# The housing problem and revealed preference theory: duality and an application

aa r X i v : . [ ec on . GN ] F e b The Housing Problem and Revealed PreferenceTheory: Duality and an Application

Ivar Ekeland(Universit´e Paris-Dauphine) Alfred Galichon(Ecole polytechnique)

August 28, 2012 * . Abstract

This paper exhibits a duality between the theory of Revealed Preference of Afriat andthe housing allocation problem of Shapley and Scarf. In particular, it is shown that Afriat’stheorem can be interpreted as a second welfare theorem in the housing problem. Using thisduality, the revealed preference problem is connected to an optimal assignment problem,and a geometrical characterization of the rationalizability of experiment data is given. Thisallows in turn to give new indices of rationalizability of the data, and to deﬁne weakernotions of rationalizability, in the spirit of Afriat’s efﬁciency index.

JEL codes : primary D11, secondary C60, C78.Keywords: revealed preferences, Afriat’s theorem, optimal assignments, indivisible allocations. * Economic Theory . The ﬁnal authenticated version is available onlineat: https://doi.org/10.1007/s00199-012-0719-x Introduction

The “Revealed Preference” problem and the “Housing problem” are two classical problems inEconomic Theory with for both a distinguished, but separate tradition. This paper is aboutconnecting them, and exploit the connection to obtain new results on Revealed Preference.• The Revealed Preference (RP) problem, posed by Samuelson at the end of the 1930’s, wassolved by Houthakker in 1950, and was given an operational solution by Sidney Afriatin 1967. This classical problem asks whether, given the observation of n consumptionsbaskets and corresponding prices, one rationalize these consumptions as the consumptionof a single consumer facing different prices.• The “housing problem” was investigated in 1974 by Shapley and Scarf. Given an initialallocation of n houses to n individuals, and assuming individuals form preferences overhouses and can trade houses, what is the core of the corresponding game? It is assumedthat houses form no preferences over owners (in sharp contrast to the “Stable matching”problem of Gale and Shapley). In this setting, they showed non-emptiness of the core,as well as an algorithm to arrive to a core allocation: the method of “top-trading cycles”,attributed to David Gale.Although both problems have generated two well established and distinct literatures, it willturn out that these problems are in fact dual in a precise sense. As we shall argue, the traditionalexpenditure/utility duality from consumer theory extends to the setting of revealed preference,and it is possible to show that the problem of Pareto efﬁciency in the housing problem andthe issue of rationalizability of consumer data are both applications of a basic mathematicalresult characterizing cyclically consistent matrices proven by Fostel, Scarf and Todd (2004),generalizing Afriat’s theorem. In particular, we show that Afriat’s theorem can be interpretedas a second welfare theorem in the housing problem . Once we have established equivalenceof both problems, we shall use the aforementioned results to give new characterization of bothproblems in terms of an optimal allocation problem. This will give us a very simple and intuitivecharacterization of the efﬁcient outcomes in the indivisible allocation problem.According to standard revealed preference theory, the data are not rationalizable when onecan exhibit a preference cycle based on the direct revealed preference relation. A strongerviolation of rationalizability occurs when there are subsets of the data where cycles exist. Aneven stronger departure from rationalizability occurs when all observations are part of such acycle. We shall characterize these three situations in terms of indexes which will measure howstrong the violations of rationalizability are. In the last part, we shall last use these results toprovide a geometric interpretation of revealed preference theory, by showing that the problem ofdetermining whether data can be rationalized or not can be recast as the problem of determiningwhether a particular point is on some part of the boundary of some convex body.2he connection between the housing problem and the revealed preference problem yieldssome insights for both theories. The important econometrics literature on revealed preferencecontains a wealth of tools to measure the intensity of departure from rationalizability. Thesetools may be successfully applied to the “housing problem” and its more applied versions suchas the “kidney problem”, which deals with assigning organ transfers to patients. These toolsmay design indices to measure how far from Pareto optimality an actual assignment is, or whatare the costs of a given constraint in terms of welfare. Conversely, the connection between thetwo problems sheds light on the role of matching theory into revealed preference theory. Thisleads to interesting research avenues, as matching theory has recently received a lot of attention,both from the empirical and computational point of views.The literature on revealed preference in consumer demand traces back to Samuelson (1938),who formulated the problem and left it open. It was solved by Afriat (1967) using nontrivialcombinatorial techniques. Diewert (1973) provided a Linear Programming proof and Varian(1982) an algorithmic solution. Fostel, Scarf and Todd (2004) and Chung-Piaw and Vohra(2003) provided alternative proofs. Matzkin (1991) and Forges and Minelli (2009) extended thetheory to nonlinear budget constraints. Geanakoplos (2006) gives a proof of Afriat’s theoremusing a minmax theorem. Although min-max formulations will appear in our paper, these aredistinct from Geanakoplos’, as it will be explained below. The setting was recently extended tocollective models of household consumptions by Cherchye, De Rock and Vermeulen in a seriesof paper (see e.g. 2007, 2010). A very recent paper by Quah (2012) extends Afriat’s Theoremin another direction n order to characterize data sets compatible with weak separability with-out concavity assumptions. The literature on the indivisible allocation problem was initiatedby Shapley and Scarf (1974), who formulated as the “housing problem” and gave an abstractcharacterization of the core. Roth et al (2004) study a related “kidney problem” and investigatemechanism design aspect. Finally, recent literature has extended revealed preference theory toclasses of matching problems. Galichon and Salani´e (2010) investigate the problem of revealedpreferences in a matching game with transferable utility, with and without unobserved hetero-geneity. Echenique et al. (2011) investigate the problem of revealed preferences in games withand without transferable utility, and without unobserved heterogeneity.This paper will be organized as follows. We ﬁrst study Pareto efﬁcient allocations, andshow the connection with the generalized theory of revealed preferences. We deduce a novelcharacterization of data rationalizability in terms of an optimal matching problem, and a geo-metric interpretation of revealed preference in terms of convex geometry. We then move on toproviding indexes of increasingly weaker forms of rationalizability.3

Revealed preference and the housing problem

Assume as in Forges and Minelli (2009) that consumer has budget constraint g i ( x ) ≤ inexperiment i where x is an L -dimensional bundle of goods, and that bundle x i is chosen where g i ( x i ) = 0 . This is a nonlinear generalization of Afriat (1969), in which g i ( x j ) = x j · p i − x i · p i ,where p i is the price vector of the bundle of goods in experiment i . Forges and Minelli askwhether there exists a utility function v ( x ) with appropriate properties such that x i ∈ argmax x { v ( x ) : g i ( x ) ≤ } in which case v is said to rationalize the data.As in the original Afriat’s paper, the ﬁrst part of their proof necessitates the existence of autility level v j associated to bundle j such that g i ( x j ) < implies v j − v i < . (1)Once the number v i ’s have been determined, the second part of their proof consists in con-structing a locally nonsatiated and continuous utility function v ( x ) such that v ( x j ) = v j . Thepreference induced by v are said to rationalize the data in the sense that i ∈ argmax j { v j : g i ( x j ) ≤ } . The existence of real numbers v i such that (1) holds is non-trivial and it turns out to beequivalent to a property of matrix R ij = g i ( x j ) called “cyclical consistency”. This is doneby appealing to an extension of Afriat’s original theorem beyond linear budget sets, proved byFostel, Scarf and Todd (2004). This result can be slightly reformulated as follows: Theorem 0 (Afriat’s theorem) . The following conditions are equivalent:(i) The matrix R ij satisﬁes “cyclical consistency” : for any cycle i , ..., i p +1 = i , ∀ k, R i k i k +1 ≤ implies ∀ k, R i k i k +1 = 0 , (2) (ii) There exist numbers ( v i , λ i ) , λ i > , such that v j − v i ≤ λ i R ij , (iii) There exist numbers v i such that R ij ≤ implies v j − v i ≤ , and R ij < implies v j − v i < . g i ( x ) deﬁned on a larger domain than the set of observed consumption bundles { x , ..., x n } . Here,the result that we give only depends on the value g i ( x j ) of the functions g i over the set of theobserved consumption bundles. Actually, even only the signs of these values matter. Here theanalysis is intrinsiquely discrete and the approach relies neither on a particular consumptionspace, nor on explicit consumption bundles x j .We now turn to the proof of Theorem 0. Proof of Theorem 0. (i) implies (ii) is proven in Fostel, Scarf and Todd (2004). (ii) immediatelyimplies (iii). We now show that (iii) implies (i). Consider a cycle i , ..., i p +1 = i , such that ∀ k, R i k i k +1 ≤ .Then by (iii) there exist numbers v i such that R ij ≤ implies v j − v i ≤ , and R ij < implies v j − v i < . thus one has v i k +1 − v i k ≤ for all k , hence all the v i k are equal. Assume now that there is a k such that R i k i k +1 < . Then v i k +1 − v i k < , which is a contradiction. Therefore, ∀ k, R i k i k +1 = 0 , which proves the cyclical consistency of matrix R , that is (i). We now turn to the problem of allocation of indivisible goods, which was initially studied byShapley and Scarf (1974). Consider n indivisible goods (eg. houses) j = 1 , ..., n to be allocatedto n individuals. Cost of allocating (eg. transportation cost) house j to individual i is c ij . Anallocation is a permutation σ of the set { , .., n } such that individual i gets house j = σ ( i ) . Let S be the set of such permutations. We assume for the moment that the initial allocation is givenby the identity permutation: good i is allocated to individual i . The problem here is to decidewhether this allocation is efﬁcient in a Pareto sense.If there are two individuals, say i and i that would both beneﬁt from swapping houses(strictly for at least one), then this allocation is not efﬁcient, as the swap would improve on thewelfare of both individuals. Thus if allocation is efﬁcient, then inequalities c i i ≤ c i i and5 i i ≤ c i i cannot hold simultaneously unless they are both equalities. More generally, Paretoefﬁciency rules out the existence of exchange rings whose members would beneﬁt (strictly forat least one) from a circular trade. In fact, we shall argue that this problem is dual to the problemof Generalized Revealed Preferences . Let us now formalize the notion of efﬁcient allocation in the previous discussion. Allocation σ ( i ) = i is Pareto efﬁcient if and only if for any σ ∈ S , inequalities c iσ ( i ) ≤ c ii cannot hold simultaneously unless these are all equalities. By the decomposition of a permuta-tion into cycles, we see that this deﬁnition is equivalent to the fact that for every “trading cycle” i , ..., i p +1 = i , ∀ k, c i k i k +1 ≤ c i k i k implies ∀ k, c i k i k +1 = c i k i k that is, introducing R ij = c ij − c ii , ∀ k, R i k i k +1 ≤ implies ∀ k, R i k i k +1 = 0 ,which is to say that allocation is efﬁcient if and only if the matrix R ij is cyclically consistent.By the equivalence between (i) and (ii) in Theorem 0 above, we have the following state-ment: Proposition 1.

In the housing problem, allocation σ ( i ) = i is efﬁcient if and only if ∃ v i and λ i > , v j − v i ≤ λ i R ij . (PARETO)Before giving an interpretation of this result, we would like to understand the link betweenefﬁciency and equilibrium in the housing problem. Assume we start from allocation σ ( i ) = i ,and we let people trade. Given a price system π where π i is the price of house i , we assume thatindividual i can sell her house for price π i , and therefore can afford any house j whose price π j is less than π i . Therefore, individual i ’s budget set B i is the set of houses that sell to a pricelower than his’ B i = { j : π j ≤ π i } We can now deﬁne the notion of a competitive equilibrium in this setting. Allocation σ ∈ S is an equilibrium supported by price system π if for each individual i , 1) house σ ( i ) is weaklypreferred by individual i among all houses that she can afford, while 2) house σ ( i ) is strictlypreferred by i among all houses she can strictly afford (i.e., that trade for prices strictly less6han her house i ). While condition 1) is a necessary requirement, reﬁnement 2) is natural froma behavioral point of view as it implies that if i is indifferent between two houses, then she isgoing to choose the cheapest house of the two.In particular, σ ( i ) = i is a No-Trade equilibrium if there is a system of prices π , where π j is the price of house j , such that: 1) whenever house j is affordable for individual i , then it isnot strictly preferred by i to i ’s house, that is π j ≤ π i implies c ij ≥ c ii , and 2) for any house j in the strict interior of i ’s budget set, i.e. that trades for strictly cheaper than house i , thenindividual i strictly prefers her own house i to house j , that is π j < π i implies c ij > c ii .Hence, σ ( i ) = i is a No-Trade equilibrium supported by prices π when conditions (E1)and (E2) below are met, where: (E1) if house j can be afforded by i , then individual i does not strictly prefer house j tohouse i , that is, π j ≤ π i implies c ij ≥ c ii ,that is π j ≤ π i implies R ij ≥ ,that is, yet equivalently: R ij < implies π j > π i . (E2) if house j is (strictly) cheaper than house i , then individual i strictly prefers house i tohouse j , that is π j < π i implies c ij > c ii ,that is π j < π i implies R ij > ,that is, yet equivalently: R ij ≤ implies π j ≥ π i . Proposition 2.

In the housing problem, allocation σ ( i ) = i is a No-Trade equilibrium sup-ported by prices π if and only if R ij < implies π j > π i , and (EQUILIBRIUM) R ij ≤ implies π j ≥ π i . But (EQUILIBRIUM) is exactly formulation (iii) of Theorem 0 with π i = − v i . By The-orem 0, we know that this statements is equivalent to statement (PARETO). Hence, we get aninterpretation of the Generalized Afriat’s Theorem as a second welfare theorem (EQUILIBRIUM) ⇐⇒ (PARETO),which we summarize in the following proposition:7 roposition 3. In the allocation problem of indivisible goods, Pareto allocations are no-tradeequilibria supported by prices, and conversely, no-trade equilibria are Pareto efﬁcient.

This is a “dual” interpretation of revealed preference, where v i (utilities in generalized RPtheory) become budgets here, and c ij (budgets in generalized RP theory) become utilities here.To summarize this duality, let us give the following table: Revealed prefs. Pareto indiv. allocs.setting consumer demand allocation problembudget sets { j : c ij ≤ c ii } {− v : − v ≤ − v i } cardinal utilities to j v j − c ij n , i ∈ { , ..., n } n oneunit of c ij dollars utilsunit of v i utils dollarsinterpretation Afriat’s theorem Welfare theorem The table suggests that there is a single agent facing n choices in revealed preference theory,while there are n agents facing one choice in the housing probem. In revealed preference theory,the cardinal utility of alternative j of the consumer in revealed preference theory is v j , while c ij expresses a budget set. Conversely, in the housing problem, − c ij is the cardinal utlity consumer i for alternative j , while − v j is interpreted as the price of bundle j . Hence, the roles of “prices”and “utilities” are exchanges in the two problems, which motivates our contention that RevealedPreference theory and the Housing problem are “dual”.It is worth remarking that the notion of domination used in our deﬁnition of Pareto efﬁciencyis different from the one used by Shapley and Scarf. Following a common use, our notion allowsfor weak domination for some individual, while Shapley and Scarf require strict domination foreverybody. The two notions are not equivalent in a setting with indivisible goods. Indeed, usingtheir notion of domination, Shapley and Scarf provide an example in which a Pareto efﬁcientallocation cannot be sustained as a competitive allocation The connection of both the revealed preference problem and the housing problem with assign-ment problems is clear; we shall now show that there is a useful connection with the optimalassignment problem (recalled below), where the sum of the individual utilities is maximized.Recall that the data are called rationalizable there are scalars v i such that R ij = g i ( x j ) < implies v j < v i , and (EQUILIBRIUM) R ij = g i ( x j ) ≤ implies v j ≤ v i . Theorem 4.

In the revealed preference problem, the data are rationalizable if and only if thereexist weights λ i > such that min σ ∈ S n X i =1 λ i R iσ ( i ) = 0 , (3) that is min σ ∈ S n X i =1 λ i c iσ ( i ) = n X i =1 λ i c ii . (4) Proof of Theorem 4.

Start from (4): ∃ λ i > , min σ ∈ S P ni =1 λ i R iσ ( i ) = 0 ⇐⇒ ∃ λ i > , min σ ∈ S P ni =1 λ i R iσ ( i ) is reached for σ = Id This problem is to ﬁnd the assignment σ ∈ S which minimizes the utilitarian welfare losscomputed as the sum of the individual costs K ij = λ i R ij , setting weight one to each individuals.This problem is therefore min σ ∈ S n X i =1 K iσ ( i ) . This problem was reformulated as a Linear Programming problem by Dantzig in the 1930s; seeShapley and Shubik 1971 for a game-theoretic interpretation, and Gretsky, Ostroy and Zame1992 for the continuous limit), and by the standard Linear Programming duality of the optimalassignment problem (Dantzig 1939; Shapley-Shubik 1971; see Ekeland 2010 for a recent reviewusing the theory of Optimal Transportation) min σ ∈ S n X i =1 K iσ ( i ) = max u i + v j ≤ K ij n X i =1 u i + n X j =1 v j , (5)where S is the set of permutations of { , ..., n } . It is well-known that for a σ ∈ S solution tothe optimal assignment problem, there is a pair ( u, v ) solution to the dual problem such that u i + v j ≤ K ij if j = σ ( i ) , then u i + v j = K ij .Hence, (4) ⇐⇒ ∃ λ i > , u, v ∈ R n u i + v j ≤ λ i R ij u i + v i = 0 ⇒ ∃ λ i > , v ∈ R n v j − v i ≤ λ i R ij , which is (ii) in Theorem 0.The previous result leads to the following two remarks.First, recall that matrix ( M ij ) is called cyclically monotone if for any cycle ( i , ..., i p ) , onehas p X k =1 M i k i k +1 − M i k i k ≥ . It is well known (see e.g. Villani, 2003) that cyclical monotonicity is equivalent to (3).Therefore, Theorem (4) states that ( R ij ) is cyclically consistent if and only if there are weights λ i > such that λ i R ij is cyclically monotone.As a second remark, introduce F = conv (cid:16)(cid:0) c iσ ( i ) (cid:1) i =1 ,...,n : σ ∈ S (cid:17) ⊂ R n which is a convex polytope. Note that min σ ∈ S n X i =1 λ i c iσ ( i ) = min C ∈F n X i =1 λ i C i . We have the following geometric characterization of the fact that the data are rationalizable:

Proposition 5.

The data are rationalizable if and only if is an extreme point of F with acomponentwise positive vector in the normal cone.Proof. Introduce W ( λ ) = min C ∈F P ni =1 λ i C i . This concave function is the support functionof C . C i = c iσ ( i ) is an extreme point of F with a componentwise positive vector in the normalcone if and only is in the superdifferential of W at such a vector λ . This holds if and only if W ( λ ) = P ni =1 λ i C i . 10 Strong and weak rationalizability

Rationalizability of the data by a single consumer, as tested by Afriat’s inequalities, is an im-portant empirical question. Hence it is of interest to introduce measures of how close the datais from being rationalizable. Since Afriat’s original work on the topic, many authors have setout proposals to achieve this. Afriat’s original “efﬁciency index” is the largest e ≤ such that R eij = R ij + (1 − e ) b i is cyclically consistent, where b i > is a ﬁxed vector with positivecomponents. In Afriat’s setting, R ij = x j · p i − x i · p i , and R eij = x j · p i − ex i · p i , so b i = x i · p i .Many other tests and empirical approaches have followed and are discussed in Varian’s (2006)review paper. A recent proposal is given in Echenique, Lee and Shum (2011), who introducethe “money pump index” as the amount of money that could be extracted from a consumer withnon-rationalizable preferences.In the same spirit, we shall introduce indices that will measure how far the dataset is frombeing rationalizable. The indices we shall build are connected to the dual interpretation ofrevealed preferences we have outlined above. Our measures of departure from rationalizabilitywill be connected to measures of departure from Pareto efﬁciency in that problem. One measureof departure from efﬁciency is the welfare gains that one would gain from moving to the efﬁcientfrontier; this is interpretable as Debreu’s (1951) coefﬁcient of resource utilization , in the caseof a convex economy. This is only an analogy, as we are here in an indivisible setting, but thisexactly the idea we shall base the construction of our indices on.It seems natural to introduce index A as A = max λ ∈ ∆ min σ ∈ S n X i =1 λ i R iσ ( i ) where ∆ = { λ ≥ , P ni =1 λ i = 1 } is the simplex of R n . Indeed, we have A ≤ , and by compacity of ∆ , equality holds if and only if there exist λ ∈ ∆ such that min σ ∈ S n X i =1 λ i R iσ ( i ) = 0 . Of course, this differs from our characterization of rationalizability in Theorem 4, as therethe weights λ i ’s need to be all positive, not simply nonnegative. It is easy to construct exampleswhere (4) holds with some zero λ i ’s and σ ( i ) = i is not efﬁcient. For example, in the housing We thank Don Brown for suggesting this interpretation to us. i = 1 has his most preferred option, then λ = 0 and all the other λ i ’s arezero, and A = 0 , thus (4) holds. However, allocation may not be Pareto because there may beinefﬁciencies among the rest of the individuals.Hence imposing λ > is crucial. Fortunately, it turns out that one can restrict the simplex ∆ to a subset which is convex, compact and away from zero, as shown in the next lemma. Lemma 6.

There is ε > (dependent only on the entries of matrix c , but independent on thematching considered) such that the following disjuction holds • either there exist no scalars λ i > satisfying the condtions in Theorem 4 • or there exist scalars λ i > satisfying the condtions in Theorem 4 and such that (cid:26) λ i ≥ ε for all i, P ni =1 λ i = 1 .Proof. Standard construction (see (

Fostel et al. (2004) )) of the λ i ’s and the v i ’s provides a de-terministic procedure that returns strictly positive λ i ≥ within a ﬁnite and bounded number ofsteps, with only the entries of R ij as input; hence λ , if it exists, is bounded, so there exists M depending only on R such that P ni =1 λ i ≤ M . Normalizing λ so that P ni =1 λ i = 1 , one seesthat one can choose ε = 1 /M .We denote ∆ ε the set of such vectors λ , and we recall that ∆ is the set of λ such that λ i ≥ for all i and P ni =1 λ i = 1 . Recall R ij = c ij − c ii , and introduce A ∗ = max λ ∈ ∆ ε min σ ∈ S n X i =1 λ i R iσ ( i ) , so that we have A ∗ ≤ and equality if and only if the data are rationalizable (as in this case there exist λ i > such thatthe characterization of rationalizability in 4 is met). Further, as ∆ ε ⊂ ∆ , one gets max λ ∈ ∆ ε min σ ∈ S n X i =1 λ i R iσ ( i ) | {z } A ∗ ≤ max λ ∈ ∆ min σ ∈ S n X i =1 λ i R iσ ( i ) | {z } A ≤ . A leads naturally to introduce a new index which comesfrom the dual min-max program B = min σ ∈ S max λ ∈ ∆ n X i =1 λ i R iσ ( i ) = min σ ∈ S max i ∈{ ,...,n } R iσ ( i ) Note that the inequality max min ≤ min max always hold, so A ≤ B , and further, we have B ≤ max i ∈{ ,...,n } R ii = 0 . Therefore we have max λ ∈ ∆ ε min σ ∈ S n X i =1 λ i R iσ ( i ) | {z } A ∗ ≤ max λ ∈ ∆ min σ ∈ S n X i =1 λ i R iσ ( i ) | {z } A ≤ min σ ∈ S max λ ∈ ∆ n X i =1 λ i R iσ ( i ) | {z } B ≤ . We shall come back to the interpretation of A ∗ = 0 , A = 0 and B = 0 as stronger or weakerforms of rationalizability of the data. Before we do that, we summarize the above results. Proposition 7.

We have:(i) A ∗ = 0 if and only if there exist scalars v i and weights λ i > such that v j − v i ≤ λ i R ij . (ii) A = 0 if and only if there exist scalars v i and weights λ i ≥ , not all zero, such that v j − v i ≤ λ i R ij . (iii) B = 0 if and only if ∀ σ ∈ S, ∃ i ∈ { , ..., n } : R iσ ( i ) ≥ . (iv) One has A ∗ ≤ A ≤ B ≤ . (6)Note that it is easy to ﬁnd examples where these inequalities hold strict for n ≥ . Proof. (i) follows from Lemma 6. To see (ii), note that A = 0 is equivalent to the existenceof λ ∈ ∆ such that min σ ∈ S P ni =1 λ i R iσ ( i ) = 0 , and the rest follows from Theorem 4. Thecondition in (iii) is equivalent to the fact that for all σ ∈ S , max i ∈{ ,...,n } R iσ ( i ) ≥ , that is forall σ ∈ S , there exists i ∈ { , ..., n } such that R iσ ( i ) ≥ . The inequalities in (iv) were explainedabove. 13n an unpublished manuscript ( ( Geanakoplos (2006) )) that he kindly communicated to uson our request, John Geanakoplos introduces the following minmax problem, which in ournotations can be deﬁned as G = max λ ∈ ∆ min σ ∈ C n X i =1 λ i R iσ ( i ) where C ⊂ S is the set of permutations that have only one cycle, ie. such that there exist a cycle i , ..., i p +1 = i such that σ ( i k ) = i k +1 and σ ( i ) = i if i / ∈ { i , ..., i p } . As C ⊂ S one has A ≤ G ≤ . Geanakoplos uses index G and von Neuman’s minmax theorem in order to provide an in-teresting alternative proof of Afriat’s theorem. However, it seems that index G is not directlyconnected with the assignment problem. A ∗ , A , B As seen above, the index A ∗ was constructed so that A ∗ = 0 if and only if the data are rational-izable. The indexes A and B will both be equal to if the data are rationalizable; hence A < or B < imply that the data is not rationalizable; however the converse is not true, so theseindexes can be interpreted as measures of weaker form of rationalizability. Hence we wouldlike to give a meaningful interpretation of the situations where A = 0 and B = 0 . It turnsout that A = 0 is equivalent to the fact that a subset of the observations can be rationalized,the subset having to have a property of coherence that we now deﬁne. B = 0 is equivalentto the fact that one cannot partition the set of observations into increasing cycles , a notion wewill now deﬁne. In other words, the condition B < means that any observation is part of apreference cycles–indeed a very strong departure from rationalizability. These indexes may beused to compute identiﬁed regions in models with partial identiﬁcation; see Ekeland, Galichonand Henry (2010) and references therein.Throughout this subsection it will be assumed that no individual is indifferent between twodistinct consumptions for the direct revealed preference relation, that is: Assumption A.

In this subsection, R is assumed to verify R ij = 0 for i = j . We ﬁrst deﬁne the notion of coherent subset of observations.

Deﬁnition 8 (Coherent subset) . In the revealed preference problem, a subset of observationsincluded in { , ..., n } is said to be coherent when i ∈ I and i directly revealed preferred to j implies j ∈ I . Namely, I is coherent when i ∈ I and R ij < implies j ∈ I.

14n particular, { , ..., n } is coherent; any subset of observations where each observation isdirectly revealed preferred to no other one is also coherent. Next, we deﬁne the notion ofincreasing cycles. Deﬁnition 9 (Increasing cycles) . A cycle i , ..., i p +1 = i is called increasing when each obser-vation is strictly directly revealed preferred to its predecessor. Namely, cycle i , ..., i p +1 = i isincreasing when R i k i k +1 < for all k ∈ { , ..., p } . Of course, the existence of an increasing cycle implies that the matrix R is not cyclicallyconsistent, hence it implies in the revealed preference problem that the data are not rationaliz-able. It results from the deﬁnition that an increasing cycle has length greater than one.We now state the main result of this section, which provides an economic interpretation forthe indexes A ∗ , A and B . Theorem 10.

We have:(i) A ∗ = 0 iff the data are rationalizable,(ii) A = 0 iff a coherent subset of the data is rationalizable,(iii) B = 0 iff there is no partition of { , ..., n } in increasing cycles.and (i) implies (ii), which implies (iii).Proof. ( i ) was proved in Theorem 4 above.Let us show the equivalence in ( ii ) . From Proposition 7 A = 0 is equivalent to the existenceof ∃ λ i ≥ , P ni =1 λ i = 1 and v ∈ R n such that v j − v i ≤ λ i R ij , so deﬁning I as the set of i ∈ { , ..., n } such that λ i > , this implies a subset I of the data isrationalizable. We now show that I is coherent. Indeed, for any two k and l not in I and i in I ,one has v k = v l ′ ≥ v i ; thus if i ∈ I and R ij < , then v j < v i , hence j ∈ I , which show that I is coherent.Conversely, assume a coherent subset of the data I is rationalizable. Then there exist ( u i ) i ∈ I and ( µ i ) i ∈ I such that µ i > and u j − u i ≤ µ i R ij for i, j ∈ I . Complete by u i = max k ∈ I u k for i / ∈ I , and introduce ˜ R ij = 1 { i ∈ I } R ij . One has ˜ R ij < implies i ∈ I and R ij < hence j ∈ I by the coherence property of I , thus u j − u i < .15ow ˜ R ij = 0 implies either i / ∈ I or R ij = 0 thus i = j ; in both cases, u j ≤ u i . Therefore, onehas ˜ R ij < implies u j − u i < , and ˜ R ij ≤ implies u j − u i ≤ .Hence by Theorem 0, there exist v i and ¯ λ i > such that v j − v i ≤ ¯ λ i ˜ R ij and deﬁning λ i = ¯ λ i { i ∈ I } , one has v j − v i ≤ λ i R ij which is equivalent to A = 0 . ( iii ) Now Proposition 7 implies that

B < implies that there is σ ∈ S such that ∀ i ∈{ , ..., n } , R iσ ( i ) < . The decomposition of σ in cycles gives a partition of { , ..., n } inincreasing cycles. ( iv ) The implication ( i ) ⇒ ( ii ) ⇒ ( iii ) results from inequality A ∗ ≤ A ≤ B ≤ . To conclude, we make a series or remarks.First, we have shown how our dual interpretation of Afriat’s theorem in its original RevealedPreference context as well as in a less traditional interpretation of efﬁciency in the housingproblem could shed new light and give new tools for both problems. It would be interesting toundestand if there is a similar duality for problems of revealed preferences in matching markets,recently investigated by Galichon and Salani´e (2010) and Echenique et al. (2011).Also, we argue that it makes sense to investigate “weak rationalizability” (ie. A = 0 or B = 0 ) instead of “strong rationalizability” (ie. A ∗ = 0 ), or equivalently, it may make senseto allow some λ i ’s to be zero. In the case of A = 0 , recall the λ i ’s are interpreted in Afriat’stheory as the Lagrange multiplier of the budget constraints. Allowing for λ = 0 corresponds toexcluding wealthiest individuals as outliers. It is well-known that when taken to the data, strongrationalizability is most often rejected. It would be interesting to test econometrically for weakrationalizability, namely whether A = 0 . B < is a very strong measure of nonrationalizability, as B < means that one can ﬁnda partition of the observation set into increasing cycles, which seems a very strong violations16f the Generalized Axiom of Revealed Preference. But this may arise in some cases, especiallywith a limited number of observations.We should emphasize on the fact that indexes A ∗ , A and B provide a measure of how closethe data is from being rationalizable, in the spirit of Afriat’s efﬁciency index. It is clear for everyempirical researcher that the relevant question about revealed preference in consumer demand isnot whether the data satisfy GARP, it is how much they violate it. These indexes are an answerto that question. Also the geometric interpretation of rationalizable is likely to provide usefulinsights for handling unobserved heterogeneity. We plan to investigate this question in furtherresearch.Last, it seems that an interesting research avenue (not pursued in the present paper) dealswith exploring similar connections as in this paper, and understand what would be the analogueof variants of the theories discussed here. For instance, in the light of the duality exposed in thepresent paper, it would be interesting to characterize the dual theory to the Revealed Preferencetheory for collective models of household consumption as in Cherchye, De Rock and Vermeulen(2007, 2010). Similarly, it would also be of interest to understand the Revealed Preference dualof the two-sided version of the housing problem, namely the marriage problem, where bothsides of the market have preferences over the other side. In these two cases as in others, whatbecomes of the connection stressed in the current paper? this question, too, is left for futureresearch. References and Notes

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