference exercise. To fix ideas, let R(c) ≡ {(P11 , P21 ), ..., (P1n , P2n )} denote the collection of all rationale pairs that represent a choice function c. Then: Definition 1 For any RSM-representable choice function c, define: the revealed 1-rationale P1c by aP1c b if aP1j b for all (P1j , P2j ) ∈ R(c); and the revealed 2-rationale P2c by aP2c b if aP2j b for all (P1j , P2j ) ∈ R(c).9 Similar to Masatlioglu et al. [2012], we take a conservative stance on revealed preference. According to our definition, the revealed rationales reflect only those features which are common to every RSM-representation of behavior. In our view, this approach is faithful to Samuelson’s [1938, 1950] idea of providing a direct link between choice behavior and the parameters of the model. Our first result shows that both revealed rationales in Definition 1 are straightforward to determine from choices on nested menus. Generalizing the analysis in Example 1, one can infer aP1c b by observing a “switch” from b to a third alternative when a is added to the menu. In turn, one can infer aP2c b by observing a “reversal” from a to b when other options are added to the menu: Proposition 1 Suppose c is RSM-representable. Then: / {a, b} for some B ⊂ X. (i) aP1c b if and only if c(B) = b and c(B ∪ {a}) ∈ c (ii) aP2 b if and only if c(A) = a and c(B) = b for some {b} ⊂ A ⊂ B ⊆ X. This result establishes the connection with some preference relations defined in prior work on the RSM model. In particular, the revealed 1-rationale coincides with the dominance relation defined by Rubinstein and Salant [2008] while the revealed 2-rationale coincides with the progressive knowledge relation defined by Houy [2008] (see also Cherepanov et al. [2013] who study this relation in the context of a more general model). b. Representations of Behavior 9

Since the revealed rationales are closely related to the revealed preference for RSMrepresentable choice (i.e. P1c , P2c ⊆ P c – see Proposition 2(i) below), we use similar notation.

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In Example 1, we use the revealed rationales to construct representations of behavior. These representations have the appealing feature that they avoid any redundancy—in terms of duplication or conflict—between the two rationales. Definition 2 An RSM-representation (P1 , P2 ) of c is minimal if there are no alternatives a, b ∈ X such that: (i) aP1 b and aP2 b; or, (ii) aP1 b and bP2 a. Our next result characterizes the entire class of minimal representations in terms of the revealed rationales. The result establishes that the logic used to construct representations in Example 1 can be generalized to construct all of the minimal representation for any RSM-representable choice function. To state the result, some definitions are required. Given a choice function c and a relation P , define P c \ P to be the P c -complement of P . Since Manzini and Mariotti’s rationale P1∗ (defined in Section II) is the P c -complement of P2c (i.e. P1∗ = P c \ P2c ), we similarly denote the P c -complement of P1c by P2∗ . Let Pi (c) ≡ {P : Pic ⊆ P ⊆ Pi∗ } denote the rationales (if any) nested between the revealed i-rationale and the P c -complement of the other revealed rationale. Finally, given a rationale Pi from (P1 , P2 ), let P−i denote the other rationale. Proposition 2 Suppose c is RSM-representable. Then, for i = 1, 2: (i) Pic ⊆ Pi∗ so that the interval of rationales Pi (c) is non-empty; and (ii) (P1 , P2 ) is minimal if and only if Pi ∈ Pi (c) and P−i = P c \ Pi . Taken together, (i) and (ii) ensure that there exists a minimal representation for every RSM-representable choice function c.10 What is more, they guarantee that this representation is unique if and only if P1c ∪ P2c = P c .11 Generalizing the insight of Example 1, this result also shows that the pairs c (P1 , P2∗ ) and (P1∗ , P2c ) minimally represent any choice function c consistent with the model. As in Example 1(ii), there is a unique minimal representation (i.e. P1c ∪ P2c = P c ) when these representations coincide. Otherwise, P1c ∪ P2c ⊂ 10

Without (i), Pi (c) may be empty—which means that no representation satisfies (ii). Since P1c , P2c ⊆ P c for RSM-representable choice, P1c ∪P2c = P c is equivalent to P1c ∪P2c ⊇ c P . To put this set inclusion more explicitly in terms of choice: if c(a, b) = a, then a and b must either be involved in a choice “switch” (i.e. aP1c b) or a choice “reversal” (i.e. aP2c b). 11

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P c and, just as in Example 1(i), there is a range of minimal representations between (P1c , P2∗ ) and (P1∗ , P2c ).12 In this case, any assignment of each pair in P in ≡ P c \(P1c ∪P2c ) to exactly one rationale defines a minimal representation.13 Intuitively, P in reflects the indeterminacy associated with identification and thus determines the scope of possible minimal representations. Besides the minimal representations, there are representations involving duplication and/or conflict between the rationales. Given Proposition 2, it is straightforward to characterize the class of all representations: every RSMrepresentation is a minimal representation with some duplication and/or conflict “added” to the second rationale. To illustrate, recall the representation (P1∗ , P c ) due to Manzini and Mariotti. Effectively, this representation “adds” P1∗ = P c \ P2c to the second rationale of the minimal representation (P1∗ , P2c ).14 c. Choice-Invariant Transformations: Given a minimal representation (P1 , P2 ), the induced choice behavior c(P1 ,P2 ) may be affected by moving preference pairs from one rationale to the other. Our last result identifies the class of such transformations that leave choices unchanged. Importantly, these transformations can be determined directly from the representation without considering the induced choice behavior. Let P denote the collection of rationale pairs (P1 , P2 ) such that P1 ∪ P2 is a total15 asymmetric binary relation on X and P1 ∩ P2 = ∅. By Proposition 2, any RSM-representable choice function c is minimally represented by some pair (P1 , P2 ) ∈ P.16 Indeed, P contains every minimal representation of every choice function consistent with the RSM model. Now, suppose (P1 , P2 ) ∈ P is a minimal RSM-representation of c such that (a, b) ∈ P2 . The following 12

In fact, there is even more structure here than indicated. For any RSM-representable c, the collection of minimal representations forms a lattice with meet and join operations defined by (P1 , P2 )∧(Pe1 , Pe2 ) ≡ (P1 ∩ Pe1 , P2 ∪ Pe2 ) and (P1 , P2 )∨(Pe1 , Pe2 ) ≡ (P1 ∪ Pe1 , P2 ∩ Pe2 ). 13 That is not to say that every (P1 , P2 ) such that Pi ∈ Pi (c) for i = 1, 2 defines a minimal representation. As per Proposition 2, it must be that Pi ∈ Pi (c) and P−i = P c \ Pi . 14 In other words, the second rationale duplicates every preference pair in the first. A similar comment applies to the representation (P1c , P c ) proposed by Rubinstein and Salant. 15 A binary relation P on X is said to be total if aP b or bP a for all a, b ∈ X. 16 Given an RSM-representable c that is minimally represented by (P1 , P2 ), it is straightforward to show that P1 ∪ P2 is a total asymmetric binary relation such that P1 ∩ P2 = ∅.

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redundancy condition captures those circumstances where (a, b) can be moved to the first rationale without affecting the induced choice behavior: Definition 3 Given (P1 , P2 ) ∈ P, (a, b) ∈ P2 is (P1 , P2 )-redundant if, for all {ai }ni=1 ⊆ X such that an = a and (ai , ai+1 ) ∈ P1 for i = 1, ..., n − 1: (aj , b) 6∈ P1 for j = 2, ..., n implies (a1 , b) ∈ P1 ∪ P2 . In words, a pair (a, b) ∈ P2 is (P1 , P2 )-redundant if, for any P1 -chain ha1 , ..., ai, the fact that b is not eliminated in the first stage by any alternative in the sub-chain ha2 , ..., ai implies that b is not preferred to a1 by either rationale. A simple example will help to illustrate this condition: Example 2 Consider the following minimal representation (P1 , P2 ) of c: P2

P1 y y x v

z v v w

v w w w y z

z z x y x x

In this example, the pair (z, x) is redundant. To see this, note that the only P1 -chain ending in z is hy, zi. Since (y, x) ∈ P2 , the redundancy condition is satisfied. In fact, (y, x) (which vacuously satisfies the redundancy condition) is the only other redundant pair in P2 . To see that (w, z) is not redundant, for instance, note that the P1 -chain hx, v, wi satisfies the premise of the redundancy condition. Since (x, z) ∈ / P1 ∪ P2 however, the condition is violated.17 It is easy to check that a pair in P2 can be moved to P1 without affecting choice behavior precisely when it satisfies the redundancy condition. In other 17

Similar reasoning establishes that none of the other pairs in P2 is redundant. For (v, z), consider the P1 -chain hx, vi. And, for (w, y) and (w, x), consider the P1 -chain hv, wi.

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words, choice is unaffected by moving (z, x) and (y, x) to P1 . On the other hand, choice is affected by moving any other pair in P2 . An appealing feature of the redundancy condition is that it also determines which pairs can be moved in the other direction. To move a pair (a, b) ∈ P1 to P2 without affecting choice, it must be that, after moving the pair to P2 , choice behavior is unchanged when it is moved back to P1 . In other words, (a, b) must be (P1 \ (a, b), P2 ∪ (a, b))-redundant.18 In fact, the only other requirement is that (P1 \ (a, b), P2 ∪ (a, b)) represents a choice function. In general, there are rationale pairs in P that induce “empty choice” for some menus. This occurs when there are cycles in P1 or cycles in P2 among the alternatives that “survive” P1 . To rule out empty choice, it is enough that P1 is acyclic and P1 “breaks” any cycle in P2 . Formally, P2 is P1 -acyclic if, for every cycle {ai }ni=1 in P2 , there exist aj , ak ∈ {ai }ni=1 such that aj P1 ak . Let PRSM ≡ {(P1 , P2 ) ∈ P : P1 is acyclic and P2 is P1 -acyclic} denote the sub-collection of such rationale pairs in P. Lemma 2 of the Appendix shows that these are precisely the pairs in P which induce choice functions. Returning to Example 2, it is easy to see that (y, z) is (P1 \(y, z), P2 ∪(y, z))redundant and (P1 \ (y, z), P2 ∪ (y, z)) ∈ PRSM . The only other such pair in P1 is (y, v). As above, it is easy to check that choice behavior is unaffected by moving (y, z) and/or (y, v) from P1 to P2 but not by moving (x, v) or (v, w). Proposition 3 Suppose (P1 , P2 ) is a minimal representation of c. Then: (i) For (a, b) ∈ P2 , the rationale pair (P1 ∪ (a, b), P2 \ (a, b)) is a minimal representation of c if and only if (a, b) is (P1 , P2 )-redundant. (ii) For (a, b) ∈ P1 , the rationale pair (P1 \ (a, b), P2 ∪ (a, b)) is a minimal representation of c if and only if (a, b) is (P1 \ (a, b), P2 ∪ (a, b))-redundant and (P1 \ (a, b), P2 ∪ (a, b)) ∈ PRSM . Like the “uniqueness up to affine transformations” result for models with cardinal utility representations, Proposition 3 specifies how to transform any minimal representation without affecting choice behavior. More generally, one 18

To avoid clutter, we use (P1 \ (a, b), P2 ∪ (a, b)) to denote (P1 \ {(a, b)}, P2 ∪ {(a, b)}).

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may be interested in choice-invariant transformations of RSM-representations that involve duplication or conflict between the rationales.19 Proposition 3 provides the key to characterizing choice-invariance in this setting as well. Given an RSM-representation (P1 , P2 ), first strip away the duplication/conflict from P2 and consider the minimal representation (P1 , P2/1 ) where P2/1 ≡ P2 \ (P1 ∪ P1−1 ). By applying Proposition 3 to (P1 , P2/1 ), one can then determine which transformations of (P1 , P2 ) are choice-invariant. d. Discussion To close, we emphasize that Proposition 3 provides the same insights as Propositions 1 and 2 when the representation itself is treated as primitive—a practice which is relatively common. In applied theory, for example, it is often more practical to work with representations directly rather than choice behavior. In this case, it is easier to appeal to Proposition 3. For this reason, we are inclined to view this result as complementary to Propositions 1 and 2. To see how Proposition 3 can be used to identify the revealed rationales from any minimal representation, consider the representation (P1 , P2 ) in Example 2. To identify the revealed rationales using Proposition 1, one must first determine the induced choice behavior c(P1 ,P2 ) . Then, it follows that P1c = {(x, v), (v, w)} and P2c = {(v, z), (w, z), (w, y), (w, x)}. In this example, the pairs in Pic (for i = 1, 2) are exactly those pairs in Pi which, according to Proposition 3, cannot be moved to P−i without affecting choice. The relationship is not particular to this example. In other words, a preference pair belongs to one of the revealed rationales if and only if the transformation related to that pair is not choice-invariant. This shows that, when one has a representation of behavior, Proposition 3 can be used to determine the revealed rationales without first passing through choices. This, in turn, provides a different way to characterize the minimal representations. It is clear that any choice-invariant transformation of a minimal representation gives another minimal representation (of the same choices). The 19

Here, one may also want to consider transformations that add or delete preference pairs.

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result stated in the last paragraph ensures that every minimal representation can be obtained in this way. In particular, a pair of rationales is a minimal representation if and only if it can be obtained from any other minimal representation through a combination of choice-invariant transformations. This shows that, when one has a representation of behavior, Proposition 3 can be used to determine all of the minimal representations directly.

IV. Implications: Inferences and Policy In this section, we examine the implications of our results for drawing inferences from behavior and for using these inferences to evaluate policy. To keep the discussion relatively brief, we focus on Examples I-V of the Introduction. a. Inferences from Identification Examples I-III address situations where the analyst wants to draw inferences about the rationales that are consistent with RSM-representable behavior. When the analyst is an economist or a market participant, the objective may simply be to develop a better understanding of a decision-maker who is believed to follow a rational shortlist method. When the analyst is a policy maker, the exercise is more likely to be motivated by welfare concerns. When there is “good reason”20 to believe that the decision-maker follows a rational shortlist method and, moreover, that one of the rationales reflects the decision-maker’s “true” preference (as in Examples II-III), identifying the features of this rationale will likely help the policy maker make better policy judgments. Broadly, this is the model-based approach to welfare advocated in some recent papers (see e.g. Rubinstein and Salant [2012]). In contrast, some have advocated a model-free concept of welfare. One such proposal is to adopt a welfare criterion based on Pareto-dominance (Bernheim and Rangel [2007, 2009]). Given the sub-domain D ⊆ 2X \ ∅ of choice situations judged relevant for welfare analysis, the Bernheim-Rangel (BR) welfare relation W is defined by aW b if c(A) 6= b for all A ∈ D such that a ∈ A. 20

The quotation marks emphasize that we take no position on what counts as good reason.

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Recently, there has been considerable debate about which approach is more appropriate for welfare analysis when choice behavior is inconsistent with preference maximization (see Rubinstein-Salant [2012] and Bernheim [2009] for an overview). The debate involves subtle issues that extend well beyond the scope of our work and we take no position on the matter. We simply wish to illustrate how identification results can help inform the discussion. Using Proposition 1, one can show the following: Remark 1 Suppose (P1 , P2 ) is an RSM-representation of c and D = 2X \ ∅. (i) P1 ⊆ W = P1∗ so that the BR welfare relation over-estimates P1 ; and (ii) P2c ∩ W = ∅ so that the BR welfare relation need not overlap with P2 . Part (ii) shows that there may be no relationship between the BR welfare relation and the “true” preference parameter of a plausible model consistent with behavior. Several papers (Manzini and Mariotti [2012b]; Masatlioglu et al. [2012]; Rubinstein and Salant [2012]) illustrate this point for models of choice related to rational shortlist methods. The novelty of Remark 1(ii) is to show that, for the RSM model, the disconnect is not limited to carefully contrived examples (as in the cited papers). Indeed, there is never any connection between the BR welfare relation and the revealed 2-rationale. Part (i) makes a related point. One criticism of the model-based approach is that it relies on a particular interpretation of what constitutes “true” preference in the model (Bernheim [2009]). As illustrated by Examples II-III, there are plausible interpretations of the RSM model where either rationale may represent the “true” preference. While this is certainly a word of warning against the model-based approach, we do not see it as an endorsement of the model-free approach. To the contrary, Remark 1(i) shows that the model-free approach cannot escape questions of interpretation simply by avoiding them. For choice data consistent with the RSM model, using the BR relation for policy necessarily attaches welfare significance to the first rationale. Setting aside the issue of welfare, Examples I-III illustrate that the analyst may have good reason to be interested in the rationales that represent behavior. This raises the question of what can be justifiably inferred from behavior. As 17

is clear from our analysis, we generally favor the conservative approach to revealed preference captured by Proposition 1. In our view, the analyst should be cautious about inferring more than what is directly revealed by behavior. At the same time, the analyst may sometimes be justified in taking a more liberal approach. One situation is where the analyst believes that “boundedly rational” distortions play a minimal role in explaining behavior (as in Cherepanov et al. [2013]). In Example II, for instance, the analyst may be confident that the tie-breaking rule has a limited influence on behavior. In that case, it might be argued that P1∗ provides a more complete understanding of “true” preference than the revealed rationale P1c . Similarly, an analyst who takes a dim view of the role played by consideration sets in Example III might favor P2∗ over P2c . The real value of Proposition 2 is that it characterizes the range of possible inferences about either rationale. b. Policy from Identification Examples IV-V address situations where a seller wants to determine the impact of advertising on consumer behavior. When consumers choose by rational shortlist methods, a firm may become more competitive not only by making its product more appealing but also by affecting the composition of the rationales used by consumers. If the first rationale reflects the product comparisons that are more salient to consumers and the second rationale those that are less salient, a firm may conceivably “promote” comparisons to the first rationale with more aggressive comparative advertising or, conversely, “demote” comparisons by obfuscating the relationship with competing products. When the firm only has access to choice data (as in Example IV), Proposition 2 provides a straightforward way to determine every pair of rationales that is consistent with the data. This makes it possible for the firm to evaluate whether “promoting” a particular comparison by comparative advertising will have an impact on consumer choices. When the firm has an accurate picture of the rationales (as in Example V), Proposition 3 provides a simple way to directly infer all modifications of the rationales that leave consumer choices unchanged. To determine whether it can save on advertising costs in this case,

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there is no need for the firm to consider consumer choices directly. As this discussion illustrates, our results make it possible to study comparative advertising in the context of a simple model. Since this practice is widespread but poorly understood in terms of theory (Grewal et al. [1997]), we feel that this application of the RSM model merits further investigation. In the same vein, we believe that Examples II-III suggest interesting applications of the RSM model where our results might provide considerable insight. In these examples, the tie-breaking rule and the consideration set were viewed as nuisance parameters. While this is certainly the case for the analyst interested in “true” preference, sellers may take a more positive view. Like comparative advertising, both are non-price mechanisms that can be leveraged to “nudge” consumers into making product choices that are more profitable for the seller (Thaler and Sunstein [2008]; Manzini et al. [2011]). The forced choice model in Example II might be used to study seller behavior when consumers have a limited ability to compare products. In this model, sellers would vie to influence the tie-breaking rules that consumers use to choose among incomparable products. This complements the approach of Piccione and Spiegler [2012], who study the influence of sellers on consumers’ ability to compare products (but not their tie-breaking rules).21 In turn, the limited consideration model in Example III might be used to study aspects of competitive advertising not captured by the model discussed in Examples IV-V or the models studied by Eliaz and Spiegler [2011a, 2011b].

V. Conclusion We study identification in Manzini and Mariotti’s [2007] model of procedural choice. We provide simple definitions of revealed preference (Proposition 1), characterize the set of representations (Proposition 2), and determine the choice-invariant transformations of every representation (Proposition 3). Each of our results has a direct analog in the setting of utility maximization. Indeed, 21

In their model, a consumer’s purchase is randomly assigned among the “tied” sellers.

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part of our objective was to show that some procedural models are amenable to the same kind of identification exercise as more traditional models. As we emphasized, our results are useful for applied work on the RSM model. To close, we mention several implications for more theoretical work: Axiomatics: Our results help shed light on the axiomatics of rational shortlist methods and related two-stage choice procedures. Regarding the RSM model, Horan [2013] establishes that one of Manzini and Mariotti’s axioms (WWARP ) amounts to the incompatibility between choice “switches” and “reversals” involving the same alternatives.22 In light of his result, Proposition 1 provides a deeper understanding of the role played by this axiom: it guarantees that the revealed rationales are disjoint. In turn, Proposition 2 provides key insights into RSM-type models where the rationales have additional structure. Two papers (Horan [2013]; Matsuki and Tadenuma [2013]) use this result to show that—by adding appropriate axioms to those of Manzini and Mariotti—it is possible to transform a minimal representation into one with the desired structure. In fact, this approach might be used for any special case of the RSM model. Thus, Proposition 2 effectively reduces the characterization of these models to the search for axioms that allow one to make the desired transformations to minimal representations.23 Identification in Other Models: Our approach to identification can be extended to other models of procedural choice. In recent work, Horan [2013] uses it to similar effect for the case of two transitive rationales; and, it seems possible to extend this to the other special cases of the RSM model cited above. More broadly, our approach extends to models where the representations have a lattice-like structure (see footnote 12). For the RSM model, this is the 22

As described in Section III.a (and before Proposition 1): adding a to A causes a “switch” from b if c(A) = b and c(A ∪ {a}) 6∈ {a, b}; and, adding alternatives to A ⊃ {b} causes a “reversal” from a to b if c(A) = a and c(B) = b for some B ⊃ A. 23 For the acyclic RSM model (Houy [2008]), the task is particularly straightforward because no transformation is even required. Since P1 is acyclic in every RSM-representation, the only concern is the acyclicity of P2 . Given a choice function c, it is enough to consider the minimal RSM-representation (P1∗ , P2c ). Since no RSM-representation allows for less in P2 (by Proposition 2), the acyclicity of P2c is necessary for c to have an acyclic RSMrepresentation. Since this condition is also sufficient, one obtains Houy’s characterization.

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key feature that allowed us to characterize revealed preference and the class of representations. A related model with this structure is the RSM model where choice may be multi-valued (Alcantud and Garc´ıa-Sanz [2010]). By extending Propositions 1 and 2 to this setting, it is possible to generalize a result of Tyson [2013], who provides a characterization of revealed preference (but not the representations) in the special case where P2 is negative transitive. Having said this, some related two-stage models, like those of Lleras et al. [2011] or Masatlioglu et al. [2012], are not amenable to our approach. For these models, the issue is that the representations do not have a lattice structure. We suspect that this could make it difficult to characterize the class of representations in terms of revealed preference (along the lines of Proposition 2).24 For other related models, like the extension of the RSM model to more than two rationales (Apesteguia and Ballester [2010]; Manzini and Mariotti [2012a]), it is not entirely clear whether our approach can be applied. Identification with Limited Data: While our work assumes that the choices from every menu in the domain 2X \ ∅ are observable, it can be extended to the case where some choices are unobserved. Importantly, the “if” direction of Proposition 1 continues to hold with limited data (i.e. choice “switches” and “reversals” still permit the same inferences about the content of the two rationales). Taking this as a point of departure, we have identified conditions (available on request) that are necessary and sufficient for choices from any collection of menus D ( 2X \ ∅ to be consistent with the RSM model (in the sense of de Clippel and Rozen [2014]). In related work, these authors characterize limited data consistent with a more general model (due to Cherepanov et al. [2013]; Manzini and Mariotti [2012b]; and, Spears [2011]). While their condition is also necessary for the RSM model, it applies only when the data is sufficiently “rich” (e.g. by including the choices from all two-element sets). In contrast, our conditions apply to every limited dataset.

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In turn, this could explain why neither of these papers provides such a characterization.

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IV. References Alcantud, Jos´ e Carlos, and Mar´ıa Garc´ıa-Sanz. 2010. “Rational Choice by Two Sequential Criteria.” Universidad de Salamanca working paper. Apesteguia, Jos´ e, and Miguel Ballester. 2010. “Choice by Sequential Procedures.” Universitat Pompeu Fabra working paper version. Au, Pak Hung, and Keiichi Kawai. 2011. “Sequentially Rationalizable Choice with Transitive Rationales.” Games and Economic Behavior 73(2) 608–614. Bernheim, Douglas. 2009. “Behavioral Welfare Economics.” Journal of the European Economic Association 7(2-3) 267–319. Bernheim, Douglas, and Antonio Rangel. 2007. “Toward Choice-Theoretic Foundations for Behavioral Welfare Economics.” American Economic Review (Papers and Proceedings) 97(2) 464–470. Bernheim, Douglas, and Antonio Rangel. 2009. “Beyond Revealed Preference: Choice-Theoretic Foundations for Behavioral Welfare Economics.” Quarterly Journal of Economics 124(1) 51–104. Cherepanov, Vadim, Timothy Feddersen, and Alvaro Sandroni. 2013. “Rationalization.” Theoretical Economics 8 775–800. de Clippel, Geoffroy, and Kareen Rozen. 2014. “Bounded Rationality and Limited Datasets: Testable Implications, Identifiability, and Out-of-Sample Prediction.” Brown University working paper. Dekel, Eddie, and Barton Lipman. 2010. “How (Not) to Do Decision Theory.” Annual Review of Economics 2 257–282. Dulleck, Uwe, Franz Hackl, Bernhard Weiss, and Rudolf Winter-Ebmer. 2011. “Buying Online: An Analysis of Shopbot Visitors.” German Economic Review 12(4) 395–408. Eliaz, Kfir, and Ran Spiegler. 2011a. “Consideration Sets and Competitive Marketing.” Review of Economic Studies 79(4) 235–262. Eliaz, Kfir, and Ran Spiegler. 2011b. “On the Strategic Use of Attention Grabbers.” Theoretical Economics 6 127–155. Grewal, Dhruv, Sukumar Kavanoor, Edward Fern, Carolyn Costley, and James Barnes. 1997. “Comparative versus Noncomparative Advertising: A MetaAnalysis.” Journal of Marketing 61(4) 1–15.

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Horan, Sean. 2013. “A Simple Model of Biased Choice.” UQAM working paper. Houy, Nicolas. 2008. “Progressive Knowledge Revealed Preferences and Sequen´ tial Rationalizability.” Ecole Polytechnique working paper. Llerars, Juan Sebastian, Yusufcan Masatlioglu, Daisuke Nakajima, and Erkut Ozbay. 2011. “When More is Less: Limited Consideration.” University of Maryland working paper. Manzini, Paola, and Marco Mariotti. 2007. “Sequentially Rationalizable Choice.” American Economic Review 97(5) 1824–1839. Manzini, Paola, and Marco Mariotti. 2012a. “Choice by Lexicographic Semiorders.” Theoretical Economics 7 1–23. Manzini, Paola, and Marco Mariotti. 2012b. “Categorize then Choose: Boundedly Rational Choice and Welfare.” Journal of the European Economic Association 10(5) 1141–1165. Manzini, Paola, Marco Mariotti, and Christopher Tyson. 2011. “Manipulation of Choice Behavior.” University of St. Andrews working paper version. Masatlioglu, Yusufcan, Daisuke Nakajima, and Erkut Ozbay. 2012. “Revealed Attention.” American Economic Review 102(5) 2183–2205. Matsuki, Jun, and Koichi Tadenuma. 2013. “Choice via Grouping Procedures.” Hitotsubashi University working paper. Piccione, Michele, and Ran Spiegler. 2012. “Price Competition under Limited Comparability.” Quarterly Journal of Economics 127(1) 97–135. Rubinstein, Ariel, and Yuval Salant. 2008. “Some Thoughts on the Principle of Revealed Preference.” In Caplin and Schotter (Eds.), The Foundations of Positive and Normative Economics: A Handbook. New York: Oxford University Press. Rubinstein, Ariel, and Yuval Salant. 2012. “Eliciting Welfare Preferences from Behavioural Data Sets.” Review of Economic Studies 79(1) 375–387. Samuelson, Paul. 1938. “A Note on the Pure Theory of Consumer’s Behaviour.” Economica (New Series) 5(17) 61–71. Samuelson, Paul. 1950. “The Problem of Integrability in Utility Theory.” Economica (New Series) 17(68) 355–385. Sen, Amartya. 1971. “Choice Functions and Revealed Preferences.” Review of Economic Studies 38(3) 307–317.

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Spears, Dean. 2011. “Intertemporal Bounded Rationality as Consideration Sets with Contraction Consistency.” B.E. Journal of Theoretical Economics 11(1). Spiegler, Ran. 2008. “On Two Points of View Regarding Revealed Preferences and Behavioral Economics.” In Caplin and Schotter (Eds.), The Foundations of Positive and Normative Economics. New York: Oxford University Press. Spiegler, Ran. 2011. Bounded Rationality and Industrial Organization. New York: Oxford University Press. Thaler, Richard, and Cass Sunstein. 2008. Nudge: Improving Decisions about Health, Wealth, and Happiness. New York: Yale University Press. Tversky, Amos, Shmuel Sattath, and Paul Slovic. 1988. “Contingent Weighting in Judgment and Choice.” Psychological Review 95(3) 371–384. Tyson, Christopher. 2013. “Behavioral Implications of Shortlisting Procedures.” Social Choice and Welfare 41(4) 941–963. Wright, Peter, and Fredrick Barbour. 1977. “Phased Decision Strategies: Sequels to an Initial Screening.” In Starr and Zeleny (Eds.), Studies in Management Sciences, Multiple Criteria Decision Making. Amsterdam: North-Holland.

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V. Mathematical Appendix For convenience, we restate the result of Manzini and Mariotti (M&M) [2007]: Theorem 1 (M&M) c is RSM-representable if and only if it satisfies: Expansion

If c(A) = x = c(B), then c(A ∪ B) = x; and

WWARP If c(A) = x = c(x, y), then c(B) 6= y for any B s.t. {x, y} ⊂ B ⊂ A.

a. Proof of Propositions 1 to 3 To simplify the notation in the proof of Proposition 1, define: - Pe1c by aPe1c b if c(B) = b and c(B ∪ {a}) 6∈ {a, b} for some B ⊂ X; and, - Pe2c by aPe2c b if c(A) = a and c(B) = b for some {b} ⊂ A ⊂ B ⊆ X. Lemma 1 If c is RSM-representable, then (i) Pe1c ⊆ P c and (ii) Pe2c ⊆ P c . Proof. (i) Suppose aPe1c b. By definition, c(B) = b and c(B ∪ {a}) 6∈ {a, b} for some B ⊂ X. If c(a, b) = b, then c(B ∪ {a}) = b by Expansion, which is a contradiction. So, c(a, b) = a. (ii) Suppose aPe2c b. By definition, c(A) = a and c(B) = b for some {b} ⊂ A ⊂ B ⊆ X. By WWARP, c(a, b) = a. Proof of Proposition 1. (⇐) (ii) Fix (P1 , P2 ) ∈ R(c) and suppose aPe2c b. By definition, c(A) = a and c(B) = b for some {b} ⊂ A ⊂ B ⊆ X. So, (a, b) ∈ / P1 since c(B) = b. Similarly, (b, a) ∈ / P1 since c(A) = a. Since c(a, b) 6= ∅, either bP2 a or aP2 b. By way of contradiction, suppose bP2 a. Then, b 6∈ max(A; P1 ) since c(A) = a. Since A ⊂ B, b ∈ / max(B; P1 ) as well, contradicting c(B) = b. (i) Fix (P1 , P2 ) ∈ R(c) and suppose aPe1c b. By definition, c(B) = b and c(B ∪ {a}) ≡ z 6∈ {a, b} for some B ⊂ X. Since c(B) = b, (x, b) ∈ / P1 for all x ∈ B \ {b}. By way of contradiction, suppose (a, b) ∈ / P1 . Then, clearly b ∈ max(B ∪ {a}; P1 ). Since c(B) = b and c(B ∪ {a}) = z, bP2 z by (ii) above. Since b ∈ max(B ∪ {a}; P1 ) however, c(B ∪ {a}) 6= z, which is a contradiction. (⇒) We show that (i) (Pe1c , P c \ Pe1c ) and (ii) (P c \ Pe2c , Pe2c ) both represent c. (ii) From the proof of Theorem 1 (M&M), (Pe1∗ , P c ) represents c where Pe1∗ is the P c -complement of Pe2c . Since it only removes duplicate comparisons from 25

the second rationale, (Pe1∗ , P c \ Pe1∗ ) also represents c. Since Pe2c ⊆ P c by Lemma 1 and Pe1∗ ≡ P c \ Pe2c , P c \ Pe1∗ = Pe2c as well. So, (P c \ Pe2c , Pe2c ) represents c. (i) Suppose c(D) = a for some D ⊂ X. First, we show a ∈ max(D; Pe1c ). By way of contradiction, suppose bPe1c a for some b ∈ D. By definition, c(A) = a and c(A∪{b}) ≡ z ∈ / {a, b} for some A ⊂ X. From these choices, it follows that c(a, z) = a (by WWARP) and c(a, b) = b (by Expansion). Since c(A ∪ D) = a by Expansion, the choices on {a, z} ⊂ A ∪ {b} ⊂ A ∪ D violate WWARP. So, bPe1c a for no b ∈ D, which, in turn, establishes a ∈ max(D; Pe1c ). Next, we show a ∈ max(max(D; Pe1c ); P c \ Pe1c ). To do so, we claim that, for any b ∈ D such that (b, a) ∈ P c \ Pe1c , there exists an x ∈ D such that xPe1c b. To prove the claim, first note that c(D \ {y}) = a for some y ∈ D. Otherwise, by the pigeonhole principle, c(D \ {y 0 }) = d = c(D \ {y 00 }) for distinct y 0 , y 00 ∈ D. Then, by Expansion, c(D) = d 6= a, which is a contradiction. Extending this logic, recursively define a sequence of sets {Di }: D1 ≡ D and Di+1 ≡ Di \ {yi } where yi ∈ Di is chosen so that c(Di+1 ) = a. By the logic above, this sequence is well-defined. Assuming |D| = n + 1, Dn = {a}. Now, fix some b ∈ D such that (b, a) ∈ P c \ Pe1c and consider Di ∪ {b} for i = 1, ..., n. Since (b, a) ∈ / Pe1c (by assumption) and c(Di ) = a (by construction), c(Di ∪{b}) ∈ {a, b}. Since c(Dn ∪{b}) = c(a, b) = b and c(D1 ∪{b}) = c(D) = a by construction and, moreover, D is finite, there exists a smallest i such that c(Di ∪ {b}) = a and c(Di+1 ∪ {b}) = b. Denote this i by j. By construction, Dj \ Dj+1 = {yj }. Then, yj Pe1c b, which establishes the claim for x ≡ yj . So, a ∈ max(max(D; Pe1c ); P c \ Pe1c ). Since Pe1c ⊆ P c by Lemma 1 and P c is total, this set must be single valued, which delivers the desired result. Note: Below, we treat the equivalences from Proposition 1 as definitional. Proof of Proposition 2. (i) First, we show P1c ∩ P2c = ∅. By way of contradiction, suppose aP1c b and aP2c b. Then: c(B) = b and c(B ∪ {a}) ≡ z ∈ / {a, b} for some B ⊂ X; and, c(A) = a and c(D) = b for some {b} ⊂ A ⊂ D ⊆ X. By Expansion, c(B ∪D) = b. But this is a contradiction. By WWARP, the nested sets {b, z} ⊂ B ⊂ B ∪ {a} ⊂ B ∪ D require c(b, z) = z and c(b, z) = b. 26

To complete the proof, observe that P1c ∪ P2c ⊆ P c by Lemma 1. Since c P1c ∩ P2c = ∅, it then follows that Pic ⊆ P c \ P−i ≡ Pi∗ for i = 1, 2. (ii) (⇐) Fix any P1 ∈ P1 (c) and P2 ∈ P2 (c). The proof of Proposition 1 shows that (P1c , P2∗ ) and (P1∗ , P2c ) represent c. We now show that (P1 , P c \ P1 ) represents c. Similar reasoning establishes the same for (P c \ P2 , P2 ). Since Pi ⊆ Pi∗ ⊆ P c for i = 1, 2 (by assumption) and P c is a total asymmetric relation (by construction), it follows that these rationale pairs are minimal. Fix some A ⊆ X and suppose c(A) = y. Since c(P1∗ ,P2c ) = y, y ∈ max(A; P1∗ ). Since P1c ⊆ P1 ⊆ P1∗ , y ∈ max(A; P1 ) ≡ A1 ⊆ max(A; P1c ). Since y ∈ max(max(A; P1c ); P2∗ ), y ∈ max(A1 ; P2∗ ). And, since P c \ P1 ⊆ P2∗ , it follows that y ∈ max(A1 ; P c \ P1 ). In other words, y ∈ max(max(A; P1 ); P c \ P1 ). Since P1 ⊆ P1∗ ⊆ P c and P c is total, it follows that max(max(A; P1 ); P c \P1 ) must be single valued, which, in turn, delivers the required result. (⇒) Suppose (P1 , P2 ) is minimal. By minimality, P1 ∩ P2 = ∅. Since (P1 , P2 ) represents c, P1 ∪ P2 ⊇ P c . If P1 ∪ P2 6= P c , then there exist x, y ∈ X such that (1) xP1 y and yP1 x or (2) xP2 y and yP2 x. Since both contradict asymmetry, P1 ∪ P2 = P c . Thus, P2 = P c \ P1 and P1 = P c \ P2 . Since Pic ⊆ Pi for i = 1, 2 (by Proposition 1), the two equalities deliver the desired result. Lemma 2 If c is RSM-representable, some (P1 , P2 ) ∈ PRSM is a minimal representation of c. Conversely, c(P1 ,P2 ) is an RSM for any (P1 , P2 ) ∈ PRSM . Proof. (⇒) Fix some RSM-representable choice function c. By Proposition 2, c has a minimal representation (P1 , P2 ) in P. Since (P1 , P2 ) never induces empty choice, it must belong to PRSM (by the argument in the text). (⇐) Fix some (P1 , P2 ) ∈ PRSM . By the argument in the text, c(P1 ,P2 ) (A) is non-empty for all A ⊆ X. So, it suffices to show that c(P1 ,P2 ) is single-valued. Since PRSM ⊆ P, P1 ∪ P2 is total. To see that this rules out multi-valued choice, suppose {x, y} ⊆ c(P1 ,P2 ) (A) for some A ⊆ X and x 6= y. Then, it must be that (x, y) 6∈ P1 ∪ P2 and (y, x) 6∈ P1 ∪ P2 , which contradicts totality. Proof of Proposition 3. (i) Let (P1 , P2 ) be a minimal representation of c. 27

(⇐) Suppose (x, y) is (P1 , P2 )-redundant. Let Pe1 ≡ P1 ∪ (x, y) and Pe2 ≡ P2 \ (x, y). Since (P1 , P2 ) is minimal, (Pe1 ∪ Pe1−1 ) ∩ (Pe2 ∪ Pe2−1 ) = ∅. To prove the result, it suffices to show that (Pe1 , Pe2 ) represents c. Fix some A ⊆ X. If {x, y} 6⊆ A, then c(Pe1 ,Pe2 ) (A) = c(P1 ,P2 ) (A) = c(A). If {x, y} ⊆ A, then max(A; Pe1 ) = max(A; P1 ) \ {y}. To establish the result, we show that c(A) 6= y in this case. To see this, suppose instead that c(A) = y. First, suppose there is a sequence of n distinct elements hai ini=1 with an = x, ai ∈ A, and ai 6= y for 1 ≤ i ≤ n such that (i) (ai , ai+1 )1≤i≤n−1 ∈ P1 and (ii) (aj , y)2≤j≤n ∈ P2 .25 Call any such sequence an α sequence. Since P1 ∩ P2 = ∅, (ii) implies (aj , y)2≤j≤n 6∈ P1 . Since (x, y) is (P1 , P2 )redundant, (a1 , y) ∈ P1 ∪P2 . If (a1 , y) ∈ P1 , then clearly y 6∈ max(A; P1 ). Since c(A) = y by assumption, it must be that (a1 , y) ∈ P2 . So, for c(P1 ,P2 ) (A) = y to hold, it must be that a1 6∈ max(A; P1 ). Therefore, there exists some z ∈ A s.t. (z, a1 ) ∈ P1 . Note that z must be distinct from the elements of the α sequence. If z = ai for some 3 ≤ i ≤ n, then c(P1 ,P2 ) ({aj }ij=1 ) = ∅, which contradicts the fact that (P1 , P2 ) represents c. So, there exists an α sequence hbi in+1 i=1 of n + 1 elements with bi = ai−1 for all 2 ≤ i ≤ n + 1 and b1 = z. This establishes the following: if c(A) = y for some A ⊇ {x, y} and there exists an α sequence of n elements in A, then there exists α sequence of n + 1 elements in A. Since A is finite, the existence of any α sequence in A yields a contradiction. By the argument in the last paragraph (with x = a1 ), hz, xi is an α sequence such that (z, x) ∈ P1 and (x, y) ∈ P2 for some z ∈ A \ {x, y}. This yields the required contradiction. So, c(A) 6= y for all A ⊇ {x, y}. Since (P1 , P2 ) is minimal, P2 ∈ P2 (c) by Proposition 2. Since c(A) 6= y for all A ⊇ {x, y}, (x, y) 6∈ P2c by definition of P2c . So, P2c ⊆ Pe2 ⊂ P2 ⊆ P2∗ . Moreover, Pe1 = P c \ Pe2 (since Pe1 ∪ Pe2 = P1 ∪ P2 = P c ). By Proposition 2, this implies that (Pe1 , Pe2 ) is a minimal representation of c. (⇒) We prove the contrapositive. Suppose that (P1 , P2 ) is a minimal representation of c and fix some (x, y) ∈ P2 that is not (P1 , P2 )-redundant. We show that (Pe1 , Pe2 ) does not represent c with Pe1 ≡ P1 ∪ (x, y) and Pe2 ≡ P2 \ (x, y). Since (x, y) is not (P1 , P2 )-redundant, there is a sequence {ai }ni=1 of minimal 25

Let (ai , ai+1 )j≤i≤k ∈ P (∈ / P ) denote that (ai , ai+1 ) ∈ P (∈ / P ) for all j ≤ i ≤ k.

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length n s.t. an = x, (ai , ai+1 )1≤i≤n−1 ∈ P1 and (aj , y)2≤j≤n ∈ P2 but (a1 , y) 6∈ P1 ∪ P2 . Since (P1 , P2 ) is a minimal representation, (a1 , y) 6∈ P1 ∪ P2 implies (y, a1 ) ∈ P1 ∪ P2 . The relevant parts of the rationales are depicted in Table 2: P1 a1 a2 a3

P2 a2 a3 a4

x a2 a3

y y y

· ·

· ·

an−1 x · ·

an−1 y · ·

Table 2: A smallest sequence violating the redundancy condition

Notice that (y, a1 ) has to be added to one of the rationales. Consider the menu A = {ai }ni=1 ∪ {y}. Then, max(A; P1 ) must be {a1 , y} or {y} (depending on whether (y, a1 ) is in P1 or P2 ). In either case, c(P1 ,P2 ) (A) = y. Since (P1 , P2 ) represents c, c(A) = y. On the other hand, y 6∈ max(A; Pe1 ) since x ∈ A and (x, y) ∈ Pe1 . So, (Pe1 , Pe2 ) cannot represent c. (ii) Suppose (P1 , P2 ) is a minimal representation of c. Let Pe1 ≡ P1 \ (x, y) and Pe2 ≡ P2 ∪(x, y). (⇐) By Lemma 2, (Pe1 , Pe2 ) ∈ PRSM must represents some RSM. Since (x, y) is (Pe1 , Pe2 )-redundant, (P1 , P2 ) represents the same choice function as (Pe1 , Pe2 ) by part (i) above. So, (Pe1 , Pe2 ) represents c. (⇒) Suppose that (Pe1 , Pe2 ) represents c. By Lemma 2, (Pe1 , Pe2 ) ∈ PRSM . To see that (x, y) is (Pe1 , Pe2 )-redundant, suppose not. Then, (P1 , P2 ) does not represent c by part (i) above, which is a contradiction. b. Proof of Remarks 1 and 2 Lemma 3 If (P1 , P2 ) is an RSM-representation of c, then P1 ⊆ P1∗ . Proof. By way of contradiction, suppose xP1 y and (x, y) ∈ / P1∗ . Since (P1 , P2 ) represents c, xP1 y implies xP c y. In combination with (x, y) ∈ / P1∗ , this implies 29

xP2c y since P2c = P c \P1∗ (as shown in the proof of Proposition 1). By definition of P2c , c(A) = y for some A ⊃ {x, y}. But, this contradicts the fact that xP1 y and the assumption that (P1 , P2 ) represents c. So, P1 ⊆ P1∗ as required. Proof of Remark 1. (i) By construction, P1∗ = P c \ P2c . Thus, W = P1∗ by definition. By Lemma 3, P1 ⊆ P1∗ which gives the desired result. (ii) First, suppose xP2c y. By definition, c(A) = y for some A ∈ D ≡ 2X \ ∅ such that x ∈ A. So, (x, y) ∈ / W . Next, suppose xW y. By definition, there exists no A ∈ D such that c(A) = y and x ∈ A. So, (x, y) ∈ / P2c . Remark 2 (i) The choice behavior in Example 1 is consistent with the RSM model. (ii) For c(X) to be consistent with the RSM model, c(X) ∈ {w, x}. Proof. (i) Since the only observed choices are from two- and three-element sets, c trivially satisfies WWARP. By footnote 8, c also satisfies Expansion. (ii) First, c(X) = y violates WWARP because c(w, y) = y and c(w, x, y) = w. Similarly, c(X) = z violates WWARP because c(x, z) = z and c(x, y, z) = x. However, neither c(X) = x nor c(X) = w violates WWARP or Expansion.

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