aa r X i v : . [ ec on . T H ] J u l PREFERENCE IDENTIFICATION
CHRISTOPHER P. CHAMBERS, FEDERICO ECHENIQUE,AND NICOLAS S. LAMBERT
Abstract.
An experimenter seeks to learn a subject’s preference relation.The experimenter produces pairs of alternatives. For each pair, the subjectis asked to choose. We argue that, in general, large but finite data do notgive close approximations of the subject’s preference, even when the limiting(countably infinite) data are enough to infer the preference perfectly. Weprovide sufficient conditions on the set of alternatives, preferences, andsequences of pairs so that the observation of finitely many choices allowsthe experimenter to learn the subject’s preference with arbitrary precision.While preferences can be identified under our sufficient conditions, we showthat it is harder to identify utility functions. We illustrate our resultswith several examples, including consumer choice, expected utility, andpreferences in the Anscombe-Aumann model. Introduction
Consider a subject who forms a preference over the objects, or alternatives,in some set X . The subject participates in an experiment in which he ispresented with a sequence of pairs of alternatives. For each pair, the subject isasked to choose one of the two alternatives offered. What can an experimenterlearn about the subject’s preference from observing these binary comparisons?Suppose that, after every observation, the experimenter computes an estimate (Chambers) Department of Economics, Georgetown University (Echenique)
Division of the Humanities and Social Sciences, California Insti-tute of Technology (Lambert)
Graduate School of Business, Stanford University
Date : First version: April 10, 2017; This version: January 12, 2018.We thank seminar audiences at Boston College, Boston University, Brown, Essex, LSE,Maryland, Johns Hopkins, George Mason, Ohio State, UCL, and audiences of conferencesand workshops at Paris School of Economics, UPenn, Warwick and University of York.Echenique thanks the National Science Foundation for its support through the grants SES1558757 and CNS 1518941. Lambert gratefully acknowledges the financial support andhospitality of Microsoft Research New York and the Yale University Cowles Foundation. of the subject’s preference consistent with the data observed up that point:the experimenter chooses a preference rationalizing the choices made by thesubject. Is the estimate a good approximation of the subject’s underlyingpreference, for a large but finite experiment?In this paper, we investigate the asymptotic behavior of preference estimatesfrom finite experiments. We ask if one can fully identify the preference of asubject at the limit with finite data. It is a question of preference identification,in the classical sense of the term. To illustrate the key issues, consider the following example. Let X ⊆ R n represent a set of consumption bundles. The subject has a preference, denotedby (cid:23) ∗ , over the elements of X . Over time, the subject is asked to choose analternative from sets B k = { x k , y k } , where k is the time index. Together, thesets B , B , . . . , B k form a finite experiment. The experimenter observes thesubject’s choice of bundle for every pair. Assume the choice is consistent withthe subject’s preference, so that if x is chosen over y , then x (cid:23) ∗ y . Note thatwe can only, at best, infer the preference of the subject on the set B ≡ ∪ ∞ k =1 B k .Thus, if the subject’s preference behaves very differently outside of the set B ,there is no hope to obtain a fine approximation of the subject’s preference overthe entire set X . Two natural conditions emerge. First, we require that (cid:23) ∗ be continuous, so one can hope to approximate the preference from finitelymany samples. Second, we require that the set B is dense in X , so that theobservations are sufficiently spread out. And indeed, we show that, under theseconditions, if one can observe the preference of the subject over the whole set B , then one can infer precisely (cid:23) ∗ on X .The two conditions, continuity of (cid:23) ∗ and denseness of B , are, however, not enough to provide good approximations of (cid:23) ∗ from finitely many observations.Knowledge of the preference over the infinite set B allows the experimenter Standard decision-theoretic language reserves the term identified for a relation betweenpreference and utility. In that context, a model is identified if every preference relation isrepresented by a unique (up to some class of transformations) set of parameters. Thus,identification in this sense requires the knowledge of an entire preference relation. In thispaper, we do not assume knowledge of the entire preference relation. Instead, we ask if onecan learn the entire preference relation with a possibly large, but nonetheless finite data set.We discuss the identification of utility functions in Section 3.4, and the relation to decisiontheory in Section 4.2.
REFERENCE IDENTIFICATION 3 to exploit the continuity assumption on the subject’s preference. With finitedata, continuity does not have enough bite. To illustrate, take X = [0 , X is captured by the binaryrelation ≥ (greater numbers are always preferred). Consider the countableset of objects B = Q ∩ (0 , B , B , . . . an enumeration of pairs ofobjects of B . Then, any continuous preference that agrees with ≥ on Q has 1weakly preferred to 0. However, for any n , one can find a preference (cid:23) n thatrationalizes the choices of the subject over B , . . . , B n , and yet that ranks 0strictly above 1.In fact, one can come up with an even more startling example: we show thatno matter the subject’s preference, the experimenter may end up inferring thatthe subject is indifferent among all alternatives (see Section 3.1). And yet, as inthe example just described, she would be able to infer the subject’s preferenceperfectly, had she access to the subject’s preference over the infinite set B allat once. The example exhibits a kind of discontinuity. With infinite data inthe form of B , we must conclude that x ≻ y , but any finite data cannot ruleout that y (cid:23) x .These examples illustrate the dangers of data-driven estimation. Non-parametric estimation with finite data can behave very differently from es-timation with infinite (even countable) data. To derive meaningful estimates,one must construct a theory that disciplines the preferences, and lays downthe proper conditions for convergence of preference estimates.This paper includes three sets of results.Our first and foremost results concern non-parametric estimation. We offerfairly general conditions so that observing sufficiently many binary choicesallows one to approximate the subject’s preference arbitrarily closely with anypreference that rationalizes the finite data .We provide two notions of rationalization, a weak and a strong one. Understrong rationalization, a rationalizing preference must reflect choices perfectly.So if one alternative is chosen over another, the preference must rank the firststrictly above the second. Under weak rationalization, the first alternativemust only be ranked at least as good as the second. Weak alternatives reflect CHAMBERS, ECHENIQUE, AND LAMBERT the phenomenon of partial observability (Chambers et al., 2014) whereby onecannot infer anything from a choice that was not made.Under both notions of rationalization, it is necessary to add structure on theenvironment and on the rationalizing preferences so as to avoid the negativeresults of the example above. Importantly, we need a notion of objective ratio-nality expressed by the monotonicity of preferences. We postulate an exoge-nous partial ordering of the set of alternatives—for example, standard vectordominance when the set of alternatives represents consumption bundles, orstochastic dominance when it is the set of lotteries over monetary amounts—and we require that the subject’s preference is monotonic with respect to thatexogenous order.With this structure, finite-experiment rationalizable preferences converge tothe subject’s underlying preference, under conditions that are consistent withmany applications in decision theory, and with experimental implementationsof decision theoretic models. Stronger conditions are needed to obtain theresult for weak rationalization—conditions that hold for preferences over Eu-clidean spaces, but rule out some common applications in decision theory—yetit is remarkable that convergence is at all attained under weak rationalization.By remaining agnostic about choices that were not made, we are inferring alot less about the subject’s preferences under the assumption of weak ratio-nalization than under strong.Our results on preference identification are relevant to a wide range of con-texts. For concreteness, we illustrate their application to the special case ofpreferences over lotteries, dated rewards, consumption bundles, and Anscombe-Aumann acts (Anscombe and Aumann, 1963). In all these cases, there isa natural objective partial order and monotonicity is a sensible assumption.There are other environments in which one cannot reasonably impose any kindof monotonicity. For instance, in the literature on discrete allocation (such asin Hylland and Zeckhauser (1979) or in the recent literature on school choice,such as in Pathak and Sethuraman (2011)) in which agents are assumed tochoose among lotteries over finitely many heterogeneous objects. Monotonic-ity would require that all agents agree on a ranking of the discrete objects thatare being allocated, an unreasonable requirement.
REFERENCE IDENTIFICATION 5
Our second set of results concerns the identification of utility functions.Given a utility representation for the agent’s preference, we show that it ispossible to carefully select finite-data utility rationalizations so as to approx-imate the subject’s utility arbitrarily closely. This result also rests on mono-tonicity assumptions. However, there is a clear difference between estimatingpreferences and utilities. While any preference estimate converges to the trueunderlying preference, for utilities we only know that a certain selection con-verges. This observation is especially relevant when estimating utilities of aparticular functional form. There is no guarantee that such utility estimateshave the correct asymptotic behavior; one can only say that the preferencesthat these utilities represent do.Our third and final results concern the identification of preferences with in-finite but countable data. We show that, when the experimenter has accessto the preference of the subject over all alternatives of a countable set, thenit is possible to recover perfectly the subject’s preference over the entire set ofalternatives X under much weaker conditions than above. We further demon-strate that, under such conditions, the experimenter can, in theory, obtainthe subject’s preference directly from the observation of a single choice of thesubject when the subject is asked to select an object among a large, infiniteset.The remainder of the paper proceeds as follows. After reviewing the liter-ature, we describe the model in Section 2. We present our main results forimportant cases of collections of alternatives in Section 3 and discuss theseresults in Section 4. In Section 5, we present our main results for general col-lections of alternatives. In Section 6, we study the relation between preferenceand utility, and provide conditions under which the identification of a prefer-ence makes it possible to identify a utility, and conversely. In Section 7, wefurther show that when the set of possible utilities is compact, one can obtaina strong form of identification, which dispenses with the postulate of existenceof a data-generating preference. We discuss the question of preference iden-tification with infinite but countable data in Section 8. Finally, in Section 9,we offer interpretations on the meaning of a data-generating preference. We CHAMBERS, ECHENIQUE, AND LAMBERT relegate the proofs and more technical results to the appendices (some of theseresults may be of independent interest).
Literature Review.
Experimentalists and decision theorists have an obviousinterest in preference estimation, but we are not aware of any study of thebehavior of preference estimates from finite experiments. The long traditionof revealed preference theory from finite data that goes back to Afriat (1967)is focused on testing, not estimation. The closest work to ours is Mas-Colell(1978), who works with finitely many observations from a demand functionover a finite number of goods. Mas-Colell assumes a rational demand functionthat satisfies a boundary condition and is “income Lipschitzian.” He assumesa sufficiently rich sequence of observations, taken from an increasing sequenceof budgets. He then shows that the sequence of rationalizing preferences, eachrationalizing a finite (but increasing) set of observations, converges to theunique preference that rationalizes the demand function.There are many differences between Mas-Colell’s exercise and ours, even ifone restricts attention to choice over bundles of finitely-many, divisible, con-sumption goods. In particular, the difference in model primitives—demandinstead of binary comparisons—is crucial. One cannot generally use choicefrom linear budgets to recreate any given binary comparison. Moreover, thereis no property analogous to the boundary and Lipschitz continuity of demandin our framework. Indeed, as shown in Mas-Colell (1977), by means of anexample due to Lloyd Shapley, without these properties, preferences are notidentified from demand. In Mas-Colell’s paper, weak and strong rationaliz-ability coincide, as he works with demand functions. In this paper, we areparticularly interested in partial observability.Also working with demand functions, the recent stream of literature by Reny(2015), K¨ubler and Polemarchakis (2017) and Polemarchakis et al. (2017) pro-vide results on the limiting behavior of finite-data utility rationalizations.These papers focus on the convergence of certain utility constructions thatrationalize finite demand data. In contrast, our main results are about the Shapley’s example also appears in Rader (1972). The example poses no problem for identi-fication in our framework of binary comparisons. It generates non-identification of demandbecause two preferences have the same marginal rate of substitution at the sampled points.With binary comparisons, the differences between two such preferences are detected.
REFERENCE IDENTIFICATION 7 convergence of (any) rationalizing preferences. There are also important differ-ences between the primitives assumed in our paper and the demand functionsassumed in these papers.The recent paper by Gorno (2017) also looks at the identification of pref-erences from abstract choice behavior. A clear difference between Gorno’sexercise and ours is that we consider the limiting behavior of large, but fi-nite, experiments. His paper deals with preference identification from a givenchoice behavior on a fixed choice set. While the two papers are concernedwith related questions, the exercises are quite different and the results are notrelated.Finally, on the technical level, we use the topology on preferences introducedby Hildenbrand (1970) and Kannai (1970), building on the work of Debreu(1954). In our study of the mapping from utility to preference, we borrow ideasfrom Mas-Colell (1974) and Border and Segal (1994). In particular, the proofof the continuity of the “certainty equivalent” representation is analogous toMas-Colell’s, and we take the notion of local strictness from Border and Segal,as well as their continuity result. 2.
Model
In this section, we introduce the definitions and conventions used throughoutthe paper, and present our main model. Our focus in this section is on twoclassical environments; namely, consumption space and Anscombe-Aumannacts.2.1.
Basic definitions and notational conventions.
Let X i be a set par-tially ordered by ≥ i , for i = 1 , . . . , n . If x, y ∈ Π ni =1 X i , then x ≥ y means that x i ≥ i y i for i = 1 , . . . , n ; and x > y that x ≥ y and x = y . We write x ≫ y when x i > i y i for i = 1 , . . . , n . The order ≥ on Π ni =1 X i is called the productorder . The interval [ a, b ] in Π ni =1 X i denotes the set { z ∈ Π ni =1 X i : b ≥ z ≥ a } .An open interval ( a, b ) denotes the set { z ∈ Π ni =1 X i : b ≫ z ≫ a } . When X i = R , and ≥ i is the usual order on the real numbers, the above definitions A partial order is reflexive, transitive, and anti-symmetric, while a strict partial order isirreflexive, transitive, and asymmetric. Then ≥ is a partial order, and each of > and ≫ are strict partial orders. CHAMBERS, ECHENIQUE, AND LAMBERT constitute the familiar ordering on R n , as well as the usual notions of intervalsand open intervals.If A ⊆ R is a Borel set, we write ∆( A ) for the set of all Borel probabilitymeasures on A . We endow ∆( A ) with the weak* topology. For x, y ∈ ∆( A ), wewrite x ≥ F OSD y when x is larger than on y in the sense of first order stochas-tic dominance (meaning that R A f dx ≥ R A f dy for all monotone increasing,continuous and bounded functions f on A ). When Ω is a finite set, we shalluse the above definitions to order ∆( A ) Ω by the product order defined fromordering ∆( A ) by first-order stochastic dominance.For an integer n , [ n ] denotes the set { , . . . , n } . So ∆([ n ]) = { x ∈ R n + : P ni =1 x i = 1 } is the simplex in R n .2.1.1. Preference relations.
Let X be a set. Given a binary relation R ⊆ X × X , we write x R y when ( x, y ) ∈ R . And we say that a function u : X → R represents R if x R y iff u ( x ) ≥ u ( y ). A preference , or preference relation , is aweak order; i.e. a binary relation over X which is complete and transitive..For a partial order ≥ on X , a preference (cid:23) on X is weakly monotone (withrespect to ≥ ) if x ≥ y implies that x (cid:23) y . For a strict partial order > on X ,a preference (cid:23) on X is strictly monotone (with respect to > ) if x > y impliesthat x ≻ y . A binary relation R is continuous if R ⊆ X × X is closed (see, for example,Bergstrom et al., 1976).The set of continuous binary relations over X , when X is a topological space,is endowed with the topology of closed convergence , we provide a definition inSection 5.1. It is the natural topology for our purposes because it is the weakesttopology for which optimal behavior is continuous.Under the assumptions of our paper, the topology of closed convergence isthe smallest topology for which the sets { ( x, y, (cid:23) ) : x ≻ y } are open (see Kannai, 1970, Theorem 3.1). So it is the weakest topology withthe following property: Suppose that a subject with preferences (cid:23) chooses x The strict part of a partial order is a strict partial order, but sometimes we are interestedin other partial orders. For example, ≫ is not the strict part of ≥ , but we use it later. REFERENCE IDENTIFICATION 9 over y ( x ≻ y ), and that another subject with sufficiently close preferences (cid:23) ′ face a choice between x ′ and y ′ , where ( x ′ , y ′ ) is sufficiently close to ( x, y ),then the second subject must choose x ′ over y ′ ( x ′ ≻ ′ y ′ ). In other words, theoptimal behavior according to (cid:23) ′ approximates the optimal behavior accordingto (cid:23) . In this sense, the topology that we impose on preferences is natural inany investigation of optimal choice.2.1.2. Choice functions.
A pair (Σ , c ) is a choice function if Σ ⊆ X \ { ∅ } isa collection of nonempty subsets of X , and c : Σ → X with ∅ = c ( A ) ⊆ A for all A ∈ Σ. When Σ, the domain of c , is implied, we refer to c as a choicefunction.A choice function (Σ , c ) is generated by a preference relation (cid:23) if c ( A ) = { x ∈ A : x (cid:23) y for all y ∈ B } , for all A ∈ Σ.The notation (Σ , c (cid:23) ) means that the choice function (Σ , c (cid:23) ) is generated bythe preference relation (cid:23) on X .2.2. The model.
There is an experimenter (a female) and a subject (a male).The subject chooses among alternatives in a set X of possible alternatives.The subjects’ choices are guided by a preference (cid:23) ∗ over X , which we refer toas data-generating preference. The experimenter seeks to infer (cid:23) ∗ from thesubject’s choices in a finite experiment.In a finite experiment, the subject is presented with finitely many unorderedpairs of alternatives B k = { x k , y k } in X . For every pair B k , the subject is askedto choose one of the two alternatives: x k or y k .A sequence of experiments is a collection Σ ∞ = { B i } i ∈ N of pairs of possiblechoices presented to the subject. Let Σ k = { B , . . . , B k } , and let B = ∪ ∞ k =1 B k be the set of all alternatives that are used over all the experiments.We make two assumptions on Σ ∞ . The first is that B is dense in X . Thesecond is that, for any x, y ∈ B there is k for which B k = { x, y } . The firstassumption is obviously needed to obtain any general identification result (see Section 4.5). The second assumption means that the experimenter is able toelicit the subject’s choices over all pairs used in her experiment. The data and its rationalizations.
For each k , the subject’s preference (cid:23) ∗ generates a choice function (Σ k , c (cid:23) ∗ ). Thus the choice behavior observedby the experimenter is always consistent with (Σ k , c (cid:23) ∗ ). We term (Σ k , c (cid:23) ∗ ) the choice function of order k generated by (cid:23) ∗ , and we term the choice function(Σ ∞ , c (cid:23) ∗ ) the choice sequence of order k generated by (cid:23) ∗ .Sometimes, we may not be able to observe the subject’s entire choice func-tion. In the spirit of Afriat (1967), we want to allow for the possibility that thesubject may in principle be willing to choose x , but does not actually chooseit. In the language of Chambers et al. (2014), we want to study the concept of partial observability . To this end, a general choice function (Σ ∞ , c ) is termeda choice sequence and this induces, for every k , a choice function on Σ k .For a choice function c and a preference (cid:23) ∗ , we use the notation c ⊑ c (cid:23) ∗ to mean that for each budget B k , c ( B k ) ⊆ c (cid:23) ∗ ( B k ); i.e. , the observed choicesfrom B k are optimal for (cid:23) ∗ .In the context of partial observability, the notion of rationalization needs toaccommodate the fact that some preference maximal alternatives may not haveactually been chosen. The next concept captures such an accommodation; andis again in the spirit of Afriat (1967).A preference (cid:23) k weakly rationalizes (Σ k , c ) if, for all B i ∈ Σ k , c ( B i ) ⊆ c (cid:23) k ( B i ). A preference (cid:23) k weakly rationalizes a choice sequence (Σ ∞ , c ) if itrationalizes the choice function of order k (Σ k , c ), for all k ≥ (cid:23) k strongly rationalizes (Σ k , c ) if, for all B i ∈ Σ k , c ( B i ) = c (cid:23) k ( B i ). A preference (cid:23) k strongly rationalizes a choice sequence (Σ ∞ , c ) if itrationalizes the choice function of order k (Σ k , c ), for all k ≥ If there is a countable dense A ⊆ X , then one can always construct such a sequence ofexperiments via a standard diagonalization argument. REFERENCE IDENTIFICATION 11 Results
Motivation.
Many results on identification in economics presume accessto rich information. In decision theory, the presumption is that one can observeenough of the subject’s choices so as to effectively know the subject’s preference (cid:23) ∗ . In this section, we point to some problems with this assumption.Let (cid:23) I = X × X denote the degenerate preference relation that regards anytwo alternatives as indifferent. Proposition 1.
Let X = [ a, b ] ⊆ R n , where a ≪ b . Let the subject’s prefer-ence (cid:23) ∗ be continuous. There exists a continuous preference (cid:23) k that stronglyrationalizes the choice function of order k generated by (cid:23) ∗ , and such that (cid:23) k →(cid:23) I . The proof of Proposition 1 is relegated to Appendix B.Proposition 1 means that, absent further conditions, the sequence of ra-tionalizations can be very different from the preference (cid:23) ∗ generating thesubject’s choices. It is possible to choose rationalizations that converge tofull indifference among all alternatives, regardless of which (cid:23) ∗ really gener-ated the subject’s choices. The objective of our paper is to show how suchproblems can be avoided.Proposition 1 suggests another distinction. There is “infinite data” in theform of the data-generating preference (cid:23) ∗ , such data is commonly assumedin decision theory; there is finite data, in the form of (Σ k , c ∗(cid:23) ); and, “limitingdata,” which would be (cid:23) ∗ | B , i.e., the preference (cid:23) ∗ restricted to domain B . With limiting data one would be able to identify (cid:23) ∗ . Indeed, we show inSection 3.5 that if (cid:23) | B = (cid:23) ∗ | B then (cid:23) = (cid:23) ∗ .Therefore, Proposition 1 illustrates a sort of discontinuity. If one only hadaccess to limiting data, there would be no problem. However, with arbitrarilylarge, but finite, data, preference rationalizations can be completely wrong.3.2. Weak rationalizations.
We now present a series of simple sufficientconditions ensuring convergence of preference rationalizations to the subject’spreference. The results are discussed in Section 4.
Let X = R n + . Recall that the strict partial order ≫ on R n + refers to therelation x ≫ y if for each i , x i > y i ( i.e. the product of > ); strict monotonicityrefers to this relation. Theorem 2.
Let the subject’s preference (cid:23) ∗ be continuous and strictly mono-tone. Suppose that c ⊑ c (cid:23) ∗ . For each k ∈ N , let (cid:23) k be a continuous andstrictly monotone preference that weakly rationalizes (Σ k , c ) . Then, (cid:23) k →(cid:23) ∗ . Theorem 2 can be generalized, as (cid:23) ∗ and (cid:23) k do not need to be transitive.They only need to be continuous, strictly monotone, and complete.3.3. Strong rationalization.
Suppose that X is either(1) R n + ,(2) or ∆([ a, b ]) Ω for a finite set Ω and [ a, b ] ⊆ R .Recall that we topologize ∆([ a, b ]) with its weak* topology, and ∆([ a, b ]) Ω with the product topology. In the case of R n + , the relation ≥ refers to theproduct of ≥ on each of the coordinates, and on ∆([ a, b ]) Ω , it is the productof ≥ F OSD . Theorem 3.
Let the subject’s preference (cid:23) ∗ be weakly monotone. For each k ∈ N , let (cid:23) k be a continuous and weakly monotone preference that stronglyrationalizes the choice function of order k generated by (cid:23) ∗ . Then, (cid:23) k →(cid:23) ∗ . Utility functions.
Let X be either of the sets in Section 3.3. In the caseof ∆([ a, b ]) Ω , > references the strict part of ≥ as defined previously.Denote by R mon the set of preferences that are strictly monotone and con-tinuous, and by U the set of strictly increasing and continuous utility functionson X . The set U is endowed with the topology of uniform convergence on com-pacta.Let Φ be the function that carries each utility function in U into the pref-erence relation that it represents. So Φ : U → R mon is such that x Φ( u ) y ifand only if u ( x ) ≥ u ( y ).We regard two utility functions as equivalent if they represent the samepreference: if they are ordinally equivalent. Define an equivalence relation ≃ on U by u ≃ v if there exists ϕ : R → R strictly increasing for which u = ϕ ◦ v .Then let U / ≃ denote the set of equivalence classes of U under ≃ endowed with REFERENCE IDENTIFICATION 13 the quotient topology. The function ˆΦ : U / ≃→ R mon maps an equivalenceclass into Φ( u ), for any u member of the equivalence class. Theorem 4. ˆΦ is a homeomorphism. Theorem 4 implies that a utility representation may be chosen from a finite-experiment rationalization so as to approximate a given utility representationfor the preference generating the choices.The following Proposition adds some structure to Theorem 4. It claims alower hemicontinuity result for Φ − , in the sense that for any utility represen-tation of a strictly monotone preference, a convergent sequence of preferencespossesses a convergent sequence of utility representations. Proposition 5.
Let the subject’s preference (cid:23) ∗ be strictly monotone and con-tinuous. Let (cid:23) k be a continuous and strictly monotone preference that stronglyrationalizes the choice function of order k generated by (cid:23) ∗ . Then, for any util-ity representation u ∗ of (cid:23) ∗ , there exist utility representations u k of (cid:23) k suchthat u k → u ∗ . Limiting data.
Let X be as in Section 3.3, and let B be the dense setof alternatives used over all experiments. Theorem 6.
Suppose that (cid:23) and (cid:23) ∗ are two continuous preference relations.If (cid:23) | B × B = (cid:23) ∗ | B × B , then (cid:23) = (cid:23) ∗ . As we discussed in Section 3.1, the case of limiting data serves to illustratethe difference between a sequence of finite experiment and the limit of a count-ably infinite dataset. Theorem 6 means that one can obtain identification of (cid:23) ∗ solely from the continuity assumption (we refer the reader to Section 8 formore details). 4. Discussion
Positive results and assumptions on X . The previous section as-sumes that X is either R n + or ∆([ a, b ]) Ω , for some finite set Ω. The pointof Theorems 2 and 3 is to provide a positive result in response to the con-cerns raised in Section 3.1. The results cover some of the most widely used choice spaces in economics: R n + is consumption space in demand theory, and∆([ a, b ]) Ω is a space of Anscombe-Aumann acts over monetary lotteries.For the Anscombe-Aumann interpretation, let Ω be a finite nonempty setof states of the world , and interpret [ a, b ] as a set of monetary payoffs. Theelements of ∆([ a, b ]) are lotteries of monetary payoffs. An Anscombe-Aumannact is a state-contingent monetary lottery; it maps elements from Ω to ∆([ a, b ]).The set of alternatives ∆([ a, b ]) Ω is then the set all Anscombe-Aumann acts.The spaces R n + and ∆([ a, b ]) Ω have in common that there is an objectivenotion of monotonicity that preferences can be made to conform to. Otherspaces share this property. Section 5 includes our most general results.4.2. Identification of utility functions.
Many results on identification indecision theory can be phrased in the following terms. There are subsets U ′ ⊆ U and R ′ ⊆ R mon, and an equivalence relation ≃ ′ on U ′ such that Φ isa bijection from U ′ / ≃ ′ onto R ′ . The idea is that, with data in the form of (cid:23) ∗ , one can uniquely “back out” an equivalence class from U ′ .Our results suggest that this is not enough when data is finite. First, oneneeds to ensure that rationalizations obtained from finite data converge tothe underlying (cid:23) ∗ . Second, the space of preferences and utilities have to behomeomorphic in order to be able to obtain a limiting utility function from alarge, but finite, dataset on choices.4.3. Partial observability.
The distinction between weak and strong ratio-nalizability is important. In fact, it is rather surprising that one can obtain aresult such as Theorem 2 for weak rationalizations.A choice sequence generated by the subject’s preference reflects both strictcomparisons as well as indifferences. In practice, however, the experimentermay not be able to properly infer the indifference of the subject regardingtwo alternatives. The difficulty arises, for example, when the experimenteroffers the subject his preferred alternative. In this case, the experimenterwould typically require that the subject selects only one of the two alternativespresented to him. Such situations, in which the experimenter cannot committo being able to see all potentially chosen elements, are referred to partial
REFERENCE IDENTIFICATION 15 observability (Chambers et al., 2014), in contrast to full observability in whichthe experimenter is able to elicit the subject’s indifference between alternatives.Weak rationalizability expresses the idea that the experimenter is not willingto commit to interpreting observed choices as the only potential choices madeby the subject. For example, if the experimenter observes that the subjectchooses x when presented the pair { x, y } , she may not be willing to infer that x ≻ ∗ y , as it may be that x ∼ ∗ y but the subject simply did not choose y . This notion of weak rationalization is used, for example, by Afriat (1967)in the context of consumer theory (for more details on this notion, see, forexample, Chambers and Echenique (2016)). Weak rationalizability is partially agnostic with respect to the status ofunchosen alternatives, so it is surprising that one can ensure convergence ofpreference rationalizations to the preference that generated the choices.4.4.
Monotone Preferences.
The problem exemplified by Proposition 1 isthat one cannot hope to obtain convergence to (cid:23) ∗ if there is no disciplineplaced on the rationalizing preferences. In a sense, we need to constrain, orstructure, the theory from which rationalizations are drawn. Our results showthat a notion of objective monotonicity is enough to ensure that rationalizingpreferences in the limit approach the subject’s preferences.As discussed in the introduction, and exemplified by Proposition 1, thecontinuity assumption on the subject’s preference, and the assumption that thealternatives offered are in the limit dense, do not generally ensure convergenceto the subject’s preference. Proposition 1 shows that the failure of convergencecan be rather dramatic. We must impose structure on the subject’s preference,and on the finite-experiment rationalizations.Observe that the preferences (cid:23) k constructed in Proposition 1 cannot bemonotone. Suppose that (cid:23) ∗ is a continuous preference relation, and supposethat x ≻ ∗ y . In the construction in Proposition 1 we obtain a sequence ofrationalizations (cid:23) k such that in the limit y is at least as good as x . This cannothappen if each rationalizing preference is weakly monotone: x ≻ ∗ y implies Analogously, the hypothesis of full observability, related to what we call strong rationaliza-tion, is the notion employed by Richter (1966). A recent work showing how to obtain bothtypes of conditions as a special case is Nishimura et al. (2017). that x ′ ≻ ∗ y ′ for ( x ′ , y ′ ) close enough to ( x, y ). Thanks to the interaction ofthe order and the topology on R n we can find a k large enough such that thereare { x ′′ , y ′′ } ∈ Σ k (meaning alternatives offered in the k th finite experiment)with x ′ ≥ x ′′ and y ′′ ≥ y ′ , and where ( x ′′ , y ′′ ) is also close to ( x, y ). If (cid:23) k ismonotone then we have x ′ (cid:23) k x ′′ and y ′′ (cid:23) y ′ . But if (cid:23) k strongly rationalizesthe choices made at the k th experiment, then x ′′ ≻ k y ′′ . So we have to have x ′ ≻ k y ′ for any ( x ′ , y ′ ) close enough to ( x, y ).4.5. On the denseness of B . We assume that B , the set of all alterna-tives used in a sequence of experiments, is dense. Our paper deals with fullynonparametric identification, so it seems impossible to obtain a general resultwithout assuming denseness of B : imagine that the experimenter leaves anopen set of alternatives outside of her experimental design. Then the subject’spreferences over alternatives in that set would be very hard to gauge.In practice, one can imagine restricting attention to smaller class of fami-lies for which one does not need to elicit choices over a set that is dense in X . For example for expected utility preferences over lotteries, or homotheticpreferences in R n , one is only trying to infer a single indifference curve. So asmaller set of choices is enough: but even in that case one would need the setof alternatives in the limit to be dense in the smaller set of choices.4.6. Preference identification and utility identification.
The theoremssay, roughly speaking, that, if we assume data generated by a (well behaved)preference (cid:23) ∗ , then any “finite sample rationalization” (cid:23) k is guaranteed toconverge to the generating preference. So estimates have the correct “largesample” properties. In particular, one may be interested in a specific theoryof choice, such as max-min or Choquet expected utility. If the subject’s (cid:23) ∗ is max-min, or Choquet, one can choose rationalizing preferences to conformto the theory, and the limit will uniquely identify the subject’s max-min, orChoquet, preference. But if one incorrectly uses rationalizing preferences out-side of the theory, the asymptotic behavior will still correct the problem, anduniquely identify (cid:23) ∗ in the limit. The theorems also say that there are certainutility representations u k that will be correct asymptotically. REFERENCE IDENTIFICATION 17
Note, however, what the theorems do not say. First, the estimates (cid:23) k areguaranteed to converge to the generating preferences (cid:23) ∗ , when the generatingpreference is known to exist. If one simply estimates the preferences (cid:23) k , thesemay fail to converge to a well-behaved preference. We present two examples tothis effect in Section 9. That said, under certain conditions (that unfortunatelyare not satisfied in the Anscombe-Aumann setting), the “size” of the set ofrationalizing preferences shrinks as k growth; see Theorem 10.Second, our results do not say that one can choose u k arbitrarily. Anyestimated rationalizing preference will converge to the preferences rationalizingthe utility, but basing the estimation on utilities is more complicated becauseit is not clear that any utility representation of (cid:23) ∗ will have the right limit, oreven converge at all. 5. General Results
In Section 3, we have presented our main results for some important specialcases of the collection of alternatives X . In this section, we present our mainresults for the general case. We now assume that X is a Polish and locallycompact space, and provide conditions under which our convergence resultscontinue to hold. The conditions we provide are the weakest we know. Thesection concludes with applications to this general case. Note that this sectionfocuses on preference identification, general results for utility identification aregiven in Section 6.5.1. Convergence of Preferences.
To speak about the approximation of thesubject’s preference, one must introduce a notion of convergence on the spaceof preferences. We use closed convergence , and endow the space of preferencerelations with the associated topology. The use of closed convergence for pref-erence relations was initiated by the work of Kannai (1970) and Hildenbrand(1970), and has become standard since then.One primary reason to adopt closed convergence is to capture the propertythat agents with similar preferences should have similar choice behavior—aproperty that is necessary to be able to learn the preference from finite data.Specifically, under the assumptions we use for most of our results, the topology of closed convergence is the smallest topology for which the sets { ( x, y, (cid:23) ) : x ≻ y } are open (see Kannai (1970) Theorem 3.1). The desired continuity of choicebehavior is expressed by the fact that sets of the form { ( x, y, (cid:23) ) : x ≻ y } areopen. The topology of closed convergence being the smallest topology withthis property is a natural reason for adopting it.The following characterization of closed convergence for the context of pref-erence relations will be used throughout the paper: Lemma 7.
Let (cid:23) n be a sequence of preference relations, and let (cid:23) be a pref-erence relation. Then (cid:23) n →(cid:23) in the topology of closed convergence if and onlyif, for all x, y ∈ X , (1) x (cid:23) y implies that for any neighborhood V of ( x, y ) in X × X there is N such that for all n ≥ N , (cid:23) n ∩ V = ∅ ; (2) if, for any neighborhood V of ( x, y ) in X × X , and any N there is n ≥ N with (cid:23) n ∩ V = ∅ , then x (cid:23) y . The following lemma plays an important role in the approximation results.
Lemma 8.
The set of all continuous binary relations on X , endowed with thetopology of closed convergence, is a compact metrizable space.Proof. See Theorem 2 (Chapter B) of Hildenbrand (2015), or Corollary 3.95of Aliprantis and Border (2006). (cid:3)
In particular, we shall denote the metric which generates the closed conver-gence topology by δ C . Recall that X is metrizable. Let d be an associatedmetric. When X is compact, one can choose δ C to be the Hausdorff metric onsubsets of X × X induced by d . On the other hand, if X is only locally com-pact, then δ C may be chosen to coincide with the Hausdorff metric on subsetsof X ∞ × X ∞ , where X ∞ is the one-point compactification of X together withsome metric generating X ∞ . See Aliprantis and Border (2006) for details.5.2. Weak rationalizations.
We now present our results on the asymptoticbehavior of preference estimates based on finite data. The results generalizethose stated in Section 3.
REFERENCE IDENTIFICATION 19
For our first result, we must define two notions. We say that a preferencerelation (cid:23) is locally strict if for every x, y ∈ X with x (cid:23) y , and every neigh-borhood V of ( x, y ) in X × X there is ( x ′ , y ′ ) ∈ V with x ′ ≻ y ′ .The first main result gives conditions of convergence of preferences thatweakly rationalize the experimental observations. Note that Theorem 9 gen-eralizes Theorem 2. Theorem 9.
Suppose that (1) the subject’s preference (cid:23) ∗ is continuous and strictly monotone, (2) the strict partial order < is an open set, (3) every continuous and strictly monotone preference relation is locallystrict.Let c ⊑ c (cid:23) ∗ be a choice sequence, and let (cid:23) k be a continuous and strictlymonotone preference that weakly rationalizes c k . Then, (cid:23) k →(cid:23) ∗ in the closedconvergence topology. Note that the assumption that (cid:23) ∗ and (cid:23) k are transitive is not needed.Instead, each of these only needs to be continuous, strictly monotone, andcomplete.Note that Theorem 9 requires the existence of the data-generating preference (cid:23) ∗ . However, even if existence of this object is not supposed, we can still“bound” the set of rationalizations to an arbitrary degree of precision. This isthe point of the next result.For a choice sequence c , let P k ( c ) be the set of continuous and strictlymonotone preferences that weakly rationalize c k . For a set of binary relations S , define diam( S ) = sup ( (cid:23) , (cid:23) ′ ) ∈ S δ C ( (cid:23) , (cid:23) ′ ) to be the diameter of S accordingto the metric δ C which generates the topology on preferences. Theorem 10.
Suppose that < has open intervals. Let c be a choice sequence,and suppose that each strictly monotone continuous preference is also locallystrict. Then one of the following holds: (1) There is k such that P k ( c ) = ∅ . (2) lim k →∞ diam ( P k ( c )) → . That is, either a choice sequence is eventually not weakly rationalizable by astrictly monotone preference, or , the set of rationalizations becomes arbitrarilysmall.Note that, as for Theorem 9, Theorem 10 can dispense with the notion oftransitivity. In this case, we would define P k ( c ) to be the set of (potentiallynontransitive) complete, continuous, and strongly monotone relations weaklyrationalizing c k .5.3. Strong rationalizations.
Say that the set X , together with the col-lection of finite experiments Σ ∞ , has the countable order property if for each x ∈ X and each neighborhood V of x in X there is x ′ , x ′′ ∈ B ∩ V with x ′ ≤ x ≤ x ′′ . We say that X has the squeezing property if for any conver-gent sequence { x n } n in X , if x n → x ∗ then there is an increasing sequence { x ′ n } n , and an a decreasing sequence { x ′′ n } n , such that x ′ n ≤ x n ≤ x ′′ n , andlim n →∞ x ′ n = x ∗ = lim n →∞ x ′′ n . Theorem 11.
Suppose that (1) the subject’s preference (cid:23) ∗ is weakly monotone, (2) ( X, Σ ∞ ) has the countable order property, and X the squeezing prop-erty.Let (cid:23) k be a continuous and weakly monotone preference that strongly ratio-nalizes the choice function of order k generated by (cid:23) ∗ . Then, (cid:23) k →(cid:23) ∗ in theclosed convergence topology. The countable order and squeezing properties are technical but not vacu-ous. Importantly, as stated below in Proposition 12, they are satisfied fortwo common cases of interest discussed in Section 3.3. Therefore, Theorem 11generalizes Theorem 3.
Proposition 12.
If either (1) the set of alternatives X is R n endowed with the order of weak vectordominance, or (2) the set of alternatives X is ∆([ a, b ]) endowed with the order of weakfirst-order stochastic dominance, REFERENCE IDENTIFICATION 21 then X has the squeezing property, and there is Σ ∞ such that ( X, Σ ∞ ) has thecountable order property. One key element behind the above two results is a natural order on the setsof possible alternatives. Via monotonicity, the order adds structure to thefamilies of preferences under consideration. Crucially, the order also relates tothe topology on the set X .5.4. Applications.
We have already highlighted the application of our resultsto Euclidean consumption spaces and Anscombe-Aumann acts over monetarylotteries (see Section 3). Here we discuss two other domains of application ofour results.5.4.1.
Lotteries over a finite prize space.
Let Π be a finite prize space . Theobjects of choice are the elements of X = ∆(Π). Fix a strict ranking of theelements of Π, and enumerate the elements of Π so that π is ranked above π , which is ranked above π , and so on. Then the elements of X can beordered with respect to first-order stochastic dominance: x is larger than y inthis order if the probability of each set { π , . . . , π k } is at least as large under x than under y , for all k = 1 , . . . , | Π | . A preference over X is monotone if italways prefer larger lotteries over smaller ones. Suppose that choices are generated by an expected utility preference (cid:23) ∗ .The fact that (cid:23) ∗ belongs to the expected utility family implies that thereare rationalizing expected utility preference (cid:23) k , for each finite experiment k .Then, the above results ensure that these converge to (cid:23) ∗ . Of course the samewould be true of any (monotone and continuous) rationalizing preference: anymode mis-specification would be corrected in the limit. In other words, anyarbitrary sequence of rationalization has the data-generating preference (cid:23) ∗ asits limit.5.4.2. Dated rewards.
We can apply our theory to intertemporal choice. Specif-ically to the choice of dated rewards (Fishburn and Rubinstein (1982)). Theset of elements of choice is R . A point ( x, t ) ∈ R is interpreted as a mone-tary payment of x delivered on date t . Endow R with the order ≤ i whereby The objective order on Π is not really needed in this case; see Example 19. The point ofthe example is to illustrate Theorem 9. ( x, t ) ≤ i ( x ′ , t ′ ) if x ≤ x ′ and t ′ ≤ t . Monotonicity of preferences means thatmore money earlier is preferred to less money later.Now one can postulate a preference (cid:23) ∗ such that ( x ′ , t ′ ) (cid:23) ∗ ( x, t ) iff δ t v ( x ) ≤ δ t ′ v ( x ′ ), for some δ ∈ (0 ,
1) and a strictly increasing function v : R + → R .This means that (cid:23) ∗ follows the exponential discounting model. Again, anyfinite experiment would be rationalizable by exponential preference, and thesewould converge to the limiting (cid:23) ∗ .6. Identification of Utility Functions
In this section, we investigate the relation between preferences and util-ity. Preferences remain topologized with the closed convergence topology. Westudy continuous utility representations, and ask when the identification of apreference allows the identification of a utility (or conversely). We show that ifwe endow the set of continuous utility functions with the topology of uniformconvergence on compacta, then convergence in one sense is equivalent to con-vergence in the other. Formally, we establish that there is a homeomorphismbetween the two spaces (when we identify two utility functions representingthe same preference relation).Throughout this section, the space of possible alternatives X is connected(and remains a locally compact Polish space, as described in our model). Con-nectedness is imposed so that every continuous preference admits a continuousrepresentation, as in Debreu (1954).We denote by U the set of strictly increasing and continuous utility func-tions on X . Similarly, R mon denotes the set of preferences which are strictlymonotone and continuous.Suppose the existence of a set M ⊆ X , satisfying the following conditions: • M has at least two distinct elements; M is connected and totally or-dered by < . In other words x, y ∈ M and x = y implies x < y or y < x . • For any m ∈ M and any neighborhood U of m in X there is m, m ∈ M ,with m ∈ [ m, m ] ⊆ U. REFERENCE IDENTIFICATION 23
Moreover if m is not the largest element of M we can choose m suchthat m < m , and if m is not the smallest element we can choose m such that m < m . • Any bounded sequence in X is bounded by elements of M . That is,for any bounded sequence { x n } there are m and m and k large so that m ≤ x n ≤ m .Let Φ : U → R mon such that Φ( u ) is the preference represented by u ∈ U . We provide two examples below that demonstrate the property just men-tioned for the case of alternatives of the form X = ∆([ a, b ]) and X = ∆([ a, b ]) n . Example 13.
Let X = ∆([ a, b ]) be the set of Borel probability distributionson a real compact interval S = [ a, b ] ⊆ R . Endow X with the weak* topologyand let ≤ be first-order stochastic dominance. Observe that X is compact,metrizable, and separable (Theorems 15.11 and 15.12 of Aliprantis and Border(2006)). Observe also that X has the countable order property (see Lemma 28in Appendix C).Let < be the strict part of ≤ . Identify S with degenerate probability distri-butions, so that s ∈ S denotes the element of X that assigns probability 1 to { s } , say δ s . Let M = S . The relative topology on S coincides with the usualtopology, so S is connected. Note that a ≤ x ≤ b for any x ∈ X .Let m ∈ M and U be a neighborhood of m in X . For each x ∈ X , let F x be the cdf associated to x . Choose ε such that the ball B ε ( m ) (in the Levymetric) with center m and radius ε is contained in U . Let ε ′ < ε . Then if y ∈ [ m − ε ′ , m + ε ′ ] we have that F y ( s − ε ) − ε ≤ F m − ε ′ ( s − ε ) − ε < F m ( s ) if s − ε ≥ m − ε ′ F y ( s − ε ) − ε ≤ F m − ε ′ ( s − ε ) − ε = − ε < F m ( s ) if s − ε < m − ε ′ Similarly, F m ( s ) = 0 < F m + ε ( s + ε ) + ε ≤ F y ( s + ε ) + ε if s + ε ≤ m + ε ′ F m ( s ) < ε = F m + ε ′ ( s + ε ) + ε ≤ F y ( s + ε ) + ε if s + ε > m + ε ′ . That is, x Φ( u ) y if and only if u ( x ) ≥ u ( y ). These inequalities mean that y ∈ B ε ( m ). Thus [ m − ε ′ , m + ε ′ ] ⊆ U , as y wasarbitrary. Example 14.
Let Ω be a nonempty set such that | Ω | < + ∞ . Suppose Ωrepresents a set of states of the world . Then ∆([ a, b ]) Ω , endowed with theproduct weak* topology, and ordered by the product order, of Ω copies of firstorder stochastic dominance, represents the set of Anscombe-Aumann acts,Anscombe and Aumann (1963). Let S = { ( δ s , . . . , δ s ) : s ∈ [ a, b ] } ; the con-stant acts whose outcomes are degenerate lotteries. Let M = S , as in theprevious example; and all topological properties satisfied there are also satis-fied here.The following results generalize those derived originally by Mas-Colell (1974),who worked with R n + . Theorem 15. Φ is an open map. Theorem 16. (Border and Segal (1994) Thm 8) Let ( X, d ) be a locally com-pact and separable metric space and R be the space of continuous preferencerelations on X , endowed with the topology of closed convergence. If (cid:23) u = Φ( u ) is locally strict, then Φ is continuous at u . If M has no isolated points, and Φ is continuous at u , then (cid:23) u is locally strict. Define an equivalence relation ≃ on U by u ≃ v if there exists ϕ : R → R strictly increasing for which u = ϕ ◦ v . Then let U / ≃ denote the set ofequivalence classes of U under ≃ endowed with the quotient topology; theequivalence class of u ∈ U is written [ u ]. The map ˆΦ : U / ≃→ R mon isdefined in the natural way, via ˆΦ([ u ]) = Φ( u ). Theorem 17. ˆΦ is a homeomorphism. Given the discussion of Example 13, Theorem 17 generalizes Theorem 4.The role of M in the case of R n + is played by the equal-coordinates ray. Itis also straightforward to apply Theorem 17 to intertemporal choice by wayof the model of dated rewards (see Section 5.4.2, by letting M be the line { ( x,
0) : x ≥ } . Observe that this function is well-defined. If v ∈ [ u ], then there is strictly increasing ϕ forwhich v = ϕ ◦ u , hence v and u represent the same preference. REFERENCE IDENTIFICATION 25 Non-monotone preferences and local strictness
When the set of utility functions is compact, we can obtain a particularlystrong result that does not rely on monotonicity, or the existence of a preferencerelation generating the choices. Instead, the generating preference is obtainedendogenously as the limit of rationalizing preferences. Let V be a compact set of continuous functions in the topology of compactconvergence, and let Φ( V ) denote the image of V under Φ, so that Φ( u ) is thepreference represented by u . Theorem 18.
Suppose V is compact, and that all (cid:23)∈ Φ( V ) are locally strict.Let c be a choice sequence, and let (cid:23) k ∈ V weakly rationalize c k . Then, thereexists (cid:23) ∗ ∈ Φ( V ) such that (cid:23) k →(cid:23) ∗ in the closed convergence topology. Fur-thermore, if (cid:23) ′ k also weakly rationalizes c k , then (cid:23) ′ k →(cid:23) ∗ . Observe that knowledge of a generating preference (cid:23) ∗ is not required; butthe hypothesis that there is a weak rationalization (cid:23) k for every c k suggeststhe possibility.Theorem 18 implies that one can sometimes obtain asymptotically utilityrationalizations drawn from V . In particular, when V is compact, Φ( V ) consistsof locally strict preferences, and Φ is a homeomorphism then Φ − ( (cid:23) k ) ∈ V converges to a utility for (cid:23) ∗ in V . One application of this kind is in Example 19. Example 19.
Let X be a finite set, and let ∆( X ) be the lotteries on X (topologized as elements of Euclidean space). Consider the set of nonconstantexpected utility preferences. Then the hypotheses of Theorem 18 hold here.To see this, observe that the set of nonconstant von Neumann-Morgensternutility indices is homeomorphic to the set S = { u ∈ R X : X x u x = 0 , k u k = 1 } . It is straightforward to see that the map φ : S → C (∆( X )) given by φ ( u )( p ) = P x u x p ( x ) is continuous. So, let V = φ ( S ) which is compact; then the setΦ( V ) is the set of nontrivial expected utility preferences. Finally, observethat each nonconstant expected utility preference is locally strict. For, if (cid:23) is In particular, these results should be contrasted with the example in Section 9.1. nonconstant, then there are p, q ∈ ∆( X ) for which p ≻ q . Then for any r (cid:23) s ,for any α > αp + (1 − α ) r ≻ αq + (1 − α ) s . Choose α small to be withinany neighborhood of ( r, s ).Next, Example 20 allows for an infinite set of prices, but restricts vonNeumann-Morgenstern utilities to have lower and upper Lipschitz bounds. Example 20.
We can consider R n + , and a class of utility functions U ba , where a, b ∈ R with 0 < a < b . U ba = { u ∈ C ( R n + ) : ∀ i ∧∀ ( x i < y i ) , a ( y i − x i ) ≤ u ( y i , x − i ) − u ( x i , x − i ) ≤ b ( y i − x i ) } . Observe that U ba ⊆ U , and consists of those members satisfying a certainLipschitz property (namely, Lipschitz boundedness above and below). By theArzela-Ascoli Theorem (see Dugundji (1966), Theorem 6.4), U ba is compact.Furthermore, each (cid:23)∈ Φ( U ba ) is locally strict, as it is strictly monotonic.8. Infinite and Countable Data
In this section, we propose two sufficient conditions that enable the recoveryof the subject’s preference from its restriction to a countable set of data points.We first show below that if we can observe a subrelation of a locally strictand continuous binary relation on a dense set, then we can infer the entirebinary relation.
Theorem 21.
Suppose that (cid:23) and (cid:23) ′ are two complete, continuous, and lo-cally strict binary relations. Let B ⊆ X be dense. If (cid:23) | B × B ⊆(cid:23) ′ | B × B , then (cid:23) = (cid:23) ′ . We then make no restriction on the preferences other than continuity, butrequires the underlying space of alternatives to be connected.
Theorem 22.
Suppose that (cid:23) and (cid:23) ′ are two continuous preference relations.Suppose X is connected, and let B ⊆ X be dense. If (cid:23) | B × B = (cid:23) ′ | B × B , then (cid:23) = (cid:23) ′ . Note that Theorem 22 generalizes Theorem 6. Without connectedness, thisresult can fail. A preference (cid:23) can be increasing on (0 , ∪ (2 , REFERENCE IDENTIFICATION 27 are two possible ways to extend it to [0 , ∪ [2 , ∼
2, or2 ≻ ex-ante from each of the sets in the support inthe distribution. A decision maker who respects basic monotonicity postulates(see Azrieli et al., 2014) correctly announces each of their choices.If we can uncover an entire preference from each of these choices, thenwe are able to elicit an entire preference using one suitably chosen randomdevice. Here, we do not investigate this theory in its full generality. But ifthere is a countable dense subset of alternatives, and a continuous preferencecan be inferred from its behavior on a countable dense subset, then we canutilize the Allais mechanism to uncover an entire preference with a singlerandomized choice. For example, we would enumerate the pairs of elementsfrom the countable dense subset, say B , B , . . . , and randomize so that eachone realizes with probability 2 − k .9. On the meaning of (cid:23) ∗ Some economists are comfortable with the idea that an agent “has” a data-generating preference (cid:23) ∗ , and some are not. The former assume that thepreference is something intrinsic to the agent, and that when presented witha choice situation the agent can access his preference and choose accordingly.Under this interpretation, our paper gives conditions under which a finite ex-periment can approximate, to an arbitrary degree of precision, the underlyingpreference that the agent uses to make choices.Other economists argue that preferences are just choices. For those in thisposition, it is meaningless to speak of a preference over pairs of alternativesfrom which the agent never chooses. We are sympathetic to this view, andour paper also contributes to this interpretation. Under proper conditions—conditions that we provide in our paper—continuity “defines” preferences over X given choices over a countable subset. This is important because estimatedpreferences provide a guide for making normative recommendations and out of sample predictions. An economist may want to estimate (cid:23) ∗ so as to makepolicy recommendations that are in the agent’s interest; in fact, this is acommon use of estimated preferences in applied work. Similarly, the economistmay want to use (cid:23) ∗ as an input in a structural economic model, and therebymake predictions for different configurations of the model. The existence andmeaning of (cid:23) ∗ is then provided for by the continuity assumption.Moreover, viewed from this angle, Theorem 10 allows us to say that the setof rationalizations can be made arbitrarily small as more and more data areobserved. In this manner, one can bound errors in welfare statements or outof sample predictions to an arbitrary degree of precision.We conclude this section with two examples that illustrate the importanceof postulating existence of an agent’s preference: without the postulate, theinferred preference may otherwise fail to converge.9.1.
The set of weakly monotone preference relations is not closed.
Suppose we are interested in rationality in the form of a strictly monotoniccontinuous preference relation. Observe that Theorems 9 and 11 hypothesizethe existence of (cid:23) ∗ . If (cid:23) ∗ ∈ R mon, for example, then we know that, in thelimit, rationalizing relations will be transitive if every (cid:23) k is. Unfortunately, weshow in this section, if we do not know that (cid:23) ∗ is transitive, we cannot ensurethat it is, even if each (cid:23) k is. That is, we demonstrate a sequence (cid:23) k of strictlymonotone preferences, where (cid:23) k →(cid:23) ∗ in the closed convergence topology, but (cid:23) ∗ is not transitive.The data are rationalizable, but the rationalization requires intransitive in-difference. So the properties of the rationalizations of order k cannot be pre-served.Figure 1 exhibits a non-transitive relation. The example is taken fromGrodal (1974). The lines depict indifference curves, but all the green indif-ference curves intersect at one point: (1 / , / This is true in spite of the claim we make in Section 9.1. It is true that the set ofrationalizations may “shrink” to something which is not transitive, but this set is shrinkingnonetheless and always contains preference relations (except in the limit).
REFERENCE IDENTIFICATION 29 / / / / Figure 1.
A non-transitive preferenceNow imagine a collection of binary comparisons that do not include (1 / , / / , / / , /
2) so that transi-tivity holds.This example is not particularly troubling, however. First, with finite exper-imentation, the violation of transitivity will never be “reached.” Second, theviolation here is not particularly egregious. Only transitivity of indifference isviolated. This holds quite generally. It can be shown that any limit point ofa sequence of preference relations must be quasitransitive, so that whenever x ≻ y and y ≻ z , it follows that x ≻ z . Quasitransitive relations enjoy manyof the useful properties of preferences. For example, continuous quasitransitiverelations possess maxima on compact sets, see e.g.
Bergstrom (1975). The argument is in Grodal (1974), but to see this suppose that (cid:23) n →(cid:23) , where each (cid:23) n is a preference relation. It can be shown that (cid:23) is complete, so suppose by means ofcontradiction that there are x, y, z ∈ X for which x ≻ y , y ≻ z , and z (cid:23) x . So, there are x n , z n for which z n (cid:23) n x n , x n → x , and z n → z . For each n , either z n (cid:23) n y or y (cid:23) n x , sothat without loss, there is a sequence for which z n (cid:23) n y , i.e. z (cid:23) y , a contradiction. Figure 2.
A transitive preference9.2.
The set of locally strict relations is not closed.
Finally we presentan example to show that the set of locally strict preference relations is notclosed. Let X = [ − , − ∪ [1 , n , let u n ( x ) = − ( x + 2) + n on[ − , −
1] and u n ( x ) = ( x − − n on [1 , u n represents a locally strict relation (cid:23) n .Let u ∗ ( x ) be the pointwise limit of u n ; i.e. u ∗ ( x ) = − ( x + 2) on [ − , − u ∗ ( x ) = ( x − on [1 , u ∗ represents (cid:23) ∗ which is not locally strict. Observe that − (cid:23) ∗
2, but for small neighborhoods there is nostrict preference.However, it is also straightforward by checking cases to show that (cid:23) n →(cid:23) ∗ . REFERENCE IDENTIFICATION 31
Figure 3.
The set of locally strict preferences is not closed.
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Appendix A. About Closed Convergence
We recall below the formal definition of closed convergence, used throughoutthe results of this paper. Let F = { F n } n be a sequence of closed sets in X × X .We define Li( F ) and Ls( F ) to be closed subsets of X × X as follows: • ( x, y ) ∈ Li( F ) if and only if, for all neighborhood V of ( x, y ), thereexists N ∈ N such that F n ∩ V = ∅ for all n ≥ N . • ( x, y ) ∈ Ls( F ) if and only if, for all neighborhood V of ( x, y ), and all N ∈ N , there is n ≥ N such that F n ∩ V = ∅ .Observe that Li( F ) ⊆ Ls( F ). The definition of closed convergence is as follows. Definition 23. F n converges to F in the topology of closed convergence ifLi( F ) = F = Ls( F ). Appendix B. Proof of Proposition 1
Denote by ( a ′ , b ′ ) the open interval { z ∈ R n : a ′ ≪ z ≪ b ′ } . For each k , let u k : ∪ kl =1 B l → [0 ,
1] be a utility representation of (cid:23) ∗ on ∪ kl =1 B l .For each k , let { [ a i , b i ] } n k i =1 be a sequence of intervals in R n with the prop-erties that a) [ a, b ] ⊆ ∪ n k i =1 [ a i , b i ], b) ( a i , b i ) ∩ ( a j , b j ) = ∅ for i = j , c) eachelement of ∪ kl =1 B l is contained in a set ( a i , b i ), and no two elements of ∪ kl =1 B l are contained in the same, and d) [ a i , b i ] is contained in some ball of radius(2 k ) − . For each interval [ a i , b i ] there is a continuous function f i such that f ( x ) = 0for all x ∈ [ a i , b i ] \ ( a i , b i ), f ( x ) = u k ( x ) if x ∈ ( a i , b i ) ∩ ∪ kl =1 B l , sup { f ( x ) : x ∈ [ a i , b i ] } = 2 and inf { f ( x ) : x ∈ [ a i , b i ] } = −
2. Let u ∗ k : [ a, b ] → R bethe function that coincides with f i on each [ a i , b i ]. Let (cid:23) k be the preferencerelation represented by u ∗ k , and note that (cid:23) k strongly rationalizes the choicefunction of order k generated by (cid:23) ∗ , and is continuous.Let x, y ∈ X . For each k , suppose that x ∈ [ a i , b i ] for the k th sequence ofsubintervals. Let x k ∈ [ a i , b i ] be such that u ∗ k ( x k ) = 2. Note that k x − x k k < /k . Similarly, suppose that y ∈ [ a j , b j ] for the k th sequence of subintervals It is obvious that such a sequence exists. First, it is immediate that it exists for n = 1.For n > B k onto each of its coordinate and carry out the one-dimensionalconstruction (choosing a sufficiently small radius for the balls covering each interval). Thentake the cartesian product of each one-dimensional interval. REFERENCE IDENTIFICATION 35 and let y k ∈ [ a j , b j ] be such that u ∗ k ( y k ) = −
2. Then x k ≻ k y k . Since ( x k , y k ) → ( x, y ) and x, y ∈ X were arbitrary this means that (cid:23) k → X × X . Appendix C. Proof of Proposition 12
The proof is implied by the following lemmas.
Lemma 24.
Let X ⊆ R n . If { x ′ n } is an increasing sequence in X , and { x ′′ n } is a decreasing sequence, such that sup { x ′ n : n ≥ } = x ∗ = inf { x ′′ n : n ≥ } .Then lim n →∞ x ′ n = x ∗ = lim n →∞ x ′′ n . Proof.
This is obvously true for n = 1. For n >
1, convergence and sups andinfs are obtained component-by-component, so the result follows. (cid:3)
Lemma 25.
Let X ⊆ R n . Let { x n } be a convergent sequence in X , with x n → x ∗ . Then there is an increasing sequence { x ′ n } and an a decreasingsequence { x ′′ n } such that x ′ n ≤ x n ≤ x ′′ n , and lim n →∞ x ′ n = x ∗ = lim n →∞ x ′′ n .Proof. Suppose that x n → x ∗ . Define x ′ n and x ′′ n by x ′ n = inf { x m : n ≤ m } and x ′′ n = sup { x m : n ≤ m } Then it is clear that x ′ n ≤ x n ≤ x ′′ n , that x ′ n is increasing, and that x ′′ n isdecreasing. Moreover,lim n →∞ x ′ n = sup { inf { x m : n ≤ m } : n ≥ } = x ∗ = inf { sup { x m : n ≤ m } : n ≥ } = lim n →∞ x ′′ n by Lemma 24. (cid:3) Lemma 26.
Let X = ∆([ a, b ]) . Let { x n } be a convergent sequence in X ,with x n → x ∗ . Then there is an increasing sequence { x ′ n } and an a decreasingsequence { x ′′ n } such that x ′ n ≤ x n ≤ x ′′ n , and lim n →∞ x ′ n = x ∗ = lim n →∞ x ′′ n .Proof. The set X ordered by first order stochastic dominance is a completelattice (see, for example, Lemma 3.1 in Kertz and R¨osler (2000)). Supposethat x n → x ∗ . Define x ′ n and x ′′ n by x ′ n = inf { x m : n ≤ m } and x ′′ n = sup { x m : n ≤ m } . Clearly, { x ′ n } is an increasing sequence, { x ′′ n } is decreasing, and x ′ n ≤ x n ≤ x ′′ n .Let F x denote the cdf associated with x . Note that F x ′′ n ( r ) = inf { F x m ( r ) : n ≤ m } while F x ′ n ( r ) is the right-continuous modification of sup { F x m ( r ) : n ≤ m } . For any point of continuity r of F , F x m ( r ) → F x ∗ ( r ), so F x ( r ) = sup { inf { F x m ( r ) : n ≤ m } : n ≥ } by Lemma 24.Moreover, F x ∗ ( r ) = inf { sup { F x m ( r ) : n ≤ m } : n ≥ } . Let ε >
0. Then F x ∗ ( r − ε ) ← sup { F x m ( r − ε ) : n ≤ m } ≤ F x ′ n ( r ) ≤ sup { F x m ( r + ε ) : n ≤ m } → F x ∗ ( r + ε )Then F x ′ n ( r ) → F x ∗ ( r ), as r is a point of continuity of F x ∗ . (cid:3) The following lemma is immediate.
Lemma 27.
Let X = R n + with the standard vector order ≤ , and let B = Q n + .Then the countable order property is satisfied. Our last lemma is a direct implication of Theorem 15.11 of Aliprantis and Border(2006).
Lemma 28.
Let a, b ∈ R , where a < b . Let X = ∆([ a, b ]) , the set of Borelprobability distributions on [ a, b ] endowed with the weak* topology. Let B bethe set of probability distributions p with finite support on Q ∩ [ a, b ] , where forall q ∈ Q ∩ [ a, b ] , p ( q ) ∈ Q . Then the countable order property is satisfied. Appendix D. Proof of Theorems 9, 21, 22 and 10
In this section, we let R mon denote the set of complete, continuous, andstrictly monotonic binary relations. Members of R mon need not be transitive.Likewise, R ls is the set of complete, continuous, and locally strict binaryrelations.We record the following fact: Lemma 29.
Let (cid:23) be a continuous binary relation. If x ≻ y then there areneighborhoods V x of x and V y of y such that x ′ ≻ y ′ for all x ′ ∈ V x and y ′ ∈ V y . We now prove Theorems 21 and 22.
REFERENCE IDENTIFICATION 37
Proof of Theorem 21.
Follows directly from Lemma 32, below. (cid:3)
Proof of Theorem 22.
First, it is straightforward to show that x ≻ y implies x (cid:23) ′ y . Because otherwise there are x, y for which x ≻ y and y ≻ ′ x . Take anopen neighborhood U about ( x, y ) and a pair ( z, w ) ∈ U ∩ ( B × B ) for which z ≻ w and w ≻ ′ z , a contradiction. Symmetrically, we also have x ≻ ′ y implies x (cid:23) y .Now, without loss, suppose that there is a pair x, y for which x ≻ y and x ∼ ′ y . By connectedness and continuity, V = { z : x ≻ z ≻ y } is nonempty and bycontinuity it is open. We claim that there is a pair ( w, z ) ∈ ( V × V ) ∩ ( B × B )for which w ≻ z . By denseness of B , there is w ∈ V ∩ B for which x ≻ w ≻ y .Similarly, { z : w ≻ z ≻ y } is nonempty and open; so there is z ∈ B for which x ≻ w ≻ z ≻ y .We have shown that there is ( w, z ) ∈ ( V × V ) ∩ ( B × B ) for which w ≻ z ,so that x ≻ w ≻ z ≻ y . Further, we have hypothesized that x ∼ ′ y . By thefirst paragraph, we know that x (cid:23) ′ w (cid:23) ′ z (cid:23) ′ y . If, by means of contradiction,we have w ≻ ′ z , then x ≻ ′ y , a contradiction. So w ∼ ′ z and w ≻ z , acontradiction to (cid:23) B × B = (cid:23) ′ B × B . (cid:3) Lemma 30.
Let A ⊆ X × X . Then {(cid:23) : A ⊆(cid:23)} is closed in the closedconvergence topology.Proof. Let (cid:23) n be a convergent sequence in the set in question, where (cid:23) n →(cid:23) .Then for all ( x, y ) ∈ A , we have x (cid:23) n y , hence x (cid:23) y . So ( x, y ) ∈(cid:23) . (cid:3) Lemma 31.
Suppose X is locally compact Polish, and that < has open inter-vals. Then R mon is closed in the topology of closed convergence.Proof. By Lemma 8, since X is locally compact Polish, the topology of closedconvergence is compact metrizable.Suppose (cid:23) n →(cid:23) where each (cid:23) n is continuous, strictly monotonic, and com-plete. We know that (cid:23) is continuous by compactness. Suppose by means of The argument for nonemptiness is as follows. If, by means of contradiction, V = ∅ , then { z : x ≻ z } and { z : z ≻ y } are nonempty open sets. Further, for any z ∈ X , either x ≻ z or z ≻ y (because if ¬ ( x ≻ z ) then by completeness z (cid:23) x , which implies that z ≻ y ).Conclude that { z : x ≻ z } ∪ { z : z ≻ y } = X and each of the sets are nonempty and open(by continuity); these sets are disjoint, violating connectedness of X . contradiction that (cid:23) is not strictly monotonic, so that there are x, y ∈ X forwhich x > y and y (cid:23) x . Then there are ( x n , y n ) → ( x, y ) for which y n (cid:23) n x n .For n large, x n > y n , a contradiction to the fact that (cid:23) n is strictly monotonic.Finally, completeness follows as for each x, y , either x (cid:23) n y or y (cid:23) n x , so thereis a subsequence n k for which either x (cid:23) n k y or for which y (cid:23) n k x . (cid:3) Lemma 32.
Suppose that B is dense, (cid:23) ′ is complete, and each of (cid:23) and (cid:23) ∗ are continuous and locally strict complete relations. Then if (cid:23) ′ | B × B ⊆(cid:23) ∗ | B × B ∩ (cid:23) | B × B , it follows that (cid:23) ∗ = (cid:23) .Proof. Suppose, by means of contradiction and without loss of generality, thatthere are x, y ∈ X for which x (cid:23) ∗ y and y ≻ x . By continuity of (cid:23) and localstrictness of (cid:23) ∗ , we can without loss of generality assume that x ≻ ∗ y and y ≻ x . By continuity of each of (cid:23) and (cid:23) ∗ , there exists a, b ∈ B such that a ≻ ∗ b and b ≻ a . But by completeness of (cid:23) ′ , either a (cid:23) ′ b , contradicting (cid:23) ′ | B × B ⊆(cid:23) | B × B , or b (cid:23) ′ a , contradicting (cid:23) ′ | B × B ⊆(cid:23) ∗ | B × B . (cid:3) We now turn to the main proof of the theorem.
Proof of Theorem 9.
By Lemma 31, R mon is compact. Let (cid:23) ′ be any strictlymonotonic and complete binary relation such that for all k and all { x, y } ∈ Σ k , x ∈ c k ( { x, y } ) if and only if x (cid:23) ′ y ( (cid:23) ′ exists by the projection requirementon choice sequences, and by the fact that c ⊑ c (cid:23) ∗ ).For each k , let (cid:23) ′ k = { ( x, y ) : { x, y } ∈ { B , . . . , B k } and x (cid:23) ′ y } .For each k , let P k = {(cid:23)∈ R mon : (cid:23) ′ k ⊆(cid:23)} , the set of relations which weakly rationalize c . Observe by definition that byLemma 30, P k is closed, and hence compact. By assumption, each (cid:23)∈ P k satisfies (cid:23)∈ R ls, and obviously, for all k , (cid:23) ∗ ∈ P k . Further, observe that T k P k = {(cid:23) ∗ } , since if (cid:23)∈ T k P k , by definition (cid:23) ′ B × B ⊆(cid:23) ∗ | B × B ∩ (cid:23) | B × B andLemma 32.The result now follows as each P i is compact and T k P k = {(cid:23) ∗ } . That is, let (cid:23) k ∈ P k , which is a decreasing, nested collection of compact sets. Suppose by REFERENCE IDENTIFICATION 39 means of contradiction and without loss that (cid:23) k →(cid:23) ′ = (cid:23) ∗ , and observe thenthat it follows that (cid:23) ′ ∈ P k for all k , contradicting T i P i = {(cid:23) ∗ } . (cid:3) Proof of Theorem 10.
Observe that for any k , the set P k = {(cid:23)∈ R mon : (cid:23) weakly rationalizes c k } is closed, and hence compact by Lemma 30. Observe that P k ( c ) ⊆ P k . More-over, it is obvious that P k +1 ⊆ P k . Suppose that there is no k for which P k ( c ) = ∅ . Then, since each P k = ∅ and each P k is compact, T k P k = ∅ . Let (cid:23) ∗ ∈ T k P k .We claim that T k P k = {(cid:23) ∗ } . Suppose by means of contradiction that thereis (cid:23)6 = (cid:23) ∗ where (cid:23)∈ T k P k . Let (cid:23) ′ be any complete relation such that for all( a, b ) ∈ B × B , a (cid:23) ′ b if and only if a ∈ c k ( { a, b } ), for k such that { a, b } ∈ Σ k .Then, by definition of weak rationalization, we have (cid:23) ′ B × B ⊆(cid:23) B × B ∩ (cid:23) ∗ B × B .Appeal to Lemma 32 to conclude that (cid:23) = (cid:23) ∗ , a contradiction.Finally, since T k P k = {(cid:23) ∗ } , and each P k is compact, it follows that lim k →∞ diam( P k ) → Hence, since 0 ≤ diam( P k ( c )) ≤ diam( P k ), the result follows. (cid:3) Appendix E. Proof of Theorem 11
The set of weakly monotone and continuous binary relations is compact inthe topology of closed convergence. Suppose wlog that (cid:23) k →(cid:23) . Then (cid:23) is acontinuous binary relation. We shall prove that (cid:23) = (cid:23) ∗ .First we show that x ≻ ∗ y implies that x ≻ y . So let x ≻ ∗ y . Let U and V be neighborhoods of x and y , respectively, such that x ′ ≻ ∗ y ′ for all x ′ ∈ U and y ′ ∈ V . Such neighborhoods exist by the continuity of (cid:23) ∗ . We prove firstthat if ( x ′ , y ′ ) ∈ U × V , then there exists N such that x ′ ≻ n y ′ for all n ≥ N .By hypothesis, there exist x ′′ ∈ U ∩ B and y ′′ ∈ V ∩ B such that x ′′ ≤ x ′ and y ′ ≤ y ′′ . Each (cid:23) n is a strong rationalization of the finite experiment of order n , so if { ˜ x, ˜ y } ∈ Σ n then ˜ x ≻ n ˜ y implies that ˜ x ≻ m ˜ y for all m ≥ n . Since x ′′ , y ′′ ∈ B , there is N is such that { x ′′ , y ′′ } ∈ Σ N . Thus x ′′ ≻ ∗ y ′′ implies that x ′′ ≻ n y ′′ for all n ≥ N . So, for n ≥ N , x ′ ≻ n y ′ , as (cid:23) n is weakly monotone. Otherwise, we could choose ǫ > (cid:23) k l , (cid:23) ′ k l such that δ C ( (cid:23) k l , (cid:23) ′ k l ) ≥ ǫ and (cid:23) k l →(cid:23)∈ T k P k and (cid:23) ′ k l →(cid:23) ′ ∈ T k P k where δ C ( (cid:23) , (cid:23) ′ ) ≥ ǫ , a contradiction. Now we establish that x ≻ y . Let { ( x n , y n ) } be an arbitrary sequence with( x n , y n ) → ( x, y ). By hypothesis, there is an increasing sequence { x ′ n } , anda decreasing sequence { y ′ n } , such that x ′ n ≤ x n and y n ≤ y ′ n while ( x, y ) =lim n →∞ ( x ′ n , y ′ n ).Let N be large enough that x ′ N ∈ U and y ′ N ∈ V . Let N ′ ≥ N be suchthat x ′ N ≻ n y ′ N for all n ≥ N ′ (we established the existence of such N ′ above).Then, for any n ≥ N ′ we have that x n ≥ x ′ n ≥ x ′ N ≻ n y ′ N ≥ y ′ n ≥ y n . By the weak monotonicity of (cid:23) n , then, x n ≻ n y n . The sequence { ( x n , y n ) } was arbitrary, so ( y, x ) / ∈(cid:23) = lim n →∞ (cid:23) n . Thus ¬ ( y (cid:23) x ). Completeness of (cid:23) implies that x ≻ y .In second place we show that if x (cid:23) ∗ y then x (cid:23) y , thus completing theproof. So let x (cid:23) ∗ y . We recursively construct sequences x n k , y n k such that x n k (cid:23) n k y n k and x n k → x , y n k → y .So, for any k ≥
1, choose x ′ ∈ N x (1 /k ) ∩ B with x ′ ≥ x , and y ′ ∈ N y (1 /k ) ∩ B with y ′ ≤ y ; so that x ′ (cid:23) ∗ x (cid:23) ∗ y (cid:23) ∗ y ′ , as (cid:23) ∗ is weakly monotone. Recallthat (cid:23) n strongly rationalizes c (cid:23) ∗ for Σ n . So x ′ (cid:23) ∗ y ′ and x ′ , y ′ ∈ B imply that x ′ (cid:23) n y ′ for all n large enough. Let n k > n k − (where we can take n = 0)such that x ′ (cid:23) n k y ′ ; and let x n k = x ′ and y n k = y ′ .Then we have ( x n k , y n k ) → ( x, y ) and x n k (cid:23) n k y n k . Thus x (cid:23) y . Appendix F. Proof of Theorem 15 and Proposition 5
We begin with two useful lemmas.
Lemma 33. Φ is an open map if for any u ∗ ∈ U and any sequence (cid:23) n in R with (cid:23) n → Φ( u ∗ ) , there is a sequence { u n } in U such that u n ∈ Φ − ( (cid:23) n ) and u n → u ∗ in the topology of compact convergence.Proof. Suppose that there is V ⊆ U open, but Φ( V ) is not open. Then thereis u ∗ ∈ V and (cid:23) n / ∈ Φ( V ) such that (cid:23) n → Φ( u ∗ ) (since closed convergencetopology is metrizable). Since u ∗ ∈ V , any sequence u n ∈ Φ − ( (cid:23) n ) for which u n → u ∗ eventually has u n ∈ V . But if u n is chosen to represent (cid:23) n , thisimplies that Φ( u n ) ∈ Φ( V ) for n large, a contradiction. (cid:3) REFERENCE IDENTIFICATION 41
Lemma 34.
For any (cid:23) and x ∈ X , there is a unique m ∗ ( x ) ∈ M with x ∼ m ∗ ( x ) . Moreover, if we fix u ∈ U then the function u (cid:23) : X → R definedby u (cid:23) ( x ) = u ( m ∗ ( x )) is a continuous utility representation of (cid:23) .Proof. Consider the sets A = { m ∈ M : m (cid:23) x } and B = { m ∈ M : x (cid:23) m } . These sets are closed because (cid:23) is continuous, their union is M as (cid:23) iscomplete, and they are nonempty as (cid:23) is monotone and there exist m, m ∈ M with m ≤ x ≤ m by our hypothesis on M . M is connected, so A and B cannot be disjoint; hence there is m ∈ M with x ∼ m . This m must be uniquebecause M is totally ordered, and (cid:23) is strictly monotone.We now show that u (cid:23) is a continuous utility representation of (cid:23) . Let x (cid:23) y .Then transitivity and monotonicity of (cid:23) imply that m ∗ ( x ) ≥ m ∗ ( y ). Thus u (cid:23) ( x ) = u ∗ ( m ∗ ( x )) ≥ u ∗ ( m ∗ ( y )) = u (cid:23) ( y ). The converse implications hold aswell; thus u (cid:23) represents (cid:23) .To prove continuity, let x n → x ∗ . We shall prove that m n = m ∗ ( x n ) → m ∗ ( x ∗ ) = ˆ m . Suppose first that ˆ m is not the largest or the least element of M . For each neighborhood U of ˆ m there exists, by our hypothesis on M , m, m ∈ M with m < ˆ m < m and [ m, m ] ⊆ U . Then V = { z ∈ X : m ≻ z } ∩ { z ∈ X : z ≻ m } is a neighborhood of x ∗ , as x ∗ ∼ ˆ m and (cid:23) is continuous and monotone. Forlarge enough n then x n ∈ V , so m n ∈ [ m, m ] ⊆ U . Suppose now that ˆ m isthe largest element of M . Then, reasoning as above, x n ∈ { z ∈ X : z ≻ m } for all large enough n , so that m ≤ m n . We have m n ≤ m as m is the largestelement of M . Thus m n ∈ [ m, m ] ⊆ U . The argument when m is the leastelement of M is analogous. (cid:3) We now turn to the main proof of the theorem, which proves Proposition 5.
Proof of Theorem 15.
Let u ∗ ∈ U and {(cid:23) n } be a sequence in R with (cid:23) n → Φ( u ∗ ). By Lemma 33 it is enough to exhibit a sequence u n ∈ Φ − ( (cid:23) n ) and u n → u ∗ in the topology of compact convergence.Let u n = u (cid:23) n as defined in Lemma 34 from u ∗ . Lemma 34 implies that u n ∈ Φ − ( (cid:23) n ). By XII Theorem 7.5 p. 268 of Dugundji (1966), to establish compact convergence it is enough to show that for any convergent sequence { x n } , with x n → x ∗ , u n ( x n ) → u ∗ ( x ∗ ).To this end, let x n → x ∗ . Let ˆ m = m ∗ ( x ∗ ) and m n ∼ n x n , using the notationin Lemma 34, and U be a neighborhood of ˆ m . Let m, m ∈ M be such that m < ˆ m < m and [ m, m ] ⊆ U . Then it must be true that m n ∈ [ m, m ] forall n large enough. To see this, note that if, for example, m n ≥ m infinitelyoften then there would exist a subsequence for which x n (cid:23) n m n (cid:23) m (bymonotonicity of (cid:23) ), which would imply that x ∗ (cid:23) m > ˆ m , as (cid:23) n →(cid:23) . Butˆ m ∼ x ∗ (cid:23) m is a violation of monotonicity.Now m n ∈ [ m, m ] ⊆ U for all n large enough means that m n → ˆ m . Thus u (cid:23) n ( x n ) = u ∗ ( x n ) → u ∗ ( x ∗ ) = u (cid:23) ( x ∗ ) , as u ∗ is continuous. (cid:3) Appendix G. Proof of Theorem 18
By Theorem 8 of Border and Segal (1994), Φ( V ) is compact, and therefore (cid:23) k possesses a limit point (cid:23) ∗ ∈ Φ( V ). By Lemma 30, the set of (cid:23) k weaklyrationalizing c k is closed, and hence compact. Suppose by means of contra-diction that there is some (cid:23) ′ k also weakly rationalizing c k which converges to (cid:23)6 = (cid:23) ∗ . Observe that each of (cid:23) ∗ and (cid:23) weakly rationalize each c k .Finally, let (cid:23) ′ be any complete relation such that for all ( a, b ) ∈ B × B , a (cid:23) ′ b if and only if a ∈ c k ( { a, b } ), for k such that { a, b } ∈ Σ k . Then, bydefinition of weak rationalization, we have (cid:23) ′ B × B ⊆(cid:23) B × B ∩ (cid:23) ∗ B × B . Appealthen to Lemma 32 to conclude that (cid:23) = (cid:23) ∗∗