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Revealed preference and identification
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ScienceDirect Journal of Economic Theory 183 (2019) 698–739 www.elsevier.com/locate/jet
Revealed preference and identification Leandro Gorno FGV EPGE, Praia de Botafogo 190/1119B, Rio de Janeiro, RJ 22250-900, Brazil Received 4 July 2017; final version received 24 June 2019; accepted 3 July 2019 Available online 10 July 2019
Abstract This paper studies preference identification in a general framework that allows for partial observability of optimal choices: Decision makers select some optimal alternatives, but not necessarily all of them. While partial observability is a methodologically appealing assumption for empirical applications, it makes recovering preferences much harder. The main result provides abstract conditions on classes of preferences and decision problems ensuring identification. The result is applied to several standard settings demonstrating the power of the method. © 2019 Elsevier Inc. All rights reserved. JEL classification: C02; D11; D81 Keywords: Revealed preference; Identification; Recoverability; Indifference; Continuity; Uniqueness
1. Introduction This paper examines the problem of recovering preferences from empirically observed choices when decision makers behave optimally. The main contribution is the formulation of reasonable theoretical restrictions that uniquely identify preferences in a general setting. Most significantly, identification is achieved without assumptions on how decision makers select among indifferent alternatives. Identification is not always possible because observed decisions may be consistent with different preferences. For example, consider a couple trying to decide whether to move to Paris or stay in New York. Ann prefers to move, while Bob is indifferent. Suppose that either Ann or Bob E-mail address: [email protected]. https://doi.org/10.1016/j.jet.2019.07.003 0022-0531/© 2019 Elsevier Inc. All rights reserved.
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makes the choice, but only the outcome is observed. Clearly, if the couple stays in New York, it would be safe to conclude that Bob made the decision, because Ann would have preferred to move. In this sense, staying in New York “reveals” that Bob’s preference is the one that counted. However, if they move to Paris, the evidence available is insufficient to infer who determined the outcome. The possibility of identifying preferences from observed behavior depends on the class of preferences being entertained. For example, suppose a voter wants to infer a politician’s type by looking at his or her past policies. Assume first that the politician can only be “left-wing” or “right-wing.” There are two policies available: “liberal” and “conservative.” Left-wing politicians strictly prefer the liberal policy to the conservative policy, and vice versa. In this case, any past policy uniquely identifies the politician’s type. Next, consider an additional type of politician, a “pragmatic,” who is indifferent between the liberal and the conservative policies. Including this third type makes potential inferences less conclusive. A liberal policy rules out a right-wing politician, but does not determine whether the politician is left-wing or pragmatic. A similar indeterminacy remains when multiple previous policy choices are observed: The only type of politician that can be learned from the data is the pragmatic (after observing a mixture of liberal and conservative policies). As the aforementioned examples illustrate, indifference poses a problem to identification inasmuch as it enlarges the set of theoretical preferences that can be confounded for a given dataset. The traditional axiomatic approach to abstract revealed preference simplifies this issue by effectively assuming that every alternative the decision maker could have chosen is observed.1 However, this approach is unsatisfactory from a methodological viewpoint, because it transmutes what is supposed to be empirical data (i.e., the decision maker’s observed choices) into a theoretical object (i.e., the set of preference-maximizing alternatives).2 In contrast, this paper develops a general framework for revealed preference analysis that affords a cleaner separation between the theories being entertained and the data intended to evaluate them. This is attained by relaxing informational assumptions while keeping the basic logic of revealed preference intact: What the decision maker chooses cannot be worse than anything else she could have chosen; unchosen options, however, are not presumed to be suboptimal. Following Chambers et al. (2014), I refer to this approach as “partial observability.” Once partial observability is adopted, the fundamental challenge is to obtain identification. This is because finding preferences consistent with the available evidence becomes relatively easy. In fact, if the only assumptions on preferences are completeness and transitivity, rationalization under partial observability is trivial: Any collection of observations is rationalized by universal indifference. At the same time, this lack of restrictions makes identification harder. Of course, it is possible to obtain restrictions on observable choice behavior, and even identification, by limiting admissible preferences. 1 In the axiomatic approach, the basic primitive is a “choice correspondence” that describes all optimal options in every choice problem under consideration. See, for example, Definition 1.D.1 in Mas-Colell et al. (1995), p. 12, and Kreps (2012), p. 2. This assumption considerably simplifies identification, even though it does not immediately render the problem trivial (if the collection of choice problems is not sufficiently rich, distinct preferences may be observationally equivalent). 2 If the decision maker faces each choice problem repeatedly and randomizes among all optimal options in a nondegenerate and sufficiently independent way, every optimal option would eventually be chosen (almost surely). This paper’s underlying claim is that it is better to formulate theoretical assumptions as explicit restrictions on preferences, and not as an ad hoc hypothesis on how the observed empirical data is generated.
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The central contribution of this paper is its proposal of general theoretical assumptions that are sufficient for identification under partial observability. I then use this framework to study important classes of preferences including several types of ordinal continuous preferences, preferences satisfying the von Neumann–Morgenstern (vNM) axioms, and qualitative probabilities. Since indifference is easily disciplined within these classes, relatively weak additional conditions suffice to uniquely recover any preference driving behavior, provided that enough data is observed. It should be stressed that departing from the axiomatic approach to relax full observability not only constitutes a methodological improvement but also has significant theoretical consequences. For example, under partial observability, choice correspondences are naturally interpreted as encoding empirical datasets. Under this interpretation, preference maximization on its own does not imply the celebrated axioms of revealed preference. Therefore, violations of these axioms are not necessarily evidence of irrational behavior: They might merely reflect the limits of the available data. Overall, the message of this paper is somewhat comforting: It is possible to apply revealed preference analysis to interesting theories of behavior, while keeping the theoretical assumptions employed to obtain identification separate from the empirical data. 1.1. Recovering quasilinear preferences with a discrete good To illustrate the methodological contribution of this paper, consider the following simple example. Suppose that Ann is a competitive consumer who has quasilinear preferences over bundles of two goods: cars (x1 ) and money (x2 ). More precisely, her preferences can be represented by a utility function U of the form U (x1 , x2 ) = v(x1 ) + x2 , for some smooth, increasing, and strictly concave function v. A cars-money bundle (x1 , x2 ) is considered feasible if it lies in the consumption set X (which I have purposely left unspecified) and satisfies the budget constraint p1 x1 + p2 x2 ≤ w, where p1 > 0 and p2 > 0 are the respective prices of each good and w > 0 is Ann’s wealth at the time of making the choice.3 Given market prices, Ann chooses a bundle that maximizes her utility among all bundles in the feasible set (i.e., the intersection of her budget set with the consumption set). Imagine that the function v is unknown but a sufficiently large and varied collection of choices is observed (and the prices at which these choices were made). Is it possible to recover Ann’s preferences? In other words, are quasilinear preferences identified in this setting? Of course, providing an answer to these questions requires being more specific about the environment in which Ann makes her choices and the kind of behavioral data available. To be concrete, assume that the analysis has to be performed based on a sequence of pairs consisting of a single consumer choice and a price vector (representing the feasible set where this choice was made). To formalize the idea that large amounts of data are available, assume further that 3 The literature (e.g., Afriat (1967), Reny (2015)) typically does not consider consumer wealth as observable data and instead recovers it from observed choices and market prices by assuming that the consumer spends her entire income. Since v is increasing, this assumption holds for Ann.
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the sequence of observed price-wealth situations is dense in the set of all possible price-wealth situations. If, as usual in classic demand theory, the consumption set is taken to be X = R2+ , identification can be established with little difficulty using known results. First note that strict convexity of preferences implies that there cannot be indifference at the optimum. Hence, the Walrasian demand must be single-valued (i.e., a demand function). Even though only choices at some feasible sets are observed (i.e., those along the dense sequence given), the whole demand function can be easily obtained using continuity. Finally, Theorem 2 in Mas-Colell (1977) implies that Ann’s preferences can be recovered from her demand function. Now suppose that cars are indivisible and they can only be consumed in integer amounts. That is, the consumption set is X = Z+ × R+ instead of R2+ . In this case, two technical issues complicate identification. First, it is easy to show that, for every possible preference Ann might have, there are budget sets over which she would indeed be indifferent between two different bundles. This means that her Walrasian demand is necessarily a correspondence, which makes it more difficult to recover under partial observability (i.e., from the observation of only some optimal choices).4 Second, even if Ann’s demand correspondence was known in its entirety (i.e., under full observability), the existing literature seems to lack recoverability results that can be applied when choices are restricted by this disconnected consumption set. This paper develops a general framework to analyze identification problems under partial observability like the one described above and provides several technical tools to tackle the associated difficulties. In particular, my results imply that Ann’s preferences are identified in this case as well. That is, at most one preference in this class can be consistent with any countable dataset {(x n , p n , w n )}n∈N such that the observed price-wealth situations {(p n , w n )}n∈N are dense in the set of all possible price-wealth situations. The specific details are contained in Section 4.1. 1.2. Overview of the paper The main results in this paper can be divided in two groups: 1) general results, and 2) results about continuous preferences. The general results are contained in Section 2 and concern the relation between identification and a simpler separation condition that requires every pair of preferences to display a strict preference reversal. Theorem 1 shows that every identified theory must be “separated” in this sense (assuming the collection of choice problems satisfies a weak technical condition). Theorem 2 provides a partial converse to Theorem 1: Every separated theory is identified on “rich” collections of choice problems. Richness here is a substantive structural requirement, but it is purely theoretical and does not impose implicit or explicit assumptions on empirical choices. Unfortunately, Theorem 2 falls short from a full characterization of identification because the richness condition employed is not necessary. Theorem 3 shows that there is a sense in which this is unavoidable: No criterion selecting collections of choice problems on the basis of individual preferences can be necessary for identification and simultaneously yield a sufficiency result for all separated classes of preferences. The main results for continuous preferences are contained in Section 3. Theorem 4 singles out a large separated class of continuous preferences and shows that this class is maximal with 4 In addition, for each feasible set that allows the purchase of at least one car, there exist uncountably many quasilinear preferences that makes multiple bundles optimal. As a result, the possibility of having indifference at the optimum persists even in a quasi-experimental setting in which prices can be selected by the analyst.
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respect to the separation property. Theorem 5 shows that, under continuity, choice behavior in limit problems does not contain additional information about preferences. This result is useful because it allows to obtain identification using countable collections of problems in settings with uncountably many alternatives. Moreover, it can be used to relax the general richness condition, as illustrated by Theorem 6. Finally, applications of these results to consumer theory, expected utility preferences, preferences over menus, and random choice correspondences are developed in Section 4. Section 5 analyses two additional theories that are not covered by the previous results: vNM preferences and qualitative probabilities. Section 6 provides some closing remarks. Appendix A contains additional material that further develops some issues discussed in Section 2 and Section 3. All proofs are in Appendix B. 1.3. Related literature The main motivation for this paper is a discomfort with embedding the assumption that all optimal choices are observed in the foundation of abstract revealed preference analysis. This discomfort is not new, and neither is the idea of dropping the problematic assumption while restricting preferences in reasonable ways. However, to the best of my knowledge, this is the first attempt to obtain general identification conditions under partial observability. The revealed preference approach was originally proposed by Samuelson (1938) as a way of freeing the theory of consumer behavior from arbitrary assumptions on utility functions. Houthakker (1950) constitutes the definitive elaboration of Samuelson’s theory taking a demand system as the primitive, while Afriat (1967) shows how to construct a utility function consistent with a finite set of observed consumer choices at given prices. Afriat’s approach is extended to general budget sets by Forges and Minelli (2009), while Reny (2015) allows for infinite datasets. The problem of eliciting preferences from market demand behavior goes back to Antonelli (1886). Samuelson (1950) provides a historical survey. Modern versions of integrability results for the Walrasian demand are given by Hurwicz and Uzawa (1971). The problem of recoverability (i.e., the uniqueness of the rationalizing preference)—which is the focus of the present contribution—is studied in the context of demand theory by Mas-Colell (1977), who provides results for the case in which one observes the Walrasian demand in its entirety (i.e., under full observability). Dybvig and Polemarchakis (1981), Kübler and Polemarchakis (2017), and others focus on identification of characteristics starting from complete knowledge of demand correspondences in other specific environments. A related strand of literature that starts with Mas-Colell (1978) studies the limiting behavior of increasingly informative revealed preference exercises, as the consumer is observed choosing in more and more price-wealth situations. However, Mas-Colell (1978) assumes that the underlying preference generates a demand function, thus rendering the question of partial observability moot. The abstract approach in this paper aims to be general: It is not confined to consumer theory and allows for a wide range of potential applications. The axiomatic structure of the theory of revealed preference was first investigated by Uzawa (1956), Arrow (1959) and Sen (1971). A modern textbook treatment of abstract revealed preference can be found in Mas-Colell et al. (1995), while a systematic presentation of the theory is provided by Chambers and Echenique (2016). As the present paper, Nishimura et al. (2017) also extend the applicability of Afriat’s approach beyond consumer theory. However, their contribution focuses on characterizing rationalizability
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as well as unifying different approaches to revealed preference, rather than studying identification. In particular, their discussion of recoverability is confined to the classical demand setting. The recent work of Chambers et al. (2018) follows the spirit of Mas-Colell (1978)’s exercise and studies the possibility of learning a decision maker’s preference in an abstract experimental setting. Both Chambers et al. (2018) and the present paper are concerned with what can be learned about preferences in general environments and both allow for partial observability. However, while Chambers et al. (2018) focus on the large-sample properties of finite-sample rationalizing preferences, this paper intends to address a more basic identification question: Can the underlying preference be recovered when the decision maker has been observed making choices in all (admissible) choice problems? This is an important distinction and an econometric analogy can be useful to clarify it. Two central econometric problems are “identification” and “inference”. While the identification problem concerns the population properties of a given theoretical structure, the inference problem is about the possibility of learning aspects of such population from a sample. For example, in a parametric model for a random object X, the typical identification question is whether the value of the parameter, say θ , can be recovered from the conditional distribution f (X|θ ). In contrast, the inference problem concerns what can be said about θ on the basis of a random sample {X1 , ..., XN } for some N ∈ N. The prevalent view among econometricians is that the identification question logically precedes inference, as a sample cannot be more informative than the population it is taken from. In this sense, the contributions of Chambers et al. (2018) and the present paper are somewhat complementary. On the one hand, Chambers et al. (2018) develop the inference aspect of preference maximization in an experimental setting that simplifies the identification problem. On the other hand, the present paper explores identification in greater detail and without assuming an experimental setting, while basically ignoring inference. Finally, this paper is also related to a body of work that investigates the empirical content of economic theories in very general terms. An example is the contribution by Chambers et al. (2014), who develop a framework to discern the nontestable claims of a given theory. Although partial observability also plays a prominent role in that paper, the focus is on theory falsifiability rather than preference identification. 2. General theory This section presents a general framework for performing revealed preference analysis under partial observability. 2.1. Basic definitions Let X be the set of alternatives faced by the decision maker. A preference is a complete and transitive binary relation on X. For any preference , denote the symmetric and asymmetric parts with ∼ and , respectively. As usual, x y and (x, y) ∈ are equivalent statements, x y means y x, and x y means that x y does not hold. (Similar notational conventions apply to other binary relations such as ∼ and .) A set I ⊆ X is an indifference set for if x ∼ y holds for all x, y ∈ I . An indifference class for is an indifference set for that is maximal with respect to set inclusion. A preference is trivial if X is an indifference set for . (Otherwise is nontrivial.) A function U : X → R is said to represent whenever x y is equivalent to U (x) ≥ U (y). An alternative x ∈ X is a -best element in B ⊆ X if x ∈ B and x y for every y ∈ B. 2X denotes the collection of all subsets of X.
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2.2. Choice models A choice model is a specification of the environment in which the decision maker (she) operates together with theoretical restrictions on her possible choice behavior. Formally, let B denote a fixed collection of nonempty subsets of X. A set B ∈ B is called a choice problem. Choice problems capture all possible ways through which alternatives in X can be presented to the decision maker in the social situation being modeled. For example, in standard consumer theory, B is the collection of all linear budget sets; whereas in an experimental setting where a decision maker is confronted with binary alternatives, B is the collection of all two-element subsets of X. In this paper, a theory of behavior or, more succinctly, a theory is a class of preferences on X. The underlying interpretation is that every time a decision maker is confronted with a choice problem, she publicly selects one of the preference-maximizing alternatives contained in it. Consequently, a theory represents an hypothetical restriction that the analyst entertains when trying to make sense of behavioral data. A choice model is then a triple (X, B, P), where X is a set of alternatives, B is a collection of choice problems, and P is a theory of behavior. It should be stressed that, as theoretical primitives, B and P are not data. Moreover, B need not contain all subsets of X nor P need contain all preference relations on X. 2.3. Rationalizing empirical choices A choice correspondence is a function c : B → 2X that satisfies c(B) ⊆ B for all B ∈ B. The inclusion of the empty set in the co-domain permits encoding “absence of data,” and thus interpreting c as a collection of observed choices. The graph of an arbitrary choice correspondence c : B → 2X is the set gr(c) := {(B, x) ∈ B × X|x ∈ c(B)}. In this paper, the word dataset is treated as equivalent to the graph of a choice correspondence. The following definition embodies the notion of rationalization I employ. Definition 1. The choice correspondence c is consistent with a preference if, for all B ∈ B, every alternative in c(B) is a -best element in B. The theory P rationalizes c if c is consistent with some ∈ P. Note that this notion of rationalization is weaker than that employed in the traditional approach to abstract revealed preference, precisely because it does not require that c(B) contains all -best elements in B. As a result, revealed preference axioms are not necessary consequences of preference maximizing behavior under partial observability. In fact, any testable restrictions must follow from additional theoretical assumptions.5 2.4. Full domain choice correspondences To qualitatively assess the informational content of different choice correspondences, consider the following definition. 5 Under partial observability, the preference that declares all alternatives indifferent rationalizes every choice correspondence. In contrast, if a choice correspondence can be rationalized in the traditional (stronger) sense, then the weak axiom of revealed preference must hold.
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Definition 2. The domain of a choice correspondence c is the collection of choice problems in B that satisfy c(B) = ∅. The choice correspondence c has full domain if its domain is B. In a typical application of this framework, a choice correspondence with full domain represents a dataset containing information about the decision maker’s behavior when confronted with a large and diverse enough set of choice problems. Note that this notion does not conflict with partial observability, which just means that the dataset at hand does not necessarily contain every optimal option, but only those that the decision maker actually chose while under observation. 2.5. Identification Fix a choice model (X, B, P). Suppose that observed choices are encoded in the choice correspondence c. There are three mutually exclusive possibilities. First, c may not be consistent with any preference in P. In this case, if the model is taken literally, one should reject theory P. Second, there may be a unique preference in P consistent with c. In this scenario, the rationalizing preference can be pinned down uniquely with the available data, so that the “parameters” of theory P can be fully recovered. Third, P may rationalize c through multiple preferences. This last case naturally occurs when c does not contain much information, either because the decision maker has not been confronted with many choice problems or because the class of choice problems itself is rather limited. However, due to the adoption of partial observability, multiplicity can persist even when B = 2X and c has full domain. This possibility is excluded by the following definition. Definition 3. A theory P is identified on B if every choice correspondence with full domain is consistent with at most one preference in P. Within an identified theory, one can always recover a preference driving behavior if enough data is available. Note that starting with smaller theories makes identification easier.6 Similarly, since Definition 3 requires at least one choice recorded for every choice problem in B, considering more choice problems facilitates identification. In fact, it is straightforward to establish that, if P is identified on B, then P is identified on B for every P ⊆ P and B ⊇ B. Two additional definitions may be helpful to discuss identification. Both definitions are relative to a fixed collection of choice problems B. First, preferences and are confoundable if there exists a choice correspondence with full domain that is simultaneously consistent with both preferences. It is clear that a theory P is identified if and only if no two distinct preferences in P are confoundable. Second, and are observationally equivalent if {} and { } rationalize exactly the same set of choice correspondences. Obviously, a theory that contains distinct preferences that are observationally equivalent is not identified, but the converse is not necessarily true. In fact, if B contains all two-element subsets of X, some theories are not identified even though there cannot be distinct observationally equivalent preferences. 6 In particular, every theory P containing a single preference is identified on every collection of choice problems. Despite its triviality, this observation is important, because it suggests simple identified theories that are consistent with datasets that violate the weak axiom of revealed preference. For example, if X = {x, y, z}, the theory P = {} where = {(x, x), (x, y), (x, z), (y, x), (y, y), (y, z), (z, z)} is identified and is consistent with the dataset {({x, y}, {x}) , ({x, y, z}, {y})}.
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2.6. A necessary condition for identification Checking identification using its definition is not very convenient. In order to develop a more operational characterization, this section studies a simple condition restricting the class of preferences under consideration. Before introducing the main condition, there is a technical obstacle in the path toward a necessity result: Infinite choice problems may not have -best elements for some ∈ P. This possibility is troublesome because it potentially weakens identification too much.7 The assumption embedded in the following definition circumvents this difficulty and helps simplify the analysis. Definition 4. A collection of choice problems B is compatible with P if there exists a -best element in B for every preference ∈ P and every choice problem B ∈ B. Note that, if B is compatible with P, then B is compatible with P for every B ⊆ B and ⊆ P. Moreover, B is automatically compatible with P if X is finite or, more generally, if X is a topological space, all problems in B are compact, and all preferences in P are continuous. The following is the key condition to obtain a necessity result: P
Definition 5. A theory P is separated if, for every pair of different preferences , ∈ P, there exist alternatives x, y ∈ X such that x y and y x. Thus, a separated theory exhibits at least one strict preference reversal for every pair of preferences. One should expect identified theories to be separated. In fact, from an empirical perspective, two preference relations are only different to the extent that they are revealed distinct through data. Since partial observability does not impose any restrictions on how indifferences are resolved, preference reversal arises as a natural condition to avoid confounding different preferences. The next theorem formalizes this intuition and shows that separability is indeed a necessary condition for preference identification under partial observability. Theorem 1. Every identified theory on a compatible collection of choice problems is separated. As a consequence of Theorem 1, a theory that fails to be separated is not identified on any compatible collection of choice problems. 2.7. A sufficient condition for identification Since compatibility is a rather weak structural condition, a theory essentially needs to be separated in order to be identified. However, this separation property is not generally sufficient. Example 1. Let X = {x, y} and B = {{x}, {y}}. Consider a theory P = , , where x y and y x. Clearly, P is separated and B is compatible with P. However, P is not identified on B. 7 Without compatibility, some theories are inconsistent with every choice correspondence with full domain. As a result, identification can hold vacuously. See Example 15 in Appendix A.3.
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The example above illustrates that identification requires the set of choice problems to be sufficiently rich. The following condition ensures that there are enough choice problems to theoretically distinguish between strict and weak preference: Definition 6. A collection of choice problems B is P-rich if, for every ∈ P and x, y ∈ X, x y implies there are B1 , ..., BN ∈ B and z1 , ..., zN+1 ∈ X such that: (1) z1 = x, zN+1 = y, (2) zn+1 ∈ Bn for all n ∈ {1, ..., N }, (3) zn is the unique -best element in Bn for all n ∈ {1, ..., N }. This condition allows one to obtain a simple theory of revealed preference under partial observability. Importantly, P-richness is a theoretical property of the choice model (X, B, P) that does not involve in any way the observed behavior of the decision maker, which, in turn, is captured solely by the choice correspondence c. This is in stark contrast with the traditional axiomatic approach that assumes full observability, the implicit requirement that c be interpreted as containing all preference maximizing options. It is easy to find examples that satisfy P-richness. In fact, any collection of choice problems containing all two-element subsets of X is P-rich for every theory P. However, collections of choice problems that satisfy Definition 6 only for some theories are often important in applications. For example, in classic demand analysis, it is natural to consider the collection of all (compact) linear budget sets on RN + , which is P-rich only under substantial assumptions regarding the preferences in P.8 The next result establishes that separated theories are identified on P-rich collections of choice problems. Theorem 2. Let P be a separated theory and let B be a P-rich collection of choice problems. Then P is identified on B. Under the compatibility and richness conditions, Theorem 1 and Theorem 2 together reduce the identification problem to checking a transparent condition for the theory under consideration. Importantly, whether or not P is separated does not depend on the specific problems contained in B. This means that whenever P is identified on some compatible collection of choice problems, it is automatically identified on every P-rich collection of choice problems. Example 2 (Identification with experimental data). Consider an experimental setting in which the decision maker may, in principle, be confronted with all possible pairs of alternatives. Formally, suppose X has at least 2 elements and B = {{x, y}|x, y ∈ X, x = y}. In this case, both compatibility with P and P-richness are trivially satisfied for any theory P. As a result, Theorem 1 and Theorem 2 together imply that separation fully characterizes identification. That is, a theory P is identified on B if and only if it is separated. 8 Mas-Colell (1977) provides an explicit example of two different preferences that satisfy most standard economic assumptions and nevertheless are observationally equivalent on the collection of linear budget sets. See Appendix A.1 for a discussion of this example and Section 4.2 for sufficient conditions for P-richness in the classic demand setting.
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The qualified equivalence established by Theorem 1 and Theorem 2 suggests to study what separation entails in concrete environments, as a means to operationalize further the notion of identification. This is the route I follow in Section 3, Section 4, and Section 5, establishing sufficient conditions for separation in a variety of settings with additional structure. 2.8. On weaker sufficient conditions As the following examples make clear, P-richness is not a necessary condition for identification. Example 3. Let X = {x, y}, B = {{x}} and P = , where x y. Clearly, P is identified on B even though B fails to be P-rich. Example 4. Let X = {x, y, z}, B = {{x, y}, {y, z}} and P = , , where x y ∼ z and y x ∼ z. Clearly, P is identified on B even though B fails to be P-rich. It therefore seems natural to seek weaker sufficient conditions for identification. However, this is very challenging. In fact, there is a formal sense in which it is impossible to obtain a useful necessary and sufficient condition at this level of generality. To see this, note that, given a theory P, a collection of choice problems B is P-rich if and only if B is {}-rich for every ∈ P. It follows that whether B is P-rich or not depends exclusively on a series of tests performed using each possible preference in P. This observation suggests to study all abstract criteria (selecting certain collections of choice problems) that satisfy this “pointwise” property. let A be an arbitrary function mapping each possible theory P into More precisely, B ⊆ 2X . The function A is said to be a pointwise criterion if it satisfies A(P) = A(P) ⊆ B ∈P A({}). Obviously, P-richness is a pointwise criterion, but by no means the only one. Consider next the following two definitions. First, A is necessary for identification if the inclusion {B|P is identified on B} ⊆ A(P) holds for every theory the function A is P. Second, said to place no restrictions on choice problems if A(P) = B B ⊆ 2X for every P. The next theorem relates these two properties for pointwise criteria. Theorem 3. If a pointwise criterion is necessary for identification, then it places no restrictions on choice problems. Theorem 3 shows that a pointwise criterion that is weak enough to select every collection of choice problems over which the theory supplied as argument is identified is in fact too weak to provide any restriction. In particular, every such criterion must admit collections of choice problems over which some separated theory fails to be identified. It follows that there does not exist a pointwise criterion that, together with separability, provides a necessary and sufficient condition for identification. This impossibility result suggests to abandon necessity and settle for a pointwise criterion weaker than P-richness, but that preserves enough strength to ensure that separated theories are
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identified (over those collections of choice problems satisfying the criterion). However, this is far from trivial (if possible at all).9 An alternative path forward is to seek conditions that are not pointwise criteria (i.e., violate the property A(P) = ∈P A({})), but are simultaneously sufficient for identification of separated theories and weaker than P-richness. Such conditions do exist but are hardly instructive and, in my view, adopting them would reduce the methodological appeal of this paper’s method.10 Finally, one can also attempt to obtain weaker criteria for specific theories. This seems to be the most promising approach and it is pursued to some extent in the following section. For example, Theorem 6 relaxes P-richness when preferences are assumed to be continuous. 3. Continuous preferences: theory Suppose that the space of alternatives X is endowed with some topology. In this section, I investigate identification under the following standard continuity condition used in Debreu (1959): Axiom 1 (Continuity). For every x ∈ X, the sets {z ∈ X|x z} and {z ∈ X|z x} are closed. This assumption is standard in economics and is necessary for the existence of a continuous utility representation of . 3.1. Regular preferences A class of continuous preferences need not be identified. This section presents a weak additional condition which ensures that a class of continuous preferences is separated. This condition combines with Theorems 1 and 2 to provide powerful identification (and non-identification) results for continuous preferences. Developing this approach requires a few additional definitions. Fix a preference . An alternative x ∈ X is a local best element for if there is an open set V ⊆ X such that x ∈ V and x y for all y ∈ V . A local worst element for is defined similarly by only substituting y x for x y in the previous sentence. x ∈ X is a local extremum for if it is either a local best element or a local worst element. A set of alternatives is -extremal if every alternative it contains is a local extremum for . With these definitions at hand, consider the following: Axiom 2 (Regularity). No -extremal indifference class for contains a local best element for and a (different) local worst element for . 9 The discussion in Appendix A.2 illustrates this difficulty showing that a minimally weaker criterion fails to guarantee that all separated theories are identified. 10 A central advantage of this paper’s approach is that the conditions for identification are structurally simple and do not restrict more than two preferences simultaneously. For example, P-richness is a condition individually imposed on each preference in theory P. Similarly, whether a theory is separated solely constraints pairs of preferences within the theory.
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Fig. 1. Representations of preferences in Example 5.
Note that an indifference class may consist of a single alternative that is both a local best and a local worst element without violating the definition. To illustrate Axiom 2, it is useful to consider simple examples of regular and non-regular preferences. Example 5. Let X = [0, 1] and consider functions U, V : X → R defined by setting: ⎧ ⎪ x ∈ [0, 1/4] , ⎨2x 1 − |2x − 1| x ∈ [0, 3/4] , U (x) := V (x) := 1/2 x ∈ (1/4, 3/4] , ⎪ 2x − 1 x ∈ (3/4, 1] , ⎩ 2x − 1 x ∈ (3/4, 1] . The graph of U is displayed in Panel A of Fig. 1. The extremal indifference classes are {0} (which is a singleton) and {1/2, 1} (which contains local best elements and no local worst elements). Thus, the preference represented by U is regular. Similarly, Panel B of Fig. 1 displays the graph of V . In this case, [1/4, 3/4] is an extremal indifference class that contains both 1/4 (a local best element) and 3/4 (a local worst element). It follows that the preference represented by V is not regular. The restrictions on indifference that regularity imposes are useful to obtain identified theories through the results in Section 2. The following theorem demonstrates this in the case of continuous preferences. Theorem 4. The class of regular continuous preferences is separated. Moreover, if X is a separable metric space, no larger class of continuous preferences is separated. Together with Theorem 2, the first part of this result implies that identification under partial observability is attainable for regular continuous preferences. Moreover, via Theorem 1, the second part of Theorem 4 implies that regularity is an optimal condition to obtain identification of continuous preferences through the method proposed in this paper. That is, if X is a separable
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metric space, additionally allowing any non-regular continuous preference necessarily precludes identification (over compatible choice problems). 3.2. Sufficient conditions for regularity Regularity is a very weak condition. This fact is important because, combined with the second part of Theorem 4, it implies that regular continuous preferences constitute a large theory that is maximal with respect to the property of being separated.11 For example, it is customary in consumer theory to assume that preferences are not only continuous, but also “monotone” or, at least, “locally nonsatiated”. These assumptions preclude the existence of local best elements and therefore imply regularity. Formally, is said to be locally nonsatiated at x ∈ X if every open set containing x also contains y such that y x. is locally nonsatiated if it is locally nonsatiated at every x ∈ X. Chambers et al. (2018) define the following weaker notion: is locally strict if V ∩ = ∅ implies V ∩ = ∅ for every open set V ⊆ X × X. The following lemma relates these notions with regularity: Lemma 1. Let X be an arbitrary topological space and let be any preference on X. Then, (1) If is locally nonsatiated, then it is locally strict. (2) If is locally strict, then it is regular. The converse implications do not hold (even if X = [0, 1] and is continuous). Lemma 1 shows that regularity should be expected to hold in most economic applications. But regularity is in fact significantly weaker than local nonsatiation or local strictness. The following result illustrates this by providing an appealing alternative sufficient condition for regularity: Lemma 2. Let X be a connected topological space and let be a continuous preference on X that is generically locally nonsatiated (i.e. local nonsatiation holds on an open dense subset of X). Then, is regular. Lemma 2 becomes false if the requirement that X be connected is removed. This is shown by Example 18 in Appendix A.3. Similarly, the statement also becomes false if regularity is substituted by local strictness in the conclusion. The following example displays a smooth preference that is generically locally nonsatiated but fails to be locally strict: Example 6. Let X = R and let be represented by U (x) = (cos(x) −1)x. Then, is continuous (in fact, U is C ∞ ) and generically locally nonsatiated (thus regular by Lemma 2), but fails to be locally strict. To see that the latter claim is true, consider any pair of options of the form (x, y) = (2πm, −2πn) for m, n ∈ N. Clearly, U (x) = U (y) = 0 implies x ∼ y. However, for all 11 Maximality with respect to separation is not a compelling argument for regularity per se. For instance, the theory that only contains universal indifference is also maximal with respect to separation, but it is clearly too small to be useful.
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sufficiently small > 0, |x − x| < and |y − y| < imply x y . Fig. 2 below illustrates the example:
Fig. 2. Utility representation of a generically locally nonsatiated preference that is not locally strict.
Finally, it is worth stressing that regularity is even weaker than generic local nonsatiation because it is not a local property. This is illustrated by the following example that can be used to construct regular preferences with a continuum of satiation points. Example 7. Let X = RN and be the preference represented by a continuous function U . Further, assume that there are two extreme alternatives x∗ , x ∗ ∈ X such that U (x∗ ) < U (x) < U (x ∗ ) for all x ∈ X \ {x∗ , x ∗ } and a path α : [0, 1] → X joining x∗ and x ∗ along which U is strictly increasing.12 Then every indifference class for must contain a point along the path α. It follows that {x∗ } and {x ∗ } are the only -extremal indifference classes for . As a result, any such is regular, independently of how the preference is defined outside of α([0, 1]). 3.3. Identification on restricted domains This paper aims to relate observable behavior to different theories of choice. In many applications, it is natural to model a continuum of alternatives even though empirical observations are usually discrete. For this reason, it is useful to obtain identification theorems based on choice correspondences with restricted domains. In this section, I provide results that relate identification on a choice model (X, B, P) with identification on a “restricted” choice model Y, B , P such that Y is a subset of X, B contains less and/or smaller choice problems, and P contains preferences over the limited domain Y . X of c as the function cl(c) : For any choice correspondence c : B → 2 , define the closure X B → 2 , given by cl(c)(B) := x ∈ X (B, x) ∈ clB ×X (gr(c)) for each B ∈ B (for some topology on choice problems). The following result exploits the structure of the class of continuous preferences to establish a foundation for identification conditions over potentially countable collections of choice problems. 12 Formally, α is a continuous function such that α(0) = x , α(1) = x ∗ and, for all t, s ∈ [0, 1], t > s implies U (α(t)) > ∗ U (α(s)).
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Theorem 5. Assume that X is a metric space, B is a collection of nonempty compact subsets of X (endowed with Hausdorff metric topology), and P is a class of continuous preferences on X. Then, (1) A choice correspondence c is consistent with ∈ P if and only if its closure cl(c) is consistent with . (2) If P is identified on B, then P is identified on every dense subset of B. (3) If P is separated, Y is a dense subset of X, and , ∈ P coincide on Y × Y , then = . Part (1) in Theorem 5 reflects the fact that, under continuity, choice behavior on limit problems does not contain additional information. In particular, it implies that, whenever two choice correspondences have the same closure, they are rationalized by the same set of continuous preferences. In the same spirit, Part (2) shows that identification over a dense subcollection of choice problems (which may be countable even with an uncountable set of alternatives) is not harder than when all choice problems are available. Part (3) shows that limiting the domain of a separated theory to a dense subset of alternatives does not throw away any information. As a result, identification of restricted preferences is sufficient to identify unrestricted (separated) preferences. In particular, this part of Theorem 5 applies to regular preferences (due to Theorem 4). Note that the results presented in Section 2 fall short of a full characterization of the identification property, because the richness condition employed is not necessary. Theorem 5 is useful to establish weaker sufficient conditions for identification of continuous preferences. Theorem 6. Assume that the conditions in Theorem 5 hold. Suppose further that there exists a set Y ⊆ X (not necessarily dense) satisfying: such that B ∈ B implies that the -best (1) For every ∈ P, B has a dense subset B elements in B are contained in Y . (2) The mapping carrying into ∩ (Y × Y ) is one-to-one. (3) PY := ∩(Y × Y ) ∈ P is separated. (4) BY := {B ∩ Y |B ∈ B} is PY -rich.
Then P is identified on B. A few remarks about Theorem 6 are in order. First, on the one hand, a trivial sufficient condition for (1) is that B ⊆ Y for all B ∈ B. On the other hand, the possibility of using dense subsets of B allows the exclusion of preference-specific “degenerate” feasible sets that would induce best elements outside of Y . Second, (2) and (3) automatically hold whenever P is separated and Y is dense in X (this follows from Part (3) of Theorem 5). However, sometimes it is convenient to use sets Y that are not dense in X in order to relax the restrictions imposed by (4). For a concrete illustration of these points and, more importantly, of the usefulness of Theorem 6 to get extra mileage out of the central method of this paper, I defer the reader to Example 8 in Section 4.1 (identification of Cobb-Douglas preferences with a discrete good) and Example 10 in Section 4.2 (identification of linear preferences in a classic demand setting).
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4. Continuous preferences: applications This section applies some of the results presented in Section 2 and Section 3 to more specific settings. 4.1. Demand theory with a discrete good This application presents an identification result for a large class of consumer preferences with a discrete good, providing formal support for the claims made in the quasilinear example discussed in Section 1.1. As it was in that case, suppose Ann consumes bundles of two goods. The second good is infinitely divisible, but the first one can only be consumed in discrete quantities. As in the case of classical demand theory that is developed immediately after this section, an observation consists of market prices and quantities bought by the consumer.13 Formally, let X = Z+ × R+ be the consumption set (with the natural topology). For every price-wealth situation (p, w) ∈ R2++ × R++ , define the following feasible set on X: Bp,w := {x ∈ X|px ≤ w} . L be the class of all such feasible sets. Note that most of these feasible sets are neither Let BD convex nor connected. For any two vectors u, v ∈ RN , let (u, v) := {αu + (1 − α)v|α ∈ (0, 1)}. This notation will be used in what follows as a shorthand to denote strict convex combinations of commodity bundles and price vectors. Relaxing the quasilinear assumption of Section 1.1, assume that Ann’s preferences are continuous and satisfy the following additional axioms:
Axiom 3 (Strict monotonicity). For every x, y ∈ X, x ≥ y and x = y imply x y. Axiom 4 (Strict convexity). For every x, y, z ∈ X, x y, x = y, and z ∈ (x, y) imply z y. If both goods were infinitely divisible, Axiom 3 and Axiom 4 would be standard. In particular, both axioms are satisfied by the restrictions to X of CES and quasilinear preferences on R2+ . Let PD be the class of all preferences on X satisfying Axiom 1, Axiom 3, and Axiom 4. It should be stressed that, in contrast with strictly convex preferences in the infinitely divisible case, all preferences in PD are consistent with indifference at the optimum. This makes partial observability both a compelling methodological assumption and a difficult challenge: As natural as this exercise is, the literature seems to lack an adequate recoverability result for this setting. However, combined with the general results presented in Section 2 and Section 3, the assumptions defining PD imply the following: L. Proposition 1. PD is identified on every dense subset of BD
Note that every dense sequence of price-wealth situations generates a countable dense subset L . The proof of Proposition 1 (in the Appendix) showcases the central method in this paper. of BD First, separability of PD is obtained as a consequence of regularity through Theorem 4. Second, 13 Individual wealth is typically considered to be unobservable and is inferred from prices and quantities by assuming that the consumer spends all its budget.
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L is established through a specific argument that exploits the additional strucPD -richness of BD ture of this environment. Then, Theorem 2 combines these two properties to obtain identification L . Finally, as an additional refinement, Theorem 5 yields identification using any sequence on BD L. of choice problems that is dense in BD Even though the class PD contains several common classes of preferences, it also excludes some important cases such as Cobb-Douglas. CD be the class of preferExample 8 (Cobb-Douglas preferences with a discrete good). Let PD ences on X = Z+ × R+ that can be represented by
U (x1 , x2 ) = x1α1 x2α2 , CD violates both strict monotonicity and strict conwhere α1 , α2 > 0. Note that every ∈ PD vexity because (0, x2 ) ∼ (0, 0) for every x2 ≥ 0. Moreover, no such is regular. However, CD is identified on B L using Theorem 6. To see this, set it is possible to establish that PD D L . Condition (1) of Theorem 6 Y = N × R++ ⊂ X and B = Bp,w w > p1 > 0, p2 > 0 ⊂ BD holds trivially because the B employed excludes all feasible sets that yield an optimal choice indifferent to (0, 0). Condition (2) holds because the bundles in X \ Y are deemed worst eleCD . Conditions (3) and (4) hold because, when restricted to Y , ments by every preference in PD CD are strictly monotonic and strictly convex, so similar arguments to the the preferences in PD ones used in the proof of Proposition 1 apply.
4.2. Classical demand theory What about the classical demand case in which all goods are infinitely divisible? Even though some new insights can be obtained applying the results of Section 2 and Section 3, the central method of this paper enjoys significantly less thrust without an indivisible good because P-richness becomes especially restrictive. L Formally, take X = RN and + (with the natural topology) let B be the collection of compact N linear budget sets (that is, all sets of the form x ∈ R+ px ≤ w for some wealth level w > 0 and a price vector p ∈ RN ++ ). Mas-Colell (1977) formalizes an idea attributed to Shapley and provides an explicit example of two different continuous, strictly monotone, and strictly convex preferences on RN + that generate the same Walrasian demand function (i.e., are observationally equivalent on B L ). This means that identification is not trivial in this setting, even if one observes all optimal choices. However, Mas-Colell (1977) also proves that upper semicontinuous, monotone, and convex preferences that are representable by a lipschitzian utility function can be (uniquely) recovered from observing the entire demand correspondence. A natural question is whether this theorem can be strengthened to obtain identification under partial observability. The results presented in Section 2 suggest to seek sufficient conditions for P-richness. In addition to the axioms formally introduced in Section 4.1 (now applied to X = RN + ), consider the following: Axiom 5 (Positive smooth monotonicity). is locally nonsatiated and admits a C 1 representation U such that U (x) >> 0 for every x 0. Axiom 6 (Positive strict convexity). For every x, y, z ∈ X, x = y, x y 0, and z ∈ (x, y) imply z y.
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Axiom 5 implies continuity, regularity, and monotonicity (x >> y implies x y), but it is independent of strict monotonicity. Axiom 6 is a slight weakening of strict convexity. Both axioms are satisfied by standard preferences, such as Cobb-Douglas in RN + , which fail strict monotonicity and strict convexity. The following proposition uses these axioms to study P-richness in this environment. Proposition 2. Let be a preference over X = RN +. (1) If satisfies positive smooth monotonicity and positive strict convexity, then B L is {}-rich. (2) If satisfies strict monotonicity and B L is {}-rich, then is strictly convex. The first part of Proposition 2 is useful to obtain identification. The following example illustrates the application of the main method of this paper to the familiar class of Cobb-Douglas preferences: Example 9 (Cobb-Douglas preferences). Let P CD be the class of preferences on X = RN + that can be represented by U (x1 , ..., xN ) =
N
xnαn ,
n=1
where αn > 0 for n = 1, ..., N . Every ∈ P CD satisfies positive smooth monotonicity and positive strict convexity. Thus, by Proposition 2, B L is P CD -rich. Moreover, positive smooth monotonicity implies regularity by Lemma 1. Thus, by Theorem 4, P CD is separated. Since P CD is separated and B L is P CD -rich, Theorem 2 implies that P CD is identified on B L . However, the sufficient conditions in the first part of Proposition 2 are quite strong. In fact, local nonsatiation and positive strict convexity together imply that optimal choices are unique on linear budget sets, leaving no room for partial observability. Moreover, the second part of Proposition 2 shows that this difficulty can hardly be avoided. The conclusion is that the central method of this paper, based on Theorem 2, needs to be modified to be useful in this setting.14 It is indeed possible to obtain identification without strict convexity by means of more specific arguments, such as Theorem 6, as the following example illustrates. Example 10 (Linear preferences). Let P L be the class of preferences on X = RN + that can be represented by U (x1 , ..., xN ) =
N
αn xn ,
n=1
where αn > 0 for n = 1, ..., N . Note that every bundle in RN ++ is strictly preferred to the zero vector but there is no linear budget set such that the demand contains only an interior point. This 14 The assumptions in the first part of Proposition 2 preclude multiple optimal choices and are stronger than those of Mas-Colell (1977)’s main result. As a consequence, Proposition 2 falls short of advancing what is known about identification in this setting.
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means that B L is not P L -rich. However, Theorem 6 implies that P L is identified on B L . To see this, take Y=
∀m = n : x = 0 . (x1 , ..., xN ) ∈ RN m +
N n=1
In words, Y contains all bundles in which at most one good is consumed. It is easy to check that conditions (2), (3), and (4) of Theorem 6 are satisfied. To verify condition (1), pick any preference ∈ P. Note that each indifference class for is the restriction of some hyperplane to RN + . Let be the set of all linear budget sets with a boundary that is not parallel to the indifference B
is dense in B L . Moreover, for every B ∈ B , the -best elements classes for . Clearly, B in B are corner solutions in which the full budget is exclusively spent in one good and, thus, are contained in Y , as required by condition (1) of Theorem 6.
4.3. Expected utility Denote by Z a separable metric space of prizes and let X the set of all Borel probability measures on Z (endowed with the topology of weak convergence). A preference on X is an expected utility preference if it can be represented by a function U : X → R of the form U (x) = udx for some bounded and continuous vNM index u : Z → R. It is well known that expected Z utility preferences are fully characterized by Axiom 1 together with Axiom 7 (Independence). For all x, y, z ∈ X and α ∈ (0, 1), x y implies αx + (1 − α)z αy + (1 − α)z. It can be shown that nontrivial expected utility preferences are regular. Thus, Theorem 4 implies the following result: Proposition 3. The class of nontrivial expected utility preferences is separated. This means that nontrivial expected utility preferences can always be recovered from sufficiently rich choice data. In particular, it is possible to recover expected utility preferences by observing choice over pairs of lotteries. The next result allows to generalize this observation by providing other interesting rich collections of choice problems in this setting. Proposition 4. The following collections of choice problems are rich with respect to the class of expected utility preferences: (1) All choice problems with exactly N ≥ 2 lotteries. (2) All convex compact choice problems. 4.4. Preferences over menus Consider the standard model of preferences over menus of lotteries: let Z be a compact metric space, be the set of Borel probability measures over Z metrized by the Lévy-Prokhorov distance, and X be the set of nonempty compact subsets of metrized with the Hausdorff distance.
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In this setting, the objects of choice are lottery menus (i.e., sets of lotteries – the elements of X), while choice problems are menus of lottery menus. For x, y ∈ X and α ∈ [0, 1], denote by αx + (1 − α)y the set {αp + (1 − α)q|p ∈ x, q ∈ y} ∈ X. Consider the following axiom: Axiom 8 (Set independence). For every x, y, z ∈ X and α ∈ (0, 1), x y implies αx + (1 − α)z αy + (1 − α)z. This section focuses on preferences on X satisfying continuity and set independence. This class includes the temptation and self-control preferences studied by Gul and Pesendorfer (2001), namely any preference with a representation of the form U (x) := max u(p) − max {v(q) − v(p)} p∈x
q∈x
for some linear and continuous functions u, v : → R. The class also includes preferences with the integral representation of Dekel et al. (2001): U (x) := sup u(p, s)μ(ds) p∈x
S
for some state space S, some signed measure μ and some function u : × S → R linear over . The following result shows that this class of preferences is within the reach of Theorem 2. Proposition 5. The class of nontrivial preferences over lottery menus that satisfy continuity and set independence is separated. It follows that continuity and set independence are strong enough to provide identification over rich collections of choice problems. As in the case of expected utility, it is possible to provide a simple result describing rich collections of choice problems in this setting. Proposition 6. The following collections of choice problems are rich with respect to the class of preferences over lottery menus satisfying continuity and set independence: (1) All choice problems with exactly N ≥ 2 lottery menus. (2) All convex compact choice problems. Of course, the resemblance to Proposition 4 stems from the structural similarity between independence and set independence. 4.5. Random datasets The exercise of recovering preferences from observed choices is relatively simple in experimental settings for two different reasons. First, if choice data is collected through an experimental design, it may be possible for the analyst to affect the shape (or even determine) the family of choice problems the decision maker will face. As shown in Example 2, this may considerably simplify the characterization of identified theories. Second, even if the family of choice problems is fixed and beyond the analyst’s control, an experimental protocol might be configured to produce a sequence of choice problems that guarantees identification without confronting the decision maker with all conceivable choice problems.
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In principle, the absence of these two features makes identification more challenging in nonexperimental settings. In this sense, conditions such as P-richness deal with the first issue by paving the way for identification results that do not require binary choice problems and, thus, can be applied to certain non-experimental settings. In this section, I explore the second difficulty: What happens when choice problems composing the observed dataset are chosen randomly, instead of being purposely selected to ensure identification? To investigate this question, consider a decision maker who faces a random sequence of choice problems, selects one optimal alternative for each problem, and this choice is observed. While observed behavior is assumed to be preference-maximizing, the decision maker is free to resolve indifferences as she deems fit (it is even possible to condition on the true state of the world). Although this tie-breaking process is not restricted in any way, the ensuing Proposition 7 establishes that, if the underlying theory is identified and the process sampling the choice problems is iid, then any continuous preference driving observed behavior can be recovered almost surely. In order to state the result precisely, I need some additional machinery. Let (, , μ) be a probability space, where is a set of states of the world, is a σ -algebra, and μ is a probability measure. Assuming B is endowed with some topology, a random problem is a measurable function Bˆ : → B. The support of a random problem Bˆ is the intersection of all closed subsets of B that contain Bˆ with probability 1. A sequence of random problems {Bˆ n }n∈N is said to be iid if, for each M ∈ N and each finite subset {Bˆ n1 , ..., Bˆ nM }, M μ Bˆ n1 ∈ A1 , ..., Bˆ nM ∈ AM = μ Bˆ ni ∈ Ai i=1
for all Borel sets A1 , ..., AM ⊆ B. A random dataset is a sequence of pairs {(Bˆ n , xˆn )}n∈N such that, for each n ∈ N, Bˆ n is a random problem and xˆn : → X is a measurable function that satisfies xˆn (ω) ∈ Bˆ n (ω) for all ω ∈ . A random dataset {(Bˆ n , xˆn )}n∈N is said to be iid if {Bˆ n }n∈N is iid, and is said to have full support if the support of Bˆ n is B for every n ∈ N. Every random dataset (Bˆ n , xˆn )n∈N naturally induces a random choice correspondence cˆ mapping each state of the world into the choice correspondence associated with the realiza15 tion choice correspondence cˆ is consistent with ∈ P if of the random dataset. A random μ ω ∈ c(ω) ˆ is consistent with = 1. Finally, let P (c, P) stand for the set of preferences in theory P that are consistent with choice correspondence c. The following proposition shows that the underlying preference is recovered almost surely. Proposition 7. Assume that X is a separable metric space, B is a collection of nonempty compact subsets of X (endowed with the Hausdorff metric topology), and P is a class of continuous preferences on X that is identified on B. Then, for every preference ∈ P and every random choice correspondence cˆ consistent with and induced by an iid random dataset with full support, P (c, ˆ P) = {} almost surely. In other words, Proposition 7 establishes that the event ω ∈ P (c(ω), ˆ P) = {} has probability 1. It should be stressed that, if the theory P is not identified on B, the conclusion of Proposition 7 fails for some random choice correspondence that satisfies the remaining conditions. 15 That is, for each ω ∈ , c(ω) ˆ is the choice correspondence with graph
Bˆ n (ω), xˆn (ω) n ∈ N .
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5. Other classes of preferences This section provides sufficient conditions for separation within two specific theories which are economically meaningful but fall outside the scope of Section 3, as they are not framed in a topological space. 5.1. vNM preferences This section studies the class of vNM preferences, which generalize expected utility preferences. For discussions of the basic framework, I refer the reader to Herstein and Milnor (1953) and Kreps (1988). Let X be a general mixture space and consider a vNM preference , which is a preference that satisfies Axiom 7 and also the following weaker form of continuity: Axiom 9 (vNM continuity). For all x, y, z ∈ X, x y z implies that there exist α, β ∈ (0, 1), such that αx + (1 − α)z y βx + (1 − β)z. It is well known that every vNM preference admits a linear representation: there is a function U : X → R that represents and satisfies U (αx + (1 − α)y) = αU (x) + (1 − α)U (y) for every x, y ∈ X and α ∈ [0, 1]. The following result shows that the vNM axioms provide a solid ground for revealed preference analysis. Proposition 8. The class of nontrivial vNM preferences is separated. Proposition 8 reveals a uniqueness property hidden within the vNM axioms: If two vNM preferences do not exhibit any strict preference reversal, at least one must be trivial. Combining this observation with Theorem 2, it follows that nontrivial vNM preferences can always be recovered from sufficiently rich choice data. Although it can be shown that nontrivial vNM preferences are regular, Proposition 8 cannot be derived from Theorem 4, because a vNM preference in a general mixture space need not be continuous (in fact, a mixture space need not have a topology defined on it). In the topological setting of Section 4.3, when X is the space of Borel probability measures over some separable metric space of prizes Z, the class of expected utility preferences is a subset of the class of vNM preferences.16 5.2. Qualitative probabilities This section discusses identification of qualitative probabilities—a central component in the theory of decisions under uncertainty, as formulated by Savage (1972) and other authors. I focus on a particular class of qualitative probabilities: Those consistent with an atomless probability measure. The importance of this theory stems from noting that every probability space on which a continuous random variable can be defined is necessarily atomless (see Villegas (1964) for a proof of this fact). 16 The inclusion is strict if and only if Z is infinite (continuity does follow from the vNM axioms when Z is finite).
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The main result of this section is that the class of qualitative probabilities consistent with an atomless probability measure constitutes a separated theory. In light of Theorem 2, this means that (probabilistic) beliefs within this theory can be uniquely recovered from observable betting behavior. Let X be a σ -algebra on some set of simple events . A preference on X is called an atomless continuous qualitative probability if the following axioms hold: Axiom 10 (Nondegeneracy). ∅ and x ∅ for all x ∈ X. Axiom 11 (Monotony). x1 ∩ x2 = ∅, x1 y1 (resp. x1 y1 ) and x2 y2 imply x1 ∪ x2 y1 ∪ y2 (resp. x1 ∪ x2 y1 ∪ y2 ) for all x1 , x2 , y1 , y2 ∈ X. Axiom 12 (Monotonic continuity). For every x, y ∈ X and every increasing sequence {yn }n∈N ↑ y, such that x yn ∈ X for all n ∈ N, x y. Axiom 13 (No atoms). For every x ∈ X that satisfies x ∅, there exists y ∈ X, such that y ⊂ x and x y ∅. Axioms 10, 11, 12, and 13 are known to be necessary and sufficient for the existence of an atomless probability measure that represents . This probability measure turns out to be unique and countably additive. The following result shows that these axioms define a separated theory. Proposition 9. The class of atomless continuous qualitative probabilities is separated. This means that the observation of enough simple bets suffices to uniquely recover the probability measure that represents the decision maker’s subjective uncertainty.17 The following example shows that Proposition 9 is not true without Axiom 13. Example 11. Consider = {a, b}, X = 2 and , , defined by: {a} ∼ {b} ∅
{a} {b} ∅.
Clearly, = and both and satisfy Axioms 10, 11, and 12, but fail Axiom 13. Moreover, the conclusion of Proposition 9 does not hold, since there are no strict preference reversals between and . Since it cannot be dispensed, a natural question is whether the requirement that there are no atoms can be weakened in some useful way. Unfortunately, I have no positive result to offer at this time. 6. Discussion In this paper, I develop a theory of abstract revealed preference that does not assume that all the optimal choices of the decision maker under analysis are observed. The proposed approach follows the spirit of Afriat’s work in that it attempts to maintain a clear separation between 17 This follows from combining Theorem 2, Proposition 9, and Theorem 4.3 in Villegas (1964).
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theoretical assumptions and empirical data, but focuses on the question of uniqueness of a rationalizing preference rather than its existence. I investigate identified theories: classes of preferences for which sufficiently rich datasets are enough to uniquely pin down the unobserved preference driving behavior. Theorem 1 and Theorem 2 combine a richness condition on choice problems with a simple preference separation property to characterize this notion of identification. To provide concrete meaning to the theory, I provide sufficient conditions for identification under standard sets of economic axioms on preferences. The strategy to recover preferences followed in this paper relies heavily on P-richness. This structural condition is useful because every separated theory P is identified on any P-rich collection of problems. However, this is an admittedly restrictive assumption that might be perceived as too strong. Two observations about this issue seem pertinent. First, Theorem 3 shows the structure of any necessary and sufficient condition in the most general setting must be significantly more convoluted and, thus, such conditions are likely to be less useful. Second, by restricting attention to continuous preferences on a metric space, Theorem 6 provides a sufficient condition for identification under somewhat weaker assumptions on choice problems. Although the present paper focuses on the identification problem, investigating the uniqueness of rationalizing preferences, partial observability also has implications for the related problem of determining the existence of such preferences. For example, one important feature of the proposed framework is that the axioms of revealed preference are no longer necessary for finding a preference that is consistent with observed behavior. Thus, the adoption of partial observability requires caution when interpreting violations of the weak axiom, as they need not be an indication of irrationality. Of course, the additional assumptions invoked in different applications do have observable implications that, in principle, allow one to falsify specific theories of behavior. Although the full characterization of these implications is beyond the objectives of this paper, it is worth noting that any sufficient condition for rationalizability in the traditional axiomatic approach to revealed preference (in which all optimal choices are observed) must also be sufficient in the present framework. Acknowledgments I thank the editor, Marciano Siniscalchi, as well as an associate editor and two referees for their constructive feedback. I also thank Andrés Carvajal, Carlos da Costa, Faruk Gul, Lucas Maestri, Ignacio Monzón, Marcelo Moreira, Paulo Natenzon, Philipp Sadowski, Bernard Salanié, and participants at various seminars for their helpful comments. Finally, I am very grateful to João Vitor Granja de Almeida, Rodrigo Naumann, and Alessandro Rivello for their outstanding research assistance. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. Appendix A. Additional material A.1. Full observability This section formalizes the notion of “full observability” and presents results comparable to those I obtain for partial observability in the body of the paper. It should come as no surprise that adopting full observability permits a relaxation of the theoretical assumptions employed to
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obtain identification. This is not for free, however, as full observability is an assumption and, as I argue in the introduction, not a methodologically appealing one. ∗ (B) denotes the set of all -best elements Given a preference and a choice problem B, c in B. The following definition extends the concept of identification to full observability: ∗ (B) = Definition 7. A theory P is identified under full observability on B if , ∈ P and c
∗ (B) for all B ∈ B imply = . c
The following definition weakens the P-richness condition in the main text: Definition 8. A collection of problems B is weakly P-rich if, for every ∈ P and (x, y) ∈ , there exist B1 , ..., BN ∈ B, z1 , ..., zN+1 ∈ X such that z1 = x, zN+1 = y, and (zn , zn+1 ) ∈ ∗ (B ) × B for all n ∈ {1, ..., N }. c n n The following result is a full observability analogue to Theorem 2. Theorem 7. If P is separated and B is weakly P-rich, then P is identified under full observability on B. It would be desirable to have a converse. However, similarly to the case of P-richness under partial observability, weak P-richness is not necessary for identification under full observability: Example 12. X = {x, y, z}, B = {{x, y}}, P = 1 , 2 , where x 1 y 1 z and y 2 x 2 z. Clearly, P is identified under full observability on B and P is separated. However, B is not weakly P-rich. Moreover, in contrast with the case of partial observability, the following example shows that separation is not necessary either: Example 13. X = {x, y}, B = {{x, y}}, P = 1 , 2 , where x 1 y and x ∼2 y. Clearly, P is identified under full observability on B and B is weakly P-rich, but P is not separated. To close this section, consider the central counterexample in Mas-Colell (1977): There exists a pair of distinct continuous, strictly monotone, strictly convex preference relations that yield the same Walrasian demand function.18 Any such preference must necessarily be regular (due to strict monotonicity). It follows that if P is the class of continuous, strictly monotone, strictly convex preferences, then P is a separated theory. Thus, in light of Theorem 7 above, the collection of linear budget sets, B L , must necessarily fail to be weakly P-rich. Let P MCL be the class of monotone and convex preferences that admit a lipschitzian representation. Remark 12 in Mas-Colell (1977) (p. 1414) implies that B L is weakly P MCL -rich. Since P MCL is separated due to monotonicity, Theorem 7 establishes that P MCL is identified under full observability on B L , a fact already known from Theorem 2’ in Mas-Colell (1977). 18 In the language of Section 2.5, the preferences of this counterexample are observationally equivalent on the class of linear budget sets.
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In contrast, whether P MCL is identified on B L under partial observability seems to be an open problem. A.2. Almost P-richness Considering that P-richness is a strong condition, this section discusses a minimal weakening: to allow the -best element zN to be non-unique. Definition 9 (Almost P-richness). A collection of choice problems B is almost P-rich if, for every ∈ P and x, y ∈ X, x y implies there are B1 , ..., BN ∈ B and z1 , ..., zN+1 ∈ X such that: (1) (2) (3) (4)
z1 = x, zN+1 = y, zn+1 ∈ Bn for all n ∈ {1, ..., N }, zn is the unique -best element in Bn for all n ∈ {1, ..., N − 1}. zN is a -best element in BN .
Like P-richness, almost P-richness is a pointwise criterion. Moreover, if B is P-rich, then B is almost P-rich, while the converse does not generally hold. However, the following example shows that even this mild relaxation is too weak to yield identification of all separated theories under partial observability. Example 14. Let X = {x, y, z}, B = {{x, y}, {y, z}} and P = , , where x y ∼ z and z x ∼ y. It is straightforward to verify that B is almost P-rich, but fails to be P-rich. Moreover, the choice correspondence c : B → 2X given by c({x, y}) = {x} and c({y, z}) = {z} has full domain and is consistent with both preferences. Hence, P is not identified on B even though P is separated. Of course, the fact that Definition 9 does not work as intended does not prove that P-richness cannot be usefully weakened in some other way. Unfortunately, I am presently unaware of any such condition. A.3. Additional examples This section contains examples complementing several passages of the body of the paper. The following example shows that separation is not necessary for identification without compatibility: Example 15. X = [0, 1), B = {X}, and P = 1 , 2 , where 1 and 2 are represented by u1 (x) = x and 1 x ∈ [0, 1/2), u2 (x) = 2x x ∈ [1/2, 1), respectively. Then, P is not separated. However, P is identified on B. The following example shows that local strictness does not imply local nonsatiation:
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Example 16. X = [0, 1], is represented by u(x) = x. is continuous and locally strict, but has a satiation point at x = 1. The following example shows that regularity does not imply local strictness: Example 17. X = [0, 4], is represented by ⎧ ⎪ x ∈ [0, 2], ⎨x u(x) = 4 − x x ∈ (2, 3], ⎪ ⎩ 1 x ∈ (3, 4]. is continuous and regular, but fails to be locally strict (consider any pair of different points x, y ∈ (3, 4]). The following example shows that connectedness cannot be dispensed with in Lemma 2: Example 18. X = [0, 1] ∪ [2, 3], is represented by x x ∈ [0, 1], u(x) = x − 1 x ∈ [2, 3]. is continuous and generically locally non satiated. However, it is not regular (consider the pair of points (1, 2)). Moreover, generic local nonsatiation is sufficient, but not necessary in Lemma 2: Example 19. X = [0, 4], is represented by ⎧ ⎪ x ∈ [0, 2], ⎨x u(x) = 4 − x x ∈ (2, 3], ⎪ ⎩ 1 x ∈ (3, 4]. is continuous and regular, but fails generic local non satiation. The following example shows that generic local nonsatiation does not imply local strictness: Example 20. X = [0, 6], is represented by ⎧ ⎪ x x ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2 − x x ∈ (1, 2], u(x) = x − 2 x ∈ (2, 4], ⎪ ⎪ ⎪ 6 − x x ∈ (4, 5], ⎪ ⎪ ⎪ ⎩x − 4 x ∈ [5, 6]. is continuous and generically locally nonsatiated (thus regular), but fails to be locally strict (consider the pair (1, 5)).
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Appendix B. Proofs This appendix contains the proofs of the theorems and propositions in the main text. The following abstract characterization of separated theories will be useful throughout the proofs. Define a binary relation on the class of all preferences by ⇐⇒ ⊆ ∧ ⊆ . Lemma 3. A theory P is separated if and only if the restriction of to P is antisymmetric. Proof. I will establish the equivalence of the contrapositives. P is not separated if and only if there are , ∈ P, such that = and, for all x, y ∈ X, x y implies y x and x y implies y x. These implications are equivalent to ⊆ ⊀ = and ⊆ ⊀ = , where the equalities follow from the completeness of and , respectively. In turn, these two inclusions amount to the definition of . Therefore, P is not separated if and only if restricted to P is not antisymmetric, as claimed. 2 B.1. Proofs of Section 2 Proof of Theorem 1. Assume that B is compatible with P, but P is not separated. Then, by Lemma 3, there are , ∈ P such that = and . Consider the choice correspon∗ (B) ∩ c∗ (B). Clearly, c is consistent dence c : B → 2X defined for each B ∈ B by c(B) := c
with both of and . I claim that c has full domain. To see this, pick any choice problem ∗ (B) = ∅. This means that there is x ∈ B such that B ∈ B. Since B is compatible with P, c ∗ (B), then x ∈ c(B). Second, x y for all y ∈ B. There are two cases to consider. First, if x ∈ c
∗ (B), then c∗ (B) = ∅ implies that there must be z ∈ c∗ (B) such that z x. However, if x ∈ / c
∗ (B), and so z ∈ c(B). In either case, c(B) = ∅ since , it must be that z x. Thus, z ∈ c holds. It follows that c has full domain. The preceding argument establishes that c has full domain and is, nevertheless, consistent with different preferences in P. This means that P is not identified. I conclude that identification over a compatible collection of choice problems implies separation. 2
Proof of Theorem 2. Assume that B is P-rich, but P is not identified on B. This means that there exist different preferences , ∈ P and a choice correspondence c with full domain that is consistent with both and . Select any x, y ∈ X, such that x y. Since B is P-rich, there is a sequence of N problems B1 , ..., BN ∈ B and a sequence of N + 1 alternatives z1 , ..., zN+1 ∈ ∗ (B ) = {z } hold. X, such that z1 = x, zN+1 = y, and, for all n ∈ {1, ..., N }, zn+1 ∈ Bn and c n n Pick any n ∈ {1, ..., N }. On the one hand, since Bn ∈ B and c has full domain, c(Bn ) = ∅. On ∗ (B ) because c is consistent with . Thus, c(B ) = {z }. Since c the other hand, c(Bn ) ⊆ c n n n
is consistent with and zn+1 ∈ Bn , one obtains zn zn+1 . Considering that this argument holds for each n ∈ {1, ..., N }, the transitivity of implies x = z1 zN+1 = y. It follows that y x. Since the choice of x, y was arbitrary, this means that P is not separated. I conclude that separation implies identification on B. 2 Proof of Theorem 3. Suppose A is necessary for identification. By definition of pointwise criterion,
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A(P) =
A({}) ⊇
∈P
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B {} is identified on B . ∈P
Theories containing on every collection of choice problems. a single preference are identified It follows that B {} is identified on B = B B ⊆ 2X holds for every preference ∈ P. Thus, A(P) ⊇ B B ⊆ 2X . Since B B ⊆ 2X is the codomain of A, the desired equality is obtained. 2 B.2. Proofs of Section 3 The proof of Theorem 4 requires some additional definitions. A preference is a tie-breaker for if ⊂ . It is easy to see that is a tie-breaker for if and only if ∼ ⊂ ∼ and ⊃ . A theory P is said to be closed to tie-breaking if S ∈ P holds for every subset S ⊆ P totally ordered by ⊆. A preference is ultimate in P if it admits no tie-breakers in P. If is a tie-breaker for , then . The following abstract result describes the space of tie-breakers for theories closed to tie-breaking. Lemma 4. Let P be a theory closed to tie-breaking. Then every ∈ P is either ultimate in P or admits an ultimate tie-breaker in P. Proof. Let P() be the set of all preferences in P that are contained in . Note that ⊆ is the reflexive closure of the property of being a tie-breaker. Moreover, ⊆ partially orders P(), and every totally ordered subset of P() is bounded by the intersection of all its elements (which belongs to P() because the theory is closed to tie-breaking). Thus, by Zorn’s lemma, P() must be ultimate; otherwise, it could not be possesses a ⊆-maximal element . Clearly, ⊆-maximal in P(). If P() = , then = . Otherwise, is a tie-breaker for . 2 The following notions are easy to depict and provide an equivalent definition of regularity that sometimes is more convenient. A pair (x, y) ∈ X × X is said to be -locally isolated if x = y and there exist open sets V x and W y, such that {z ∈ X|x z} ∩ V = {z ∈ X|z y} ∩ W = ∅. A point x ∈ X is said to be -transversal if every open set V containing x also contains x ∗ , x∗ ∈ V , such that x ∗ x x∗ . Then, it is straightforward to show that the preference is regular if and only if every indifference class for with a -locally isolated pair contains a -transversal point. (See Fig. 3.) Lemma 5. If and are regular continuous preferences satisfying , then = . Proof. Suppose, seeking a contradiction, that = . Without loss of generality, I can assume that there exists x, y ∈ X, such that y x and y x. By completeness, x y. Since , x y implies x y. It follows that y ∼ x. I will now show that y ∼ x y contradicts the regularity of . In preparation for the ar gument, define the sets V := z ∈ X z y and W := z ∈ X x z . Note that V and W are nonempty since x ∈ V and y ∈ W . Moreover, both sets are open by continuity and completeness of . Now suppose, seeking a contradiction, that is regular. Let I denote the indifference class for containing {x, y}. There are two cases to consider: Either I is -extremal or not. I treat each of these cases separately.
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Fig. 3. Characterizing regularity.
If I is -extremal, then (x, y) is not -locally isolated. It follows that either there exists x ∈ V such that x x or there exists y ∈ W such that y y. In the first case, y ∼ x x and transitivity of imply y x . Thus, yields y x , which contradicts x ∈ V . The second case is symmetric: y y ∼ x implies y x, so yields y x, thus contradicting y ∈ W . If I is not -extremal, there must be a -transversal point w ∈ I . Since x y, the completeness of implies w ∈ V ∪ W . This means that either w ∈ V or w ∈ W . Suppose first that w ∈ V . Since V is open, the -transversality of w implies that there exists x ∈ V , such that w x . Since x ∼ w, transitivity of implies that x x . If w ∈ W , a symmetric argument yields the existence of y ∈ W , such that y y. It follows that there exists either x ∈ V such that x x , or y ∈ W such that y y. Thus, the argument in the previous paragraph for the case in which (x, y) is not locally isolated applies. 2 Lemma 6. Let X be a separable metric space. For every continuous preference that is not regular, there exists a continuous preference , such that ⊂ . Proof. Fix a continuous preference that is not regular. I claim that there exists a continuous preference , such that ⊂ . The argument below proves this by breaking some indifferences without violating continuity. Since is not regular, there is a -extremal indifference class for , say I , containing points x0 , y0 , such that (x0 , y0 ) is a -locally isolated pair. Define P0 (resp. N0 ) as the set of all points x ∈ I , such that there exists an open set V x such that every y ∈ V satisfies y x (resp. x y). Since (x0 , y0 ) is a -locally isolated pair, x0 ∈ P0 and y0 ∈ N0 . Thus, P0 and N0 are nonempty. Since there are no -transversal points in I , I = P0 ∪ N0 and (I \ P0 ) ∩ (I \ N0 ) = ∅ (although P0 and N0 need not be disjoint). Moreover, note that both P0 and N0 are open in I by construction. Define P := {x ∈ X \ N0 |x x0 } and N := {x ∈ X \ P0 |x0 x}. If P or N turns out to be empty according to this definition, redefine it as P := {x0 } or N := {y0 }, respectively. P and N are clearly disjoint. Moreover, they are both closed in X. To see this, let {xn } be a sequence in P that converges to xˆ ∈ X and suppose, seeking a contradiction, that xˆ ∈ / P . It follows from / N0 for all n. Moreover, since xn x0 for all n, continuity implies the definition of P that xn ∈ / P , it must be that xˆ ∈ N0 . Then, considering the definition of N0 , xn ∼ x0 xˆ x0 . But, since xˆ ∈ must hold for all sufficiently large n. However, since xn ∈ / N0 , this means that xn ∈ I \ N0 for all sufficiently large n. Since I \ N0 is closed in I , I conclude that xˆ ∈ I \ N0 , providing the desired contradiction. This means that P is closed. The argument showing that N is closed is symmetric. Since X is a separable metric space and is continuous, there exists a continuous function U : X → R that represents (see Debreu (1954)). Since X is a metric space, it is also a normal
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space and, by Urysohn’s lemma, there exists a continuous function h : X → [0, 1], such that h(x) = 1 for x ∈ P and h(x) = 0 for x ∈ N (see Munkres (2000)). Finally, define := {(x, y) ∈ X × X|U (x) + h(x) ≥ U (y) + h(y)}. It is easy to verify that is a continuous preference and that ⊂ . Thus, ⊂ as desired. 2 Proof of Theorem 4. The first claim follows directly from Lemma 3 and Lemma 5. To prove the second claim, note that of continuous preferences is closed to tie the class breaking. To see this, suppose S ≡ i i ∈ I is an arbitrary set of continuous preferences totally ordered by ⊆ and define ∗ := ∩i∈I i . Clearly, ∗ is transitive and continuous, since these properties are preserved by arbitrary intersections. To see that it must also be complete, suppose—seeking a contradiction—that there are x, y ∈ X, such that x ∗ y ∗ x. Since ∗ = ∪i∈I i = ∪i∈I ≺i , it must be that x i1 y i2 x for some indexes i1 , i2 ∈ I , such that i1 = i2 . Since S is totally ordered, either i1 ⊂ i2 or i2 ⊂ i1 . Consider first the case in which i1 ⊂ i2 . Then, since i1 is a tie-breaker for i2 and y i2 x, it follows that y i1 x. This contradicts x i1 y. The case in which i2 ⊂ i1 is handled by a symmetric argument. Now, suppose is a continuous preference that is not regular. Since I have just proved that continuous preferences are closed to tie-breaking, Lemma 4 and Lemma 6 together imply that there exists a continuous preference that is an ultimate tie-breaker for . Suppose, seeking a contradiction, that is not regular. Then Lemma 6 implies that there is a continuous tie-breaker for . It follows that is not ultimate, providing the desired contradiction. I conclude that must be regular, completing the proof. 2 Proof of Lemma 1. Suppose first that is locally nonsatiated everywhere. Let (x, y) ∈ V ∩ for any arbitrary open set V . By the definition of the product topology, there must be an open rectangle V1 × V2 ⊆ V such that x ∈ V1 and y ∈ V2 . By local nonsatiation, there is x ∈ V1 such that x x. It follows that (x , y) ∈ V ∩ , establishing local strictness. For the second implication, it suffices to prove that local strictness of precludes the existence of an indifference class for containing a -locally isolated pair. I show this by establishing the contrapositive. If x, y are two different points such that x ∼ y, x is a local worst element, and y a local best element (for ), then there is an open rectangle V = V1 × V2 containing (y, x) such that y ∈ V1 implies y y and x ∈ V2 implies x x. It follows that V ∩ = ∅, contradicting local strictness. Finally, that the converse implications do not hold follows directly from Example 16 and Example 17, which feature continuous preferences over a compact interval. 2 Before proving Lemma 2, I need to establish the following lemma: Lemma 7. Let X be an arbitrary topological space and let P : X → {0, 1} be a pointwise property that holds generically on X (i.e. P (x) = 1 for all x on an open dense subset of X). Then, 1 ∈ P (V ) for every nonempty open set V ⊆ X. Proof. Define the set Y := {x ∈ X|P (x) = 1} and take any nonempty open set V ⊆ X. I claim that V ∩ Y = ∅. Suppose, seeking a contradiction, that V ∩ Y = ∅. Then, Y ⊆ X \ V , which implies that clX (Y ) ⊆ clX (X \ V ) = X \ V ⊂ X, contradicting the fact that Y is dense in X by assumption. Since V ∩ Y = ∅ and P (Y ) = {1} by definition, the result follows immediately. 2
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Proof of Lemma 2. Suppose, seeking a contradiction, that is not regular. Then, there must exist a set I ⊆ X and different points x0 , y0 ∈ I such that I is a -extremal indifference class for , x0 is a local worst element for , and y0 is a local best element for . The argument in the proof of Lemma 6 above implies that there exist disjoint nonempty closed sets P and N such that x0 ∈ P and y0 ∈ N . Define the open set V := X \ (P ∪ N ). Clearly, z ∈ V implies z ∈ I (that is, z ∼ x0 ∼ y0 ). Moreover, by Lemma 7 and the assumption that local nonsatiation holds generically on X, if V was nonempty there would be z0 ∈ V such that satisfies local nonsatiation at z0 . This cannot happen by definition of V (otherwise z0 would belong to P ). It follows that V must be empty. But then, {P , N } is a closed nontrivial partition of X, contradicting the assumption that X is connected. 2 The following two lemmas are used in the proof of Theorem 5. Before proceeding, I need some additional definitions and topological structure. Let (X, d) be a metric space. Denote by FX (resp. KX ) the collection of all nonempty closed (resp. compact) subsets of X equipped with the Hausdorff metric topology. Lemma 8. Consider a sequence {(An , xn )}n∈N in FX × X, such that An converges to A ∈ KX and xn ∈ An for all n ∈ N. Then (xn )n∈N has a subsequence converging to a point in A. Proof. Since limn→∞ An = A, there is a sequence {yh }h∈N in A and subsequence {xnh }h∈N , such that d(yh , xnh ) < 1/ h for all h ∈ N. Since yh ∈ A for all h ∈ N and A is compact, there is a subsequence {y˜k }k∈N ≡ {yhk }k∈N converging in A. Let x ∗ := limk→∞ y˜k . I will now show that the sequence {x˜k }k∈N ≡ {xnhk }k∈N converges to x ∗ . Choose any > 0. Since {y˜k }k∈N converges to x ∗ , there is k1∗ sufficiently large, such that d(x ∗ , y˜k ) < /2 for all k ≥ k1∗ . Let k2∗ := min {k ∈ N|hk > 2}. Clearly, d(y˜k , x˜k ) ≡ d(yhk , xnhk ) < 1/ hk < /2 for all k ≥ k2∗ . Define k ∗ := max{k1∗ , k2∗ }. It thus follows from the triangle inequality that d(x ∗ , x˜k ) ≤ d(x ∗ , y˜k ) + d(y˜k , x˜k ) < /2 + /2 = for all k ≥ k ∗ . This concludes the proof.
2
Lemma 9. Fix a continuous preference and consider any choice correspondence c consistent with . Then dom(cl(c)) = clB (dom(c)) and cl(c) is consistent with . Proof. Clearly, dom(cl(c)) ⊆ clB (dom(c)). To show equality, consider any B ∈ clB (dom(c)). Since B is a metric space, it is sequential. Hence, there is a sequence {Bn }n∈N in dom(c), such that limn→∞ Bn = B. Since Bn ∈ dom(c) for all n ∈ N, there is a sequence {xn }n∈N , such that xn ∈ c(Bn ) for all n ∈ N. Since B is compact, Lemma 8 implies that {xn }n∈N has a subsequence {x˜k }k∈N ≡ {xnk }k∈N converging to a point in B. Thus, if x ∗ is such a point, then x ∗ ∈ cl(c)(B). This shows that cl(c)(B) = ∅ for all B ∈ clB (dom(c)). Hence, clB (dom(c)) ⊆ dom(cl(c)). I now claim that cl(c) is consistent with . To prove this, select any (B, x) ∈ B × X, such that ∗ (B). Because x ∈ cl(c)(B), there are sequences {B } x ∈ cl(c)(B). I need to show x ∈ c n n∈N → B and {xn }n∈N → x, such that xn ∈ c(Bn ) for all n ∈ N. Now suppose, seeking a contradiction, that there is x ∈ B, such that x x . Continuity and completeness together imply that is a closed subset of X × X. Hence, there are open sets V , W , such that (x, x ) ∈ V × W and y z for all (y, z) ∈ V × W . Since {xn }n∈N converges to x, xn ∈ V must hold for all sufficiently large n. Since {Bn }n∈N converges to B, there is a sequence {xn }n∈N , such that xn ∈ Bn ∩ W for
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all sufficiently large n. However, this means that xn xn . Since xn ∈ c(Bn ) and xn ∈ Bn , this contradicts c being consistent with . 2 Proof of Theorem 5. For Part (1), let c be an arbitrary choice correspondence. I need to show that c and cl(c) are rationalized by the same subset of P. This statement can be expressed using the notation introduced in Section 4.5 as P (c, P) = P (cl(c), P). To establish the equality, I will ∗ (B) for prove both inclusions. Suppose first that ∈ P (cl(c), P). Then c(B) ⊆ cl(c)(B) ⊆ c all B ∈ B. Hence, ∈ P (c, P). I conclude that P (cl(c), P) ⊆ P (c, P). Suppose now that ∈ P (c, P). By definition of P (c, P), this means that c is consistent with . Lemma 9 thus implies that cl(c) is also consistent with . It follows—this time by definition of P (cl(c), P)—that ∈ P (cl(c), P). I conclude that P (c, P) ⊆ P (cl(c), P). This completes the proof of this part. For Part (2), assume that P is identified on B, let B be a dense subset of B, and pick any , ∈ P. Consider a choice correspondence c : B → 2X with domain B that is simultaneously consistent with and . I need to show that = . By Lemma 9, cl(c) is also consistent with both and . Moreover, dom(cl(c)) = clB (dom(c)) = clB (B ) = B. Since P is identified on B, it follows that = , as desired. To establish Part (3), I shall prove that = implies Y = Y (where Y := ∩ (Y × Y ) and Y := ∩ (Y × Y )). Suppose that = . Then, since P is separated, there exist points x, y ∈ X such that x y x. Because both and are continuous, there must be open sets V x and W y such that v w v for all v ∈ V and w ∈ W . The assumption that Y is dense in X implies that V ∩ Y = ∅ and W ∩ Y = ∅. It follows that there exist two alternatives v ∗ , w ∗ ∈ Y such that v ∗ w ∗ v ∗ . But this, in turn, implies Y = Y , concluding the proof. 2 Proof of Theorem 6. Fix any ∈ P and consider a choice correspondence c that is consistent with and has full domain. To prove the result, I need to establish that P (c, P) = . ∗ (B) ⊆ Y for every B ∈ Condition (1) in the statement of the result implies that c(B) ⊆ c . Define c : B → 2Y by setting c (B ∩ Y ) := c(B) ∩ Y for every B ∈ B. Notice that B Y Y Y
and cl (B ) = B dom(clY (cY )) = clBY (dom(cY )) by Lemma 9. Moreover, dom(cY ) ⊇ B B by Condition (1). It follows that dom(clY (cY )) = clBY (dom(cY )) ⊇ clBY (B ) = clB (B ) ∩ BY = BY .
This means that clY (cY ) has full domain. Since PY is separated and BY is PY -rich, Theorem 2 implies that PY is identified on BY . Combining this with Part (1) in Theorem 5 yields P (cY , PY ) = P (clY (cY ), PY ) = ∩ (Y × Y ) . Let π : P → PY be the projection defined by setting π() := ∩ (Y × Y ) for each ∈ P. By Condition (2), P (c, P) = π −1 (P (cY , PY )) = π −1 ∩ (Y × Y ) = . I conclude that P is identified on B, completing the proof. 2 B.3. Proofs of Section 4 The following three lemmas are used in the proof of Proposition 1.
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Lemma 10. PD is a separated theory. Proof. Pick ∈ PD . Axiom 3 implies that is locally nonsatiated, thus regular. Since all preferences in PD are continuous, Theorem 4 establishes the desired result. 2 ∗ (B Lemma 11. For any x ∈ X such that x 0, there exists p ∈ R2++ such that c p,px ) = {x}.
Proof. First consider the case in which x2 = 0. Then, it must be that x1 > 0 because x 0. In that case, define γ ∗ := inf {γ > 0|(x1 − 1, γ ) x}. Since (x1 , 0) (x1 − 1, 0), γ ∗ = 0 would imply (x1 , 0) ∼ (x1 − 1, 0). This contradicts Axiom 4, considering that (x1 + 1, 0) (x1 − 1, 0) and (x1 , 0) = (1/2)(x1 + 1, 0) + (1/2)(x1 − 1, 0). It follows that γ ∗ > 0. Then, setting p := (1, 2/γ ∗ ) gives the result. Now consider the case in which x2 > 0. Define the price vector 1 pR := 1, , x2 − γ ∗ where γ ∗ := inf {γ > 0|(x1 + 1, γ ) x}. Continuity of implies (x1 + 1, γ ∗ ) x and also pR x = pR (x1 + 1, γ ∗ ). In fact, γ ∗ > 0 implies (x1 + 1, γ ∗ ) ∼ x. However, it is possible that (x1 + 1, γ ∗ ) x when γ ∗ = 0. Define another price vector pL as follows. If x1 = 0, pL = (1, 0). If x1 > 0, then 1 pL := 1, ∗ , δ − x2 where δ ∗ := sup {δ > 0|x (x1 − 1, δ)}. Here, in the case x1 > 0 and δ ∗ < +∞, (x1 − 1, δ ∗ ) ∼ x (by continuity) and pL (x1 − 1, δ ∗ ) = pL x. Note that, due to Axiom 4, δ ∗ − x2 > x2 − γ ∗ > 0 and, so, pR and pL satisfy pR ≥ pL and pR = pL , so, in particular, (pL , pR ) = ∅. In the typical situation in which x1 > 0, δ ∗ < +∞, and γ ∗ > 0, the following characterization of the best bundle for a price vector p = (1, α) and w = px holds: (1) (2) (3) (4)
∗ (B ∗ If α = 1/(δ ∗ − x2 ), then c p,w ) = {x, (x1 − 1, δ )}. ∗ ∗ ∗ If 1/(x2 − γ ) > α > 1/(δ − x2 ), then c (Bp,w ) = {x}. ∗ (B ∗ If α = 1/(x2 − γ ∗ ), then c p,w ) = {x, (x1 + 1, γ )}. ∗ (B Else, x ∈ / c p,w ).
The other cases, in which either δ ∗ = +∞, γ ∗ = 0, or x1 = 0, are dealt with in a similar ∗ (B fashion. In each of these cases, it is possible to show that c p,w ) = {x} whenever 1/(x2 − ∗ (B ∗ ∗ γ ) > α > 1/(δ − x2 ). I conclude that, for any price vector p ∈ (pL , pR ), c p,px ) = {x}. 2 L is P -rich. Lemma 12. BD D
Proof. Pick ∈ PD and x, y ∈ X such that x y. Letting x ≡ (x1 , x2 ) and y ≡ (y1 , y2 ), there are three cases: x1 = y1 , x1 < y1 , and x1 > y1 . In the first case, it must be that x2 > y2 by Axiom 3. Then, Lemma 11 shows that there exists ∗ (B a price vector p ∈ R2++ such that c p,px ) = {x}. In the second case, x1 < y1 , so let N := y1 − x1 + 1 ≥ 2. Fix some small > 0. Following the ideas in the proof of Lemma 11, define bundles z1 , ..., zN ∈ X by setting z1 := x, zN+1 := y,
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and zn+1 := (z1n + 1, α n − ) for each n ∈ {1, ..., N − 1}, where α n is such that zn ∼ (z1n + 1, α n ) holds. Note that z1 z2 ... zN . Let p n ∈ R2++ be defined as 1 n p := 1, n+1 . z2 − z2n Clearly, p n zn = p n zn+1 holds by construction. It is easy to see that z1N = y1 and, if is small enough, z2N > y2 , which means that pN zN > p N y. Moreover, the same argument used in the ∗ (B n n n ) = {zn } for each proof of Lemma 11 establishes that, for all sufficiently small , c p ,p z n ∈ {1, ..., N}. Finally, in the third case, the proof proceeds almost as in the second case. Now let N := x1 − y1 + 1 ≥ 2 and define zn+1 := (z1n − 1, α n − ), where again zn ∼ (z1n+1 , α n ) zn+1 . The L is P -rich. 2 rest of the argument is identical. The three cases above establish that BD D L . By Lemma 10, P is a separated Proof of Proposition 1. Let B0 be a dense subset of BD D L . Since B theory. By Lemma 12, B is PD -rich. Then, by Theorem 2, PD is identified on BD 0 L is dense in BD , PD is identified on B0 by Part (2) of Theorem 5. 2
Proof of Proposition 2. For (1), let U : X → R be a C 1 representation of such that U (x) >> 0 whenever x 0. The existence of such a representation implies that is continuous, monotone (x >> y implies x y) and weakly monotone (x ≥ y implies x y). Pick any two points x, y ∈ X such that x y. Consider first the case in which y >> 0. Note that y 0 by monotonicity. I claim that, for each t ∈ [0, 1], there exists a unique λt ∈ R+ such that U (λt [(1 − t)x + ty]) = (1 − t)U (x) + tU (y). If t ∈ {0, 1}, the claim holds as a trivial consequence of monotonicity (with λ0 = λ1 = 1). If t ∈ (0, 1), define F : R++ × (0, 1) → R by setting F (λ, t) := U (λ [(1 − t)x + ty]) − (1 − t)U (x) − tU (y). Note that F (λ, t) is increasing in λ. Fixing t ∈ (0, 1) and using U (x) > U (y) > U (0), lim F (λ, t) = U (0) − (1 − t)U (x) − tU (y) < (1 − t) [U (y) − U (x)] < 0. λ↓0
Moreover, setting λ := (1 − t)−1 , U (x) > U (y), y >> 0, and monotonicity imply that F λ, t = U (x + λty) − (1 − t)U (x) − tU (y) > U (x + λty) − U (x) > 0.
Since the function F (·, t) is increasing, continuous, and satisfies limλ↓0 F (λ, t) < 0 < F λ, t , there exists a unique λt ∈ (0, λ) such that F (λt , t) = 0, as claimed. The function F defined above is C 1 and satisfies ∂F (λ, t) U (zt )zt = > 0, ∂λ λ=λt λt where zt := λt [(1 − t)x + ty]. It follows by the implicit function theorem that there is a C 1 function t → λ˜ t satisfying F (λ˜ t˜, t˜) = 0 for all t˜ in a neighborhood of t . Since, for each t ∈ [0, 1],
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the equation F (λ, t) = 0 has a unique solution λ = λt , it must be that λ˜ t = λt . It follows that the mapping t → λt constructed in the previous paragraph is C 1 . Since the mapping t → λt is C 1 , the mapping t → zt is C 1 as well. Moreover, z0 = x, z1 = y, and also dU (zt ) = U (y) − U (x) < 0, dt which means that utility decreases along the smooth trajectory zt connecting x and y. Defining pt z ≤ pt zt ∈ B L , Axiom 6 implies that c∗ (Bt ) = {zt }. pt := U (zt ) >> 0 and Bt := z ∈ RN + Moreover, dzt dzt dU (zt ) = U (zt ) = = U (y) − U (x) < 0. dt dt dt Now, for each N ∈ N, define tn,N := n/N and consider the grid {t0,N , t1,N , ..., tN,N }. The mapping t → zt is C 1 and the mapping t → pt is continuous. It follows that, whenever t = n/N , ztn+1,N − ztn,N dzt lim ptn,N < 0. = pt N→+∞ tn+1,N − tn,N dt pt
As a consequence, if N is large enough, ztn+1,N − ztn,N 1 ptn,N ztn+1,N − ztn,N = ptn,N <0 N tn+1,N − tn,N holds for every n ∈ {0, ..., N }. Hence, ptn,N ztn+1,N ≤ ptn,N ztn,N . Summarizing, for all sufficiently large N , the sequence {ztn,N }N n=0 satisfies zt0,N = x, ztN,N = ∗ (B y, ztn+1,N ∈ Btn,N , and c ) = {z } for all n ∈ {0, ..., N − 1}. This shows that the condition tn,N tn,N required by {}-richness holds for x and y, provided y >> 0. Now suppose that y >> 0 does not hold. In that case, take some y >> y such that x y . (This is always possible due to the continuity of .) Since y ≥ 0 implies that y >> 0, the previous argument can be applied to verify the {}-richness requirement on x and y . In this way, N one obtains a sequence of choices {zn }N+1 n=1 and budget sets {Bn }n=1 such that z1 = x, zN+1 = y , ∗ zn+1 ∈ Bn , and c (Bn ) = {zn } for all n ∈ {1, ...N }. Since y >> y ensures that y ∈ BN , if one
uses zN+1 = y instead of zN+1 = y the requirements imposed by the definition of {}-richness remain fulfilled. This completes the proof of this case and establishes that the collection of linear budget sets B L is {}-rich. For (2), suppose, seeking a contradiction, that B L is {}-rich but is not strictly convex. Then there are different alternatives x, y ∈ X = RN + such that x y, and α ∈ (0, 1) such that z := αx + (1 − α)y y. By completeness, x y z. Since z is a strict convex combination of ∗ (B) = {z}. Since B L is {}-rich, this implies z w for x and y, there is no B ∈ B L such that c
all w ∈ RN + , so w z for all w ∈ X (by completeness). Because is strictly monotone, it follows that z = 0. Since 0 is the unique extreme point of RN + and z is a strict convex combination of x and y, it follows that x = y = 0, contradicting x = y. I conclude that must be strictly convex. 2
Proof of Proposition 3. As mentioned in the main text, expected utility preferences are continuous (this follows from the fact that the expected utility representations are continuous by the Portmanteau lemma). To verify regularity, consider any nontrivial expected utility preference . Since is nontrivial, there must be lotteries x, y ∈ X such that x y. Now let I ⊆ X be an
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arbitrary indifference class for , and consider lotteries z, z ∈ I and open sets V , W such that z ∈ V and z ∈ W . By completeness, either x z ∼ z or z ∼ z y. Suppose that x z ∼ z . Since mixing lotteries is a continuous operation, there must be sufficiently small > 0 such that x + (1 − )z ∈ V and x + (1 − )z ∈ W . But, by independence, x + (1 − )z z and x + (1 − )z z . It follows that (z, z ) cannot be a locally isolated pair. A symmetric argument leads to the same conclusion in the case z ∼ z y. Since I and z, z were arbitrary, is regular. 2 Proof of Proposition 4. Fix an expected utility preference and suppose x y. To prove (1), let N ≥ 2 be a fixed number and consider the choice problem B = {x, y} ∪ 2−n x + (1 − 2−n )y n = 1, ..., N − 2 . Clearly, B has exactly N elements. Moreover, by independence, x z for all z ∈ B \ {x}. It ∗ (B) = {x}. I conclude that Definition 6 is satisfied by the collection of all choice follows that c problems with N elements. To prove (2), consider the segment [x, y] := {αx + (1 − α)y|α ∈ [0, 1]}. Clearly, [x, y] is ∗ ([x, y]) = {x}. As a result, to establish the desired richness, it suffices to show convex and c that [x, y] is a compact subset of X. To see this, consider the projection π : [0, 1] → [x, y] defined by setting π(α) := αx + (1 − α)y for each α ∈ [0, 1]. Every sequence {αn }n∈N such that limn→∞ αn → α verifies lim udπ(αn ) = lim αn udx + lim (1 − αn ) udy, n→∞
n→∞
X
=α =
n→∞
X
udx + (1 − α) X
X
udy, X
udπ(α) X
for all bounded continuous functions u : X → R. Thus, the Portmanteau lemma implies that π is continuous. Since [x, y] = π ([0, 1]) and [0, 1] is compact, [x, y] must be compact as well. 2 Proof of Proposition 5. Let P be the class of preferences on X that are nontrivial, continuous and satisfy set independence. By Theorem 4, it suffices to show that all preferences in P are regular. The argument for this is structured exactly as the one used in the proof of Proposition 3. The only additional difficulty is that it is not immediate to show that every open set around a menu contains strict convex combinations with every other menu. Consider any preference ∈ P. Since is nontrivial, there must be menus x, y ∈ X such that x y. Let I ⊆ X be an arbitrary indifference class for , and consider menus z, z ∈ I and open sets V , W such that z ∈ V and z ∈ W . By completeness, either x z ∼ z or z ∼ z y. Suppose that x z ∼ z . I claim that there exists > 0 such that x + (1 − )z ∈ V . To see this, let dH : X × X → [0, ∞) be the Hausdorff distance on X and let dL : × → [0, ∞) be the Lévy-Prokhorov distance on . Define functions f, g : [0, 1) → X by setting f () := max min dL (p, (1 − )q + r) p∈z q∈z,r∈x
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and g() := max min dL (p, (1 − )q + r) q∈z,r∈x p∈z
for each ∈ [0, 1). Note that f (0) = g(0) = 0. Given that x and z are compact, is convex and dL is continuous (on × ), I conclude that dH (z, x + (1 − )z) = max{f (), g()} and that the claim is equivalent to lim↓0 f () = lim↓0 g() = 0. But this continuity follows from applying Berge’s maximum theorem twice, since x and z are fixed nonempty compact menus and (1 − )q + r depends continuously on . A similar argument shows that there exists > 0 such that x + (1 − )z ∈ W . Thus, ∗ x + (1 − ∗ )z ∈ V and ∗ x + (1 − ∗ )z ∈ W hold simultaneously for ∗ := min{, } > 0. Since ∗ x + (1 − ∗ )z z and ∗ x + (1 − ∗ )z z by set independence, (z, z ) cannot be a locally isolated pair. A symmetric argument leads to the same conclusion in the case z ∼ z y. Since I and z, z were arbitrary, must be regular. 2 Proof of Proposition 6. The argument is similar to the one establishing Proposition 4. The only relevant difference lies in establishing that every segment joining two menus is compact in part (2). Following the idea in the proof of Proposition 5, it can be shown that segments are closed subsets of X. To complete the proof, note that, since X is a compact metric space (equipped with the Hausdorff distance), every closed subset of X is compact. 2 In preparation for the proof of Proposition 7, I need to establish the intuitive result that the image of an iid sequence of random elements in a separable metric space is dense in its support with probability 1. For any topological space Y , let clY denote the closure operator that maps any subset of Y into the intersection of all closed sets containing it. Lemma 13. Let (X, d) be a separable metric space and let (, , μ) be a probability space with an iid sequence {xn }n∈N of random elements xn : → X for each n ∈ N. Define the support of each xn as Sn := A ∈ FX μ xn−1 (A) = 1 , and let S := n∈N Sn . Define the random set R : → 2X by setting R(ω) := {xn (ω)|n ∈ N} for each ω ∈ . Then clX (R) = S almost surely. Proof. First, I show that clX (R) ⊆ S almost surely. Since S is closed in X by definition, it is sufficient to prove that μ(R ⊆ S) = 1. It is easy to show that μ(xn ∈ Sn ) = 1 for each n ∈ N (see Theorem 2.1 in Parthasarathy (1967)). Note that since the sequence {xn }n∈N is iid, S = Sn and, thus, μ(xn ∈ S) = 1 for every n ∈ N. Hence,
μ(R ⊆ S) = μ (∀n ∈ N : xn ∈ S) = μ (xn ∈ S) = 1. n∈N
Next, I show that clX (R) ⊇ S almost surely. Define V := V ∈ 2X \ FX V ∩ S = ∅ and consider any open set V ∈ V . By definition of S, it must be that μ (xn ∈ V ) > 0 for all n ∈ N. Thus, since the sequence {xn }n∈N is iid, there exists a number p ∈ [0, 1) such that μ (xn ∈ / V)= p for all n ∈ N. Thus,
μ (V ∩ R = ∅) = μ (xn ∈ / V)= p = lim p n = 0. n→∞
n∈N
n∈N
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Since X is a separable metric space, it is second-countable. This means that there exists a count able basis of open sets W C . Define W := W ∈ W C W ∩ S = ∅ . W is clearly countable and satisfies W ⊆ V . Hence, μ(W ∩ R = ∅) = 0 for every W ∈ W . Moreover, since each V ∈ V is the union of the open sets in W C contained in V , the following equality holds: {ω ∈ |V ∩ R(ω) = ∅} = {ω ∈ |W ∩ R(ω) = ∅} . V ∈V
W ∈W
Therefore, μ clX (R) S = μ (S \ clX (R) = ∅) = μ ∃V ∈ 2X \ FX : V ∩ S = ∅ ∧ V ∩ R = ∅ , ⎛ ⎞ {ω ∈ |V ∩ R(ω) = ∅}⎠ , = μ (∃V ∈ V : V ∩ R = ∅) = μ ⎝ ⎛ = μ⎝
V ∈V
⎞
{ω ∈ |W ∩ R(ω) = ∅}⎠ ≤
W ∈W
μ (W ∩ R = ∅) = 0,
W ∈W
where the inequality is a consequence of the countable subadditivity of μ. It follows that μ (clX (R) ⊇ S) = 1 − μ clX (R) S = 1 − 0 = 1. It is easy to see that μ (clX (R) ⊇ S) = μ (clX (R) ⊆ S) = 1 implies μ (clX (R) = S) = 1.
2
I now proceed to prove Proposition 7. c(ω))) ˆ = B . Since the Proof of Proposition 7. Consider the event 1 := ω ∈ clB (dom( ˆ = Bˆ n (ω)n ∈ N for every ω ∈ , support of Bˆ n is B for every n ∈ N and dom(c(ω)) Lemma 13 implies that μ(1 ) = 1. Consider also the event 2 := ω ∈ | c(ω) ˆ is consistent 2 ∗ 1 2 with . Since cˆ is consistent with , μ( ) = 1. Define := ∩ . Clearly, μ(∗ ) = 1. Select ω ∈ ∗ . By Lemma 9, cl(c(ω)) ˆ has full domain and cl(c(ω)) ˆ is consistent with . Since P is identified on B, P (cl(c(ω)), ˆ P) = {}. Combining this with Part (1) in Theorem 5 and the fact that all preferences in P are continuous by construction, I conclude that P (c(ω), ˆ P) = {}. Since μ(∗ ) = 1, it follows that P (c, ˆ P) = {} almost surely, as claimed. 2 B.4. Proofs of Section 5 Proof of Proposition 8. Let , be nontrivial vNM preferences, such that . Suppose, seeking a contradiction, that = . Then, without loss of generality, there should be z, y ∈ X, such that z y and z y. Completeness of implies y z. Moreover, and y z imply y z. It follows that y ∼ z and y z. Now, because is nontrivial, there must be x y. Since y ∼ z, it follows that either x y ∼ z or y ∼ z x. The treatment of these two cases is symmetric, so I focus exclusively on the first one. Suppose x y ∼ z. Because , it follows that x y and x z. Since y z, mixed transitivity implies that x z. In summary, thus far, I have x y ∼ z and x y z. Since is a vNM preference and x z, there exists a linear representation U , such that U (x) = 1 and U (z) = 0. Let α := U (y) ∈ (0, 1], and define p := αx + (1 − α)z. Clearly, p ∼ y by construction. Now define
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1 1 q := p + z = 2 2
1 1 α x + 1 − α z. 2 2
On the one hand, q z ∼ y, and so q y. On the other hand, y ∼ p q, and so y q. Thus, q y q. However, since , q y implies q y. This is inconsistent with y q, thus yielding the desired contradiction. I conclude that = , which means that the class of nontrivial vNM preferences is separated, as claimed. 2 Proof of Proposition 9. Let and be atomless continuous qualitative probabilities, such that . Suppose, seeking a contradiction, that = . This implies without loss of generality that there are events x, y ∈ X, such that y x and y x. By completeness, x y. Since , this implies x y. Thus, y ∼ x y. If x ∼ ∅, define x := \ y and y := \ x, which satisfy x ∼ y and x y by Axiom 11. Thus, assume without loss that x ∅. I now construct an increasing sequence of events {xn }n∈N ↑ x, such that x xn for all n ∈ N. Let x be the restriction of to Xx := {z ∩ x|z ∈ X}. Clearly, Xx is a σ -algebra and x is an atomless continuous qualitative probability on Xx . By Theorem 3.5 in Villegas (1964), there is a x -uniform random variable f : x → [0, 1]. This means that, for any two intervals I, J contained in [0, 1], f −1 (I ) x f −1 (J ) if and only if λ(I ) ≥ λ(J ), where λ is the Lebesgue measure. Define the sequence {xn }n∈N as xn := f
−1
!
n−1 ∪ {1} 0, n
Clearly, {xn } is increasing and satisfies lim xn =
n→∞
xn = f −1 ([0, 1]) = x.
n∈N
Since f is x -uniform, x x xn holds for all n ∈ N. Since xn ⊂ x, x x xn is equivalent to x xn . Moreover, since y ∼ x, transitivity implies y xn . Then, because , it follows that y xn ↑ x. Monotonic continuity therefore implies y x, which poses a contradiction. I conclude that = . This means that the class of atomless continuous qualitative probabilities is a separated theory, as claimed. 2 B.5. Proofs of Appendix A Proof of Theorem 7. Let P be separated and let B be weakly P-rich. Suppose, seeking a contradiction, that P is not identified under full observability on B. Then, by definition, there ∗ (B) = c∗ (B) for all B ∈ B. Since P is separe distinct preferences , ∈ P such that c arated and = , there must be x, y ∈ X such that x y x. Moreover, since B is weakly P-rich and x y, there exist B1 , ..., BN ∈ B, z1 , ..., zN+1 ∈ X such that z1 = x, zN+1 = y, ∗ (B ) × B for all n ∈ {1, ..., N }. The hypothesis that c∗ (B) = c∗ (B) for all and (zn , zn+1 ) ∈ c n n ∗ (B ) × B for all n ∈ {1, ..., N } and, so, z z B ∈ B implies that (zn , zn+1 ) ∈ c n n n n+1 for all
n ∈ {1, ..., N}. Finally, since is transitive, it follows that x = z1 zN+1 = y, contradicting y x. 2
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References Afriat, S.N., 1967. The construction of utility functions from expenditure data. Int. Econ. Rev. 8 (1), 67–77. Antonelli, G., 1886. Sulla Teoria Matematica della Economica Politica. Arrow, K.J., 1959. Rational choice functions and orderings. Economica 26 (102), 121–127. Chambers, C.P., Echenique, F., 2016. Revealed Preference Theory. Cambridge University Press. Chambers, C.P., Echenique, F., Lambert, N.S., 2018. Preference Identification. Working paper. Chambers, P.C., Echenique, F., Shmaya, E., 2014. The axiomatic structure of empirical content. Am. Econ. Rev. 104 (8), 2303–2319. Debreu, G., 1954. Decision processes. In: Representation of a Preference Ordering by a Numerical Function. John Wiley and Sons. Debreu, G., 1959. Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Yale University Press. Dekel, E., Lipman, B.L., Rustichini, A., 2001. Representing preferences with a unique subjective state space. Econometrica 69 (4), 891–934. Dybvig, P., Polemarchakis, H.M., 1981. Recovering cardinal utility. Rev. Econ. Stud. 48 (1), 159–166. Forges, F., Minelli, E., 2009. Afriat’s theorem for general budget sets. J. Econ. Theory 144 (1), 135–145. Gul, F., Pesendorfer, W., 2001. Temptation and self-control. Econometrica 69 (6), 1403–1435. Herstein, I.N., Milnor, J., 1953. An axiomatic approach to measurable utility. Econometrica 21 (2), 291–297. Houthakker, H.S., 1950. Revealed preference and the utility function. Economica 17 (66), 159–174. Hurwicz, L., Uzawa, H., 1971. On the integrability of demand functions. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H.F. (Eds.), Preferences, Utility and Demand. Harcourt, Brace, Jovanovich, pp. 174–214. Kreps, D.M., 1988. Notes on the Theory of Choice. Westview Press. Kreps, D.M., 2012. Microeconomic Foundations I: Choice and Competitive Markets. Princeton University Press. Kübler, F., Polemarchakis, H.M., 2017. The identification of beliefs from asset demand. Econometrica 85 (4), 1219–1238. Mas-Colell, A., 1977. The recoverability of consumers’ preferences from market demand behavior. Econometrica 45 (6), 1409–1430. Mas-Colell, A., 1978. On revealed preference analysis. Rev. Econ. Stud. 45 (1), 121–131. Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University Press. Munkres, J.R., 2000. Topology. Prentice Hall. Nishimura, H., Ok, E.A., Quah, J.K.-H., 2017. A comprehensive approach to revealed preference theory. Am. Econ. Rev. 107 (4), 1239–1263. Parthasarathy, K.R., 1967. Probability Measures on Metric Spaces. Academic Press. Reny, P.J., 2015. Rationalizable consumer behavior. Econometrica 83 (1), 175–192. Samuelson, P.A., 1938. A note on the pure theory of consumer’s behavior. Economica 5, 61–71. Samuelson, P.A., 1950. The problem of integrability in utility theory. Economica 17 (68), 355–385. Savage, L.J., 1972. The Foundations of Statistics. Dover Publications. Sen, A.K., 1971. Choice functions and revealed preference. Rev. Econ. Stud. 38 (3), 307–317. Uzawa, H., 1956. Note on preference and axioms of choice. Ann. Inst. Stat. Math. 8, 35–40. Villegas, C., 1964. On qualitative probability σ -algebras. Ann. Math. Stat. 35, 1787–1796.
ScienceDirect Journal of Economic Theory 183 (2019) 698–739 www.elsevier.com/locate/jet
Revealed preference and identification Leandro Gorno FGV EPGE, Praia de Botafogo 190/1119B, Rio de Janeiro, RJ 22250-900, Brazil Received 4 July 2017; final version received 24 June 2019; accepted 3 July 2019 Available online 10 July 2019
Abstract This paper studies preference identification in a general framework that allows for partial observability of optimal choices: Decision makers select some optimal alternatives, but not necessarily all of them. While partial observability is a methodologically appealing assumption for empirical applications, it makes recovering preferences much harder. The main result provides abstract conditions on classes of preferences and decision problems ensuring identification. The result is applied to several standard settings demonstrating the power of the method. © 2019 Elsevier Inc. All rights reserved. JEL classification: C02; D11; D81 Keywords: Revealed preference; Identification; Recoverability; Indifference; Continuity; Uniqueness
1. Introduction This paper examines the problem of recovering preferences from empirically observed choices when decision makers behave optimally. The main contribution is the formulation of reasonable theoretical restrictions that uniquely identify preferences in a general setting. Most significantly, identification is achieved without assumptions on how decision makers select among indifferent alternatives. Identification is not always possible because observed decisions may be consistent with different preferences. For example, consider a couple trying to decide whether to move to Paris or stay in New York. Ann prefers to move, while Bob is indifferent. Suppose that either Ann or Bob E-mail address: [email protected]. https://doi.org/10.1016/j.jet.2019.07.003 0022-0531/© 2019 Elsevier Inc. All rights reserved.
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makes the choice, but only the outcome is observed. Clearly, if the couple stays in New York, it would be safe to conclude that Bob made the decision, because Ann would have preferred to move. In this sense, staying in New York “reveals” that Bob’s preference is the one that counted. However, if they move to Paris, the evidence available is insufficient to infer who determined the outcome. The possibility of identifying preferences from observed behavior depends on the class of preferences being entertained. For example, suppose a voter wants to infer a politician’s type by looking at his or her past policies. Assume first that the politician can only be “left-wing” or “right-wing.” There are two policies available: “liberal” and “conservative.” Left-wing politicians strictly prefer the liberal policy to the conservative policy, and vice versa. In this case, any past policy uniquely identifies the politician’s type. Next, consider an additional type of politician, a “pragmatic,” who is indifferent between the liberal and the conservative policies. Including this third type makes potential inferences less conclusive. A liberal policy rules out a right-wing politician, but does not determine whether the politician is left-wing or pragmatic. A similar indeterminacy remains when multiple previous policy choices are observed: The only type of politician that can be learned from the data is the pragmatic (after observing a mixture of liberal and conservative policies). As the aforementioned examples illustrate, indifference poses a problem to identification inasmuch as it enlarges the set of theoretical preferences that can be confounded for a given dataset. The traditional axiomatic approach to abstract revealed preference simplifies this issue by effectively assuming that every alternative the decision maker could have chosen is observed.1 However, this approach is unsatisfactory from a methodological viewpoint, because it transmutes what is supposed to be empirical data (i.e., the decision maker’s observed choices) into a theoretical object (i.e., the set of preference-maximizing alternatives).2 In contrast, this paper develops a general framework for revealed preference analysis that affords a cleaner separation between the theories being entertained and the data intended to evaluate them. This is attained by relaxing informational assumptions while keeping the basic logic of revealed preference intact: What the decision maker chooses cannot be worse than anything else she could have chosen; unchosen options, however, are not presumed to be suboptimal. Following Chambers et al. (2014), I refer to this approach as “partial observability.” Once partial observability is adopted, the fundamental challenge is to obtain identification. This is because finding preferences consistent with the available evidence becomes relatively easy. In fact, if the only assumptions on preferences are completeness and transitivity, rationalization under partial observability is trivial: Any collection of observations is rationalized by universal indifference. At the same time, this lack of restrictions makes identification harder. Of course, it is possible to obtain restrictions on observable choice behavior, and even identification, by limiting admissible preferences. 1 In the axiomatic approach, the basic primitive is a “choice correspondence” that describes all optimal options in every choice problem under consideration. See, for example, Definition 1.D.1 in Mas-Colell et al. (1995), p. 12, and Kreps (2012), p. 2. This assumption considerably simplifies identification, even though it does not immediately render the problem trivial (if the collection of choice problems is not sufficiently rich, distinct preferences may be observationally equivalent). 2 If the decision maker faces each choice problem repeatedly and randomizes among all optimal options in a nondegenerate and sufficiently independent way, every optimal option would eventually be chosen (almost surely). This paper’s underlying claim is that it is better to formulate theoretical assumptions as explicit restrictions on preferences, and not as an ad hoc hypothesis on how the observed empirical data is generated.
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The central contribution of this paper is its proposal of general theoretical assumptions that are sufficient for identification under partial observability. I then use this framework to study important classes of preferences including several types of ordinal continuous preferences, preferences satisfying the von Neumann–Morgenstern (vNM) axioms, and qualitative probabilities. Since indifference is easily disciplined within these classes, relatively weak additional conditions suffice to uniquely recover any preference driving behavior, provided that enough data is observed. It should be stressed that departing from the axiomatic approach to relax full observability not only constitutes a methodological improvement but also has significant theoretical consequences. For example, under partial observability, choice correspondences are naturally interpreted as encoding empirical datasets. Under this interpretation, preference maximization on its own does not imply the celebrated axioms of revealed preference. Therefore, violations of these axioms are not necessarily evidence of irrational behavior: They might merely reflect the limits of the available data. Overall, the message of this paper is somewhat comforting: It is possible to apply revealed preference analysis to interesting theories of behavior, while keeping the theoretical assumptions employed to obtain identification separate from the empirical data. 1.1. Recovering quasilinear preferences with a discrete good To illustrate the methodological contribution of this paper, consider the following simple example. Suppose that Ann is a competitive consumer who has quasilinear preferences over bundles of two goods: cars (x1 ) and money (x2 ). More precisely, her preferences can be represented by a utility function U of the form U (x1 , x2 ) = v(x1 ) + x2 , for some smooth, increasing, and strictly concave function v. A cars-money bundle (x1 , x2 ) is considered feasible if it lies in the consumption set X (which I have purposely left unspecified) and satisfies the budget constraint p1 x1 + p2 x2 ≤ w, where p1 > 0 and p2 > 0 are the respective prices of each good and w > 0 is Ann’s wealth at the time of making the choice.3 Given market prices, Ann chooses a bundle that maximizes her utility among all bundles in the feasible set (i.e., the intersection of her budget set with the consumption set). Imagine that the function v is unknown but a sufficiently large and varied collection of choices is observed (and the prices at which these choices were made). Is it possible to recover Ann’s preferences? In other words, are quasilinear preferences identified in this setting? Of course, providing an answer to these questions requires being more specific about the environment in which Ann makes her choices and the kind of behavioral data available. To be concrete, assume that the analysis has to be performed based on a sequence of pairs consisting of a single consumer choice and a price vector (representing the feasible set where this choice was made). To formalize the idea that large amounts of data are available, assume further that 3 The literature (e.g., Afriat (1967), Reny (2015)) typically does not consider consumer wealth as observable data and instead recovers it from observed choices and market prices by assuming that the consumer spends her entire income. Since v is increasing, this assumption holds for Ann.
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the sequence of observed price-wealth situations is dense in the set of all possible price-wealth situations. If, as usual in classic demand theory, the consumption set is taken to be X = R2+ , identification can be established with little difficulty using known results. First note that strict convexity of preferences implies that there cannot be indifference at the optimum. Hence, the Walrasian demand must be single-valued (i.e., a demand function). Even though only choices at some feasible sets are observed (i.e., those along the dense sequence given), the whole demand function can be easily obtained using continuity. Finally, Theorem 2 in Mas-Colell (1977) implies that Ann’s preferences can be recovered from her demand function. Now suppose that cars are indivisible and they can only be consumed in integer amounts. That is, the consumption set is X = Z+ × R+ instead of R2+ . In this case, two technical issues complicate identification. First, it is easy to show that, for every possible preference Ann might have, there are budget sets over which she would indeed be indifferent between two different bundles. This means that her Walrasian demand is necessarily a correspondence, which makes it more difficult to recover under partial observability (i.e., from the observation of only some optimal choices).4 Second, even if Ann’s demand correspondence was known in its entirety (i.e., under full observability), the existing literature seems to lack recoverability results that can be applied when choices are restricted by this disconnected consumption set. This paper develops a general framework to analyze identification problems under partial observability like the one described above and provides several technical tools to tackle the associated difficulties. In particular, my results imply that Ann’s preferences are identified in this case as well. That is, at most one preference in this class can be consistent with any countable dataset {(x n , p n , w n )}n∈N such that the observed price-wealth situations {(p n , w n )}n∈N are dense in the set of all possible price-wealth situations. The specific details are contained in Section 4.1. 1.2. Overview of the paper The main results in this paper can be divided in two groups: 1) general results, and 2) results about continuous preferences. The general results are contained in Section 2 and concern the relation between identification and a simpler separation condition that requires every pair of preferences to display a strict preference reversal. Theorem 1 shows that every identified theory must be “separated” in this sense (assuming the collection of choice problems satisfies a weak technical condition). Theorem 2 provides a partial converse to Theorem 1: Every separated theory is identified on “rich” collections of choice problems. Richness here is a substantive structural requirement, but it is purely theoretical and does not impose implicit or explicit assumptions on empirical choices. Unfortunately, Theorem 2 falls short from a full characterization of identification because the richness condition employed is not necessary. Theorem 3 shows that there is a sense in which this is unavoidable: No criterion selecting collections of choice problems on the basis of individual preferences can be necessary for identification and simultaneously yield a sufficiency result for all separated classes of preferences. The main results for continuous preferences are contained in Section 3. Theorem 4 singles out a large separated class of continuous preferences and shows that this class is maximal with 4 In addition, for each feasible set that allows the purchase of at least one car, there exist uncountably many quasilinear preferences that makes multiple bundles optimal. As a result, the possibility of having indifference at the optimum persists even in a quasi-experimental setting in which prices can be selected by the analyst.
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respect to the separation property. Theorem 5 shows that, under continuity, choice behavior in limit problems does not contain additional information about preferences. This result is useful because it allows to obtain identification using countable collections of problems in settings with uncountably many alternatives. Moreover, it can be used to relax the general richness condition, as illustrated by Theorem 6. Finally, applications of these results to consumer theory, expected utility preferences, preferences over menus, and random choice correspondences are developed in Section 4. Section 5 analyses two additional theories that are not covered by the previous results: vNM preferences and qualitative probabilities. Section 6 provides some closing remarks. Appendix A contains additional material that further develops some issues discussed in Section 2 and Section 3. All proofs are in Appendix B. 1.3. Related literature The main motivation for this paper is a discomfort with embedding the assumption that all optimal choices are observed in the foundation of abstract revealed preference analysis. This discomfort is not new, and neither is the idea of dropping the problematic assumption while restricting preferences in reasonable ways. However, to the best of my knowledge, this is the first attempt to obtain general identification conditions under partial observability. The revealed preference approach was originally proposed by Samuelson (1938) as a way of freeing the theory of consumer behavior from arbitrary assumptions on utility functions. Houthakker (1950) constitutes the definitive elaboration of Samuelson’s theory taking a demand system as the primitive, while Afriat (1967) shows how to construct a utility function consistent with a finite set of observed consumer choices at given prices. Afriat’s approach is extended to general budget sets by Forges and Minelli (2009), while Reny (2015) allows for infinite datasets. The problem of eliciting preferences from market demand behavior goes back to Antonelli (1886). Samuelson (1950) provides a historical survey. Modern versions of integrability results for the Walrasian demand are given by Hurwicz and Uzawa (1971). The problem of recoverability (i.e., the uniqueness of the rationalizing preference)—which is the focus of the present contribution—is studied in the context of demand theory by Mas-Colell (1977), who provides results for the case in which one observes the Walrasian demand in its entirety (i.e., under full observability). Dybvig and Polemarchakis (1981), Kübler and Polemarchakis (2017), and others focus on identification of characteristics starting from complete knowledge of demand correspondences in other specific environments. A related strand of literature that starts with Mas-Colell (1978) studies the limiting behavior of increasingly informative revealed preference exercises, as the consumer is observed choosing in more and more price-wealth situations. However, Mas-Colell (1978) assumes that the underlying preference generates a demand function, thus rendering the question of partial observability moot. The abstract approach in this paper aims to be general: It is not confined to consumer theory and allows for a wide range of potential applications. The axiomatic structure of the theory of revealed preference was first investigated by Uzawa (1956), Arrow (1959) and Sen (1971). A modern textbook treatment of abstract revealed preference can be found in Mas-Colell et al. (1995), while a systematic presentation of the theory is provided by Chambers and Echenique (2016). As the present paper, Nishimura et al. (2017) also extend the applicability of Afriat’s approach beyond consumer theory. However, their contribution focuses on characterizing rationalizability
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as well as unifying different approaches to revealed preference, rather than studying identification. In particular, their discussion of recoverability is confined to the classical demand setting. The recent work of Chambers et al. (2018) follows the spirit of Mas-Colell (1978)’s exercise and studies the possibility of learning a decision maker’s preference in an abstract experimental setting. Both Chambers et al. (2018) and the present paper are concerned with what can be learned about preferences in general environments and both allow for partial observability. However, while Chambers et al. (2018) focus on the large-sample properties of finite-sample rationalizing preferences, this paper intends to address a more basic identification question: Can the underlying preference be recovered when the decision maker has been observed making choices in all (admissible) choice problems? This is an important distinction and an econometric analogy can be useful to clarify it. Two central econometric problems are “identification” and “inference”. While the identification problem concerns the population properties of a given theoretical structure, the inference problem is about the possibility of learning aspects of such population from a sample. For example, in a parametric model for a random object X, the typical identification question is whether the value of the parameter, say θ , can be recovered from the conditional distribution f (X|θ ). In contrast, the inference problem concerns what can be said about θ on the basis of a random sample {X1 , ..., XN } for some N ∈ N. The prevalent view among econometricians is that the identification question logically precedes inference, as a sample cannot be more informative than the population it is taken from. In this sense, the contributions of Chambers et al. (2018) and the present paper are somewhat complementary. On the one hand, Chambers et al. (2018) develop the inference aspect of preference maximization in an experimental setting that simplifies the identification problem. On the other hand, the present paper explores identification in greater detail and without assuming an experimental setting, while basically ignoring inference. Finally, this paper is also related to a body of work that investigates the empirical content of economic theories in very general terms. An example is the contribution by Chambers et al. (2014), who develop a framework to discern the nontestable claims of a given theory. Although partial observability also plays a prominent role in that paper, the focus is on theory falsifiability rather than preference identification. 2. General theory This section presents a general framework for performing revealed preference analysis under partial observability. 2.1. Basic definitions Let X be the set of alternatives faced by the decision maker. A preference is a complete and transitive binary relation on X. For any preference , denote the symmetric and asymmetric parts with ∼ and , respectively. As usual, x y and (x, y) ∈ are equivalent statements, x y means y x, and x y means that x y does not hold. (Similar notational conventions apply to other binary relations such as ∼ and .) A set I ⊆ X is an indifference set for if x ∼ y holds for all x, y ∈ I . An indifference class for is an indifference set for that is maximal with respect to set inclusion. A preference is trivial if X is an indifference set for . (Otherwise is nontrivial.) A function U : X → R is said to represent whenever x y is equivalent to U (x) ≥ U (y). An alternative x ∈ X is a -best element in B ⊆ X if x ∈ B and x y for every y ∈ B. 2X denotes the collection of all subsets of X.
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2.2. Choice models A choice model is a specification of the environment in which the decision maker (she) operates together with theoretical restrictions on her possible choice behavior. Formally, let B denote a fixed collection of nonempty subsets of X. A set B ∈ B is called a choice problem. Choice problems capture all possible ways through which alternatives in X can be presented to the decision maker in the social situation being modeled. For example, in standard consumer theory, B is the collection of all linear budget sets; whereas in an experimental setting where a decision maker is confronted with binary alternatives, B is the collection of all two-element subsets of X. In this paper, a theory of behavior or, more succinctly, a theory is a class of preferences on X. The underlying interpretation is that every time a decision maker is confronted with a choice problem, she publicly selects one of the preference-maximizing alternatives contained in it. Consequently, a theory represents an hypothetical restriction that the analyst entertains when trying to make sense of behavioral data. A choice model is then a triple (X, B, P), where X is a set of alternatives, B is a collection of choice problems, and P is a theory of behavior. It should be stressed that, as theoretical primitives, B and P are not data. Moreover, B need not contain all subsets of X nor P need contain all preference relations on X. 2.3. Rationalizing empirical choices A choice correspondence is a function c : B → 2X that satisfies c(B) ⊆ B for all B ∈ B. The inclusion of the empty set in the co-domain permits encoding “absence of data,” and thus interpreting c as a collection of observed choices. The graph of an arbitrary choice correspondence c : B → 2X is the set gr(c) := {(B, x) ∈ B × X|x ∈ c(B)}. In this paper, the word dataset is treated as equivalent to the graph of a choice correspondence. The following definition embodies the notion of rationalization I employ. Definition 1. The choice correspondence c is consistent with a preference if, for all B ∈ B, every alternative in c(B) is a -best element in B. The theory P rationalizes c if c is consistent with some ∈ P. Note that this notion of rationalization is weaker than that employed in the traditional approach to abstract revealed preference, precisely because it does not require that c(B) contains all -best elements in B. As a result, revealed preference axioms are not necessary consequences of preference maximizing behavior under partial observability. In fact, any testable restrictions must follow from additional theoretical assumptions.5 2.4. Full domain choice correspondences To qualitatively assess the informational content of different choice correspondences, consider the following definition. 5 Under partial observability, the preference that declares all alternatives indifferent rationalizes every choice correspondence. In contrast, if a choice correspondence can be rationalized in the traditional (stronger) sense, then the weak axiom of revealed preference must hold.
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Definition 2. The domain of a choice correspondence c is the collection of choice problems in B that satisfy c(B) = ∅. The choice correspondence c has full domain if its domain is B. In a typical application of this framework, a choice correspondence with full domain represents a dataset containing information about the decision maker’s behavior when confronted with a large and diverse enough set of choice problems. Note that this notion does not conflict with partial observability, which just means that the dataset at hand does not necessarily contain every optimal option, but only those that the decision maker actually chose while under observation. 2.5. Identification Fix a choice model (X, B, P). Suppose that observed choices are encoded in the choice correspondence c. There are three mutually exclusive possibilities. First, c may not be consistent with any preference in P. In this case, if the model is taken literally, one should reject theory P. Second, there may be a unique preference in P consistent with c. In this scenario, the rationalizing preference can be pinned down uniquely with the available data, so that the “parameters” of theory P can be fully recovered. Third, P may rationalize c through multiple preferences. This last case naturally occurs when c does not contain much information, either because the decision maker has not been confronted with many choice problems or because the class of choice problems itself is rather limited. However, due to the adoption of partial observability, multiplicity can persist even when B = 2X and c has full domain. This possibility is excluded by the following definition. Definition 3. A theory P is identified on B if every choice correspondence with full domain is consistent with at most one preference in P. Within an identified theory, one can always recover a preference driving behavior if enough data is available. Note that starting with smaller theories makes identification easier.6 Similarly, since Definition 3 requires at least one choice recorded for every choice problem in B, considering more choice problems facilitates identification. In fact, it is straightforward to establish that, if P is identified on B, then P is identified on B for every P ⊆ P and B ⊇ B. Two additional definitions may be helpful to discuss identification. Both definitions are relative to a fixed collection of choice problems B. First, preferences and are confoundable if there exists a choice correspondence with full domain that is simultaneously consistent with both preferences. It is clear that a theory P is identified if and only if no two distinct preferences in P are confoundable. Second, and are observationally equivalent if {} and { } rationalize exactly the same set of choice correspondences. Obviously, a theory that contains distinct preferences that are observationally equivalent is not identified, but the converse is not necessarily true. In fact, if B contains all two-element subsets of X, some theories are not identified even though there cannot be distinct observationally equivalent preferences. 6 In particular, every theory P containing a single preference is identified on every collection of choice problems. Despite its triviality, this observation is important, because it suggests simple identified theories that are consistent with datasets that violate the weak axiom of revealed preference. For example, if X = {x, y, z}, the theory P = {} where = {(x, x), (x, y), (x, z), (y, x), (y, y), (y, z), (z, z)} is identified and is consistent with the dataset {({x, y}, {x}) , ({x, y, z}, {y})}.
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2.6. A necessary condition for identification Checking identification using its definition is not very convenient. In order to develop a more operational characterization, this section studies a simple condition restricting the class of preferences under consideration. Before introducing the main condition, there is a technical obstacle in the path toward a necessity result: Infinite choice problems may not have -best elements for some ∈ P. This possibility is troublesome because it potentially weakens identification too much.7 The assumption embedded in the following definition circumvents this difficulty and helps simplify the analysis. Definition 4. A collection of choice problems B is compatible with P if there exists a -best element in B for every preference ∈ P and every choice problem B ∈ B. Note that, if B is compatible with P, then B is compatible with P for every B ⊆ B and ⊆ P. Moreover, B is automatically compatible with P if X is finite or, more generally, if X is a topological space, all problems in B are compact, and all preferences in P are continuous. The following is the key condition to obtain a necessity result: P
Definition 5. A theory P is separated if, for every pair of different preferences , ∈ P, there exist alternatives x, y ∈ X such that x y and y x. Thus, a separated theory exhibits at least one strict preference reversal for every pair of preferences. One should expect identified theories to be separated. In fact, from an empirical perspective, two preference relations are only different to the extent that they are revealed distinct through data. Since partial observability does not impose any restrictions on how indifferences are resolved, preference reversal arises as a natural condition to avoid confounding different preferences. The next theorem formalizes this intuition and shows that separability is indeed a necessary condition for preference identification under partial observability. Theorem 1. Every identified theory on a compatible collection of choice problems is separated. As a consequence of Theorem 1, a theory that fails to be separated is not identified on any compatible collection of choice problems. 2.7. A sufficient condition for identification Since compatibility is a rather weak structural condition, a theory essentially needs to be separated in order to be identified. However, this separation property is not generally sufficient. Example 1. Let X = {x, y} and B = {{x}, {y}}. Consider a theory P = , , where x y and y x. Clearly, P is separated and B is compatible with P. However, P is not identified on B. 7 Without compatibility, some theories are inconsistent with every choice correspondence with full domain. As a result, identification can hold vacuously. See Example 15 in Appendix A.3.
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The example above illustrates that identification requires the set of choice problems to be sufficiently rich. The following condition ensures that there are enough choice problems to theoretically distinguish between strict and weak preference: Definition 6. A collection of choice problems B is P-rich if, for every ∈ P and x, y ∈ X, x y implies there are B1 , ..., BN ∈ B and z1 , ..., zN+1 ∈ X such that: (1) z1 = x, zN+1 = y, (2) zn+1 ∈ Bn for all n ∈ {1, ..., N }, (3) zn is the unique -best element in Bn for all n ∈ {1, ..., N }. This condition allows one to obtain a simple theory of revealed preference under partial observability. Importantly, P-richness is a theoretical property of the choice model (X, B, P) that does not involve in any way the observed behavior of the decision maker, which, in turn, is captured solely by the choice correspondence c. This is in stark contrast with the traditional axiomatic approach that assumes full observability, the implicit requirement that c be interpreted as containing all preference maximizing options. It is easy to find examples that satisfy P-richness. In fact, any collection of choice problems containing all two-element subsets of X is P-rich for every theory P. However, collections of choice problems that satisfy Definition 6 only for some theories are often important in applications. For example, in classic demand analysis, it is natural to consider the collection of all (compact) linear budget sets on RN + , which is P-rich only under substantial assumptions regarding the preferences in P.8 The next result establishes that separated theories are identified on P-rich collections of choice problems. Theorem 2. Let P be a separated theory and let B be a P-rich collection of choice problems. Then P is identified on B. Under the compatibility and richness conditions, Theorem 1 and Theorem 2 together reduce the identification problem to checking a transparent condition for the theory under consideration. Importantly, whether or not P is separated does not depend on the specific problems contained in B. This means that whenever P is identified on some compatible collection of choice problems, it is automatically identified on every P-rich collection of choice problems. Example 2 (Identification with experimental data). Consider an experimental setting in which the decision maker may, in principle, be confronted with all possible pairs of alternatives. Formally, suppose X has at least 2 elements and B = {{x, y}|x, y ∈ X, x = y}. In this case, both compatibility with P and P-richness are trivially satisfied for any theory P. As a result, Theorem 1 and Theorem 2 together imply that separation fully characterizes identification. That is, a theory P is identified on B if and only if it is separated. 8 Mas-Colell (1977) provides an explicit example of two different preferences that satisfy most standard economic assumptions and nevertheless are observationally equivalent on the collection of linear budget sets. See Appendix A.1 for a discussion of this example and Section 4.2 for sufficient conditions for P-richness in the classic demand setting.
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The qualified equivalence established by Theorem 1 and Theorem 2 suggests to study what separation entails in concrete environments, as a means to operationalize further the notion of identification. This is the route I follow in Section 3, Section 4, and Section 5, establishing sufficient conditions for separation in a variety of settings with additional structure. 2.8. On weaker sufficient conditions As the following examples make clear, P-richness is not a necessary condition for identification. Example 3. Let X = {x, y}, B = {{x}} and P = , where x y. Clearly, P is identified on B even though B fails to be P-rich. Example 4. Let X = {x, y, z}, B = {{x, y}, {y, z}} and P = , , where x y ∼ z and y x ∼ z. Clearly, P is identified on B even though B fails to be P-rich. It therefore seems natural to seek weaker sufficient conditions for identification. However, this is very challenging. In fact, there is a formal sense in which it is impossible to obtain a useful necessary and sufficient condition at this level of generality. To see this, note that, given a theory P, a collection of choice problems B is P-rich if and only if B is {}-rich for every ∈ P. It follows that whether B is P-rich or not depends exclusively on a series of tests performed using each possible preference in P. This observation suggests to study all abstract criteria (selecting certain collections of choice problems) that satisfy this “pointwise” property. let A be an arbitrary function mapping each possible theory P into More precisely, B ⊆ 2X . The function A is said to be a pointwise criterion if it satisfies A(P) = A(P) ⊆ B ∈P A({}). Obviously, P-richness is a pointwise criterion, but by no means the only one. Consider next the following two definitions. First, A is necessary for identification if the inclusion {B|P is identified on B} ⊆ A(P) holds for every theory the function A is P. Second, said to place no restrictions on choice problems if A(P) = B B ⊆ 2X for every P. The next theorem relates these two properties for pointwise criteria. Theorem 3. If a pointwise criterion is necessary for identification, then it places no restrictions on choice problems. Theorem 3 shows that a pointwise criterion that is weak enough to select every collection of choice problems over which the theory supplied as argument is identified is in fact too weak to provide any restriction. In particular, every such criterion must admit collections of choice problems over which some separated theory fails to be identified. It follows that there does not exist a pointwise criterion that, together with separability, provides a necessary and sufficient condition for identification. This impossibility result suggests to abandon necessity and settle for a pointwise criterion weaker than P-richness, but that preserves enough strength to ensure that separated theories are
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identified (over those collections of choice problems satisfying the criterion). However, this is far from trivial (if possible at all).9 An alternative path forward is to seek conditions that are not pointwise criteria (i.e., violate the property A(P) = ∈P A({})), but are simultaneously sufficient for identification of separated theories and weaker than P-richness. Such conditions do exist but are hardly instructive and, in my view, adopting them would reduce the methodological appeal of this paper’s method.10 Finally, one can also attempt to obtain weaker criteria for specific theories. This seems to be the most promising approach and it is pursued to some extent in the following section. For example, Theorem 6 relaxes P-richness when preferences are assumed to be continuous. 3. Continuous preferences: theory Suppose that the space of alternatives X is endowed with some topology. In this section, I investigate identification under the following standard continuity condition used in Debreu (1959): Axiom 1 (Continuity). For every x ∈ X, the sets {z ∈ X|x z} and {z ∈ X|z x} are closed. This assumption is standard in economics and is necessary for the existence of a continuous utility representation of . 3.1. Regular preferences A class of continuous preferences need not be identified. This section presents a weak additional condition which ensures that a class of continuous preferences is separated. This condition combines with Theorems 1 and 2 to provide powerful identification (and non-identification) results for continuous preferences. Developing this approach requires a few additional definitions. Fix a preference . An alternative x ∈ X is a local best element for if there is an open set V ⊆ X such that x ∈ V and x y for all y ∈ V . A local worst element for is defined similarly by only substituting y x for x y in the previous sentence. x ∈ X is a local extremum for if it is either a local best element or a local worst element. A set of alternatives is -extremal if every alternative it contains is a local extremum for . With these definitions at hand, consider the following: Axiom 2 (Regularity). No -extremal indifference class for contains a local best element for and a (different) local worst element for . 9 The discussion in Appendix A.2 illustrates this difficulty showing that a minimally weaker criterion fails to guarantee that all separated theories are identified. 10 A central advantage of this paper’s approach is that the conditions for identification are structurally simple and do not restrict more than two preferences simultaneously. For example, P-richness is a condition individually imposed on each preference in theory P. Similarly, whether a theory is separated solely constraints pairs of preferences within the theory.
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Fig. 1. Representations of preferences in Example 5.
Note that an indifference class may consist of a single alternative that is both a local best and a local worst element without violating the definition. To illustrate Axiom 2, it is useful to consider simple examples of regular and non-regular preferences. Example 5. Let X = [0, 1] and consider functions U, V : X → R defined by setting: ⎧ ⎪ x ∈ [0, 1/4] , ⎨2x 1 − |2x − 1| x ∈ [0, 3/4] , U (x) := V (x) := 1/2 x ∈ (1/4, 3/4] , ⎪ 2x − 1 x ∈ (3/4, 1] , ⎩ 2x − 1 x ∈ (3/4, 1] . The graph of U is displayed in Panel A of Fig. 1. The extremal indifference classes are {0} (which is a singleton) and {1/2, 1} (which contains local best elements and no local worst elements). Thus, the preference represented by U is regular. Similarly, Panel B of Fig. 1 displays the graph of V . In this case, [1/4, 3/4] is an extremal indifference class that contains both 1/4 (a local best element) and 3/4 (a local worst element). It follows that the preference represented by V is not regular. The restrictions on indifference that regularity imposes are useful to obtain identified theories through the results in Section 2. The following theorem demonstrates this in the case of continuous preferences. Theorem 4. The class of regular continuous preferences is separated. Moreover, if X is a separable metric space, no larger class of continuous preferences is separated. Together with Theorem 2, the first part of this result implies that identification under partial observability is attainable for regular continuous preferences. Moreover, via Theorem 1, the second part of Theorem 4 implies that regularity is an optimal condition to obtain identification of continuous preferences through the method proposed in this paper. That is, if X is a separable
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metric space, additionally allowing any non-regular continuous preference necessarily precludes identification (over compatible choice problems). 3.2. Sufficient conditions for regularity Regularity is a very weak condition. This fact is important because, combined with the second part of Theorem 4, it implies that regular continuous preferences constitute a large theory that is maximal with respect to the property of being separated.11 For example, it is customary in consumer theory to assume that preferences are not only continuous, but also “monotone” or, at least, “locally nonsatiated”. These assumptions preclude the existence of local best elements and therefore imply regularity. Formally, is said to be locally nonsatiated at x ∈ X if every open set containing x also contains y such that y x. is locally nonsatiated if it is locally nonsatiated at every x ∈ X. Chambers et al. (2018) define the following weaker notion: is locally strict if V ∩ = ∅ implies V ∩ = ∅ for every open set V ⊆ X × X. The following lemma relates these notions with regularity: Lemma 1. Let X be an arbitrary topological space and let be any preference on X. Then, (1) If is locally nonsatiated, then it is locally strict. (2) If is locally strict, then it is regular. The converse implications do not hold (even if X = [0, 1] and is continuous). Lemma 1 shows that regularity should be expected to hold in most economic applications. But regularity is in fact significantly weaker than local nonsatiation or local strictness. The following result illustrates this by providing an appealing alternative sufficient condition for regularity: Lemma 2. Let X be a connected topological space and let be a continuous preference on X that is generically locally nonsatiated (i.e. local nonsatiation holds on an open dense subset of X). Then, is regular. Lemma 2 becomes false if the requirement that X be connected is removed. This is shown by Example 18 in Appendix A.3. Similarly, the statement also becomes false if regularity is substituted by local strictness in the conclusion. The following example displays a smooth preference that is generically locally nonsatiated but fails to be locally strict: Example 6. Let X = R and let be represented by U (x) = (cos(x) −1)x. Then, is continuous (in fact, U is C ∞ ) and generically locally nonsatiated (thus regular by Lemma 2), but fails to be locally strict. To see that the latter claim is true, consider any pair of options of the form (x, y) = (2πm, −2πn) for m, n ∈ N. Clearly, U (x) = U (y) = 0 implies x ∼ y. However, for all 11 Maximality with respect to separation is not a compelling argument for regularity per se. For instance, the theory that only contains universal indifference is also maximal with respect to separation, but it is clearly too small to be useful.
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sufficiently small > 0, |x − x| < and |y − y| < imply x y . Fig. 2 below illustrates the example:
Fig. 2. Utility representation of a generically locally nonsatiated preference that is not locally strict.
Finally, it is worth stressing that regularity is even weaker than generic local nonsatiation because it is not a local property. This is illustrated by the following example that can be used to construct regular preferences with a continuum of satiation points. Example 7. Let X = RN and be the preference represented by a continuous function U . Further, assume that there are two extreme alternatives x∗ , x ∗ ∈ X such that U (x∗ ) < U (x) < U (x ∗ ) for all x ∈ X \ {x∗ , x ∗ } and a path α : [0, 1] → X joining x∗ and x ∗ along which U is strictly increasing.12 Then every indifference class for must contain a point along the path α. It follows that {x∗ } and {x ∗ } are the only -extremal indifference classes for . As a result, any such is regular, independently of how the preference is defined outside of α([0, 1]). 3.3. Identification on restricted domains This paper aims to relate observable behavior to different theories of choice. In many applications, it is natural to model a continuum of alternatives even though empirical observations are usually discrete. For this reason, it is useful to obtain identification theorems based on choice correspondences with restricted domains. In this section, I provide results that relate identification on a choice model (X, B, P) with identification on a “restricted” choice model Y, B , P such that Y is a subset of X, B contains less and/or smaller choice problems, and P contains preferences over the limited domain Y . X of c as the function cl(c) : For any choice correspondence c : B → 2 , define the closure X B → 2 , given by cl(c)(B) := x ∈ X (B, x) ∈ clB ×X (gr(c)) for each B ∈ B (for some topology on choice problems). The following result exploits the structure of the class of continuous preferences to establish a foundation for identification conditions over potentially countable collections of choice problems. 12 Formally, α is a continuous function such that α(0) = x , α(1) = x ∗ and, for all t, s ∈ [0, 1], t > s implies U (α(t)) > ∗ U (α(s)).
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Theorem 5. Assume that X is a metric space, B is a collection of nonempty compact subsets of X (endowed with Hausdorff metric topology), and P is a class of continuous preferences on X. Then, (1) A choice correspondence c is consistent with ∈ P if and only if its closure cl(c) is consistent with . (2) If P is identified on B, then P is identified on every dense subset of B. (3) If P is separated, Y is a dense subset of X, and , ∈ P coincide on Y × Y , then = . Part (1) in Theorem 5 reflects the fact that, under continuity, choice behavior on limit problems does not contain additional information. In particular, it implies that, whenever two choice correspondences have the same closure, they are rationalized by the same set of continuous preferences. In the same spirit, Part (2) shows that identification over a dense subcollection of choice problems (which may be countable even with an uncountable set of alternatives) is not harder than when all choice problems are available. Part (3) shows that limiting the domain of a separated theory to a dense subset of alternatives does not throw away any information. As a result, identification of restricted preferences is sufficient to identify unrestricted (separated) preferences. In particular, this part of Theorem 5 applies to regular preferences (due to Theorem 4). Note that the results presented in Section 2 fall short of a full characterization of the identification property, because the richness condition employed is not necessary. Theorem 5 is useful to establish weaker sufficient conditions for identification of continuous preferences. Theorem 6. Assume that the conditions in Theorem 5 hold. Suppose further that there exists a set Y ⊆ X (not necessarily dense) satisfying: such that B ∈ B implies that the -best (1) For every ∈ P, B has a dense subset B elements in B are contained in Y . (2) The mapping carrying into ∩ (Y × Y ) is one-to-one. (3) PY := ∩(Y × Y ) ∈ P is separated. (4) BY := {B ∩ Y |B ∈ B} is PY -rich.
Then P is identified on B. A few remarks about Theorem 6 are in order. First, on the one hand, a trivial sufficient condition for (1) is that B ⊆ Y for all B ∈ B. On the other hand, the possibility of using dense subsets of B allows the exclusion of preference-specific “degenerate” feasible sets that would induce best elements outside of Y . Second, (2) and (3) automatically hold whenever P is separated and Y is dense in X (this follows from Part (3) of Theorem 5). However, sometimes it is convenient to use sets Y that are not dense in X in order to relax the restrictions imposed by (4). For a concrete illustration of these points and, more importantly, of the usefulness of Theorem 6 to get extra mileage out of the central method of this paper, I defer the reader to Example 8 in Section 4.1 (identification of Cobb-Douglas preferences with a discrete good) and Example 10 in Section 4.2 (identification of linear preferences in a classic demand setting).
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4. Continuous preferences: applications This section applies some of the results presented in Section 2 and Section 3 to more specific settings. 4.1. Demand theory with a discrete good This application presents an identification result for a large class of consumer preferences with a discrete good, providing formal support for the claims made in the quasilinear example discussed in Section 1.1. As it was in that case, suppose Ann consumes bundles of two goods. The second good is infinitely divisible, but the first one can only be consumed in discrete quantities. As in the case of classical demand theory that is developed immediately after this section, an observation consists of market prices and quantities bought by the consumer.13 Formally, let X = Z+ × R+ be the consumption set (with the natural topology). For every price-wealth situation (p, w) ∈ R2++ × R++ , define the following feasible set on X: Bp,w := {x ∈ X|px ≤ w} . L be the class of all such feasible sets. Note that most of these feasible sets are neither Let BD convex nor connected. For any two vectors u, v ∈ RN , let (u, v) := {αu + (1 − α)v|α ∈ (0, 1)}. This notation will be used in what follows as a shorthand to denote strict convex combinations of commodity bundles and price vectors. Relaxing the quasilinear assumption of Section 1.1, assume that Ann’s preferences are continuous and satisfy the following additional axioms:
Axiom 3 (Strict monotonicity). For every x, y ∈ X, x ≥ y and x = y imply x y. Axiom 4 (Strict convexity). For every x, y, z ∈ X, x y, x = y, and z ∈ (x, y) imply z y. If both goods were infinitely divisible, Axiom 3 and Axiom 4 would be standard. In particular, both axioms are satisfied by the restrictions to X of CES and quasilinear preferences on R2+ . Let PD be the class of all preferences on X satisfying Axiom 1, Axiom 3, and Axiom 4. It should be stressed that, in contrast with strictly convex preferences in the infinitely divisible case, all preferences in PD are consistent with indifference at the optimum. This makes partial observability both a compelling methodological assumption and a difficult challenge: As natural as this exercise is, the literature seems to lack an adequate recoverability result for this setting. However, combined with the general results presented in Section 2 and Section 3, the assumptions defining PD imply the following: L. Proposition 1. PD is identified on every dense subset of BD
Note that every dense sequence of price-wealth situations generates a countable dense subset L . The proof of Proposition 1 (in the Appendix) showcases the central method in this paper. of BD First, separability of PD is obtained as a consequence of regularity through Theorem 4. Second, 13 Individual wealth is typically considered to be unobservable and is inferred from prices and quantities by assuming that the consumer spends all its budget.
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L is established through a specific argument that exploits the additional strucPD -richness of BD ture of this environment. Then, Theorem 2 combines these two properties to obtain identification L . Finally, as an additional refinement, Theorem 5 yields identification using any sequence on BD L. of choice problems that is dense in BD Even though the class PD contains several common classes of preferences, it also excludes some important cases such as Cobb-Douglas. CD be the class of preferExample 8 (Cobb-Douglas preferences with a discrete good). Let PD ences on X = Z+ × R+ that can be represented by
U (x1 , x2 ) = x1α1 x2α2 , CD violates both strict monotonicity and strict conwhere α1 , α2 > 0. Note that every ∈ PD vexity because (0, x2 ) ∼ (0, 0) for every x2 ≥ 0. Moreover, no such is regular. However, CD is identified on B L using Theorem 6. To see this, set it is possible to establish that PD D L . Condition (1) of Theorem 6 Y = N × R++ ⊂ X and B = Bp,w w > p1 > 0, p2 > 0 ⊂ BD holds trivially because the B employed excludes all feasible sets that yield an optimal choice indifferent to (0, 0). Condition (2) holds because the bundles in X \ Y are deemed worst eleCD . Conditions (3) and (4) hold because, when restricted to Y , ments by every preference in PD CD are strictly monotonic and strictly convex, so similar arguments to the the preferences in PD ones used in the proof of Proposition 1 apply.
4.2. Classical demand theory What about the classical demand case in which all goods are infinitely divisible? Even though some new insights can be obtained applying the results of Section 2 and Section 3, the central method of this paper enjoys significantly less thrust without an indivisible good because P-richness becomes especially restrictive. L Formally, take X = RN and + (with the natural topology) let B be the collection of compact N linear budget sets (that is, all sets of the form x ∈ R+ px ≤ w for some wealth level w > 0 and a price vector p ∈ RN ++ ). Mas-Colell (1977) formalizes an idea attributed to Shapley and provides an explicit example of two different continuous, strictly monotone, and strictly convex preferences on RN + that generate the same Walrasian demand function (i.e., are observationally equivalent on B L ). This means that identification is not trivial in this setting, even if one observes all optimal choices. However, Mas-Colell (1977) also proves that upper semicontinuous, monotone, and convex preferences that are representable by a lipschitzian utility function can be (uniquely) recovered from observing the entire demand correspondence. A natural question is whether this theorem can be strengthened to obtain identification under partial observability. The results presented in Section 2 suggest to seek sufficient conditions for P-richness. In addition to the axioms formally introduced in Section 4.1 (now applied to X = RN + ), consider the following: Axiom 5 (Positive smooth monotonicity). is locally nonsatiated and admits a C 1 representation U such that U (x) >> 0 for every x 0. Axiom 6 (Positive strict convexity). For every x, y, z ∈ X, x = y, x y 0, and z ∈ (x, y) imply z y.
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Axiom 5 implies continuity, regularity, and monotonicity (x >> y implies x y), but it is independent of strict monotonicity. Axiom 6 is a slight weakening of strict convexity. Both axioms are satisfied by standard preferences, such as Cobb-Douglas in RN + , which fail strict monotonicity and strict convexity. The following proposition uses these axioms to study P-richness in this environment. Proposition 2. Let be a preference over X = RN +. (1) If satisfies positive smooth monotonicity and positive strict convexity, then B L is {}-rich. (2) If satisfies strict monotonicity and B L is {}-rich, then is strictly convex. The first part of Proposition 2 is useful to obtain identification. The following example illustrates the application of the main method of this paper to the familiar class of Cobb-Douglas preferences: Example 9 (Cobb-Douglas preferences). Let P CD be the class of preferences on X = RN + that can be represented by U (x1 , ..., xN ) =
N
xnαn ,
n=1
where αn > 0 for n = 1, ..., N . Every ∈ P CD satisfies positive smooth monotonicity and positive strict convexity. Thus, by Proposition 2, B L is P CD -rich. Moreover, positive smooth monotonicity implies regularity by Lemma 1. Thus, by Theorem 4, P CD is separated. Since P CD is separated and B L is P CD -rich, Theorem 2 implies that P CD is identified on B L . However, the sufficient conditions in the first part of Proposition 2 are quite strong. In fact, local nonsatiation and positive strict convexity together imply that optimal choices are unique on linear budget sets, leaving no room for partial observability. Moreover, the second part of Proposition 2 shows that this difficulty can hardly be avoided. The conclusion is that the central method of this paper, based on Theorem 2, needs to be modified to be useful in this setting.14 It is indeed possible to obtain identification without strict convexity by means of more specific arguments, such as Theorem 6, as the following example illustrates. Example 10 (Linear preferences). Let P L be the class of preferences on X = RN + that can be represented by U (x1 , ..., xN ) =
N
αn xn ,
n=1
where αn > 0 for n = 1, ..., N . Note that every bundle in RN ++ is strictly preferred to the zero vector but there is no linear budget set such that the demand contains only an interior point. This 14 The assumptions in the first part of Proposition 2 preclude multiple optimal choices and are stronger than those of Mas-Colell (1977)’s main result. As a consequence, Proposition 2 falls short of advancing what is known about identification in this setting.
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means that B L is not P L -rich. However, Theorem 6 implies that P L is identified on B L . To see this, take Y=
∀m = n : x = 0 . (x1 , ..., xN ) ∈ RN m +
N n=1
In words, Y contains all bundles in which at most one good is consumed. It is easy to check that conditions (2), (3), and (4) of Theorem 6 are satisfied. To verify condition (1), pick any preference ∈ P. Note that each indifference class for is the restriction of some hyperplane to RN + . Let be the set of all linear budget sets with a boundary that is not parallel to the indifference B
is dense in B L . Moreover, for every B ∈ B , the -best elements classes for . Clearly, B in B are corner solutions in which the full budget is exclusively spent in one good and, thus, are contained in Y , as required by condition (1) of Theorem 6.
4.3. Expected utility Denote by Z a separable metric space of prizes and let X the set of all Borel probability measures on Z (endowed with the topology of weak convergence). A preference on X is an expected utility preference if it can be represented by a function U : X → R of the form U (x) = udx for some bounded and continuous vNM index u : Z → R. It is well known that expected Z utility preferences are fully characterized by Axiom 1 together with Axiom 7 (Independence). For all x, y, z ∈ X and α ∈ (0, 1), x y implies αx + (1 − α)z αy + (1 − α)z. It can be shown that nontrivial expected utility preferences are regular. Thus, Theorem 4 implies the following result: Proposition 3. The class of nontrivial expected utility preferences is separated. This means that nontrivial expected utility preferences can always be recovered from sufficiently rich choice data. In particular, it is possible to recover expected utility preferences by observing choice over pairs of lotteries. The next result allows to generalize this observation by providing other interesting rich collections of choice problems in this setting. Proposition 4. The following collections of choice problems are rich with respect to the class of expected utility preferences: (1) All choice problems with exactly N ≥ 2 lotteries. (2) All convex compact choice problems. 4.4. Preferences over menus Consider the standard model of preferences over menus of lotteries: let Z be a compact metric space, be the set of Borel probability measures over Z metrized by the Lévy-Prokhorov distance, and X be the set of nonempty compact subsets of metrized with the Hausdorff distance.
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In this setting, the objects of choice are lottery menus (i.e., sets of lotteries – the elements of X), while choice problems are menus of lottery menus. For x, y ∈ X and α ∈ [0, 1], denote by αx + (1 − α)y the set {αp + (1 − α)q|p ∈ x, q ∈ y} ∈ X. Consider the following axiom: Axiom 8 (Set independence). For every x, y, z ∈ X and α ∈ (0, 1), x y implies αx + (1 − α)z αy + (1 − α)z. This section focuses on preferences on X satisfying continuity and set independence. This class includes the temptation and self-control preferences studied by Gul and Pesendorfer (2001), namely any preference with a representation of the form U (x) := max u(p) − max {v(q) − v(p)} p∈x
q∈x
for some linear and continuous functions u, v : → R. The class also includes preferences with the integral representation of Dekel et al. (2001): U (x) := sup u(p, s)μ(ds) p∈x
S
for some state space S, some signed measure μ and some function u : × S → R linear over . The following result shows that this class of preferences is within the reach of Theorem 2. Proposition 5. The class of nontrivial preferences over lottery menus that satisfy continuity and set independence is separated. It follows that continuity and set independence are strong enough to provide identification over rich collections of choice problems. As in the case of expected utility, it is possible to provide a simple result describing rich collections of choice problems in this setting. Proposition 6. The following collections of choice problems are rich with respect to the class of preferences over lottery menus satisfying continuity and set independence: (1) All choice problems with exactly N ≥ 2 lottery menus. (2) All convex compact choice problems. Of course, the resemblance to Proposition 4 stems from the structural similarity between independence and set independence. 4.5. Random datasets The exercise of recovering preferences from observed choices is relatively simple in experimental settings for two different reasons. First, if choice data is collected through an experimental design, it may be possible for the analyst to affect the shape (or even determine) the family of choice problems the decision maker will face. As shown in Example 2, this may considerably simplify the characterization of identified theories. Second, even if the family of choice problems is fixed and beyond the analyst’s control, an experimental protocol might be configured to produce a sequence of choice problems that guarantees identification without confronting the decision maker with all conceivable choice problems.
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In principle, the absence of these two features makes identification more challenging in nonexperimental settings. In this sense, conditions such as P-richness deal with the first issue by paving the way for identification results that do not require binary choice problems and, thus, can be applied to certain non-experimental settings. In this section, I explore the second difficulty: What happens when choice problems composing the observed dataset are chosen randomly, instead of being purposely selected to ensure identification? To investigate this question, consider a decision maker who faces a random sequence of choice problems, selects one optimal alternative for each problem, and this choice is observed. While observed behavior is assumed to be preference-maximizing, the decision maker is free to resolve indifferences as she deems fit (it is even possible to condition on the true state of the world). Although this tie-breaking process is not restricted in any way, the ensuing Proposition 7 establishes that, if the underlying theory is identified and the process sampling the choice problems is iid, then any continuous preference driving observed behavior can be recovered almost surely. In order to state the result precisely, I need some additional machinery. Let (, , μ) be a probability space, where is a set of states of the world, is a σ -algebra, and μ is a probability measure. Assuming B is endowed with some topology, a random problem is a measurable function Bˆ : → B. The support of a random problem Bˆ is the intersection of all closed subsets of B that contain Bˆ with probability 1. A sequence of random problems {Bˆ n }n∈N is said to be iid if, for each M ∈ N and each finite subset {Bˆ n1 , ..., Bˆ nM }, M μ Bˆ n1 ∈ A1 , ..., Bˆ nM ∈ AM = μ Bˆ ni ∈ Ai i=1
for all Borel sets A1 , ..., AM ⊆ B. A random dataset is a sequence of pairs {(Bˆ n , xˆn )}n∈N such that, for each n ∈ N, Bˆ n is a random problem and xˆn : → X is a measurable function that satisfies xˆn (ω) ∈ Bˆ n (ω) for all ω ∈ . A random dataset {(Bˆ n , xˆn )}n∈N is said to be iid if {Bˆ n }n∈N is iid, and is said to have full support if the support of Bˆ n is B for every n ∈ N. Every random dataset (Bˆ n , xˆn )n∈N naturally induces a random choice correspondence cˆ mapping each state of the world into the choice correspondence associated with the realiza15 tion choice correspondence cˆ is consistent with ∈ P if of the random dataset. A random μ ω ∈ c(ω) ˆ is consistent with = 1. Finally, let P (c, P) stand for the set of preferences in theory P that are consistent with choice correspondence c. The following proposition shows that the underlying preference is recovered almost surely. Proposition 7. Assume that X is a separable metric space, B is a collection of nonempty compact subsets of X (endowed with the Hausdorff metric topology), and P is a class of continuous preferences on X that is identified on B. Then, for every preference ∈ P and every random choice correspondence cˆ consistent with and induced by an iid random dataset with full support, P (c, ˆ P) = {} almost surely. In other words, Proposition 7 establishes that the event ω ∈ P (c(ω), ˆ P) = {} has probability 1. It should be stressed that, if the theory P is not identified on B, the conclusion of Proposition 7 fails for some random choice correspondence that satisfies the remaining conditions. 15 That is, for each ω ∈ , c(ω) ˆ is the choice correspondence with graph
Bˆ n (ω), xˆn (ω) n ∈ N .
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5. Other classes of preferences This section provides sufficient conditions for separation within two specific theories which are economically meaningful but fall outside the scope of Section 3, as they are not framed in a topological space. 5.1. vNM preferences This section studies the class of vNM preferences, which generalize expected utility preferences. For discussions of the basic framework, I refer the reader to Herstein and Milnor (1953) and Kreps (1988). Let X be a general mixture space and consider a vNM preference , which is a preference that satisfies Axiom 7 and also the following weaker form of continuity: Axiom 9 (vNM continuity). For all x, y, z ∈ X, x y z implies that there exist α, β ∈ (0, 1), such that αx + (1 − α)z y βx + (1 − β)z. It is well known that every vNM preference admits a linear representation: there is a function U : X → R that represents and satisfies U (αx + (1 − α)y) = αU (x) + (1 − α)U (y) for every x, y ∈ X and α ∈ [0, 1]. The following result shows that the vNM axioms provide a solid ground for revealed preference analysis. Proposition 8. The class of nontrivial vNM preferences is separated. Proposition 8 reveals a uniqueness property hidden within the vNM axioms: If two vNM preferences do not exhibit any strict preference reversal, at least one must be trivial. Combining this observation with Theorem 2, it follows that nontrivial vNM preferences can always be recovered from sufficiently rich choice data. Although it can be shown that nontrivial vNM preferences are regular, Proposition 8 cannot be derived from Theorem 4, because a vNM preference in a general mixture space need not be continuous (in fact, a mixture space need not have a topology defined on it). In the topological setting of Section 4.3, when X is the space of Borel probability measures over some separable metric space of prizes Z, the class of expected utility preferences is a subset of the class of vNM preferences.16 5.2. Qualitative probabilities This section discusses identification of qualitative probabilities—a central component in the theory of decisions under uncertainty, as formulated by Savage (1972) and other authors. I focus on a particular class of qualitative probabilities: Those consistent with an atomless probability measure. The importance of this theory stems from noting that every probability space on which a continuous random variable can be defined is necessarily atomless (see Villegas (1964) for a proof of this fact). 16 The inclusion is strict if and only if Z is infinite (continuity does follow from the vNM axioms when Z is finite).
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The main result of this section is that the class of qualitative probabilities consistent with an atomless probability measure constitutes a separated theory. In light of Theorem 2, this means that (probabilistic) beliefs within this theory can be uniquely recovered from observable betting behavior. Let X be a σ -algebra on some set of simple events . A preference on X is called an atomless continuous qualitative probability if the following axioms hold: Axiom 10 (Nondegeneracy). ∅ and x ∅ for all x ∈ X. Axiom 11 (Monotony). x1 ∩ x2 = ∅, x1 y1 (resp. x1 y1 ) and x2 y2 imply x1 ∪ x2 y1 ∪ y2 (resp. x1 ∪ x2 y1 ∪ y2 ) for all x1 , x2 , y1 , y2 ∈ X. Axiom 12 (Monotonic continuity). For every x, y ∈ X and every increasing sequence {yn }n∈N ↑ y, such that x yn ∈ X for all n ∈ N, x y. Axiom 13 (No atoms). For every x ∈ X that satisfies x ∅, there exists y ∈ X, such that y ⊂ x and x y ∅. Axioms 10, 11, 12, and 13 are known to be necessary and sufficient for the existence of an atomless probability measure that represents . This probability measure turns out to be unique and countably additive. The following result shows that these axioms define a separated theory. Proposition 9. The class of atomless continuous qualitative probabilities is separated. This means that the observation of enough simple bets suffices to uniquely recover the probability measure that represents the decision maker’s subjective uncertainty.17 The following example shows that Proposition 9 is not true without Axiom 13. Example 11. Consider = {a, b}, X = 2 and , , defined by: {a} ∼ {b} ∅
{a} {b} ∅.
Clearly, = and both and satisfy Axioms 10, 11, and 12, but fail Axiom 13. Moreover, the conclusion of Proposition 9 does not hold, since there are no strict preference reversals between and . Since it cannot be dispensed, a natural question is whether the requirement that there are no atoms can be weakened in some useful way. Unfortunately, I have no positive result to offer at this time. 6. Discussion In this paper, I develop a theory of abstract revealed preference that does not assume that all the optimal choices of the decision maker under analysis are observed. The proposed approach follows the spirit of Afriat’s work in that it attempts to maintain a clear separation between 17 This follows from combining Theorem 2, Proposition 9, and Theorem 4.3 in Villegas (1964).
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theoretical assumptions and empirical data, but focuses on the question of uniqueness of a rationalizing preference rather than its existence. I investigate identified theories: classes of preferences for which sufficiently rich datasets are enough to uniquely pin down the unobserved preference driving behavior. Theorem 1 and Theorem 2 combine a richness condition on choice problems with a simple preference separation property to characterize this notion of identification. To provide concrete meaning to the theory, I provide sufficient conditions for identification under standard sets of economic axioms on preferences. The strategy to recover preferences followed in this paper relies heavily on P-richness. This structural condition is useful because every separated theory P is identified on any P-rich collection of problems. However, this is an admittedly restrictive assumption that might be perceived as too strong. Two observations about this issue seem pertinent. First, Theorem 3 shows the structure of any necessary and sufficient condition in the most general setting must be significantly more convoluted and, thus, such conditions are likely to be less useful. Second, by restricting attention to continuous preferences on a metric space, Theorem 6 provides a sufficient condition for identification under somewhat weaker assumptions on choice problems. Although the present paper focuses on the identification problem, investigating the uniqueness of rationalizing preferences, partial observability also has implications for the related problem of determining the existence of such preferences. For example, one important feature of the proposed framework is that the axioms of revealed preference are no longer necessary for finding a preference that is consistent with observed behavior. Thus, the adoption of partial observability requires caution when interpreting violations of the weak axiom, as they need not be an indication of irrationality. Of course, the additional assumptions invoked in different applications do have observable implications that, in principle, allow one to falsify specific theories of behavior. Although the full characterization of these implications is beyond the objectives of this paper, it is worth noting that any sufficient condition for rationalizability in the traditional axiomatic approach to revealed preference (in which all optimal choices are observed) must also be sufficient in the present framework. Acknowledgments I thank the editor, Marciano Siniscalchi, as well as an associate editor and two referees for their constructive feedback. I also thank Andrés Carvajal, Carlos da Costa, Faruk Gul, Lucas Maestri, Ignacio Monzón, Marcelo Moreira, Paulo Natenzon, Philipp Sadowski, Bernard Salanié, and participants at various seminars for their helpful comments. Finally, I am very grateful to João Vitor Granja de Almeida, Rodrigo Naumann, and Alessandro Rivello for their outstanding research assistance. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. Appendix A. Additional material A.1. Full observability This section formalizes the notion of “full observability” and presents results comparable to those I obtain for partial observability in the body of the paper. It should come as no surprise that adopting full observability permits a relaxation of the theoretical assumptions employed to
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obtain identification. This is not for free, however, as full observability is an assumption and, as I argue in the introduction, not a methodologically appealing one. ∗ (B) denotes the set of all -best elements Given a preference and a choice problem B, c in B. The following definition extends the concept of identification to full observability: ∗ (B) = Definition 7. A theory P is identified under full observability on B if , ∈ P and c
∗ (B) for all B ∈ B imply = . c
The following definition weakens the P-richness condition in the main text: Definition 8. A collection of problems B is weakly P-rich if, for every ∈ P and (x, y) ∈ , there exist B1 , ..., BN ∈ B, z1 , ..., zN+1 ∈ X such that z1 = x, zN+1 = y, and (zn , zn+1 ) ∈ ∗ (B ) × B for all n ∈ {1, ..., N }. c n n The following result is a full observability analogue to Theorem 2. Theorem 7. If P is separated and B is weakly P-rich, then P is identified under full observability on B. It would be desirable to have a converse. However, similarly to the case of P-richness under partial observability, weak P-richness is not necessary for identification under full observability: Example 12. X = {x, y, z}, B = {{x, y}}, P = 1 , 2 , where x 1 y 1 z and y 2 x 2 z. Clearly, P is identified under full observability on B and P is separated. However, B is not weakly P-rich. Moreover, in contrast with the case of partial observability, the following example shows that separation is not necessary either: Example 13. X = {x, y}, B = {{x, y}}, P = 1 , 2 , where x 1 y and x ∼2 y. Clearly, P is identified under full observability on B and B is weakly P-rich, but P is not separated. To close this section, consider the central counterexample in Mas-Colell (1977): There exists a pair of distinct continuous, strictly monotone, strictly convex preference relations that yield the same Walrasian demand function.18 Any such preference must necessarily be regular (due to strict monotonicity). It follows that if P is the class of continuous, strictly monotone, strictly convex preferences, then P is a separated theory. Thus, in light of Theorem 7 above, the collection of linear budget sets, B L , must necessarily fail to be weakly P-rich. Let P MCL be the class of monotone and convex preferences that admit a lipschitzian representation. Remark 12 in Mas-Colell (1977) (p. 1414) implies that B L is weakly P MCL -rich. Since P MCL is separated due to monotonicity, Theorem 7 establishes that P MCL is identified under full observability on B L , a fact already known from Theorem 2’ in Mas-Colell (1977). 18 In the language of Section 2.5, the preferences of this counterexample are observationally equivalent on the class of linear budget sets.
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In contrast, whether P MCL is identified on B L under partial observability seems to be an open problem. A.2. Almost P-richness Considering that P-richness is a strong condition, this section discusses a minimal weakening: to allow the -best element zN to be non-unique. Definition 9 (Almost P-richness). A collection of choice problems B is almost P-rich if, for every ∈ P and x, y ∈ X, x y implies there are B1 , ..., BN ∈ B and z1 , ..., zN+1 ∈ X such that: (1) (2) (3) (4)
z1 = x, zN+1 = y, zn+1 ∈ Bn for all n ∈ {1, ..., N }, zn is the unique -best element in Bn for all n ∈ {1, ..., N − 1}. zN is a -best element in BN .
Like P-richness, almost P-richness is a pointwise criterion. Moreover, if B is P-rich, then B is almost P-rich, while the converse does not generally hold. However, the following example shows that even this mild relaxation is too weak to yield identification of all separated theories under partial observability. Example 14. Let X = {x, y, z}, B = {{x, y}, {y, z}} and P = , , where x y ∼ z and z x ∼ y. It is straightforward to verify that B is almost P-rich, but fails to be P-rich. Moreover, the choice correspondence c : B → 2X given by c({x, y}) = {x} and c({y, z}) = {z} has full domain and is consistent with both preferences. Hence, P is not identified on B even though P is separated. Of course, the fact that Definition 9 does not work as intended does not prove that P-richness cannot be usefully weakened in some other way. Unfortunately, I am presently unaware of any such condition. A.3. Additional examples This section contains examples complementing several passages of the body of the paper. The following example shows that separation is not necessary for identification without compatibility: Example 15. X = [0, 1), B = {X}, and P = 1 , 2 , where 1 and 2 are represented by u1 (x) = x and 1 x ∈ [0, 1/2), u2 (x) = 2x x ∈ [1/2, 1), respectively. Then, P is not separated. However, P is identified on B. The following example shows that local strictness does not imply local nonsatiation:
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Example 16. X = [0, 1], is represented by u(x) = x. is continuous and locally strict, but has a satiation point at x = 1. The following example shows that regularity does not imply local strictness: Example 17. X = [0, 4], is represented by ⎧ ⎪ x ∈ [0, 2], ⎨x u(x) = 4 − x x ∈ (2, 3], ⎪ ⎩ 1 x ∈ (3, 4]. is continuous and regular, but fails to be locally strict (consider any pair of different points x, y ∈ (3, 4]). The following example shows that connectedness cannot be dispensed with in Lemma 2: Example 18. X = [0, 1] ∪ [2, 3], is represented by x x ∈ [0, 1], u(x) = x − 1 x ∈ [2, 3]. is continuous and generically locally non satiated. However, it is not regular (consider the pair of points (1, 2)). Moreover, generic local nonsatiation is sufficient, but not necessary in Lemma 2: Example 19. X = [0, 4], is represented by ⎧ ⎪ x ∈ [0, 2], ⎨x u(x) = 4 − x x ∈ (2, 3], ⎪ ⎩ 1 x ∈ (3, 4]. is continuous and regular, but fails generic local non satiation. The following example shows that generic local nonsatiation does not imply local strictness: Example 20. X = [0, 6], is represented by ⎧ ⎪ x x ∈ [0, 1], ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2 − x x ∈ (1, 2], u(x) = x − 2 x ∈ (2, 4], ⎪ ⎪ ⎪ 6 − x x ∈ (4, 5], ⎪ ⎪ ⎪ ⎩x − 4 x ∈ [5, 6]. is continuous and generically locally nonsatiated (thus regular), but fails to be locally strict (consider the pair (1, 5)).
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Appendix B. Proofs This appendix contains the proofs of the theorems and propositions in the main text. The following abstract characterization of separated theories will be useful throughout the proofs. Define a binary relation on the class of all preferences by ⇐⇒ ⊆ ∧ ⊆ . Lemma 3. A theory P is separated if and only if the restriction of to P is antisymmetric. Proof. I will establish the equivalence of the contrapositives. P is not separated if and only if there are , ∈ P, such that = and, for all x, y ∈ X, x y implies y x and x y implies y x. These implications are equivalent to ⊆ ⊀ = and ⊆ ⊀ = , where the equalities follow from the completeness of and , respectively. In turn, these two inclusions amount to the definition of . Therefore, P is not separated if and only if restricted to P is not antisymmetric, as claimed. 2 B.1. Proofs of Section 2 Proof of Theorem 1. Assume that B is compatible with P, but P is not separated. Then, by Lemma 3, there are , ∈ P such that = and . Consider the choice correspon∗ (B) ∩ c∗ (B). Clearly, c is consistent dence c : B → 2X defined for each B ∈ B by c(B) := c
with both of and . I claim that c has full domain. To see this, pick any choice problem ∗ (B) = ∅. This means that there is x ∈ B such that B ∈ B. Since B is compatible with P, c ∗ (B), then x ∈ c(B). Second, x y for all y ∈ B. There are two cases to consider. First, if x ∈ c
∗ (B), then c∗ (B) = ∅ implies that there must be z ∈ c∗ (B) such that z x. However, if x ∈ / c
∗ (B), and so z ∈ c(B). In either case, c(B) = ∅ since , it must be that z x. Thus, z ∈ c holds. It follows that c has full domain. The preceding argument establishes that c has full domain and is, nevertheless, consistent with different preferences in P. This means that P is not identified. I conclude that identification over a compatible collection of choice problems implies separation. 2
Proof of Theorem 2. Assume that B is P-rich, but P is not identified on B. This means that there exist different preferences , ∈ P and a choice correspondence c with full domain that is consistent with both and . Select any x, y ∈ X, such that x y. Since B is P-rich, there is a sequence of N problems B1 , ..., BN ∈ B and a sequence of N + 1 alternatives z1 , ..., zN+1 ∈ ∗ (B ) = {z } hold. X, such that z1 = x, zN+1 = y, and, for all n ∈ {1, ..., N }, zn+1 ∈ Bn and c n n Pick any n ∈ {1, ..., N }. On the one hand, since Bn ∈ B and c has full domain, c(Bn ) = ∅. On ∗ (B ) because c is consistent with . Thus, c(B ) = {z }. Since c the other hand, c(Bn ) ⊆ c n n n
is consistent with and zn+1 ∈ Bn , one obtains zn zn+1 . Considering that this argument holds for each n ∈ {1, ..., N }, the transitivity of implies x = z1 zN+1 = y. It follows that y x. Since the choice of x, y was arbitrary, this means that P is not separated. I conclude that separation implies identification on B. 2 Proof of Theorem 3. Suppose A is necessary for identification. By definition of pointwise criterion,
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A(P) =
A({}) ⊇
∈P
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B {} is identified on B . ∈P
Theories containing on every collection of choice problems. a single preference are identified It follows that B {} is identified on B = B B ⊆ 2X holds for every preference ∈ P. Thus, A(P) ⊇ B B ⊆ 2X . Since B B ⊆ 2X is the codomain of A, the desired equality is obtained. 2 B.2. Proofs of Section 3 The proof of Theorem 4 requires some additional definitions. A preference is a tie-breaker for if ⊂ . It is easy to see that is a tie-breaker for if and only if ∼ ⊂ ∼ and ⊃ . A theory P is said to be closed to tie-breaking if S ∈ P holds for every subset S ⊆ P totally ordered by ⊆. A preference is ultimate in P if it admits no tie-breakers in P. If is a tie-breaker for , then . The following abstract result describes the space of tie-breakers for theories closed to tie-breaking. Lemma 4. Let P be a theory closed to tie-breaking. Then every ∈ P is either ultimate in P or admits an ultimate tie-breaker in P. Proof. Let P() be the set of all preferences in P that are contained in . Note that ⊆ is the reflexive closure of the property of being a tie-breaker. Moreover, ⊆ partially orders P(), and every totally ordered subset of P() is bounded by the intersection of all its elements (which belongs to P() because the theory is closed to tie-breaking). Thus, by Zorn’s lemma, P() must be ultimate; otherwise, it could not be possesses a ⊆-maximal element . Clearly, ⊆-maximal in P(). If P() = , then = . Otherwise, is a tie-breaker for . 2 The following notions are easy to depict and provide an equivalent definition of regularity that sometimes is more convenient. A pair (x, y) ∈ X × X is said to be -locally isolated if x = y and there exist open sets V x and W y, such that {z ∈ X|x z} ∩ V = {z ∈ X|z y} ∩ W = ∅. A point x ∈ X is said to be -transversal if every open set V containing x also contains x ∗ , x∗ ∈ V , such that x ∗ x x∗ . Then, it is straightforward to show that the preference is regular if and only if every indifference class for with a -locally isolated pair contains a -transversal point. (See Fig. 3.) Lemma 5. If and are regular continuous preferences satisfying , then = . Proof. Suppose, seeking a contradiction, that = . Without loss of generality, I can assume that there exists x, y ∈ X, such that y x and y x. By completeness, x y. Since , x y implies x y. It follows that y ∼ x. I will now show that y ∼ x y contradicts the regularity of . In preparation for the ar gument, define the sets V := z ∈ X z y and W := z ∈ X x z . Note that V and W are nonempty since x ∈ V and y ∈ W . Moreover, both sets are open by continuity and completeness of . Now suppose, seeking a contradiction, that is regular. Let I denote the indifference class for containing {x, y}. There are two cases to consider: Either I is -extremal or not. I treat each of these cases separately.
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Fig. 3. Characterizing regularity.
If I is -extremal, then (x, y) is not -locally isolated. It follows that either there exists x ∈ V such that x x or there exists y ∈ W such that y y. In the first case, y ∼ x x and transitivity of imply y x . Thus, yields y x , which contradicts x ∈ V . The second case is symmetric: y y ∼ x implies y x, so yields y x, thus contradicting y ∈ W . If I is not -extremal, there must be a -transversal point w ∈ I . Since x y, the completeness of implies w ∈ V ∪ W . This means that either w ∈ V or w ∈ W . Suppose first that w ∈ V . Since V is open, the -transversality of w implies that there exists x ∈ V , such that w x . Since x ∼ w, transitivity of implies that x x . If w ∈ W , a symmetric argument yields the existence of y ∈ W , such that y y. It follows that there exists either x ∈ V such that x x , or y ∈ W such that y y. Thus, the argument in the previous paragraph for the case in which (x, y) is not locally isolated applies. 2 Lemma 6. Let X be a separable metric space. For every continuous preference that is not regular, there exists a continuous preference , such that ⊂ . Proof. Fix a continuous preference that is not regular. I claim that there exists a continuous preference , such that ⊂ . The argument below proves this by breaking some indifferences without violating continuity. Since is not regular, there is a -extremal indifference class for , say I , containing points x0 , y0 , such that (x0 , y0 ) is a -locally isolated pair. Define P0 (resp. N0 ) as the set of all points x ∈ I , such that there exists an open set V x such that every y ∈ V satisfies y x (resp. x y). Since (x0 , y0 ) is a -locally isolated pair, x0 ∈ P0 and y0 ∈ N0 . Thus, P0 and N0 are nonempty. Since there are no -transversal points in I , I = P0 ∪ N0 and (I \ P0 ) ∩ (I \ N0 ) = ∅ (although P0 and N0 need not be disjoint). Moreover, note that both P0 and N0 are open in I by construction. Define P := {x ∈ X \ N0 |x x0 } and N := {x ∈ X \ P0 |x0 x}. If P or N turns out to be empty according to this definition, redefine it as P := {x0 } or N := {y0 }, respectively. P and N are clearly disjoint. Moreover, they are both closed in X. To see this, let {xn } be a sequence in P that converges to xˆ ∈ X and suppose, seeking a contradiction, that xˆ ∈ / P . It follows from / N0 for all n. Moreover, since xn x0 for all n, continuity implies the definition of P that xn ∈ / P , it must be that xˆ ∈ N0 . Then, considering the definition of N0 , xn ∼ x0 xˆ x0 . But, since xˆ ∈ must hold for all sufficiently large n. However, since xn ∈ / N0 , this means that xn ∈ I \ N0 for all sufficiently large n. Since I \ N0 is closed in I , I conclude that xˆ ∈ I \ N0 , providing the desired contradiction. This means that P is closed. The argument showing that N is closed is symmetric. Since X is a separable metric space and is continuous, there exists a continuous function U : X → R that represents (see Debreu (1954)). Since X is a metric space, it is also a normal
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space and, by Urysohn’s lemma, there exists a continuous function h : X → [0, 1], such that h(x) = 1 for x ∈ P and h(x) = 0 for x ∈ N (see Munkres (2000)). Finally, define := {(x, y) ∈ X × X|U (x) + h(x) ≥ U (y) + h(y)}. It is easy to verify that is a continuous preference and that ⊂ . Thus, ⊂ as desired. 2 Proof of Theorem 4. The first claim follows directly from Lemma 3 and Lemma 5. To prove the second claim, note that of continuous preferences is closed to tie the class breaking. To see this, suppose S ≡ i i ∈ I is an arbitrary set of continuous preferences totally ordered by ⊆ and define ∗ := ∩i∈I i . Clearly, ∗ is transitive and continuous, since these properties are preserved by arbitrary intersections. To see that it must also be complete, suppose—seeking a contradiction—that there are x, y ∈ X, such that x ∗ y ∗ x. Since ∗ = ∪i∈I i = ∪i∈I ≺i , it must be that x i1 y i2 x for some indexes i1 , i2 ∈ I , such that i1 = i2 . Since S is totally ordered, either i1 ⊂ i2 or i2 ⊂ i1 . Consider first the case in which i1 ⊂ i2 . Then, since i1 is a tie-breaker for i2 and y i2 x, it follows that y i1 x. This contradicts x i1 y. The case in which i2 ⊂ i1 is handled by a symmetric argument. Now, suppose is a continuous preference that is not regular. Since I have just proved that continuous preferences are closed to tie-breaking, Lemma 4 and Lemma 6 together imply that there exists a continuous preference that is an ultimate tie-breaker for . Suppose, seeking a contradiction, that is not regular. Then Lemma 6 implies that there is a continuous tie-breaker for . It follows that is not ultimate, providing the desired contradiction. I conclude that must be regular, completing the proof. 2 Proof of Lemma 1. Suppose first that is locally nonsatiated everywhere. Let (x, y) ∈ V ∩ for any arbitrary open set V . By the definition of the product topology, there must be an open rectangle V1 × V2 ⊆ V such that x ∈ V1 and y ∈ V2 . By local nonsatiation, there is x ∈ V1 such that x x. It follows that (x , y) ∈ V ∩ , establishing local strictness. For the second implication, it suffices to prove that local strictness of precludes the existence of an indifference class for containing a -locally isolated pair. I show this by establishing the contrapositive. If x, y are two different points such that x ∼ y, x is a local worst element, and y a local best element (for ), then there is an open rectangle V = V1 × V2 containing (y, x) such that y ∈ V1 implies y y and x ∈ V2 implies x x. It follows that V ∩ = ∅, contradicting local strictness. Finally, that the converse implications do not hold follows directly from Example 16 and Example 17, which feature continuous preferences over a compact interval. 2 Before proving Lemma 2, I need to establish the following lemma: Lemma 7. Let X be an arbitrary topological space and let P : X → {0, 1} be a pointwise property that holds generically on X (i.e. P (x) = 1 for all x on an open dense subset of X). Then, 1 ∈ P (V ) for every nonempty open set V ⊆ X. Proof. Define the set Y := {x ∈ X|P (x) = 1} and take any nonempty open set V ⊆ X. I claim that V ∩ Y = ∅. Suppose, seeking a contradiction, that V ∩ Y = ∅. Then, Y ⊆ X \ V , which implies that clX (Y ) ⊆ clX (X \ V ) = X \ V ⊂ X, contradicting the fact that Y is dense in X by assumption. Since V ∩ Y = ∅ and P (Y ) = {1} by definition, the result follows immediately. 2
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Proof of Lemma 2. Suppose, seeking a contradiction, that is not regular. Then, there must exist a set I ⊆ X and different points x0 , y0 ∈ I such that I is a -extremal indifference class for , x0 is a local worst element for , and y0 is a local best element for . The argument in the proof of Lemma 6 above implies that there exist disjoint nonempty closed sets P and N such that x0 ∈ P and y0 ∈ N . Define the open set V := X \ (P ∪ N ). Clearly, z ∈ V implies z ∈ I (that is, z ∼ x0 ∼ y0 ). Moreover, by Lemma 7 and the assumption that local nonsatiation holds generically on X, if V was nonempty there would be z0 ∈ V such that satisfies local nonsatiation at z0 . This cannot happen by definition of V (otherwise z0 would belong to P ). It follows that V must be empty. But then, {P , N } is a closed nontrivial partition of X, contradicting the assumption that X is connected. 2 The following two lemmas are used in the proof of Theorem 5. Before proceeding, I need some additional definitions and topological structure. Let (X, d) be a metric space. Denote by FX (resp. KX ) the collection of all nonempty closed (resp. compact) subsets of X equipped with the Hausdorff metric topology. Lemma 8. Consider a sequence {(An , xn )}n∈N in FX × X, such that An converges to A ∈ KX and xn ∈ An for all n ∈ N. Then (xn )n∈N has a subsequence converging to a point in A. Proof. Since limn→∞ An = A, there is a sequence {yh }h∈N in A and subsequence {xnh }h∈N , such that d(yh , xnh ) < 1/ h for all h ∈ N. Since yh ∈ A for all h ∈ N and A is compact, there is a subsequence {y˜k }k∈N ≡ {yhk }k∈N converging in A. Let x ∗ := limk→∞ y˜k . I will now show that the sequence {x˜k }k∈N ≡ {xnhk }k∈N converges to x ∗ . Choose any > 0. Since {y˜k }k∈N converges to x ∗ , there is k1∗ sufficiently large, such that d(x ∗ , y˜k ) < /2 for all k ≥ k1∗ . Let k2∗ := min {k ∈ N|hk > 2}. Clearly, d(y˜k , x˜k ) ≡ d(yhk , xnhk ) < 1/ hk < /2 for all k ≥ k2∗ . Define k ∗ := max{k1∗ , k2∗ }. It thus follows from the triangle inequality that d(x ∗ , x˜k ) ≤ d(x ∗ , y˜k ) + d(y˜k , x˜k ) < /2 + /2 = for all k ≥ k ∗ . This concludes the proof.
2
Lemma 9. Fix a continuous preference and consider any choice correspondence c consistent with . Then dom(cl(c)) = clB (dom(c)) and cl(c) is consistent with . Proof. Clearly, dom(cl(c)) ⊆ clB (dom(c)). To show equality, consider any B ∈ clB (dom(c)). Since B is a metric space, it is sequential. Hence, there is a sequence {Bn }n∈N in dom(c), such that limn→∞ Bn = B. Since Bn ∈ dom(c) for all n ∈ N, there is a sequence {xn }n∈N , such that xn ∈ c(Bn ) for all n ∈ N. Since B is compact, Lemma 8 implies that {xn }n∈N has a subsequence {x˜k }k∈N ≡ {xnk }k∈N converging to a point in B. Thus, if x ∗ is such a point, then x ∗ ∈ cl(c)(B). This shows that cl(c)(B) = ∅ for all B ∈ clB (dom(c)). Hence, clB (dom(c)) ⊆ dom(cl(c)). I now claim that cl(c) is consistent with . To prove this, select any (B, x) ∈ B × X, such that ∗ (B). Because x ∈ cl(c)(B), there are sequences {B } x ∈ cl(c)(B). I need to show x ∈ c n n∈N → B and {xn }n∈N → x, such that xn ∈ c(Bn ) for all n ∈ N. Now suppose, seeking a contradiction, that there is x ∈ B, such that x x . Continuity and completeness together imply that is a closed subset of X × X. Hence, there are open sets V , W , such that (x, x ) ∈ V × W and y z for all (y, z) ∈ V × W . Since {xn }n∈N converges to x, xn ∈ V must hold for all sufficiently large n. Since {Bn }n∈N converges to B, there is a sequence {xn }n∈N , such that xn ∈ Bn ∩ W for
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all sufficiently large n. However, this means that xn xn . Since xn ∈ c(Bn ) and xn ∈ Bn , this contradicts c being consistent with . 2 Proof of Theorem 5. For Part (1), let c be an arbitrary choice correspondence. I need to show that c and cl(c) are rationalized by the same subset of P. This statement can be expressed using the notation introduced in Section 4.5 as P (c, P) = P (cl(c), P). To establish the equality, I will ∗ (B) for prove both inclusions. Suppose first that ∈ P (cl(c), P). Then c(B) ⊆ cl(c)(B) ⊆ c all B ∈ B. Hence, ∈ P (c, P). I conclude that P (cl(c), P) ⊆ P (c, P). Suppose now that ∈ P (c, P). By definition of P (c, P), this means that c is consistent with . Lemma 9 thus implies that cl(c) is also consistent with . It follows—this time by definition of P (cl(c), P)—that ∈ P (cl(c), P). I conclude that P (c, P) ⊆ P (cl(c), P). This completes the proof of this part. For Part (2), assume that P is identified on B, let B be a dense subset of B, and pick any , ∈ P. Consider a choice correspondence c : B → 2X with domain B that is simultaneously consistent with and . I need to show that = . By Lemma 9, cl(c) is also consistent with both and . Moreover, dom(cl(c)) = clB (dom(c)) = clB (B ) = B. Since P is identified on B, it follows that = , as desired. To establish Part (3), I shall prove that = implies Y = Y (where Y := ∩ (Y × Y ) and Y := ∩ (Y × Y )). Suppose that = . Then, since P is separated, there exist points x, y ∈ X such that x y x. Because both and are continuous, there must be open sets V x and W y such that v w v for all v ∈ V and w ∈ W . The assumption that Y is dense in X implies that V ∩ Y = ∅ and W ∩ Y = ∅. It follows that there exist two alternatives v ∗ , w ∗ ∈ Y such that v ∗ w ∗ v ∗ . But this, in turn, implies Y = Y , concluding the proof. 2 Proof of Theorem 6. Fix any ∈ P and consider a choice correspondence c that is consistent with and has full domain. To prove the result, I need to establish that P (c, P) = . ∗ (B) ⊆ Y for every B ∈ Condition (1) in the statement of the result implies that c(B) ⊆ c . Define c : B → 2Y by setting c (B ∩ Y ) := c(B) ∩ Y for every B ∈ B. Notice that B Y Y Y
and cl (B ) = B dom(clY (cY )) = clBY (dom(cY )) by Lemma 9. Moreover, dom(cY ) ⊇ B B by Condition (1). It follows that dom(clY (cY )) = clBY (dom(cY )) ⊇ clBY (B ) = clB (B ) ∩ BY = BY .
This means that clY (cY ) has full domain. Since PY is separated and BY is PY -rich, Theorem 2 implies that PY is identified on BY . Combining this with Part (1) in Theorem 5 yields P (cY , PY ) = P (clY (cY ), PY ) = ∩ (Y × Y ) . Let π : P → PY be the projection defined by setting π() := ∩ (Y × Y ) for each ∈ P. By Condition (2), P (c, P) = π −1 (P (cY , PY )) = π −1 ∩ (Y × Y ) = . I conclude that P is identified on B, completing the proof. 2 B.3. Proofs of Section 4 The following three lemmas are used in the proof of Proposition 1.
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Lemma 10. PD is a separated theory. Proof. Pick ∈ PD . Axiom 3 implies that is locally nonsatiated, thus regular. Since all preferences in PD are continuous, Theorem 4 establishes the desired result. 2 ∗ (B Lemma 11. For any x ∈ X such that x 0, there exists p ∈ R2++ such that c p,px ) = {x}.
Proof. First consider the case in which x2 = 0. Then, it must be that x1 > 0 because x 0. In that case, define γ ∗ := inf {γ > 0|(x1 − 1, γ ) x}. Since (x1 , 0) (x1 − 1, 0), γ ∗ = 0 would imply (x1 , 0) ∼ (x1 − 1, 0). This contradicts Axiom 4, considering that (x1 + 1, 0) (x1 − 1, 0) and (x1 , 0) = (1/2)(x1 + 1, 0) + (1/2)(x1 − 1, 0). It follows that γ ∗ > 0. Then, setting p := (1, 2/γ ∗ ) gives the result. Now consider the case in which x2 > 0. Define the price vector 1 pR := 1, , x2 − γ ∗ where γ ∗ := inf {γ > 0|(x1 + 1, γ ) x}. Continuity of implies (x1 + 1, γ ∗ ) x and also pR x = pR (x1 + 1, γ ∗ ). In fact, γ ∗ > 0 implies (x1 + 1, γ ∗ ) ∼ x. However, it is possible that (x1 + 1, γ ∗ ) x when γ ∗ = 0. Define another price vector pL as follows. If x1 = 0, pL = (1, 0). If x1 > 0, then 1 pL := 1, ∗ , δ − x2 where δ ∗ := sup {δ > 0|x (x1 − 1, δ)}. Here, in the case x1 > 0 and δ ∗ < +∞, (x1 − 1, δ ∗ ) ∼ x (by continuity) and pL (x1 − 1, δ ∗ ) = pL x. Note that, due to Axiom 4, δ ∗ − x2 > x2 − γ ∗ > 0 and, so, pR and pL satisfy pR ≥ pL and pR = pL , so, in particular, (pL , pR ) = ∅. In the typical situation in which x1 > 0, δ ∗ < +∞, and γ ∗ > 0, the following characterization of the best bundle for a price vector p = (1, α) and w = px holds: (1) (2) (3) (4)
∗ (B ∗ If α = 1/(δ ∗ − x2 ), then c p,w ) = {x, (x1 − 1, δ )}. ∗ ∗ ∗ If 1/(x2 − γ ) > α > 1/(δ − x2 ), then c (Bp,w ) = {x}. ∗ (B ∗ If α = 1/(x2 − γ ∗ ), then c p,w ) = {x, (x1 + 1, γ )}. ∗ (B Else, x ∈ / c p,w ).
The other cases, in which either δ ∗ = +∞, γ ∗ = 0, or x1 = 0, are dealt with in a similar ∗ (B fashion. In each of these cases, it is possible to show that c p,w ) = {x} whenever 1/(x2 − ∗ (B ∗ ∗ γ ) > α > 1/(δ − x2 ). I conclude that, for any price vector p ∈ (pL , pR ), c p,px ) = {x}. 2 L is P -rich. Lemma 12. BD D
Proof. Pick ∈ PD and x, y ∈ X such that x y. Letting x ≡ (x1 , x2 ) and y ≡ (y1 , y2 ), there are three cases: x1 = y1 , x1 < y1 , and x1 > y1 . In the first case, it must be that x2 > y2 by Axiom 3. Then, Lemma 11 shows that there exists ∗ (B a price vector p ∈ R2++ such that c p,px ) = {x}. In the second case, x1 < y1 , so let N := y1 − x1 + 1 ≥ 2. Fix some small > 0. Following the ideas in the proof of Lemma 11, define bundles z1 , ..., zN ∈ X by setting z1 := x, zN+1 := y,
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and zn+1 := (z1n + 1, α n − ) for each n ∈ {1, ..., N − 1}, where α n is such that zn ∼ (z1n + 1, α n ) holds. Note that z1 z2 ... zN . Let p n ∈ R2++ be defined as 1 n p := 1, n+1 . z2 − z2n Clearly, p n zn = p n zn+1 holds by construction. It is easy to see that z1N = y1 and, if is small enough, z2N > y2 , which means that pN zN > p N y. Moreover, the same argument used in the ∗ (B n n n ) = {zn } for each proof of Lemma 11 establishes that, for all sufficiently small , c p ,p z n ∈ {1, ..., N}. Finally, in the third case, the proof proceeds almost as in the second case. Now let N := x1 − y1 + 1 ≥ 2 and define zn+1 := (z1n − 1, α n − ), where again zn ∼ (z1n+1 , α n ) zn+1 . The L is P -rich. 2 rest of the argument is identical. The three cases above establish that BD D L . By Lemma 10, P is a separated Proof of Proposition 1. Let B0 be a dense subset of BD D L . Since B theory. By Lemma 12, B is PD -rich. Then, by Theorem 2, PD is identified on BD 0 L is dense in BD , PD is identified on B0 by Part (2) of Theorem 5. 2
Proof of Proposition 2. For (1), let U : X → R be a C 1 representation of such that U (x) >> 0 whenever x 0. The existence of such a representation implies that is continuous, monotone (x >> y implies x y) and weakly monotone (x ≥ y implies x y). Pick any two points x, y ∈ X such that x y. Consider first the case in which y >> 0. Note that y 0 by monotonicity. I claim that, for each t ∈ [0, 1], there exists a unique λt ∈ R+ such that U (λt [(1 − t)x + ty]) = (1 − t)U (x) + tU (y). If t ∈ {0, 1}, the claim holds as a trivial consequence of monotonicity (with λ0 = λ1 = 1). If t ∈ (0, 1), define F : R++ × (0, 1) → R by setting F (λ, t) := U (λ [(1 − t)x + ty]) − (1 − t)U (x) − tU (y). Note that F (λ, t) is increasing in λ. Fixing t ∈ (0, 1) and using U (x) > U (y) > U (0), lim F (λ, t) = U (0) − (1 − t)U (x) − tU (y) < (1 − t) [U (y) − U (x)] < 0. λ↓0
Moreover, setting λ := (1 − t)−1 , U (x) > U (y), y >> 0, and monotonicity imply that F λ, t = U (x + λty) − (1 − t)U (x) − tU (y) > U (x + λty) − U (x) > 0.
Since the function F (·, t) is increasing, continuous, and satisfies limλ↓0 F (λ, t) < 0 < F λ, t , there exists a unique λt ∈ (0, λ) such that F (λt , t) = 0, as claimed. The function F defined above is C 1 and satisfies ∂F (λ, t) U (zt )zt = > 0, ∂λ λ=λt λt where zt := λt [(1 − t)x + ty]. It follows by the implicit function theorem that there is a C 1 function t → λ˜ t satisfying F (λ˜ t˜, t˜) = 0 for all t˜ in a neighborhood of t . Since, for each t ∈ [0, 1],
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the equation F (λ, t) = 0 has a unique solution λ = λt , it must be that λ˜ t = λt . It follows that the mapping t → λt constructed in the previous paragraph is C 1 . Since the mapping t → λt is C 1 , the mapping t → zt is C 1 as well. Moreover, z0 = x, z1 = y, and also dU (zt ) = U (y) − U (x) < 0, dt which means that utility decreases along the smooth trajectory zt connecting x and y. Defining pt z ≤ pt zt ∈ B L , Axiom 6 implies that c∗ (Bt ) = {zt }. pt := U (zt ) >> 0 and Bt := z ∈ RN + Moreover, dzt dzt dU (zt ) = U (zt ) = = U (y) − U (x) < 0. dt dt dt Now, for each N ∈ N, define tn,N := n/N and consider the grid {t0,N , t1,N , ..., tN,N }. The mapping t → zt is C 1 and the mapping t → pt is continuous. It follows that, whenever t = n/N , ztn+1,N − ztn,N dzt lim ptn,N < 0. = pt N→+∞ tn+1,N − tn,N dt pt
As a consequence, if N is large enough, ztn+1,N − ztn,N 1 ptn,N ztn+1,N − ztn,N = ptn,N <0 N tn+1,N − tn,N holds for every n ∈ {0, ..., N }. Hence, ptn,N ztn+1,N ≤ ptn,N ztn,N . Summarizing, for all sufficiently large N , the sequence {ztn,N }N n=0 satisfies zt0,N = x, ztN,N = ∗ (B y, ztn+1,N ∈ Btn,N , and c ) = {z } for all n ∈ {0, ..., N − 1}. This shows that the condition tn,N tn,N required by {}-richness holds for x and y, provided y >> 0. Now suppose that y >> 0 does not hold. In that case, take some y >> y such that x y . (This is always possible due to the continuity of .) Since y ≥ 0 implies that y >> 0, the previous argument can be applied to verify the {}-richness requirement on x and y . In this way, N one obtains a sequence of choices {zn }N+1 n=1 and budget sets {Bn }n=1 such that z1 = x, zN+1 = y , ∗ zn+1 ∈ Bn , and c (Bn ) = {zn } for all n ∈ {1, ...N }. Since y >> y ensures that y ∈ BN , if one
uses zN+1 = y instead of zN+1 = y the requirements imposed by the definition of {}-richness remain fulfilled. This completes the proof of this case and establishes that the collection of linear budget sets B L is {}-rich. For (2), suppose, seeking a contradiction, that B L is {}-rich but is not strictly convex. Then there are different alternatives x, y ∈ X = RN + such that x y, and α ∈ (0, 1) such that z := αx + (1 − α)y y. By completeness, x y z. Since z is a strict convex combination of ∗ (B) = {z}. Since B L is {}-rich, this implies z w for x and y, there is no B ∈ B L such that c
all w ∈ RN + , so w z for all w ∈ X (by completeness). Because is strictly monotone, it follows that z = 0. Since 0 is the unique extreme point of RN + and z is a strict convex combination of x and y, it follows that x = y = 0, contradicting x = y. I conclude that must be strictly convex. 2
Proof of Proposition 3. As mentioned in the main text, expected utility preferences are continuous (this follows from the fact that the expected utility representations are continuous by the Portmanteau lemma). To verify regularity, consider any nontrivial expected utility preference . Since is nontrivial, there must be lotteries x, y ∈ X such that x y. Now let I ⊆ X be an
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arbitrary indifference class for , and consider lotteries z, z ∈ I and open sets V , W such that z ∈ V and z ∈ W . By completeness, either x z ∼ z or z ∼ z y. Suppose that x z ∼ z . Since mixing lotteries is a continuous operation, there must be sufficiently small > 0 such that x + (1 − )z ∈ V and x + (1 − )z ∈ W . But, by independence, x + (1 − )z z and x + (1 − )z z . It follows that (z, z ) cannot be a locally isolated pair. A symmetric argument leads to the same conclusion in the case z ∼ z y. Since I and z, z were arbitrary, is regular. 2 Proof of Proposition 4. Fix an expected utility preference and suppose x y. To prove (1), let N ≥ 2 be a fixed number and consider the choice problem B = {x, y} ∪ 2−n x + (1 − 2−n )y n = 1, ..., N − 2 . Clearly, B has exactly N elements. Moreover, by independence, x z for all z ∈ B \ {x}. It ∗ (B) = {x}. I conclude that Definition 6 is satisfied by the collection of all choice follows that c problems with N elements. To prove (2), consider the segment [x, y] := {αx + (1 − α)y|α ∈ [0, 1]}. Clearly, [x, y] is ∗ ([x, y]) = {x}. As a result, to establish the desired richness, it suffices to show convex and c that [x, y] is a compact subset of X. To see this, consider the projection π : [0, 1] → [x, y] defined by setting π(α) := αx + (1 − α)y for each α ∈ [0, 1]. Every sequence {αn }n∈N such that limn→∞ αn → α verifies lim udπ(αn ) = lim αn udx + lim (1 − αn ) udy, n→∞
n→∞
X
=α =
n→∞
X
udx + (1 − α) X
X
udy, X
udπ(α) X
for all bounded continuous functions u : X → R. Thus, the Portmanteau lemma implies that π is continuous. Since [x, y] = π ([0, 1]) and [0, 1] is compact, [x, y] must be compact as well. 2 Proof of Proposition 5. Let P be the class of preferences on X that are nontrivial, continuous and satisfy set independence. By Theorem 4, it suffices to show that all preferences in P are regular. The argument for this is structured exactly as the one used in the proof of Proposition 3. The only additional difficulty is that it is not immediate to show that every open set around a menu contains strict convex combinations with every other menu. Consider any preference ∈ P. Since is nontrivial, there must be menus x, y ∈ X such that x y. Let I ⊆ X be an arbitrary indifference class for , and consider menus z, z ∈ I and open sets V , W such that z ∈ V and z ∈ W . By completeness, either x z ∼ z or z ∼ z y. Suppose that x z ∼ z . I claim that there exists > 0 such that x + (1 − )z ∈ V . To see this, let dH : X × X → [0, ∞) be the Hausdorff distance on X and let dL : × → [0, ∞) be the Lévy-Prokhorov distance on . Define functions f, g : [0, 1) → X by setting f () := max min dL (p, (1 − )q + r) p∈z q∈z,r∈x
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and g() := max min dL (p, (1 − )q + r) q∈z,r∈x p∈z
for each ∈ [0, 1). Note that f (0) = g(0) = 0. Given that x and z are compact, is convex and dL is continuous (on × ), I conclude that dH (z, x + (1 − )z) = max{f (), g()} and that the claim is equivalent to lim↓0 f () = lim↓0 g() = 0. But this continuity follows from applying Berge’s maximum theorem twice, since x and z are fixed nonempty compact menus and (1 − )q + r depends continuously on . A similar argument shows that there exists > 0 such that x + (1 − )z ∈ W . Thus, ∗ x + (1 − ∗ )z ∈ V and ∗ x + (1 − ∗ )z ∈ W hold simultaneously for ∗ := min{, } > 0. Since ∗ x + (1 − ∗ )z z and ∗ x + (1 − ∗ )z z by set independence, (z, z ) cannot be a locally isolated pair. A symmetric argument leads to the same conclusion in the case z ∼ z y. Since I and z, z were arbitrary, must be regular. 2 Proof of Proposition 6. The argument is similar to the one establishing Proposition 4. The only relevant difference lies in establishing that every segment joining two menus is compact in part (2). Following the idea in the proof of Proposition 5, it can be shown that segments are closed subsets of X. To complete the proof, note that, since X is a compact metric space (equipped with the Hausdorff distance), every closed subset of X is compact. 2 In preparation for the proof of Proposition 7, I need to establish the intuitive result that the image of an iid sequence of random elements in a separable metric space is dense in its support with probability 1. For any topological space Y , let clY denote the closure operator that maps any subset of Y into the intersection of all closed sets containing it. Lemma 13. Let (X, d) be a separable metric space and let (, , μ) be a probability space with an iid sequence {xn }n∈N of random elements xn : → X for each n ∈ N. Define the support of each xn as Sn := A ∈ FX μ xn−1 (A) = 1 , and let S := n∈N Sn . Define the random set R : → 2X by setting R(ω) := {xn (ω)|n ∈ N} for each ω ∈ . Then clX (R) = S almost surely. Proof. First, I show that clX (R) ⊆ S almost surely. Since S is closed in X by definition, it is sufficient to prove that μ(R ⊆ S) = 1. It is easy to show that μ(xn ∈ Sn ) = 1 for each n ∈ N (see Theorem 2.1 in Parthasarathy (1967)). Note that since the sequence {xn }n∈N is iid, S = Sn and, thus, μ(xn ∈ S) = 1 for every n ∈ N. Hence,
μ(R ⊆ S) = μ (∀n ∈ N : xn ∈ S) = μ (xn ∈ S) = 1. n∈N
Next, I show that clX (R) ⊇ S almost surely. Define V := V ∈ 2X \ FX V ∩ S = ∅ and consider any open set V ∈ V . By definition of S, it must be that μ (xn ∈ V ) > 0 for all n ∈ N. Thus, since the sequence {xn }n∈N is iid, there exists a number p ∈ [0, 1) such that μ (xn ∈ / V)= p for all n ∈ N. Thus,
μ (V ∩ R = ∅) = μ (xn ∈ / V)= p = lim p n = 0. n→∞
n∈N
n∈N
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Since X is a separable metric space, it is second-countable. This means that there exists a count able basis of open sets W C . Define W := W ∈ W C W ∩ S = ∅ . W is clearly countable and satisfies W ⊆ V . Hence, μ(W ∩ R = ∅) = 0 for every W ∈ W . Moreover, since each V ∈ V is the union of the open sets in W C contained in V , the following equality holds: {ω ∈ |V ∩ R(ω) = ∅} = {ω ∈ |W ∩ R(ω) = ∅} . V ∈V
W ∈W
Therefore, μ clX (R) S = μ (S \ clX (R) = ∅) = μ ∃V ∈ 2X \ FX : V ∩ S = ∅ ∧ V ∩ R = ∅ , ⎛ ⎞ {ω ∈ |V ∩ R(ω) = ∅}⎠ , = μ (∃V ∈ V : V ∩ R = ∅) = μ ⎝ ⎛ = μ⎝
V ∈V
⎞
{ω ∈ |W ∩ R(ω) = ∅}⎠ ≤
W ∈W
μ (W ∩ R = ∅) = 0,
W ∈W
where the inequality is a consequence of the countable subadditivity of μ. It follows that μ (clX (R) ⊇ S) = 1 − μ clX (R) S = 1 − 0 = 1. It is easy to see that μ (clX (R) ⊇ S) = μ (clX (R) ⊆ S) = 1 implies μ (clX (R) = S) = 1.
2
I now proceed to prove Proposition 7. c(ω))) ˆ = B . Since the Proof of Proposition 7. Consider the event 1 := ω ∈ clB (dom( ˆ = Bˆ n (ω)n ∈ N for every ω ∈ , support of Bˆ n is B for every n ∈ N and dom(c(ω)) Lemma 13 implies that μ(1 ) = 1. Consider also the event 2 := ω ∈ | c(ω) ˆ is consistent 2 ∗ 1 2 with . Since cˆ is consistent with , μ( ) = 1. Define := ∩ . Clearly, μ(∗ ) = 1. Select ω ∈ ∗ . By Lemma 9, cl(c(ω)) ˆ has full domain and cl(c(ω)) ˆ is consistent with . Since P is identified on B, P (cl(c(ω)), ˆ P) = {}. Combining this with Part (1) in Theorem 5 and the fact that all preferences in P are continuous by construction, I conclude that P (c(ω), ˆ P) = {}. Since μ(∗ ) = 1, it follows that P (c, ˆ P) = {} almost surely, as claimed. 2 B.4. Proofs of Section 5 Proof of Proposition 8. Let , be nontrivial vNM preferences, such that . Suppose, seeking a contradiction, that = . Then, without loss of generality, there should be z, y ∈ X, such that z y and z y. Completeness of implies y z. Moreover, and y z imply y z. It follows that y ∼ z and y z. Now, because is nontrivial, there must be x y. Since y ∼ z, it follows that either x y ∼ z or y ∼ z x. The treatment of these two cases is symmetric, so I focus exclusively on the first one. Suppose x y ∼ z. Because , it follows that x y and x z. Since y z, mixed transitivity implies that x z. In summary, thus far, I have x y ∼ z and x y z. Since is a vNM preference and x z, there exists a linear representation U , such that U (x) = 1 and U (z) = 0. Let α := U (y) ∈ (0, 1], and define p := αx + (1 − α)z. Clearly, p ∼ y by construction. Now define
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1 1 q := p + z = 2 2
1 1 α x + 1 − α z. 2 2
On the one hand, q z ∼ y, and so q y. On the other hand, y ∼ p q, and so y q. Thus, q y q. However, since , q y implies q y. This is inconsistent with y q, thus yielding the desired contradiction. I conclude that = , which means that the class of nontrivial vNM preferences is separated, as claimed. 2 Proof of Proposition 9. Let and be atomless continuous qualitative probabilities, such that . Suppose, seeking a contradiction, that = . This implies without loss of generality that there are events x, y ∈ X, such that y x and y x. By completeness, x y. Since , this implies x y. Thus, y ∼ x y. If x ∼ ∅, define x := \ y and y := \ x, which satisfy x ∼ y and x y by Axiom 11. Thus, assume without loss that x ∅. I now construct an increasing sequence of events {xn }n∈N ↑ x, such that x xn for all n ∈ N. Let x be the restriction of to Xx := {z ∩ x|z ∈ X}. Clearly, Xx is a σ -algebra and x is an atomless continuous qualitative probability on Xx . By Theorem 3.5 in Villegas (1964), there is a x -uniform random variable f : x → [0, 1]. This means that, for any two intervals I, J contained in [0, 1], f −1 (I ) x f −1 (J ) if and only if λ(I ) ≥ λ(J ), where λ is the Lebesgue measure. Define the sequence {xn }n∈N as xn := f
−1
!
n−1 ∪ {1} 0, n
Clearly, {xn } is increasing and satisfies lim xn =
n→∞
xn = f −1 ([0, 1]) = x.
n∈N
Since f is x -uniform, x x xn holds for all n ∈ N. Since xn ⊂ x, x x xn is equivalent to x xn . Moreover, since y ∼ x, transitivity implies y xn . Then, because , it follows that y xn ↑ x. Monotonic continuity therefore implies y x, which poses a contradiction. I conclude that = . This means that the class of atomless continuous qualitative probabilities is a separated theory, as claimed. 2 B.5. Proofs of Appendix A Proof of Theorem 7. Let P be separated and let B be weakly P-rich. Suppose, seeking a contradiction, that P is not identified under full observability on B. Then, by definition, there ∗ (B) = c∗ (B) for all B ∈ B. Since P is separe distinct preferences , ∈ P such that c arated and = , there must be x, y ∈ X such that x y x. Moreover, since B is weakly P-rich and x y, there exist B1 , ..., BN ∈ B, z1 , ..., zN+1 ∈ X such that z1 = x, zN+1 = y, ∗ (B ) × B for all n ∈ {1, ..., N }. The hypothesis that c∗ (B) = c∗ (B) for all and (zn , zn+1 ) ∈ c n n ∗ (B ) × B for all n ∈ {1, ..., N } and, so, z z B ∈ B implies that (zn , zn+1 ) ∈ c n n n n+1 for all
n ∈ {1, ..., N}. Finally, since is transitive, it follows that x = z1 zN+1 = y, contradicting y x. 2
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