Rational Choice with Categories

Transcription

1 Rational Choice with Categories Preliminary version. Bruno A. Furtado Leandro Nascimento Gil Riella June 25, 2015 Abstract We propose a model of rational choice in the presence of categories. Given a subjective categorization of the choice set, the agent, when faced with a choice problem, picks the best elements available from each category. The model explains certain important deviations from the Weak Axiom of Revealed Preferences, while being fully characterized by other observable properties of the agent s choice behaviour. In the more general framework, our representation generalizes the maximization of incomplete preferences. For the specific case in which categories are disjoint, we prove that it is equivalent to the maximization of incomplete preferences plus a somewhat intuitive property. JEL classification: D01 1 Introduction Categorization is one of the most fundamental and pervasive cognitive procedures. Its importance lies in the fact that it permits us to understand and make predictions about objects and events in our world (Medin and Aguilar, 1999; Smith and Medin, 1981), as well as to choose among the sometimes overwhelming array of alternatives available. When one wishes to acquire a cell phone plan, odds are she won t consider the whole array of products, sort them in decreasing order of preference, and then pick her favorite one. She will first categorize the plans according to certain characteristics - such as quality and size of data packages, whether they are prepaid or postpaid, size of the minutes allowance, and so on -, and only then choose the best products for each category. Since categorization and concept formation have such central roles in the study of cognition, and therefore of rationality, it is peculiar that they are seldom explicitly taken into account in economic decision theory. In fact, a recurring criticism of the theory of rational choice is that it fails to take into account the role of category formation in the human decision making process. 1 We therefore develop a model that departs from the traditional rational choice framework precisely in that it allows for categorization to play a part on the choice process. Ministério da Fazenda do Brasil. bruno.furtado@fazenda.gov.br. Departamento de Economia, Universidade de Brasília. lgnascimento@unb.br Departamento de Economia, Universidade de Brasília. riella@unb.br 1 For an example of such a critique, see Schwartz (2000). 1

2 A decision maker is assumed to be endowed with incomplete preferences over the set of all possible alternatives she might be asked to choose from. By incomplete we mean that there are pairs of alternatives in the choice set which cannot be ranked as either better, worse or indifferent to one another. For a given categorization of the choice set, the agent only compares elements within categories, and then chooses the best ones in each of them. The agent s final choice will simply be the union, over every category, of the elements chosen within the categories. The only restriction that will be imposed upon the categorization is that preferences be complete inside each one of the categories. The model we propose also has testable implications for observable behaviour. It is a well established fact in choice theory that if full rationality is understood to be behaviour which is compatible with the maximization of transitive and complete preferences, then it is equivalent to behaviour abiding by Samuelson s Weak Axiom of Revealed Preferences (WARP). Despite its apparent simplicity, the model we put forth can account for many important violations of WARP, while being fully characterized by other observable properties of the agent s choice behaviour, including choice axioms that are well known in the literature. More recently, many weakenings of WARP have been proposed, leading to different models of bounded rationality. The one that most closely resembles our work is Manzini and Mariotti (2012). Their model, called Categorize-Then-Choose (CTC), consists of a two-stage process: in the first stage, the agent ranks all categories within the feasible set, eliminating all alternatives belonging to a dominated category; in the second stage, she picks her most preferred alternative among the remaining ones. Hence, the crucial difference between CTC and our model is that CTC assumes that the agent can make comparisons between categories, as well as within them. It has been suggested that choices between categories can be problematic, 2 and therefore we restrict our model to choices inside categories. No priority is given to any category over another. In section 2 we present some key concepts. Section 3 defines what we mean by rational choice with categories, and describes and interprets the main representation results of the paper. In section 4 we explore a special case of our model, when categories are disjoint sets. Section 5 places our model within the wider context of rational choice theory. Finally, in section 6 are some concluding remarks, including suggestions for future studies. 2 Setup The nonempty set X represents the set of all mutually exclusive alternatives available to the decision maker. By a choice field on X we mean a collection of sets Ω X 2 X \ { } that includes all singleton sets and is closed under finite unions. A choice field on X is interpreted as the family of all choice problems that the decision maker can possibly face. Any set A Ω X is referred to as a choice problem. When X 3 and Ω X is a choice field on X we call the pair (X, Ω X ) a choice space. In the event Ω X consists of all nonempty finite subsets of X we refer to (X, Ω X ) as a finite choice space. A choice correspondence is a function c : Ω X 2 X satisfying c(a) A for all A Ω X. Note that this definition allows for the possibility of the decision maker choosing more than one alternative for any given choice problem. The set c(a) is interpreted as 2 Schwartz (2000, pp. 82): Within each category, it may be relatively easy to express preferences. Between categories, however, expressing preferences is more problematic. 2

3 the collection of all choosable elements from the choice problem A, and the final choice of the decision maker will inevitably be a member of c(a). The agent might finalize her choice by subjectively randomizing between members of A. The symmetric and asymmetric parts of a reflexive binary relation X X will be denoted, respectively, by and. Hence is the disjoint union of and. Lack of preference will be represented by : x y if, and only if, neither x y nor y x. With this notation at hand, the symbol is used to signify comparability. Recall that a binary relation is acyclic when for any n N and x 1,..., x n X, if x i x i+1 for i = 1,..., n 1, then it is not true that x n x 1. A transitive and reflexive binary relation on X will be called a preference relation on X. Alternatively, we may also simply use the mathematical jargon and call such a relation a preorder on X. Both terms will be used interchangeably in this paper. For any S X and X X, the restriction of to S is denoted by S. When X is a nonempty set endowed with a preorder, and T X, we define: max(t, ) = {x T : (y x), y T }, the set of all -maximal elements of T. max(t, ) = {x T : x y, y T }, the set of all -maximum elements of T. Note that max(t, ) max(t, ), and that both sets are equal if is complete. Furthermore, if T is finite, max(t, ). On the other hand, if T is infinite, without any additional requirements on T or, the set of -maximal elements may well be empty. 3 General Categories 3.1 Categorization Given a preference relation on a set X, we define a category as a subset S of X in which is complete. This restriction on the structure of categories is both mild and intuitive, since one of the foremost reasons to group objects in the same category is the existence of some similarity among them (Hahn and Chater, 1997). It seems natural that similar alternatives are more likely to be comparable to one another, and hence our assumption about their mutual comparability is not so restrictive. Definition 1 (Categorization) Let X be a set endowed with a preorder. A cover S of X (that is, a collection of nonempty subsets of X with S = X) is a categorization of (X, ) if S is complete for every S S. Any S S is called a category of S. It will be noted that the definition imposes very little structure on the cover S. Particularly, it does not require that the categories be mutually disjoint. Therefore, this formulation allows for an individual to have {Chinese food} as a category, with {Chinese noodles} as a separate category within the first, and another category, say {Pasta}, that also contains {Chinese noodles} but is not a subset of {Chinese food}. Such intersecting categories may arise if, for example, multiple criteria are used for the categorization procedure, and these criteria overlap. 3 Furthermore, implicit in the definition of categorization is the fact that categories are, at least in part, subjective. That is, the manner in which the acting agent categorizes 3 As Manzini and Mariotti (2012, pp. 11) exemplify, think of categorizing cameras by price band and brand, or flights by cheapness and convenience. 3

4 alternatives is dependent upon her (subjective) preferences. Thus, implicit categories are not assumed to be observable, except perhaps indirectly, from the decision maker s choice behaviour. An outside observer cannot assume, for instance, that {Chinese food} and {Mexican food} will be relevant categories to the decision maker in any particular application. This is in line with recent studies in the field of cognitive psychology. 4 Perhaps the most natural way to rationalize choices in the presence of categories is simply to pick, among the alternatives presented in the choice problem, the best elements from each category. For a categorization S 2 X \ { } and a preference relation X X, c can be said to admit a representation by categorization if, for every A Ω X, c(a) = S S max(a S, ). This type of representation allows for behaviour that would not be possible if choice was simply the result of the maximization of a preference relation over the choice problem, that is, c(a) = max(a, ) for all A Ω X. We will further explore this in section 5, but the following example illustrates this point. Example 1 Let X = {x, y, z} and Ω X = 2 X \{ }. The choice correspondence defined by c({x, y}) = {x, y}, c({x, z}) = {x, z}, c({y, z}) = {y, z} and c({x, y, z}) = {y, z} cannot be rationalized as c( ) = max(, ), for any preference relation. On the other hand, such a choice correspondence can be rationalized by a maximization over categories by defining = {(x, x), (y, y), (z, z), (y, x), (z, x)} and S = {{x, y}, {x, z}}. Notice that, in the example above, x may be chosen even though every other element of X is strictly preferred to it. Although this may seem counter-intuitive at first, one may think of it as a predilection for diversity: an individual may choose a second-best option simply because she feels like having something from a certain category, even though a better alternative from a different category is available (think of going to an Italian restaurant and having pasta for a change, even though you would normally prefer having pizza). 3.2 General Model Having established that categorization is an important cognitive mechanism, and that the analysis of rational choice with categories cannot be boiled down to the maximization of an incomplete preference relation, it remains to determine what types of choice behaviour can be rationalized by maximizing over categories. The following results aim to answer this question in the general setting considered so far, where the structure imposed upon the categorization is restricted to that in Definition 1. We begin by stating a couple of postulates. Axiom 1 (Chernoff - α) For every A Ω X, if x c(a), then x c(b) for every B Ω X such that x B A. The (α) axiom implies that if an alternative is chosen in some choice problem, it will still be chosen if the choice problem shrinks. In particular, it implies that c(c(a)) = c(a), provided that c(a) is a choice problem. It was originally proposed by Chernoff (1954). 4 For example, Smith et al. (1998) find that the type of categorization process used by an individual - whether it is rule-based or similarity-based, for instance - impacts the resulting categories in a predictable manner. 4

5 Axiom 2 (Dominance) There exists an acyclic relation such that, for any A, B Ω X with A B, if x c(a) but x / c(b), then there exists y B \ A with y x and y c(a {y}). We can now prove the following result. Theorem 1 Let (X, Ω X ) be an arbitrary choice space. A choice correspondence c : Ω X 2 X satisfies (α) and Dominance if, and only if, there exists a preorder X X and a collection S 2 X \{ } such that is complete inside every S S and, for any A Ω X, c(a) = S S max(a S, ). Theorem 1 establishes the necessary and sufficient conditions for a choice correspondence to be rationalizable by a maximization with categories, but relies on the Dominance axiom, which is not given in terms of observable behavior. However, it gives us the exact conditions which a choice correspondence must satisfy in order to be representable in the desired manner. Consequently, it provides an useful shortcut to prove more interesting representation theorems in specific environments, as we will see presently. Also, Dominance makes it easier to compare the representation by categorization to other models of choice with bounded rationality, as we will show in section General model with finite choice field The weakness of the Dominance axiom is that it presupposes the existence of an acyclic relation with some desired characteristics. Ideally, such a binary relation should be elicited directly from the choice correspondence. As a candidate for the necessary relation, define P by xp y if, and only if, there exists a choice problem A such that y c(a) but y / c(a {x}). It turns out that, if the choice correspondence c can indeed be represented by c( ) = S S max(, ) for some preorder X X and S 2 X \ { }, then xp y reveals that x is strictly preferred to y with respect to. That is, we must have x y. We impose the following postulates on c: Axiom 3 (Aizerman - AA) If A, B Ω X are such that c(a) B A, then c(b) c(a). This axiom, which had its prominent role in decision theory, social choice and control theory recognized by Aizerman and Malishevski (1981), states that eliminating from a choice problem some alternatives that are not in the choice set cannot make new alternatives chosen. Axiom 4 (Acyclicity) The relation P is acyclic. Recall that the binary relation P is defined so that xp y only if there exists a situation where the addition of x to a choice problem makes y stop being chosen. In some sense, xp y means that x was revealed to be strictly preferred to y, so that it makes sense to impose that P be acyclic. The next lemma is the last necessary ingredient for the characterization of a representation by rational choice with categories when (X, Ω X ) is a finite choice space. 5

6 Lemma 1 Let (X, Ω X ) be a finite choice space and c : Ω X 2 X a choice correspondence on Ω X satisfying (α), AA and Acyclicity. Then c satisfies Dominance with P as the relevant binary relation. We are now ready to state our main result for finite choice spaces, giving us a full characterization of the sort of choice correspondences which are rationalizable by maximizing subject to a given categorization. Its proof derives directly from Theorem 1 and Lemma 1, and, consequently, is omitted. Theorem 2 Let (X, Ω X ) be a finite choice space. A choice correspondence c : Ω X 2 X satisfies (α), AA and Acyclicity if, and only if, there exists a preorder X X and a collection S 2 X \{ } such that is complete inside every S S and, for any A Ω X, c(a) = S S max(a S, ). Crucially, the characterization in Theorem 2 depends only upon observable behaviour, and is therefore easily testable, at least in principle. This is in contrast to Theorem 1, which hinges upon the unobservable Dominance axiom. 4 Disjoint Categories 4.1 Categorization with disjoint categories In many particular applications, the process of categorization gives rise to disjoint categories. That is, every element of the choice set must belong to one, and only one, category. For instance, if the only relevant categories for a decision maker are {food} and {shelter}, it is improbable that any given element of the choice set will be in both groups at once. 5 With effect, in psychology and philosophy, the classical view of categorization, which goes back to Aristotle, claims that categories are discrete entities, characterized by necessary and sufficient conditions for membership. 6 Thus, according to this view, categories are by definition mutually exclusive. 7 More recently, it has been extensively documented and experimentally tested by cognitive psychologists that children, when learning novel words, tend to naturally form disjoint categories. That is, when presented with a new label, they will be mostly inclined to map it to a novel rather than a familiar object (Markman, 1990; Markman et al., 2003). Although not entirely conclusive, such examples provide evidence that dividing the world into disjoint categories comes naturally to humans, which justifies a detailed analysis of this special case of our representation. We therefore devote this section to characterizing a representation theorem for when the categorization is a partition of X. 5 Excluding, of course, the possibility of wicked witches with gingerbread houses. 6 For a review of theories of categorization in psychology, see for example Smith and Medin (1981) and Komatsu (1992). 7 Actually, the only sort of intersection between categories allowed by the classical view is when a category is a subset, or refinement, of another. Therefore, it could be said that, for the same degree of generality - or, in the psychological jargon, for the same hierarchical level - categories are disjoint. 6

7 4.2 Axioms and Representation We begin by stating a couple of postulates that will be crucial to characterize the representation by a disjoint categorization, the first of which is already well known in the literature on social choice and decision theory (Sen, 1971; Moulin, 1985). Axiom 5 (Expansion - γ) For any collection A Ω X such that A is a choice problem, if x c(a) for every A A, then x c( A). If some alternative is chosen over all choice problems in a collection, then it must also be chosen in their union, provided that the union is itself a choice problem. Axiom 6 (Revealed Mutual Comparability - RMC) For every A Ω X such that there exists x X with c({x, y}) = 1 for every y A, we have c(b) A = c(a), for all B Ω X such that A B and c(b) A. When there exists an alternative x X such that c({x, y}) = 1 for every y A, one can, in some sense, infer that all elements of A belong to the same class of objects, since they are all strictly comparable to a common alternative. The RMC axiom then imposes that a WARP-like property holds when we compare the choices from A with the choices from choice problems larger than A. As it turns out, (α), (γ) and RMC together give us a full characterization of choice by disjoint categories. We state this formally below. Theorem 3 Let (X, Ω X ) be a choice space and c : Ω X 2 X any choice correspondence on Ω X. Then, c satisfies (α), (γ) and RMC if, and only if, there exists a preorder X X and a partition S of X such that is complete inside every S S and, for any A Ω X, c(a) = max(a S, ). S S 5 Relation to Other Models 5.1 Categorization and Pseudo-rationalization We say that a choice correspondence c is pseudo-rationalizable if there exists a collection of complete preorders such that, for any choice problem A, c chooses the best elements from each of these preorders. More formally, we have the following definition. Definition 2 (Pseudo-rationalization) A choice correspondence c on a choice field (X, Ω X ) admits a representation by pseudo-rationalization if there exists a collection R of complete preorders on X such that, for any choice problem A Ω X, c(a) = R max(a, ). Pseudo-rationalizable choice correspondences, over a finite choice space, were axiomatized by Aizerman and Malishevski (1981) (see also Moulin (1985)). It turns out that, in such choice spaces, pseudo-rationalizability is exactly characterized by the (α) and AA postulates. Therefore, Theorem 2 implies that, in a finite choice space, the existence of 7

8 a representation by categorization implies the existence of a pseudo-rational representation. Moreover, as the following example shows, the Acyclicity axiom is not redundant, which means that pseudo-rationalizability is indeed a strictly more general concept than representation by categorization. Example 2 Let X = {x, y, z, w} and consider a choice correspondence c pseudo-rationalized by the linear orders y 1 x 1 z 1 w and w 2 z 2 x 2 y. Since x c({x, y}), x c({x, z}), but x / c({x, y, z}), we have yp x and zp x. On the other hand, since z c({x, z}), z c({z, w}), but z / c({x, z, w}), we also have wp z and xp z, which violates Acyclicity. Therefore, c does not have a representation by categories. Although we are not aware of any axiomatization of pseudo-rationalizable choice correspondences for general choice spaces, we can still show that, in fact, this is true in general. Consider the following, not so basic, postulate: Axiom 7 (Preference Maximization Compatibility (PMC)) For all A Ω X and every x c(a), there exists an acyclic relation A,x such that x A,x y for every y A \ {x} and, for any other choice problem B, max(b, A,x ) c(b). We can prove the following result: Theorem 4 Let (X, Ω X ) be any choice space. A choice correspondence c : Ω X 2 X satisfies (α) and PMC if, and only if, it admits a representation by pseudo-rationalization. We can also prove the following observation: Lemma 2 Let (X, Ω X ) be any choice space. If a choice correspondence c : Ω X satisfies (α) and Dominance, then it satisfies PMC. 2 X It is now clear that any choice correspondence that admits a representation by categorization is pseudo-rationalizable. Formally, we have the following: Corollary 1 Let (X, Ω X ) be any choice space. If a choice correspondence c : Ω X 2 X admits a representation by categorization, then c is pseudo-rationalizable. 5.2 Maximization of incomplete preferences and categorization Let (X, Ω X ) be an arbitrary choice space and consider any (possibly incomplete) preference relation on X. If c is a choice correspondence on (X, Ω X ) such that c(a) = max(a, ) for every A Ω X, we say that c has a representation by the maximization of the incomplete preference. Choice correspondences that maximize incomplete preferences were studied by Eliaz and Ok (2006). Suppose now that a choice correspondence c maximizes an incomplete preference relation. It is easy to see that c satisfies the Dominance postulate with the strict part of being the acyclic relation that appears in that property. Since it is also clear that c satisfies (α), Theorem 1 implies that c has a representation by categorization. Formally: Theorem 5 Let (X, Ω X ) be any choice space. If a choice correspondence c : Ω X 2 X admits a representation by the maximization of a (possibly incomplete) preference relation, then c admits a representation by categorization. 8

9 It is easily checked that the converse to Theorem 5 is not true. With effect, as Example 1 shows, a representation by categories does not necessarily satisfy (γ), which is a necessary condition for a representation by the maximization of incomplete preferences. Therefore, our more general model of choice with categories actually generalizes the maximization of incomplete preferences, in the sense that the former can explain all choice behaviour explained by the latter, but not conversely. Suppose now that a choice correspondence c admits a representation by a categorization S in which the categories are disjoint, and the preference relation is given by. Define the binary relation ˆ X X by x ˆ y if, and only if, x y and there exists S S with x, y S. It is easy to see that ˆ is a preference relation and that c maximizes ˆ. This shows that representations by categorization with disjoint categories are particular cases of the maximization of incomplete preferences. Formally: Theorem 6 Let (X, Ω X ) be any choice space. If a choice correspondence c : Ω X 2 X admits a representation by categorization with disjoint categories, then c admits a representation by the maximization of a (possibly incomplete) preference relation. 5.3 Categorization and the decomposition of incomplete preferences Let X be any set and consider a (possibly incomplete) preference relation on X. Gorno (2015) shows that can be decomposed into a set of local preferences which are complete in their respective domains, and this is the content of the next theorem. Theorem 7 (Gorno (2015)) Let be a (possibly incomplete) preference relation on an arbitrary set X. Then there exists a collection D of subsets of X such that: (i) is complete on D for every D D; (ii) for every D D and y X \ D, there exists x D such that x and y are incomparable; (iii) = D D D. 8 Now fix a subset A of X. It is clear that if x max(a, ), then x max(a D, ) for every D D with x D. Conversely, suppose that x A, but there exists y A with y x. By (iii), there must exist D D such that x, y D. For such D, it is clear that x / max(a D, ). This suggests that we can give the following interpretation for the maximization of an incomplete preference procedure. First, the individual divides the world into categories such that two elements that belong to a common category are always comparable. Second, like in the representation in Theorem 1, she identifies the best elements in each category. Differently from the representation in Theorem 1, however, she refines her choices by eliminating from the choice set the alternatives that, although best with respect to some category, are strictly worse than some alternative in another category. This discussion can be summarized by the following result: 8 Theorem 1 in Gorno (2015) shows that D satisfies three properties in addition to the ones listed above, but they are not relevant for the discussion in this section. 9

10 Theorem 8 Let be a (possibly incomplete) preference relation on an arbitrary set X. Then there exists a collection D of subsets of X such that conditions (i), (ii) and (iii) are satisfied and, for every A X, max(a, ) = {x A : x max(a D, ) for every D D with x D} We note that if it weren t for condition (ii), Theorem 8 would be trivial, since we could simply define D by D := {{x, y} : x y}. It is also easily seen that, although we do not impose it in the statement of Theorem 3, we can assume, without loss of generality, that the pair (, S) in that theorem satisfies (ii). This suggests that we can think of the representation by categorization with disjoint categories as the particular case of the decomposition in Theorem 8 when D is a partition. In fact, such a case is characterized by the transitivity of the comparability relation induced by. To see that, consider the following postulate on : Axiom 8 (Transitive Comparability) A binary relation X X satisfies Transitive Comparability if x y and y z together imply x z. We can now show the following: Theorem 9 A preference relation on an arbitrary set X satisfies Transitive Comparability if, and only if, there exists a partition D of X such that properties (i), (ii) and (iii) are satisfied and, for every A X, max(a, ) = D D max(a D, ). 6 Conclusion We have presented a pure model of rational choice with categories. Categorization is an universal cognitive mechanism, and its importance for human understanding and decision making is widely recognized in many fields, such as psychology, philosophy and linguistics. More recently, some researchers in economics have also taken interest in this phenomenon, notably Mullainathan et al. (2008) and Manzini and Mariotti (2012). However, their models are quite different from our own. Mullainathan et al. (2008) focuses on the effects of categorization - or, as they call it, coarse thinking - on Bayesian updating, and apply the model to the study of uninformative persuasion. On the other hand, Manzini and Mariotti (2012) develop a pure theory of choice in the presence of categorization, but assume that the individual chooses both between and within categories. The representation put forth in this paper gives an accurate description of choices inside categories, but assumes that some external factor, inaccessible to the modeller, is responsible for the agent s choice of a category in any given situation. Such an approach has two major advantages: (i) from an empirical standpoint, comparisons among categories are known to be complicated, and it is not clear a stable preference among categories should exist, and (ii) it can account for some major violations of full rationality, such as some forms of menu effects. While leaving the method of choice between categories as an open question, we encourage future research on the subject. Also, our model fits nicely within the existing rational choice framework. It is a particular case of pseudo-rationalization, a representation well known in the theory of social 10

11 pseudo-rationality general categories incomplete preferences disjoint categories rational model Figure 1: Relation to other models choice, and at the same time generalizes the maximization of incomplete preferences, which in turn is a generalization of the model with disjoint categories. Finally, throughout the paper we mostly take categories as given and place very few restrictions on their possible structures. This allows our model to explain a wide range of behavioural patterns, but says nothing about what kinds of categorization procedures are actually undertaken in concrete situations. Although determining the process through which categories are formed is an important and interesting question, it lies beyond the scope of this paper and is indeed still very much a matter of contention among cognitive psychologists. 9 It seems, however, that the way in which people categorize is context dependent (Smith and Samuelson, 1997). That could imply that, in a rational choice context, the categorization structure would depend on the preferences of the agent. This makes the study of such structure a matter of considerable interest to decision theorists. A Proofs A.1 Proof of Theorem 1 It is easily seen that the representation implies (α) and Dominance, where the acyclic relation in the statement of Dominance is the strict part of. Conversely, suppose c satisfies (α) and Dominance for an acyclic relation. Let be any complete preorder that extends. 10 Fix any A Ω X and x c(a). Define S := {x} {y X \ A : y x and y c(b) for every B Ω X with A {y} B (A {z X : y z})}. Notice that A S = {x} and, consequently, {x} = max(a S, ). Now fix any choice problem B and suppose that y max(b S, ). We need the following claim: 9 For an overview of the the competing theories of categorization and concept formation, see for instance Heit (1997); Smith and Samuelson (1997); Shanks (1997). 10 Since is acyclic, the transitive closure of is a strict partial order that extends. Now the Szpilrajn Theorem (see Szpilrajn (1930)) guarantees that there exists a complete partial order that extends. 11

12 Claim A.1: For any z B \ A such that z y, we have that z / c(a {z}). Proof. Take any z B \ A such that z y. Since, by construction, y max(b S, ), we have that z / S. Hence, there exists a choice problem E with A {z} E (A {w X : z w}) and z / c(e). But then, if z c(a {z}), by Dominance, there must exist w with z w, but w z. This contradicts the construction of. Because of (α) and the claim above, we know that z / c(a {y, z}) for any z B \ A with z y. But then Dominance implies that y c(a B). By (α), this implies that y c(b). We have shown that, for every choice problem A and x c(a), there exists S x,a X such that {x} = max(a S x,a, ) and, for every other choice problem B, max(b S x,a, ) c(b). Let S = {S x,a X : x c(a) and A Ω X } be the collection of all such sets. Then, for every A Ω X, c(a) = S S max(a S, ). A.2 Proof of Lemma 1 Take any A, B Ω X with A B, and suppose there exists x c(a) \ c(b). Enumerate (B \ A) {x} = {y 1,..., y n }, where y 1 = x. Define the set N := {i {1,..., n 1} : x / c(a {y 1,..., y i+1 })}. Because N, there exists a smallest element i N. Notice that y i +1P x. Moreover, if y i +1 / c(a {y 1,..., y i +1}), then c(a {y 1,..., y i +1}) A {y 1,..., y i }, and it follows from AA that c(a {y 1,..., y i }) c(a {y 1,..., y i +1}). Hence x / c(a {y 1,..., y i }), a contradiction to the definition of i. It now follows from (α) that y i +1 c(a {y i +1}). A.3 Proof of Theorem 3 It is easily seen that the representation implies the three axioms. Conversely, suppose that c is a choice correspondence that satisfies (α), (γ) and RMC. Define the binary relation X X by x y if, and only if, x c({x, y}) and there exists z X such that c({x, z}) = c({y, z}) = 1. We note that, for any distinct x, y X, x y if, and only if, {x} = c({x, y}). Given that, it is easy to see that (α) and (γ) imply that c(a) = max(a, ) for every A Ω X. We need the following claim. Claim A.2: The relation is a preorder. Proof. It is clear that is reflexive, so we only have to show that it is transitive. Suppose first that x, y, z X are such that x y and y z. That is, {x} = c({x, y}) and y = c({y, z}). By (α), we learn that {x} = c({x, y, z}). Now RMC implies that {x} = c({x, y, z}) {x, z} = c({x, z}). That is, x z. Note that this observation implies that whenever x y, there exists z X such that either x z and y z, or z x and z y. Suppose now that x, y, z X are such that x y and y z. By RMC, if z c({x, y, z}), then we would have y c({x, y, z}), which cannot happen, because of (α). We conclude that {x} = c({x, y, z}). Now (γ) implies that {x} = c({x, z}) and, consequently, x z. Now suppose that x y and y z for some distinct x, y, z X. If there exists w X such that w x and w y, it must also be the case that w z. But then, 12

13 z c({x, z}) would imply that z x and our previous observation would give us that y x. We conclude that, in such a case, {x} = c({x, z}) and, therefore, x z. If, on the other hand, there exists w X such that x w and y w, then, since y z, we know that z and w are -comparable. If w z, then we have already seen that x w implies x z. If z w, then (α) and RMC imply that z / c({x, y, z}) and c({x, y, z}) {x, z} = c({x, z}) = {x}. Again, we learn that x z. Finally, suppose that x, y, z are distinct alternatives such that x y and y z. If there exists w X such that w x and w y, then we have already seen that w z. Consequently, RMC implies that c({x, y, z}) = {x, y, z} and c({x, z}) = c({x, y, z}) {x, z} = {x, z}. We learn that x z. If there exists w X with x w and y w, then our previous observations imply that z w. A similar reasoning to the one above then shows that x z. To complete the proof of the theorem, we observe that is comparability-transitive. To see that, suppose x, y, z X are such that x is -comparable to y and y is comparable to z. The only cases that are not immediate consequences of the transitivity of are when x y and z y, or y x and y z. In both cases, the definition of guarantees that x and z are -comparable. The theorem is then a direct implication of Theorem 9 and the fact that c(a) = max(a, ) for every choice problem A Ω X. A.4 Proof of Theorem 4 Suppose first that c admits a representation by pseudo-rationalization for some collection of complete preorders R. It is clear that c satisfies (α). Now, fix A Ω X and x c(a), and pick any complete preorder R with x max(a, ). Define the binary relation A,x X X by y A,x z if, and only if, y = x, z x and x z, or y z. It is easy to see that A,x is an acyclic relation such that x A,x y for every y A \ {x}, and, for every choice problem B, max(b, A,x ) max(b, ) c(b). This shows that c satisfies PMC Conversely, suppose that c satisfies (α) and PMC. For every choice problem A and every x c(a), let A,x be a complete preorder that extends A,x. Since A,x extends A,x, we have {x} = max(a, A,x ) and max(b, A,x ) max(b, A,x ) for every choice problem B. Define R by R := { A,x : A Ω X and x c(a)}. It is clear that R is a pseudo-rational representation of c. A.5 Proof of Lemma 2 Suppose that c satisfies (α) and Dominance, for some acyclic relation. Fix a choice problem A and some x c(a). Define the set à by à := (A \ {x}) {y X : y / c(a {y})}. Now define the relation A,x by y A,x z if, and only if, y / à and z Ã, or y, z à and y z, or y, z / à and y z. Let s first show that A,x is acyclic. For that, suppose that y 1 A,x y 2 A,x A,x y n 1 A,x y n. If y i / à and y i+1 à for some i n 1, then it is clear that y 1 / à and y n Ã. Consequently, we do not have y n A,x y 1. If this does not happen, then we have y 1 y 2 y n 1 y n. Since is acyclic, we do not have y n y 1 and, again, we do not have y n A,x y 1. This shows that A,x is acyclic. Moreover, it is clear that x A,x y for every y A \ {x}. Finally, fix a choice problem B and suppose that y / c(b). Since y c({y}), Dominance guarantees that there exists z B such that z y. If y Ã, this implies that z A,x y. Suppose, thus, that y / Ã. By (α), we know that y / c(a B). Since y c(a {y}), Dominance 13

14 implies that there exists z B \ A such that z y and z c(a {y, z}). By (α), this implies that z Ã. Consequently, we get that z A,x y. This shows that, for every choice problem B, max(b, A,x ) c(b). A.6 Proof of Theorem 9 It is clear that if there exists a partition D of X such that (i), (ii) and (iii) are satisfied,then satisfies Transitive Comparability. Conversely, suppose that is a preference relation on X that satisfies Transitive Comparability. By Theorem 8, there exists a collection D of subsets of X such that conditions (i), (ii) and (iii) are satisfied and, for every A X, max(a, ) = {x A : x max(a D, ) for every D D with x D} Thus, we will be done if we can show that D is a partition. To see that, assume that there exists distinct D 1, D 2 D and x X such that x D 1 D 2. Since D 1 and D 2 are distinct, we can assume, without loss of generality, that there exists y D 1 \ D 2. By (ii), there exists z D 2 such that y and z are incomparable. Since both y and z are comparable to x, this violates Transitive Comparability. We conclude that D is a partition. References Aizerman, M. A. and A. V. Malishevski (1981). General theory of best variants choice: Some aspects. IEEE Transactions on Automatic Control 26 (5), Chernoff, H. (1954). Rational selection of decision functions. Econometrica 22 (4), Eliaz, K. and E. Ok (2006, July). Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences. Games and Economic Behavior 56 (1), Gorno, L. (2015). On the structure of incomplete preferences. Hahn, U. and N. Chater (1997). Concepts and Similarity. In Knowledge, concepts and categories. Studies in cognition., pp Heit, E. (1997). Knowledge and Concept Learning. In Knowledge, concepts and categories. Studies in cognition., pp Komatsu, L. K. (1992). Recent views of conceptual structure. Psychological Bulletin 112 (3), Manzini, P. and M. Mariotti (2012). Categorize then choose: Boundedly rational choice and welfare. Journal of the European Economic Association 10, Markman, E. M. (1990). Constraints Children Place on Word Meanings. Cognitive Science 14 (1590), Markman, E. M., J. L. Wasow, and M. B. Hansen (2003). Use of the mutual exclusivity assumption by young word learners. 14

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