# Ordinal and cardinal solution concepts for two-sided matching

OOrdinal and cardinal solution concepts for two-sided matching Federico Echenique California Institute of Technology

Alfred Galichon Sciences Po, ParisApril 23, 2014 The authors wish to thank Juan Pereyra Barreiro and Aditya Kuvalekar formany useful suggestions, and for pointing out a mistake in one of the examples ina previous version of the paper. Address: Division of the Humanities and Social Sciences, Mail Code 228-77,Caltech, Pasadena, CA 91125, USA. E-mail: [email protected] Address: Department of Economics, Sciences Po, 75007 Paris, France. E-mail: [email protected] This research has received funding from theEuropean Research Council under the European Union’s Seventh Framework Pro-gramme (FP7/2007-2013) / ERC grant agreement n ◦ a r X i v : . [ ec on . GN ] F e b bstract We characterize solutions for two-sided matching, both in the transferableand in the nontransferable-utility frameworks, using a cardinal formulation.Our approach makes the comparison of the matching models with and with-out transfers particularly transparent. We introduce the concept of a no-tradematching to study the role of transfers in matching. A no-trade matching isone in which the availability of transfers do not aﬀect the outcome.

JEL classiﬁcation numbers:

C71,C78

Key words:

Stable matching; Afriat’s Theorem; Gale and Shapley; Assignmentgame

Introduction

We explore the role of transfers and cardinal utility in matching markets.Economists regularly use one- and two-sided models, with and without trans-fers. For example auctions allow for monetary transfers among the agents,while models of marriage, organ donation and “housing” exchanges do not.There are two-sided matching models of the labor market without transfers,such as the market for medical interns in the US; and traditional models ofthe labor market where salaries, and therefore transfers, are allowed. We seekto understand how and why transfers matter in markets for discrete goods.The question is interesting to us as theorists, but it also matters greatlyfor one of the most important applications of matching markets, namely themedical interns market. In the market for medical interns in the US (see A. E.Roth, 1984a, A. Roth and Sotomayor, 1990, and A. E. Roth, 2002), hospitalsmatch with applicants using a centralized clearinghouse that implements astable matching. We always think of this market as one without transfers,because salaries are ﬁxed ﬁrst, before the matching is established. So at thestage in which the parties “bargain” over who is to be matched to whom,salaries are already ﬁxed, and thus there are no transfers.There is a priori no reason for things to be this way. Hospitals and internscould instead bargain over salaries and employment at the same time. Thisis arguably the normal state of aﬀairs in most other labor markets; and ithas been speciﬁcally advocated for the medical interns market in the US(see Crawford, 2005). It is therefore important to understand the impact ofdisallowing transfers in a matching market. Our paper is a ﬁrst step towardsunderstanding this problem.In a two-sided matching market—for our purposes, in the Gale-Shapleymarriage market—this impact is important. We consider two canonical mod-els: the marriage market without transfers (the NTU model) and the mar-riage market with transfers (the TU model, also called the assignment game).There are Pareto eﬃcient, and even stable, matchings in the NTU modelthat a utilitarian social planner would never choose, regardless of how sheweights agents’ utilities. A utilitarian social planner has implicitly access totransfers. Our results motivate an investigation into the distance betweenthe utilitarian welfare in the presence of transfers, and the utilitarian welfarein the absence of transfers. We show that this gap can be arbitrarily large.In fact, it can grow exponentially with the size of the market. Of coursethis is established in an environment in which utilities are bounded, and the1ound is constant while the market grows (otherwise the exercise would bemeaningless). See our Proposition 16 .We present results characterizing Pareto eﬃciency and the role of transfersin marriage models. Ex-ante Pareto optimality in the model with transfers ischaracterized by the maximization of the weighted utilitarian sum of utilities,while Pareto optimality when there are no transfers is equivalent to a diﬀerentmaximization problem, one where the weighted sum of “adjusted” utilities areemployed. Each of these problems, in turn, have a formulation as a system oflinear inequalities. The results follow (perhaps unexpectedly) from Afriat’stheorem in the theory of revealed preference.In order to explore the role of transfers, we study a special kind of stablematching: A no-trade stable matching in a marriage market is a matchingthat is not aﬀected by the presence of transfers. This is the central notion inour paper. Agents are happy remaining matched as speciﬁed by the matching,even if transfers are available, and even though they do not make use oftransfers. Transfers are available, but they are not needed to support thestable matching. There is thus a clear sense in which transfers play no rolein a no-trade stable matching.The notion of no-trade stable matching is useful for the following reason.We can think of transfers as making some agents better oﬀ at the expenseof others. It is then possible to modify a market by choosing a cardinal util-ity representation of agents preferences with the property that the matchingremains stable with and without transfers (Theorem 12). Under certain cir-cumstances, namely when the stable matchings are “isolated,” we can choosea cardinal representation that will work in this way for every stable matching.So under such a cardinal representation of preferences, any stable matchingremains stable regardless of the presence of transfers. Finally (Example 14),we cannot replicate the role of transfers by re-weighting agents’ utilities. Ingeneral, to instate a no-trade stable matching, we need the full freedom ofchoosing alternative cardinal representations.It is easy to generate examples of stable matchings that cannot be sus-tained when transfers are allowed, and of stable matchings that can be sus-tained with transfers (in the sense of being utilitarian-eﬃcient), but wheretransfers are actually used to sustain stability. We present conditions un- From the viewpoint of the recent literature in computer science on the “price of anar-chy” (see e.g. Roughgarden, 2005), Proposition 16 says that the “price of no transfers” canbe arbitrarily bad, and grow super-exponentially with the size of the market. The model without transfers was introduced by Gale and Shapley, 1962.The model with transfers is due to Shapley and Shubik, 1971. Kelso andCrawford, 1982 extended the models further, and in some sense Kelso andCrawford’s is the ﬁrst paper to investigate the eﬀects of adding transfersto the Gale-Shapley marriage model. A. E. Roth, 1984b and Hatﬁeld andMilgrom, 2005 extended the model to allow for more complicated contracts,not only transfers (see Hatﬁeld and Kojima, 2010 and Echenique, 2012 for adiscussion of the added generality of contracts). We are apparently the ﬁrstto consider the eﬀect of transfers on a given market, with speciﬁed cardinalutilities, and the ﬁrst to study the notion of a no-trade stable matching.

Let M and W be ﬁnite and disjoint sets of, respectively, men and women,which are assumed to be in equal number; M ∪ W comprise the agents inour model. We can formalize the marriage “market” of M and W in twoways, depending on whether we assume that agents preferences have cardinalcontent, or that they are purely ordinal. For our results, it will be crucial tokeep in mind the diﬀerence between the two frameworks.An ordinal marriage market is a tuple ( M, W, P ), where P is a preferenceproﬁle : a list of preferences > i for every man i and > j for every woman j .Each > i is a linear order over W , and each > j is a linear order over M .Here, agents always prefer being matched with anyone rather than beingunmatched. The weak order associated with > s is denoted by ≥ s for any TU and NTU stability are not comparable in this sense. Empirically, though, theyare comparable, with TU stability having strictly more testable implications than NTUstability Echenique et al., 2011. ∈ M ∪ W . We often specify a preference proﬁle by describing instead utility functionsfor all the agents. A cardinal marriage market is a tuple (

M, W, U, V ), where U and V deﬁne the agents’ utility functions: U ( i, j ) (resp. V ( i, j )) is theamount utility derived by man i (resp. woman j ) out of his match withwoman j (resp. man i ). The utility functions U and V represent P if, forany i and i (cid:48) in M , and j and j (cid:48) in W , U ( i, j ) > U ( i, j (cid:48) ) ⇐⇒ j > i j (cid:48) , and V ( i, j ) > V ( i (cid:48) , j ) ⇐⇒ i > j i (cid:48) . We say that U and V are a cardinal representation of P . Clearly, forany cardinal marriage market ( M, W, U, V ) there is a corresponding ordinalmarket.A one-to-one function σ : M → W is called a matching . When w = σ ( m )we say that m and w are matched, or married, under σ . In our setting,under a given matching, each man or woman is married to one and only onepartner of the opposite sex. We shall denote by A the set of matchings. Weshall assume that M and W have the same number of elements, so that A isnon-empty.In our deﬁnition of matching is that agents are always married: we donot allow for the possibility of singles.Under our assumptions, we can write M = { m , . . . , m n } and W = { w , . . . , w n } . For notational convenience, we often identify m i and w j withthe numbers i and j , respectively. So when we write j = σ ( i ) we meanthat woman w j and man m i are matched under σ . This usage is a bit dif-ferent than what it standard notation in matching theory, but it makes theexposition of our results a lot simpler.We shall often ﬁx an arbitrary matching, and without loss of generalitylet this matching be the identity matching , denoted by σ . That is, σ ( i ) = i. For a matching σ , let u σ ( i ) = U ( i, σ ( i )) and v σ ( j ) = V ( σ − ( j ) , j ). When σ = σ , we shall often omit it as a subscript and just use the notation u and v . A linear order is a binary relation that is complete, transitive and antisymmetric. Theweak order ≥ s is deﬁned as a ≥ s b if a = b or if a > s b . fractional matching is a matrix π = ( π i,j ) such that π ij ≥ π ij the probability thatindividuals i and j get matched, the constraints on π are1 = n (cid:88) i (cid:48) =1 π i (cid:48) j = n (cid:88) j (cid:48) =1 π ij (cid:48) ∀ i, j ∈ { , ..., n } (i.e. π is a bistochastic matrix ). It is a celebrated result (the Birkhoﬀ von-Neumann Theorem) that such matrices result from a lottery over matchings.Let B denote the set of all fractional matchings. We describe here some commonly used solution concepts. The ﬁrst solutionscapture the notion of Pareto eﬃciency. In second place, we turn to notionof core stability for matching markets. For simplicity of exposition, we writethese deﬁnitions for the speciﬁc matching σ . Of course by relabeling we canexpress the same deﬁnitions for an arbitrary matching.A solution concept singles out certain matchings as immune to certainalternative outcomes that could be better for the agents. If we view suchalternatives as arising ex-post, after any uncertainty over which matchingarises has been resolved, then we obtain a diﬀerent solution concept than ifwe view the alternatives in an ex-ante sense. Matching σ ( i ) = i is ex-post NTU Pareto eﬃcient , or simply ex-post Paretoeﬃcient if there is no matching σ that is at least as good as σ for allagents, and strictly better for some agents. That is, such that the inequalities U ( i, σ ( i )) ≥ U ( i, i ) and V ( σ − ( j ) , j ) ≥ V ( j, j ) cannot simultaneously holdwith at least one strict inequality.In considering alternative matchings, it is easy to see that one can restrictoneself to cycles. The resulting formulation of eﬃciency is very useful, asit allows us to relate eﬃciency with standard notions in the literature onrevealed preference.Hence matching σ ( i ) = i is ex-post Pareto eﬃcient if and only if for every5ycle i , ..., i p +1 = i , inequalities U ( i k , i k +1 ) ≥ U ( i k , i k ) and V ( i k , i k +1 ) ≥ V ( i k , i k ) cannot hold simultaneously unless they all are equalities. In otherwords: Observation 1

Matching σ ( i ) = i is ex-post Pareto eﬃcient if for everycycle i , ..., i p +1 = i , and for all k , inequalities U ( i k , i k +1 ) ≥ U ( i k , i k ) and V ( i k , i k +1 ) ≥ V ( i k , i k ) ,cannot hold simultaneously unless they are all equalities. In an ex-ante setting, we can think of probabilistic alternatives to σ .As a result, we obtain the notion of ex-ante Pareto eﬃciency. To deﬁne thisnotion, we require not only that there is no other matching which is preferredby every individual, but also that there is no lottery over matchings thatwould be preferred.Formally: Matching σ ( i ) = i is ex-ante NTU Pareto eﬃcient or simply ex-ante Pareto eﬃcient , if for any π ∈ B , and for all i and j , inequalities (cid:88) j π ij U ( i, j ) ≥ U ( i, i ) and (cid:88) i π ij V ( i, j ) ≥ V ( j, j )cannot hold simultaneously unless they are all equalities.Note that the problem of ex-post eﬃciency is purely ordinal, as ex-posteﬃciency of some outcome only depends on the rank order preferences, noton the particular cardinal representation of it. In contrast, the problem ofex-ante eﬃciency is cardinal, as we are adding and comparing utility levelsacross states of the world. The concept of NTU Pareto eﬃciency is interestingfor example in the context of school choice problems, where the assignmentis often not deterministic, and transfers are not permitted. In this context,this concept provides an adequate assessment of welfare. We now assume that utility is transferable across individuals. In this case, amatching is Pareto eﬃcient if no other matching produces a higher welfare, Recall that a cycle i , ..., i p , i p +1 = i is a permutation σ such that σ ( i ) = i , σ ( i ) = i ,..., σ ( i p − ) = i p , and σ ( i p ) = i . σ ( i ) = i is TU Pareto eﬃcient if there is no matching σ forwhich n (cid:88) i =1 U ( i, σ ( i )) + V ( i, σ ( i )) > n (cid:88) i =1 ( U ( i, i ) + V ( i, i )) . Observation 2

Matching σ ( i ) = i is TU Pareto eﬃcient if for every cycle i , ..., i p +1 = i , and for all k , inequalities p (cid:88) k =1 U ( i k , i k +1 ) + V ( i k , i k +1 ) ≥ p (cid:88) k =1 U ( i k , i k ) + V ( i k +1 , i k +1 ) cannot hold simultaneously unless they are all equalities. In the previous deﬁnitions, transfers are allowed across any individuals.One may have considered the possibility of transfers only between matchedindividuals. It is however well known since Shapley and Shubik, 1971 thatthis apparently more restrictive setting leads in fact to the same notion ofeﬃciency. As we recall below, TU Pareto eﬃciency is equivalent to TUstability, so to avoid confusions, we shall systematically refer to “TU stability”instead of “TU Pareto eﬃciency,” and, in the sequel, reserve the notion ofeﬃciency for NTU eﬃciency.

We now review notions of stability. Instead of focusing on the existence ofa matching which would be an improvement for everyone (as in Pareto eﬃ-ciency), we focus on matchings which would be an improvement for a newlymatched pair of man and woman. Thus we obtain two solution concepts,depending on whether we allow for transferable utility.Our deﬁnitions are classical and trace back to Gale and Shapley, 1962and Shapley and Shubik, 1971. See A. Roth and Sotomayor, 1990 for anexposition of the relevant theory. 7atching σ ( i ) = i is stable in the nontransferable utility matching mar-ket , or NTU stable if there is no “blocking pair” ( i, j ), i.e. a pair ( i, j ) suchthat U ( i, j ) > U ( i, i ) and V ( i, j ) > V ( j, j ) simultaneously hold.Hence, using our assumptions on utility, we obtain the following: Deﬁnition 3

Matching σ ( i ) = i is NTU stable if ∀ i, j : min ( U ( i, j ) − U ( i, i ) , V ( i, j ) − V ( j, j )) ≤ . Of course, this notion is an ordinal notion and should not depend onthe cardinal representation of men and women’s preferences, only on theunderlying ordinal matching market.

Utility is transferable across pair ( i, j ) if there is the possibility of a utilitytransfer t (of either sign) from j to i such that the utility of i becomes U ( i, j ) + t , and utility of j becomes V ( i, j ) − t . When we assume that utilityis transferable, in contrast, we must allow blocking pairs to use transfers.Then a couple ( i, j ) can share, using transfers, the “surplus” U ( i, j ) + V ( i, j ).Thus we obtain the deﬁnition: Deﬁnition 4

Matching σ ( i ) = i is TU stable , if there are vectors ˜ u ( i ) and ˜ v ( j ) such that for each i and j , ˜ u ( i ) + ˜ v ( j ) ≥ U ( i, j ) + V ( i, j ) must hold with equality for i = j . By a celebrated result of Shapley and Shubik, 1971, this notion is equiv-alent to the notion of TU Pareto eﬃciency. Note that there may be multiplevectors ˜ u and ˜ v for the given matching σ . The notions of TU and NTU stability have been known and studied for a verylong time. Here, we seek to better understand the eﬀect that the possibilityof transfers has on a matching market. We introduce a solution concept thatis meant to relate the two notions. 8ote that if matching σ ( i ) = i is TU stable, then there are transfersbetween the matched partners, say from woman i to man i , equal to T i = ˜ u ( i ) − U ( i, i ) = V ( i, i ) − ˜ v ( i ) (2.1)where the payoﬀs ˜ u ( i ) and ˜ v ( j ) are those of Deﬁnition 4. We want to un-derstand the situations when matching σ ( i ) = i is TU stable but when noactual transfers are made “in equilibrium.” As a result, the matching σ isNTU stable as well as it is TU stable.We motivate the notion of a No-Trade stable matching with an example.We present a matching market with a matching which is both the uniqueTU stable matching and also the unique NTU stable matching. In order foragents to accept it, however, transfers are needed. Example 5

In this and other examples, we write the payoﬀs U and V inmatrix form. In the matrices, the payoﬀ in row i and column j is the utility U ( i, j ) for man i in matrix U , and utility V ( i, j ) for woman j in matrix V .Consider the following utilities U = , V = Note that the matching σ ( i ) = i is the unique NTU stable matching,and is also the unique TU stable matching. To sustain it in the TU game,however, requires transfers. Indeed, u ( i ) = 0 and v ( j ) = 2 cannot be TUstable payoﬀs as u ( i ) + v ( j ) < U ( i , j ) + V ( i , j ) = 3 contradicts Deﬁnition 4. Intuitively, one needs to compensate agent i = 1 in order for him to remained matched with j = 1 . Hence, even though σ is NTU-stable and TU-stable, transfers between the agents are required to sustain it as TU-stable. Anticipating the deﬁnition to follow, this means thismatching is not a No-trade stable matching. Matching σ ( i ) = i is no-trade stable when it is TU stable and thereare no actual transfers between partners at equilibrium. In other words,Equation (2.1) should hold with T i = 0. That is, U ( i, i ) = u ( i ), V ( j, j ) = v ( j ), and so: 9 eﬁnition 6 (No-Trade Matching) Matching σ ( i ) = i is no-trade stable ifand only if for all i and j , U ( i, j ) + V ( i, j ) ≤ U ( i, i ) + V ( j, j ) . Therefore in a no-trade stable matching, two matched individuals wouldhave the opportunity to operate monetary transfers, but they choose not todo so. To put this in diﬀerent terms, in a no-trade stable matching, spousesare “uncorrupted” because no monetary transfer actually takes place betweenthem, but they are not “incorruptible”, because the rules of the game wouldallow for it.

We now present simple characterizations of the solution concepts describedin Section 2.2.Our characterizations involve cardinal notions, even for the solutions thatare purely ordinal in nature. The point is to characterize all solutions usingsimilar concepts, so it is easier to understand how the solutions diﬀer. It willalso help us understand the role of transfers in matching markets.Deﬁne σ as the matching such that σ ( i ) = i . We need to introduce thefollowing notation: R ij = U ( i, i ) − U ( i, j ) S ij = V ( j, j ) − V ( i, j ) , deﬁned for each i ∈ M and j ∈ W . Note that R ij measures how much i prefers his current partner to j , and S ij measures how much j prefers hercurrent partner to i . These two concepts are dependent on the matching σ ,which we take as ﬁxed in the following result. Theorem 7

Matching σ ( i ) = i is:(a) No-trade stable iﬀ for all i and j in { , ..., n } ≤ R ij + S ij (3.1) (b) NTU stable iﬀ for all i and j in { , ..., n } ≤ max ( R ij , S ij ) (3.2)10 c) TU stable iﬀ there exists T ∈ R n such that for all i and j in { , ..., n } T j − T i ≤ R ij + S ij (3.3) (d) Ex-ante Pareto eﬃcient iﬀ there exist v i and λ i , µ j > such that forall i and j in { , ..., n } v j − v i ≤ λ i R ij + µ j S ij (3.4) (e) Ex-post Pareto iﬀ there exist v i and λ i > such that for all i and j in { , ..., n } v j − v i ≤ λ i max ( R ij , S ij ) . (3.5)Observe that (3.4) and (3.5) are “Afriat inequalities,” using the termi-nology in revealed preference theory.As a consequence of the previous characterizations, it is straightforwardto list the chains of implications between the various solution concepts. Theorem 8 (i) The two following chains of implications always hold:– No-trade Stable implies TU Stable, TU Stable implies Ex-ante ParetoEﬃcient, Ex-ante Pareto Eﬃcient implies Ex-post Pareto Eﬃcient, and– No-trade Stable implies NTU stable, NTU stable implies Ex-post ParetoEﬃcient.(ii) Assume that there are two agents on each side of the market. Thentwo additional implications hold:– Ex-Post Pareto Eﬃcient implies (and thus is equivalent to) Ex-AntePareto Eﬃcient, and– NTU stable implies Ex-Ante Pareto Eﬃcient.Any further implication which does not logically follow from those writtenis false. See Figure (1).(iii) Assume that there are at least three agents on each side of the market.Then any implication that does not logically follow from the ones stated inpart (i) of Theorem 7 above are false. See Figure (2).

The implications in Theorem 8 are illustrated in Figures (1) and (2).The proof of Theorem 8 is given in the Appendix. It relies on the followingcounterexamples. 11igure 1: Summary of the implications in Theorem 8, part (ii) when thereare only two agents on each side of the market.

Example 9

Consider U = (cid:18) −

21 0 (cid:19) and V = (cid:18) −

21 0 (cid:19) (3.6)

Then σ is TU stable (and thus Ex-Ante Pareto Eﬃcient and Ex-PostPareto Eﬃcient), but not NTU stable, and not No-trade stable. Example 10

Consider U = − − − , V = − − − Then σ is NTU stable (hence Ex-Post Pareto eﬃcient). But it is not Ex-Ante Pareto eﬃcient (hence neither No-trade stable nor TU stable). Indeedconsider the fair lottery over the 6 existing pure assignments. Under thislottery, each agent achieves a payoﬀ of 1/3, hence this lottery is ex-antepreferred by each agent to σ . Example 11

Consider now U = (cid:18) − (cid:19) and V = (cid:18) (cid:19) (3.7) Then σ is ex-ante Pareto eﬃcient, and it is ex-post Pareto eﬃcient, butit is not TU stable, and it is not NTU stable. Ex-ante Pareto eﬃciencyfollows from λ i = 5 , µ j = 1 and v = 2 , v = 1 . Ex-post Pareto eﬃciencyis clear. It is easily seen that σ is not TU stable. It is also clear that thematching is not NTU stable, as i = 1 , j = 2 form a blocking pair. As we explained above, we use no-trade stability to shed light on the role oftransfers. Given a stable NTU matching, one may ask if is there a cardinalrepresentation of the agents’ utility such that the stable matching is no-tradestable. 13s we shall see, the answer is yes if we are allowed to tailor the cardinalrepresentation to the given stable matching. If we instead want a represen-tation that works for all stable matchings in the market, we shall resort to aregularity condition: that the matchings be isolated.Finally, some statements in Theorem 7 involve the rescaling of utilities:they show how optimality can be understood through the existence of weights on the agents that satisfy certain properties. We can similarly imagine ﬁnd-ing, not an arbitrary cardinal representation of preferences, but a restrictedrescaling of utilities that captures the role of transfers. That is to say, arescaling of utilities that ensures that the matching is no-trade stable. Weshall present an example to the eﬀect that such a rescaling is not possible.

Our ﬁrst question is whether for a given NTU stable matching, there is acardinal representation of preferences under which the same matching is No-trade stable. The answer is yes.

Theorem 12

Let ( M, W, P ) be an ordinal matching market. If σ is a NTUstable matching, then there is a cardinal representation of P such that σ is ano-trade stable matching in the corresponding cardinal market. It is natural to try to strengthen this result in two directions. First, wecould expect to choose the cardinal representation of preferences as a linearrescaling of a given cardinal representation of the preferences. Given what weknow about optimality being characterized by choosing appropriate utilityweights, it makes sense to ask whether any stable matching can be obtainedas a No-Trade Stable Matching if one only weights agents in the right way.Namely:

Problem 13

Is it the case that matching σ is NTU stable if and only if thereis λ i , µ j > such that for all i and j , ≤ λ i R ij + µ j S ij ?After all, transfers favor some agents over others, and utility weights playa similar role. Our next example shows that this is impossible. It exhibits astable matching that is not No-Trade for any choice of utility weights.14 xample 14 Consider a marriage market deﬁned as follows. The sets of menand women are: M = { m , m , m , m } and W = { w , w , w , w } . Agents’preferences are deﬁned through the following utility functions: U = .

01 0 1 / −

10 1 − / / / / / / / / / , V = / −

11 0 − / / / / / / / / / The unique stable matching is underlined. Uniqueness is readily veriﬁedby running the Gale-Shapley algorithm. So u i = v j = 1 / for all i and j .Yet it is shown in the appendix that there are no λ i , µ j > such that for all ( i, j ) , λ i ( u i − U ij ) + µ j ( v j − V ij ) ≥ . This example has indiﬀerence in payoﬀs. It is simple to perturb the payoﬀsso that there are no more indiﬀerence, and the conclusion still holds. This isexplained in the proof in the appendix.

Example 14 has the following implication (which also follows from Exam-ple 5).

Corollary 1

There are cardinal matching markets that do not possess a no-trade matchings.

Given Example 14, it is clear that No-Trade can only be achieved by ap-propriate choice of agents’ utility functions. Our next question deals with theexistence of cardinal utilities such that the set of No-trade stable matchingsand NTU stable matchings will coincide for all stable matchings in a mar-ket. We show that if the stable matchings are isolated then one can choosecardinal utilities such that all stable matchings are No-Trade.Let S ( P ) denote the set of all stable matchings. A matching σ ∈ S ( P ) is isolated if σ (cid:48) ( a ) (cid:54) = σ ( a ) for all a ∈ M ∪ W and all σ (cid:48) ∈ S ( P ) \ { σ } . Theorem 15

There is a representation ( U, V ) of P such that for all σ ∈ S ( P ) , if σ is isolated then σ is no trade stable for ( U, V ) . The question whether the conclusion holds without the assumption thatthe matching is isolated remains open to investigation.15 .2 Price of no transfers

The logic of the previous subsection can be pushed further, to obtain a “Priceof Anarchy,” in the spirit of the recent literature in computer science (Rough-garden, 2005). We quantify the cost in social surplus (sum of agents’ utilities)that results from NTU stability: we can think of this cost as an eﬃciencygap inherent in the notion of NTU stable matching. The result is that thegap can be arbitrarily large, and that it grows “super exponentially” in thesize of the market (i.e. it grows at a faster rate than n g , for any g , where n is the size of the market).Let ∆ (cid:15) denote the subset of the simplex in R n in which every componentis at least (cid:15) : ∆ (cid:15) = { (( α ( i )) i ∈ M , ( β ( j )) j ∈ W ) : ∀ i ∈ M α ( i ) ≥ (cid:15), ∀ j ∈ W, β ( j ) ≥ (cid:15) } . Let S ( M, W, U, V ) denote the set of stable matchings in the cardinalmatching market (

M, W, U, V ). In the statement of the results below, wewrite the matchings in S ( M, W, U, V ) as fractional matchings π in whichevery entry in π is either 0 or 1. Proposition 16

For every (cid:15) > , n , g , and K > There is a cardinal mar-riage market ( M, W, U, V ) , with n men and women, and where utilities U and V are bounded by K such that min ( α,β ) ∈ ∆ (cid:15) (cid:26) max π ∈ Π (cid:80) ni =1 α ( i ) (cid:80) nj (cid:48) =1 π i,j (cid:48) U ( i, j (cid:48) )+ (cid:80) nj =1 β ( j ) (cid:80) ni (cid:48) =1 π i (cid:48) ,j V ( i (cid:48) , j ) (cid:27) max ( α,β ) ∈ ∆ (cid:15) (cid:26) max π ∈ S ( M,W,U,V ) (cid:80) ni =1 α ( i ) (cid:80) nj (cid:48) =1 π i,j (cid:48) U ( i, j (cid:48) )+ (cid:80) nj =1 β ( j ) (cid:80) ni (cid:48) =1 π i (cid:48) ,j V ( i (cid:48) , j ) (cid:27) is Ω( n g K )Proposition 16 shows that the gap in the sum of utilities, between themaximizing (probabilistic) matchings, and the stable matchings, is large andgrows with the size of the market at a rate that is arbitrarily large. Moreover,the gap is large regardless of how one weighs agents’ utilities.Of course, the interpretation of Proposition 16 is not completely straight-forward. It does not seem right to compare the sum of utilities in a model inwhich transfers are not allowed with the sum of utilities in the TU model. Nevertheless, we hope that Proposition 16 sheds additional light on the roleof transfers in matching markets. This problem of interpretation is present throughout the literature on the price ofanarchy. eferences Crawford, V. (2005). The ﬂexible-salary match: A proposal to increase thesalary ﬂexibility of the national resident matching program.

Journalof Economic Behavior & Organization , , 149–160.Echenique, F. (2012). Contracts versus salaries in matching. American Eco-nomic Review , (1), 594–601.Echenique, F., Lee, S., Shum, M., & Yenmez, M. B. (2011). The revealedpreference theory of stable and extremal stable matchings [WorkingPaper].Ekeland, I., & Galichon, A. (2013). The housing problem and revealed pref-erence theory: Duality and an application.

Economic Theory , (3),425–441.Fostel, A., Scarf, H., & Todd, M. (2004). Two new proofs of afriat’s theorem. Economic Theory , , 211–219.Gale, D., & Shapley, L. S. (1962). College admissions and the stability ofmarriage. The American Mathematical Monthly , (1), 9–15.Hatﬁeld, J., & Kojima, F. (2010). Substitutes and stability for matching withcontracts. Journal of Economic Theory , (5), 1704–1723.Hatﬁeld, J., & Milgrom, P. (2005). Matching with contracts. American Eco-nomic Review , (4), 913–935.Kelso, A. S., & Crawford, V. P. (1982). Job matching, coalition formation,and gross substitutes. Econometrica , , 1483–1504.Roth, A., & Sotomayor, M. (1990). Two-sided matching: A study in game-theoretic modelling and analysis (Vol. 18). Cambridge University Press,Cambridge England.Roth, A. E. (1984a). The evolution of the labor market for medical internsand residents: A case study in game theory.

The Journal of PoliticalEconomy , (6), 991–1016.Roth, A. E. (1984b). Stability and polarization of interests in job matching. Econometrica , (1), 47–57.Roth, A. E. (2002). The economist as engineer: Game theory, experimenta-tion, and computation as tools for design economics. Econometrica , (4), pp. 1341–1378.Roughgarden, T. (2005). Selﬁsh routing and the price of anarchy (Vol. 74).Shapley, L., & Shubik, M. (1971). The assignment game I: The core.

Inter-national Journal of Game Theory , (1), 111–130.17 Appendix: Proofs

A.1 Proof of Theorem 7

Proof. (i) (a) Characterization of No-trade stable matchings as in (3.1)follows directly from the deﬁnition.(b) Characterization (3.2) follows directly from the deﬁnition.(c) For characterization of TU stability in terms of (3.3), recall that ac-cording to the deﬁnition, matching σ ( i ) = i is TU Stable if there are vectors u ( i ) and v ( j ) such that for each i and j , u ( i ) + v ( j ) ≥ U ( i, j ) + V ( i, j )with equality for i = j . Hence, there exists a monetary transfer T i (of eithersign) from man i to woman i at equilibrium given by T i = u ( i ) − U ( i, i ) = V ( i, i ) − v ( i ) . The stability condition rewrites as T j − T i ≤ R ij + S ij , thus, one is led tocharacterization (3.3).(d) For characterization of Ex-ante Pareto eﬃcient matchings in termsof (3.4), the proof is an extension of the proof by Fostel et al., 2004, whichgive in full. Assume σ is Ex-ante eﬃcient. Then the Linear Programmingproblem max (cid:88) i x i + (cid:88) j y j s.t. x i = − (cid:88) j π ij R ij and y j = − (cid:88) i π ij S ij (cid:88) k π ik = (cid:88) k π ki and (cid:88) k π ik = 1 x i ≥ y j ≥ π ij ≥ . is feasible and its value is zero. Thus it coincides with the value of its dual,which is min − (cid:88) i φ i s.t. v j − v i ≤ λ i R ij + µ j S ij + φ i λ i ≥ µ j ≥ λ i , µ j , v i and φ i in the dual problem are the Lagrange multi-pliers associated to the four constraints in the primal problem, and variables π ij , x i , and y j in the primal problem are the Lagrange multipliers associatedto the three constraints in the dual problem. Hence the dual program isfeasible, and there exist vectors λ , µ , and φ , such that v j − v i ≤ λ i R ij + µ j S ij + φ i (A.1) λ i ≥ µ j ≥ (cid:88) i φ i = 0but setting j = i in inequality (A.1) implies (because R ii = S ii = 0) that φ i ≥

0, hence as (cid:80) i φ i = 0, thus φ i = 0. Therefore it exist vectors λ i > µ j >

0, such that v j − v i ≤ λ i R ij + µ j S ij . (e) For characterization of Ex-post Pareto eﬃcient matchings in terms of(3.5), assume σ is Ex-post Pareto eﬃcient, and let Q ij = max ( R ij , S ij ) , so that by deﬁnition, matrix Q ij satisﬁes “cyclical consistency” : for any cycle i , ..., i p +1 = i , ∀ k, Q i k i k +1 ≤ ∀ k, Q i k i k +1 = 0 , (A.2)By the Linear Programming proof of Afriat’s theorem in Fostel et al., 2004 ,see implication (i) implies (ii) in Ekeland and Galichon, 2013, there are scalars λ i > v i such that (3.5) holds. A.2 Proof of Theorem 8

Proof. (i) No trade stable implies TU Stable is obtained by taking with T i = 0 in (3.3).TU Stable implies Ex-ante Pareto is obtained by taking v i = T i and λ i = µ j = 1 in (3.4). The link between Afriat’s theorem and the characterization of eﬃciency in the housingproblem was ﬁrst made in Ekeland and Galichon, 2013.

19o show that Ex-ante Pareto implies Ex-post Pareto, assume there exist v i and λ i , µ j > v j − v i ≤ λ i R ij + µ j S ij . Now assume max ( R ij , S ij ) ≤

0. Then v j − v i ≤

0, and the same implication holds with strict inequalities.By implication (iii) implies (ii) in Ekeland and Galichon, 2013, there existscalars v (cid:48) i and λ (cid:48) i such that v (cid:48) j − v (cid:48) i ≤ λ (cid:48) i max ( R ij , S ij ).No-trade stable implies NTU stable follows from R ij + S ij ≤ R ij , S ij ).NTU Stable implies Ex-post Pareto is obtained by taking λ i = 1 and v i = 0 in (3.5).Part (i) of the result is proved using a series of counterexample, which forthe most part only require two agents (one can incorporate a third neutralagents, which has zero utility regardless of the outcome).We show point (iii) before point (ii). In order to show (iii), it is enoughto show the following claims, proved in Examples 9 to 11: • TU Stable does not imply NTU Stable – cf. example 9 • NTU Stable does not imply Ex-Ante Pareto – cf. example 10 • Ex-Ante Pareto does not imply TU Stable – cf. example 11 • Ex-Ante Pareto does not imply NTU Stable – cf. example 11 • Ex-Post Pareto does not imply Ex-Ante Pareto – cf. example 10 • Ex-Post Pareto does not imply NTU stable – cf. example 11.To prove part (ii), we note that in the proof of part (i), the only instancewhere we needed three agents was to disprove that NTU stable implies Ex-Ante Pareto eﬃcient and to disprove that Ex-Post Pareto eﬃcient impliesEx-Ante Pareto eﬃcient. We will show that these implications actually holdwhen there are only two agents. Indeed, when there are two agents, σ isEx-Ante Pareto eﬃcient if there are positive scalars λ , λ , µ and µ suchthat 0 ≤ λ R + λ R + µ S + µ S which is equivalent to 0 ≤ max ( R , R , S , S ) . (A.3)Therefore, if σ is Ex-Post Pareto eﬃcient, then v j − v i ≤ λ i max ( R ij , S ij ).But either v − v or v − v is nonnegative, thus (A.3) holds, and Ex-PostPareto eﬃcient implies Ex-Ante Pareto eﬃcient.20 .3 Proof of Theorem 12 Assume µ ( i ) = i (this is w.l.o.g. as can always relabel individuals). Take R ij = U ( i, j ) − U ( i, i ) and S ij = V ( i, j ) − V ( j, j ). µ is Stable iﬀ min ( R ij , S ij ) ≤ i and j , with strict inequality for j (cid:54) = i . Consider¯ U ( i, j ) = 12 − e − tR ij for i (cid:54) = j ¯ U ( i, i ) = 0one has: • ¯ U ( i, j ) > > e − tR ij that is − log 2 > − tR ij that is tR ij > log 2 hence R ij > • ¯ U ( i, j ) < tR ij < log 2 hence R ij < t > max i (cid:54) = j (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) log 2 R ij (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) log 2 S ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) and let ¯ V ( i, j ) = 12 − e − tS ij for i (cid:54) = j ¯ V ( i, i ) = 0Then ¯ U ( i, j ) ≤ V ( i, j ) ≤

0, thus¯ U ( i, j ) + ¯ V ( i, j ) ≤ U ( i, i ) + ¯ V ( j, j ) . Thus µ is a No-Trade stable Matching associated to utilities ¯ U and ¯ V . A.4 Claim in Example 14

Rephrasing, we want to know if there are α ( m ) > β ( w ) > m, w ), α ( m )( u ( m ) − U ( m, w )) + β ( w )( v ( w ) − V ( m, w )) ≥ A which has one column for each m and each w , and onerow for each pair ( m, w ) ∈ M × W . The matrix A = ( a ( m,w ) ,a ) ( m,w ) ∈ M × W,a ∈ M ∪ W

21s deﬁned as follows. The row corresponding to ( m, w ) has zeroes in all its en-tries except in the columns corresponding to m and w . It has u ( m ) − U ( m, w )in m ’s column and v ( w ) − V ( m, w ) in w ’s column.The problem is to ﬁnd x (cid:29) A · x ≥

0. We introduce thematrix B such that the i ’th row of B is the vector e i = (0 , . . . , , . . . ,

0) witha 1 only in entry i . Then we want to ﬁnd a vector x ∈ R n such that A · x ≥ B · x (cid:29)

0. By Motzkin’s Theorem of the Alternative, such a vector x exists iﬀ there is no ( y, z ), with z > z ≥ z (cid:54) = 0) such that y · A + z · B = 0 .i i (cid:48) i i j j (cid:48) j w (cid:48) j j j j j (cid:48) j j j j j j (cid:48) j j j (cid:48) j j i (cid:48) i i i (cid:48) i i i i i i (cid:48) i i i i i (cid:48) i Utilities are: i i (cid:48) i i δ / / / / / /

30 0 1 / / − − / / j j (cid:48) j j / / / / / /

30 0 1 / / − − / / A are: i i (cid:48) i i j j (cid:48) j j i, j / − (1 + δ ) 0 0 0 1 / i, j (cid:48) / / − i (cid:48) , j / − / i (cid:48) , j (cid:48) − / / A is ( − δ, , , , , , , i, j ) is: i i (cid:48) i i j j (cid:48) j j i, j / / − / , y = (1 , , , , , . . . ,

0) and z = ( δ, , . . . ,

0) exhibit a solution to thealternative system as y · A + z · B = ( − δ, , , , , , ,

0) + δ (1 , , . . . ,

0) = 0Observe that in the construction of a solution to the dual system above,we could perturb utilities by adding payoﬀs to each entry, in such a way thatwe obtain the matrix A + A (cid:48) instead of A above. By choosing the perturbationso that y · A (cid:48) = 0 the result goes through. A.5 Proof of Theorem 15

Let S ( P ) be the set of stable matchings in the ordinal matching market( M, W, P ). Suppose that there are N stable matchings, and enumerate them,so S ( P ) = { µ , . . . , µ K } .To prove the proposition we ﬁrst establish some simple lemmas. Lemma 2

For any i ∈ M and j ∈ W , (cid:12)(cid:12) { k : j > i µ k ( i ) } (cid:12)(cid:12) + (cid:12)(cid:12) { k : i > j µ k ( j ) } (cid:12)(cid:12) ≤ K − (cid:12)(cid:12) { k : j = µ k ( i ) } (cid:12)(cid:12) Proof.

Let j > i µ k ( i ); then for µ k to be stable we need that µ k ( j ) > j i . So (cid:12)(cid:12) { k : j > i µ k ( i ) } (cid:12)(cid:12) ≤ (cid:12)(cid:12) { k : µ k ( j ) > j i } (cid:12)(cid:12) .Then, (cid:12)(cid:12) { k : i > j µ k ( j ) } (cid:12)(cid:12) = K − (cid:12)(cid:12) { k : µ k ( j ) ≥ j i } (cid:12)(cid:12) ≤ K − (cid:12)(cid:12) { k : j > i µ k ( i ) } (cid:12)(cid:12) − (cid:12)(cid:12) { k : j = µ k ( i ) } (cid:12)(cid:12) , where the last inequality follows from the previous paragraph and the factthat preferences > j are strict.Let ˆ U ( i, j ) = (cid:12)(cid:12) { k : j ≥ i µ k ( i ) } (cid:12)(cid:12) and ˆ V ( i, j ) = (cid:12)(cid:12) { k : i > j µ k ( j ) } (cid:12)(cid:12) . By theprevious lemma, ˆ U ( i, j ) + ˆ V ( i, j ) ≤ K for all i and j . Lemma 3 If µ is an isolated stable matching, µ (cid:48) is a stable matching, and i, ˆ i ∈ M , then µ ( i ) > i µ (cid:48) ( i ) iﬀ µ (ˆ i ) > ˆ i µ (cid:48) (ˆ i ) . roof. Suppose (reasoning by contradiction) that µ ( i ) > i µ (cid:48) ( i ) while µ (cid:48) (ˆ i ) ≥ ˆ i µ (ˆ i ). Since µ is isolated and preferences are strict, we have µ (cid:48) (ˆ i ) > ˆ i µ (ˆ i ).Now let ˆ µ = µ ∨ µ (cid:48) , using the join operator in the lattice of stable matchings(see A. Roth and Sotomayor, 1990). Then ˆ µ ( i ) = µ ( i ) and ˆ µ (ˆ i ) = µ (cid:48) (ˆ i ). Soˆ µ ∈ S ( P ), ˆ µ ( i ) = µ ( i ), and ˆ µ (cid:54) = µ ; a contradiction of the hypothesis that µ is isolated. Lemma 4 If µ is an isolated stable matching then ˆ U ( i, µ ( i )) + ˆ V ( µ ( j ) , j ) = K. Proof.

We prove that { k : µ (cid:54) = µ k and µ ( i ) ≥ i µ k ( i ) } = { k : µ (cid:54) = µ k and µ k ( j ) ≥ j µ ( j ) } . The lemma follows then becauseˆ U ( i, µ ( i )) + ˆ V ( µ ( j ) , j ) = (cid:12)(cid:12) { n : µ ( i ) ≥ i µ k ( i ) } (cid:12)(cid:12) + (cid:12)(cid:12) { k : µ k ( j ) > j µ ( j ) } (cid:12)(cid:12) = 1 + (cid:12)(cid:12) { k : µ (cid:54) = µ k and µ ( i ) ≥ i µ k ( i ) } (cid:12)(cid:12) + ( K − (cid:12)(cid:12) { k : µ (cid:54) = µ k and µ k ( j ) ≥ j µ ( j ) } (cid:12)(cid:12) − . Let µ ( i ) ≥ i µ k ( i ) and let i = µ ( j ). Since µ (cid:54) = µ k is isolated and preferencesare strict, µ ( i ) > i µ k ( i ). Then by Lemma 3, µ ( i ) > i µ k ( i ); so j = µ ( i )implies that µ k ( j ) > j µ ( j ). Similarly, if µ k ( j ) > j µ ( j ) then µ ( i ) > i µ k ( i ). So µ ( i ) > i µ k ( i ).We are now in a position to prove the proposition.Deﬁne a representation U and V of P as follows. Fix δ such that 0 < δ < /

2. Let U ( i, j ) = ˆ U ( i, j ) and V ( i, j ) = ˆ V ( i, j ) if there is µ ∈ S ( P ) such that j = µ ( i ). Otherwise, if j is worse than i ’s partner in any stable matching,let U ( i, j ) < P ); and if there is µ ∈ S ( P ) such that j > i µ ( i ), let µ be the best such matching for i , andchoose U ( i, j ) such that U ( i, j ) − U ( i, µ ( i )) < δ . Choose V similarly.Let µ be an isolated matching. Fix a pair ( i, j ) and suppose, wlog that u µ ( i ) − U ( i, j ) < v µ ( i ) − V ( i, j ) ≥ u µ ( i ) − U ( i, j ) ≥ v µ ( i ) − V ( i, j ) ≥ < i, j ) would constitute a blocking pair).First, if i and j are matched in some matching µ (cid:48) ∈ S ( P ) then u µ ( i ) − U ( i, j ) + v µ ( i ) − V ( i, j ) = u µ ( i ) − ˆ U ( i, j ) + v µ ( i ) − ˆ V ( i, j ) so it follows that24 µ ( i ) − U ( i, j ) + v µ ( i ) − V ( i, j ) ≥ U ( i, j ) and ˆ V ( i, j ).Second, let us assume that i and j are not matched in any matching in S ( P ). Since u µ ( i ) − U ( i, j ) < i than j . Let µ be such that j > i µ (cid:48) ( i ) implies that µ ( i ) ≥ i µ (cid:48) ( i ).Thus u µ ( i ) − U ( i, j ) > − δ by deﬁnition of U ( i, j ). Since j > i µ ( i ), wealso have µ ( j ) > j i , or µ would not be stable. Then, letting µ be thebest matching in S ( P ) for j , out of those that are worse than i , we have v µ ( j ) − V ( i, j ) = v µ ( j ) − v µ ( j ) + v µ ( j ) − V ( i, j ) > − δ , as µ ( j ) > j µ ( j )implies that v µ ( j ) − v µ ( j ) ≥ V ( i, j ) implies that v µ ( j ) − V ( i, j ) > − δ .Finally, u µ ( i ) − U ( i, j ) + v µ ( i ) − V ( i, j ) = u µ ( i ) − u µ ( i ) + u µ ( i ) − U ( i, j )+ v µ ( i ) − v µ ( j ) + v µ ( j ) − V ( i, j )= ( u µ ( i ) − u µ ( i ) + v µ ( i ) − v µ ( j ))+ ( u µ ( i ) − U ( i, j )) + ( v µ ( j ) − V ( i, j )) ≥ − δ ) + (1 − δ ) > , where the ﬁrst inequality follows from the remarks in the previous para-graphs, and from the fact that K = u µ ( i ) + v µ ( i ) ≥ u µ ( i ) + v µ ( j ) by Lem-mas 2 and 4. The second inequality follows because δ < /

2. This provesthe proposition.

A.6 Proof of Proposition 16

Let n be an even positive number. Let ( M, W, U, V ) be a marriage marketwith n men and n women, deﬁned as follows. The agents ordinal preferencesare deﬁned in the following tables: 25 i i · · · i n − i n − i n j j j · · · j n − j n − j n − j j j · · · j n − j j j j j · · · j n − j j ... j n/ j n/ j n/ · · · j n/ − j n/ − j n/ − j n ... j n − j n j · · · j n j j · · · j n − j n − The table means that j is the most preferred partner for i , followed by j ,and so on. The women’s’ preferences are as follows. j j j · · · j n − j n − j n i i i · · · i n − i i i i i · · · i n − i i ... i n/ i n/ i n/ · · · i n/ − i n/ − i n/ − i n/ i n/ i n/ · · · i n/ i n/ ... i n − i n i · · · i n − i n − i n i i · · · i n − i n − i i i · · · i n − i n i i i · · · i n − i n − i n It is a routine matter to verify that there is a unique stable matching in thismarket. It has i matched to j n/ , i matched to j n/ , and so on, untilwe obtain that i n − is matched to j n/ − . We have i n matched to j n . (Thelogic of this example is that i n creates cycles in the man-proposing algorithmwhich pushes the men down in their proposals until reaching the matchingin the “middle” of their preferences; j n plays the same role in the womanproposing version of the algorithm).Deﬁne agents’ cardinal preferences as follows. Let U ( i, j ) = [ n − r m ( w )] 1 n g + max { , n − − r i ( j ) } ( K − n − n g ) , r i ( j ) is the rank of woman j in i ’ preferences. Similarly deﬁne V ( i, j ),replacing r i ( j ) with r j ( i ). Then, given the preferences deﬁned above, theagents utilities at the unique stable matching satisfy: u ( i l ) = v ( j l ) = 12 n g − , l = 1 , . . . , n − u ( i n ) = v ( j n ) = ( n/ −

1) 1 n g . So that the sum of all agents utilities at the unique stable matching is:2( n − n g − ) + 2( n/ −

1) 1 n g , and agents’ weighted sum of utilities is at mostmax { n g − , ( n/ −

1) 1 n g } . Consider the matchings µ ∗ ( i l ) = j l , l = 1 , . . . , n , and ˆ µ ( j ) = i , . . . ˆ µ ( j n − ) = i n − , ˆ µ ( j n − ) = i , ˆ µ ( j n ) = i n . Let π be the random matching that resultsfrom choosing µ ∗ and ˆ µ with equal probability. Then, for all i (cid:54) = i n and j (cid:54) = j n we have that (cid:88) j (cid:48) π i,j (cid:48) U ( i, j (cid:48) ) = (cid:88) i (cid:48) π j,i (cid:48) V ( i (cid:48) , j ) = K/ , while (cid:88) j (cid:48) π i,j (cid:48) U ( i n , j (cid:48) ) = (cid:88) i (cid:48) π j,i (cid:48) V ( i (cid:48) , j n ) = ( n/ −

1) 1 n . Then (cid:88) i ∈ M α ( i ) (cid:88) j (cid:48) ∈ W π i,j (cid:48) U ( i, j (cid:48) ) + (cid:88) j ∈ W β ( j ) (cid:88) i (cid:48) ∈ M π j,i (cid:48) V ( i (cid:48) , j ) ≥ (cid:15)nK/ . So, regardless of the values of α and β in ∆ (cid:15) , the fraction (cid:80) i ∈ M α ( i ) (cid:80) j (cid:48) ∈ W π i,j (cid:48) U ( i, j (cid:48) ) + (cid:80) j ∈ W β ( j ) (cid:80) i (cid:48) ∈ M π j,i (cid:48) V ( i (cid:48) , j ) (cid:80) i ∈ M α ( i ) u ( i ) + (cid:80) j ∈ W β ( j ) v ( j )is bounded below by (cid:15)nK/ { n g − , ( n/ − n g } , which is Ω( Kn gg