On Human Capital and Team Stability
aa r X i v : . [ ec on . GN ] F e b On Human Capital and Team Stability Pierre-Andr´e Chiappori Columbia University
Alfred Galichon NYU and Sciences Po
Bernard Salani´e Columbia UniversityAugust 13, 2017 We thank Yeon-Koo Che, Navin Kartik, the editor and referees, and especiallyArnaud Dupuy, Fuhito Kojima and Phil Reny for useful comments. This paperbuilds on material from an unpublished manuscript circulated under the name “TheRoommate Problem Is More Stable Than You Think,” which is now obsolete.Accepted for publication by the
Journal of Human Capital , Volume 13, Number2, Summer 2019. URL: https://doi.org/10.1086/702925. Address: Department of Economics, Columbia University, 1009A Interna-tional Affairs Building, 420 West 118th St., New York, NY 10027, USA. E-mail:[email protected]. Chiappori gratefully acknowledges financial support fromthe NSF (award Economics Department and Courant Institute, New York University and Eco-nomics Department, Sciences Po. Address: NYU, Department of Economics. 19W 4th Street, New York, NY 10012, USA. Email: [email protected]. Galichon grate-fully acknowledges funding from NSF grant DMS-1716489, ERC grants FP7-295298,FP7-312503, FP7-337665, and ANR grant Famineq. Address: Department of Economics, Columbia University, 1131 InternationalAffairs Building, 420 West 118th Street, New York, NY 10027, USA. E-mail:[email protected]. bstract
In many economic contexts, agents from a same population team up to bet-ter exploit their human capital. In such contexts (often called “roommatematching problems”), stable matchings may fail to exist even when utilityis transferable. We show that when each individual has a close substitute, astable matching can be implemented with minimal policy intervention. Ourresults shed light on the stability of partnerships on the labor market. More-over, they imply that the tools crafted in empirical studies of the marriageproblem can easily be adapted to many roommate problems.
Introduction
Among Gary Becker’s seminal contributions to labor economics, two areof particular importance. The most obvious one is the notion of humancapital, after which this Journal is named. A second important contributionis the development of matching models with transferable utility . Althoughthe market for marriage was Becker’s favorite field of application for thetheory, many of the insights he developed are deeply relevant for the analysisof labor issues as well. The labor market can often be fruitfully seen asmatching people to jobs—an insight that has been thoroughly exploited inthe literature, in particular in a search context .There is, however, a fundamental difference between matching on thelabor and the marriage market. In the latter case, bilateral matching is anatural framework; individuals who match mostly belong to two distinctsubpopulations. Not so, however, on the labor market. Workers match notonly to jobs, but also (and often primarily) to other workers. Lawyers gatherin law firms, doctors associate in medical practices, architects congregate inarchitectural firms. While such partnerships are typical of the professions,they extend to other services firms such as consultancies. More generally,the notion that workers, through their employment relationships, tend tomatch to other workers with similar characteristics, has received a lot of at-tention and clear empirical support. For instance, a recent paper by Ehrlichand Kim, 2015 shows that immigrants endowed with similar skills tend toteam up and/or to separate in the same sectors. In particular, higher lev-els of human capital of specific skill groups in the destination country tendto increase the immigration flows of corresponding groups from the sourcecountry. The authors convincingly argue that these effects are crucial inassessing the economic consequences of migrations.From a theoretical perspective, these features raise specific problems. Insharp contrast with the bipartite literature, the formal analysis of workers’matching on human capital must acknowledge the fact that the individualsunder consideration typically belong to the same population. It has beenknown for some time that this apparently minor difference in settings maygenerate largely divergent properties. Take, for instance, the specific casein which teams consist of exactly two people, both coming from the samepopulation: this is classically called the roommate matching problem .The standard equilibrium concept in matching is stability; a matching See for instance Becker, 1973 and Becker, 1974. See in particular Mortensen and Pissarides, 1994.
1s stable if it is robust to unilateral and bilateral deviations. We will followthis long tradition in this paper: when we say that “an equilibrium exists”,for instance, we mean that “a stable matching exists”. Can we expect thatthe roommate matching game always has a stable matching, so that thetheoretical analysis could, as in the bipartite framework, concentrate on theproperties and the comparative statics of this stable outcome? Or could it bethe case that a stable matching fails to exist, which might cast serious doubtson the relevance of matching models for the analysis of these situations?The answer to that important question has been known for a long timein the Non Transferable Utility (NTU) context; indeed, Gale and Shapley,1962 have shown that stable matchings may not exist. However, applyingthe NTU approach to a labor market requires wages to be exogenously fixed,rather than being endogenously determined at equilibrium. In most mar-kets, this is not the relevant framework. Much more adequate is a Trans-ferable Utility (TU) framework, in which any potential team generates asurplus that is (endogenously) shared by its members.The problem, however, is that roommates matching games under TUtend to have different properties than their bipartite counterparts. In abipartite setting, a stable matching exists under mild continuity and com-pactness conditions; it maximizes aggregate surplus, and the associated indi-vidual surpluses solve the dual imputation problem. A first conclusion of thepresent paper is that in the two-partner roommate matching problem underTU, on the contrary, stable matching may fail to exist. This is a potentiallydamaging conclusion, since it might require reconsidering the relevance ofmatching models in this context.Our second conclusion, however, tends to mitigate this negative resultby showing that its economic implications may be much less damaging thanone would expect. Specifically, we consider a model in which agents belongto various “types”, where each type consists of individuals of indistinguish-able characteristics and tastes. In this context, we show two main results.First, a stable matching always exists when the number of individuals ineach type is even . Second, when the number of individuals of any giventype is large enough, there always exist “quasi-stable” matchings: even ifa stable matching does not exist, existence can be restored with minimalpolicy intervention. To do this, one only needs to convince one individualto leave the game in each type with an odd number of individuals. If thisrequires a compensation to be paid, this can be done at a per capita costthat goes to zero when the population of each type goes to infinity.We refer the reader to our conclusion for the implications of these find-ings in terms of the stability of partnerships. We also show there that the2mpirical tools devised for the bipartite matching setting should carry overdirectly to the roommate context when the populations under considera-tion are large. Some of the results of the present paper are applied in thisdirection in Ciscato et al., 2015. Existing literature
Since Gale and Shapley, 1962, a few papers have stud-ied the property of NTU stable roommate matchings when they do exist.Gusfield and Irving, 1989 showed that the set of singles is the same in allstable matchings; Klaus and Klijn, 2010 study whether any of them canbe “fair”. Efficient algorithms have also been available since Irving, 1985.Necessary and sufficient existence conditions under strict preferences havebeen found by Tan, 1991 for complete stable matchings and by Sotomayor,2005 for stable matchings. Chung, 2000 shows that a condition he calls“no odd rings” is sufficient for stable matchings to exist under weak prefer-ences. Rodrigues-Neto, 2007 introduces “symmetric utilities” and Gudmun-sson, 2014 uses “weak cycles.”The TU case has been less studied in the theoretical literature, in spiteof its relevance in empirical applications. Chung, 2000 shows that whenthe division of surplus obeys an exogenous rule, odd rings are ruled outand the roommate problem has a stable matching; but that is clearly notan appealing assumption. Karlander and Eriksson, 2001 provide a graph-theoretic characterization of stable outcomes when they exist; and Klausand Nichifor, 2010 studies their properties. Talman and Yang, 2011 give acharacterization in terms of integer programming.The results of this paper are also related to those of Azevedo et al.,2013, who show the existence of a Walrasian equilibrium in an economywith indivisible goods, a continuum of agents and quasilinear utility. Unliketheir main results, ours apply in markets with finite numbers of agents. Ourmethods are also original. As is well-known, in bipartite problems all feasiblematchings that maximize social surplus are stable. This is not true in room-mate problems; but we show how any roommate problem can be “cloned”in order to construct an associated bipartite problem. We then exploit thisinsight to prove existence of stable matchings in roommate problems witheven numbers of agents within each type.To the best of our knowledge, the connection between the unipartite andbipartite problems stressed in this paper is new. See Chiappori and Salani´e, 2016 for a recent survey. A Simple Example
We start by giving the intuition of our main results on an illustrative exam-ple.
It has been known since Gale and Shapley that a stable matching may notexist for the roommate problem under non-transferable utility. As it turnsout, it is almost equally easy to construct an example of non-existence ofa stable matching with transferable utility. Here a matching defines whois matched to whom and how the corresponding surplus is divided betweenthe partners. Stability requires that • no partner would be better off by leaving the partnership • no group of individuals could break off their current match, rematchtogether, and generate a higher joint surplus than the sum of theircurrent individual utilities.Consider the following example, in which only two-member matches arepossible: Example 1
The population has three individuals. Any unmatched individualhas zero utility. The joint surplus created by the matching of any two ofthem is given by the off-diagonal terms of the matrix
Φ = − −
58 5 − (1) so that individuals 1 and 2 create, if they match, a surplus of 6; 1 and 3create a surplus of 8, etc.Assume that there exists a stable matching. A matching in which allindividuals remain single is obviously not stable; any stable matching must besuch that one person remains single and the other two are matched together.Let ( u x ) be the utility that individual of type x = 1 , , gets out of this game;stability imposes u x + u y ≥ Φ xy for all potential matches, with equality if x and y are actually matched—and u x ≥ with equality if x is single. One canreadily check, however, that no set of numbers ( u , u , u ) satisfying theserelationships for all x and y exists: whichever the married pair is, one of he matched partners would increase her utility by matching with the singleperson. Indeed, if the matched pair is { , } , then u + u = 6 , u = 0 , u ≥ contradicts u + u ≥ : agent 3, being single, is willing to give up anyamount smaller than 8 to be matched with 1, while the match between 1 and2 cannot provide 1 with more than 6. Similarly, if the married pair is { , } ,then u + u = 5 , u = 0 , u ≥ , u ≥ contradicts both u + u ≥ and u + u ≥ (so that 1 is willing to givemore than 5 and less than 6 to agent 2 to match with her, and more than 5and less than 8 to 3.) Finally, if the married pair is { , } , then u + u = 8 , u = 0 , u ≥ , u ≥ is incompatible with u + u ≥ , which follows from combining u + u ≥ and u + u ≥ with u = 0 (since agent 2 is single 1 could match with herand capture almost 6, while 3 could match with her and capture almost 5;these outside options are more attractive than anything 1 and 3 can achievetogether.) We conclude that no stable matching exists. Note that there is nothing pathological in Example 1. The surpluses caneasily be (locally) modified without changing the result. Also, the conclusiondoes not require an odd number of agents; one can readily introduce afourth individual, who generates a small enough surplus with any roommate,without changing the non-existence finding.
However, there exists a simple modification that restores existence in Exam-ple 1. Let us now duplicate the economy by “cloning” each agent; technically,we now have three types x = 1 , , x = y is as in Example 1; but we now alsoneed to define the surplus generated by the matching of two clones (twoindividuals of the same type.) Take it to be 2 for every type—more on thislater. We then have the matrix:Φ ′ = (2)5onsider the following matching µ ∗ : there is one match between a type1 and a type 2 individuals, one between type 1 and type 3, and one be-tween type 2 and type 3. Assume individuals share the surplus so that eachindividual of type 1 gets 4 .
5, each individual of type 2 gets 1 .
5, and eachindividual of type 3 gets 3 .
5. This is clearly feasible; and it is easy to verifythat it is a stable matching.Less obvious but still true is the fact (proved later on) that existencewould still obtain for any values chosen for the diagonal of the matrix,although the stable matching pattern that would emerge may be different .In other words, our cloning operation always restores the existence of a stablematch, irrespective of the values of the joint surpluses created by matchesbetween clones. Our main result is better understood when related to another, closely linkedproblem: finding a feasible matching that maximizes total surplus. Totalsurplus is simply the sum of the joint surpluses of every match (keeping toa normalized utility of zero for singles). In the standard, bipartite frame-work, the adjective “feasible” refers to the fact that each individual can onlybe matched to one partner or stay single. Roommate matching, however,introduces an additional feasibility constraint. For any two types x = y ,denote µ xy the number of matches between an individual of type x and anindividual of type y ; since a roommate matching for which µ xy and µ yx dif-fer would clearly not be feasible, it must be the case that µ xy = µ yx . Thisadditional symmetry constraint is absent from the bipartite model, wherethese two individuals would belong to two separate subpopulations and thenumber of marriages between say, a college-educated man and a womanwho is a high-school graduate may well differ (and typically does) from thenumber of marriages between a college-educated woman and a man who isa high-school graduate.This symmetry constraint is the source of the difficulty in finding stableroommate matchings; and our cloning operation addresses it. To see this onour Example 1, first go back to roommate matching with one individual ofeach type x = 1 , ,
3, and neglect the symmetry constraint. Since there isonly one individual of each type x , she cannot match with herself: µ xx ≡ For instance, if the diagonal elements are large enough, the stable matching matcheseach individual with her clone. x, X y = x µ xy ≤ y, X x = y µ xy ≤ . The two matchings µ = and µ = are feasible in this limited sense; and they both achieve the highest possiblesurplus when the symmetry conditions are disregarded. The existence oftwo solutions is not surprising: given the symmetric nature of the surplusmatrix Φ, if a matrix µ maximizes total surplus, so does its transpose µ t .Unfortunately, neither is symmetric, and therefore neither makes any sensein the roommate problem. For instance, µ has agent 1 matched both withagent 3 (in the first row) and with agent 2 (in the first column). Also, notethat a third solution to this relaxed problem is the unweighted mean of µ and µ , µ m = / / / / / / However, while this matrix is indeed symmetric, its coefficients are not in-teger and thus it is not a feasible matching either; moreover, and quite in-terestingly, it cannot be interpreted as the outcome of randomization sinceit is not a convex combination of feasible roommate matching matrices .Let us now reintroduce the symmetry constraint. The (now fully) fea-sible matching that maximizes total surplus can only have one matchedpair and one single; and the pair that should be matched clearly consists ofindividuals 1 and 3: ¯ µ = . For any stable roommate matching matrix, the sum of coefficients equals 2, reflect-ing the fact that one agent must remain single. This property is preserved by convexcombination; however, the sum of coefficients of µ m equals 3. µ is not a solution to the maximization problem without sym-metry constraint; in other words, the symmetry constraint is binding in thisexample. As we shall see below, this is characteristic of situations in whichthe roommate matching problem with transferable utility does not have astable matching. Indeed, we prove in the next section that a stable matchingexists if and only if the symmetry constraint does not bind .Now take the “cloned” version of Example 1, in which each type x hastwo individuals. It is easy to see that the solution to the relaxed problemwhich neglects the symmetry constraint is the µ ∗ of section 2.2, which issymmetric; therefore the symmetry constraint does not bind, and a stablematching exists. This is a general result: we shall see below that in anycloned roommate matching setup, at least one solution to the relaxed prob-lem is symmetric—which implies the existence of a stable match. The relaxed problem, in turn, has a natural interpretation in terms of bipar-tite matching. Start from the three-agent Example 1, and define an associ-ated bipartite matching problem as follows: clone the population again, butthis time assign a label (such as “man” or “woman”) to each of the two sub-populations. Then consider the bipartite matching problem between thesesubpopulations of “men” and “women”, with the joint surplus matrix givenby Φ ′ in (2).By standard results, there always exists a stable matching in this asso-ciated bipartite matching problem; and it maximizes the associated totalsurplus. In our example, µ and µ are the two stable matchings. Any con-vex combination such as µ m can be interpreted as a randomization betweenthese two matchings; it is natural to focus on µ m since it is the only symmet-ric one and feasible roommate matchings must be symmetric. As remarkedabove, in the original roommate problem µ m cannot be stable, since it hasnon-integer element.Now if the roommate matching problem is cloned we can proceed as inthe above paragraph, except that with twice the number of individuals weshould work with 2 µ m . As an integer symmetric matrix, reinterpreted in thecloned roommate matching setup, it defines a feasible roommate matchingwhich is stable—in fact it is the stable matching µ ∗ of section 2.2. Thisconstruction is general: we shall see below that any roommate matchingproblem in which the number of individuals in each type is even has asymmetric stable match.We now provide a formal derivation of these results.8 The Formal Setting
We consider a population of individuals who belong to a finite set of types X . Individuals of the same type are indistinguishable. We denote n x thenumber of individuals of type x ∈ X , and N = X x ∈X n x the total size of the population.Without loss of generality, we normalize the utilities of singles to be zerothroughout. A match consists of two partners of types x and y . An individual of anytype can be matched with any individual of the same or any other type, orremain single. In particular, there is no restriction that matches only involvetwo partners of different “genders.”Let a match { x, y } generate a surplus Φ xy . In principle the two partnerscould play different roles. In sections 3 and 4 we will assume that they arein fact symmetric within a match, so that Φ xy is assumed to be a symmetricfunction of ( x, y ): Assumption 1
The surplus Φ xy is symmetric in ( x, y ) . We show in section 5 that, surprising as it may seem, there is in factno loss of generality in making this assumption. The intuition is simple:if Φ xy fails to be symmetric in ( x, y ), so that the partners’ roles are notexchangeable, then they should choose their roles so to maximize output.This boils down to replacing Φ xy with the symmetric max (Φ xy , Φ yx ). Thusour results extend easily when we do not impose Assumption 1; but it iseasier to start from the symmetric case.A matching can be described by a matrix of numbers (cid:0) µ xy (cid:1) indexed by x, y ∈ X , such that • µ x is the number of singles of type x • when y = 0, µ xy is the number of matches between types x and y .9he numbers µ xy should be integers; given Assumption 1, they should besymmetric in ( x, y ); and they should satisfy the scarcity constraints. Moreprecisely, the number of individuals of type x must equal the number µ x of singles of type x , plus the number of pairs in which only one partner hastype x , plus twice the number of pairs in which the two partners are of type x —since such a same-type pair has two individuals of type x .Finally, the set of feasible roommate matchings is P ( n ) = µ = (cid:0) µ xy (cid:1) : µ xx + P y = x µ xy ≤ n x µ xy = µ yx µ xy ∈ N (3) We define an outcome ( µ, u ) as the specification of a feasible roommatematching µ and an associated vector of payoffs u x to each individual of type x . These payoffs have to be feasible: that is, the sum of payoffs across thepopulation has to be equal to the total output under the matching µ . Nowin a roommate matching µ , the total surplus created is S R ( µ ; Φ) = X x µ xx Φ xx + X x = y µ xy Φ xy . (4)This leads to the following definition of a feasible outcome: an outcome( µ, u ) is feasible if µ is a feasible roommate matching and X x ∈X n x u x = S R ( µ ; Φ) . (5)We define stability as in Gale and Shapley, 1962: an outcome ( µ, u ) isstable if it cannot be blocked by an individual or by a pair of individuals.More precisely, an outcome ( µ, u ) is stable if it is feasible, and if for any twotypes x, y ∈ X , (i) u x ≥
0, and (ii) u x + u y ≥ Φ xy . By extension, a matching µ is called stable if there exists a payoff vector ( u x ) such that the outcome( µ, u ) is stable.In bipartite matching the problem of stability is equivalent to the prob-lem of optimality : stable matchings maximize total surplus. Things are obvi-ously more complicated in roommate matchings—there always exist surplus-maximizing matchings, but they may not be stable. The maximum of the Note that in the second sum operator the pair { x, y } appears twice, one time as ( x, y )and another time as ( y, x ); but the joint surplus Φ xy it creates must only be counted once,hence the division by 2. P ( n ) is W P ( n, Φ) = max S R ( µ ; Φ) (6) s.t. µ xx + X y = x µ xy ≤ n x µ xy = µ yx µ xy ∈ N . While no stable matching may actually achieve this value, it plays an im-portant role in our argument.
We shall now see that to every roommate matching problem we can asso-ciate a bipartite matching problem which generates almost the same level ofaggregate surplus. More precisely, we will prove that for every vector of pop-ulations of types n = ( n x ) and every symmetric surplus function Φ = (Φ xy ),the highest possible surplus in the roommate matching problem is “close to”that achieved in a bipartite problem with mirror populations of men andwomen and half the surplus function: W P ( n, Φ) ≃ W B ( n, n, Φ / . where W B ( n, n, Φ /
2) is defined as the maximal surplus of the bipartitematching problem: W B ( n, n, Φ /
2) = max ν ∈ B ( n, n ) S B ( ν ; Φ) (7)where S B ( ν ; Φ) = P x,y ∈X ν xy Φ xy and B ( n, n ) is the set of feasible matchingsin the bipartite problem: B ( n, n ) = ν = ( ν xy ) : P y ν xy ≤ n x P x ν xy ≤ n y ν xy ∈ N (8)We also define stability for a feasible bipartite matching ( ν xy ) in theusual way: there must exist payoffs ( u x , v y ) such that S B ( ν ; Φ) = X x ∈X n x u x + X y ∈X n y v y (9) u x + v y ≥ Φ xy u x ≥ , v y ≥ ν , and they coincide with the solutions of (7). Moreover, the associatedpayoffs ( u, v ) solve the dual program; that is, they minimize P x ∈X n x u x + P y ∈X n y v y over the feasible set of program (9). Finally, for any stablematching, µ xy > u x + v y = Φ xy /
2, and µ x > u x = 0 . Remark 3.1
The marriage problem obviously is a particular case of the room-mate problem: if in a roommate matching problem Φ xy = −∞ whenever x and y have the same gender, then any optimal or stable matching will beheterosexual. W P and W B It is not hard to see that W P ( n, Φ) ≤ W B ( n, n, Φ / . In fact, we can boundthe difference between these two values:
Theorem 1
Under Assumption 1, W P ( n, Φ) ≤ W B ( n, n, Φ / ≤ W P ( n, Φ) + |X | Φ where Φ = sup x,y ∈X Φ xy . and |X | is the cardinal of the set X , i.e. the number of types in the popula-tion. Proof.
See appendix.
In some cases, W P ( n, Φ) and W B ( n, n, Φ /
2) actually coincide. For in-stance:
Proposition 2 If n x is even for each x ∈ X , then under Assumption 1, W P ( n, Φ) = W B ( n, n, Φ / . Proof.
See appendix.
The existence of stable roommate matchings is directly related to the diver-gence of W P ( n, Φ) and W B ( n, n, Φ / heorem 3 Under Assumption 1,(i) There exist stable roommate matchings if and only if W P ( n, Φ) = W B ( n, n, Φ / . (ii) Whenever they exist, stable roommate matchings achieve the maxi-mal aggregate surplus W P ( n, Φ) in (6).(iii) Whenever a stable roommate matching exists, individual utilities atequilibrium ( u x ) solve the following, dual program: min u,A X x u x n x (10) s.t. u x ≥ u x + u y ≥ Φ xy + A xy A xy = − A yx Proof.
See appendix.Note that while the characterization of the existence of a stable matchingin terms of equality between an integer program and a linear program is awell-known problem in the literature on matching (see Talman and Yang,2011 for the roommate problem), the link with a bipartite matching problemis new.Also note that in program (10), the antisymmetric matrix A has a naturalinterpretation: A xy is the Lagrange multiplier of the symmetry constraints µ xy = µ yx in the initial program (6). Our proof shows that if µ xy > A xy must be non-positive; but since µ yx = µ xy the multiplier A yx must also be non-positive,so that both must be zero. The lack of existence of a stable roommatematching is therefore intimately linked to a binding symmetry constraint.Given Proposition 2, Theorem 3 has an immediate corollary: with aneven number of individuals per type, there must exist a stable roommatematching. Formally: Corollary 3.1 If n x is even for each x ∈ X , then under Assumption 1, thereexists a stable roommate matching. In particular, for any roommate matching problem, its “cloned” version,in which each agent has been replaced with a couple of clones, has a sta-ble matching; and this holds irrespective of the surplus generated by the13atching of two identical individuals. Of course, in general much less thanfull cloning is needed to restore existence; we give this statement a precisemeaning in the next paragraph.Our next result shows that one can restore the existence of a stablematching by removing at most one individual of each type from the popula-tion; if these individuals have to be compensated for leaving the game, thiscan be done at limited total cost:
Theorem 4 (Approximate stability)
Under Assumption 1, in a populationof N individuals, there exists a subpopulation of at least N − |X | individualsamong which there exist a stable matching, where |X | is the number of types.The total cost for the regulator to compensate the individuals left aside isbounded above by |X | Φ . Proof.
See appendix.
We now consider the case of a “large” game, in which there are “many” agents of each type . Intuitively, even though an odd number of agents in any typemay result in non existence of a stable roommate matching, the resultinggame becomes “close” to one in which a stable matching exists. We nowflesh out this intuition by providing a formal analysis.We start with a formal definition of a large game. For that purpose, weconsider a sequence of games with the same number of types and the samesurplus matrix, but with increasing populations in each type. If n kx denotesthe population of type x in game k and N k = P x n kx is the total populationof that game, then we consider situations in which, when k → ∞ : N k → ∞ and n kx /N k −→ f x where f x are constant numbers.As the population gets larger, aggregate surplus increases proportionally;it is therefore natural to consider the average surplus , computed by dividingaggregate surplus by the size of the population. We also extend the definitionof W B in program (7) to non-integers in the obvious way so as to define thelimit average bipartite problem W B ( f, f, Φ / W B ( cn, cm, Φ /
2) = c W B ( n, m, Φ / c >
0. 14 roposition 5
In the large population limit, under Assumption 1, the averagesurplus in the roommate matching problem converges to the limit averagesurplus in the related bipartite matching problem. That is, lim k →∞ W P (cid:0) n k , Φ (cid:1) N k = lim N k →∞ W B (cid:0) n k , n k , Φ / (cid:1) N k = W B ( f, f, Φ / . Proof.
See appendix.Our approximation results crucially rely on the number of types becom-ing small relative to the total number of individuals. By definition, twoindividuals of the same type are indistinguishable in our formulation, bothin their preferences and in the way potential partners evaluate them. Thismay seem rather strong; however, a closer look at the proof of Theorem 5shows that our bound can easily be refined. In particular, we conjecturethat with a continuum of types, Theorem 5 would hold exactly.A related effect of the number of individuals becoming much larger thanthe number of types is that the costs of the policy to restore stability inTheorem 4 become negligible:
Proposition 6
In the large population limit and under Assumption 1,(i) one may remove a subpopulation of asymptotically negligible size inorder to restore the existence of stable matchings.(ii) the average cost per individual of restoring the existence of stablematchings tends to zero.
Proof.
See appendix.In particular, in the case of a continuum of individuals (that is, whenthere is a finite number of types and an infinite number of individuals ofeach type), we recover the results of Azevedo et al., 2013 (hereafter, AWW).To make the connection with this paper, the partner types in our settingtranslates into goods in AWW’s. The social welfare in our setting translatesinto the utility u of a single consumer in AWW. u is such that u ( C ) =Φ ( { x, y } ) for C = { x, y } , u ( { x } ) = 0, and u = −∞ elsewhere (or verynegative). Then it can be shown without difficulty that the existence of a TUstable matching in our setting is equivalent to the existence of a Walrasianequilibrium in the AWW setting. Thus existence and TU stability in thecase of a continuum of individuals follows from Theorem and Proposition inAWW. 15 The Nonexchangeable Roommate Problem We now investigate what happens when the surplus Φ xy is not necessarilysymmetric. This will arise when the roles played by the partners are notexchangeable. For instance, a pilot and a copilot on a commercial airplanehave dissymmetric roles, but may be both chosen from the same population.Hence, in this section, we shall assume away Assumption 1, and we referto the “nonexchangeable roommate problem”; it contains the exchangeableproblem as a special case.As it turns out, this can be very easily recast in the terms of an equivalentsymmetric roommate problem. Indeed if Φ xy > Φ yx , then any match ofan (ordered) 2-uple ( y, x ) will be dominated by a matching of a ( x, y ) 2-uple, and the partners may switch the roles they play and generate moresurplus. Therefore, in any optimal (or stable) solution there cannot be sucha ( y, x ) 2-uple. As a consequence, the nonexchangeable roommate problemis equivalent to an exchangeable problem where the surplus function is equalto the maximum joint surplus x and y may generate together, that isΦ ′ xy = max (Φ xy , Φ yx ) ;and since this is symmetric our previous results apply almost directly. De-noting π xy the number of ( x, y ) pairs (in that order), one has µ xy = π xy + π yx , x = yµ xx = π xx and obviously, π xy need not equal π yx . The population count equation is n x = X y ∈X ( π xy + π yx ) , ∀ x ∈ X and the social surplus from a matching π is X x,y ∈X π xy Φ xy . so that the optimal surplus in the nonexchangeable problem is W ′P ( n, Φ) = max X x,y ∈X π xy Φ xy s.t. n x = X y ∈X ( π xy + π yx ) , ∀ x ∈ X . We are grateful to Arnaud Dupuy for correcting a mistake in a preliminary version ofthe paper.
Theorem 7
The nonexchangeable roommate matching problem is solved byconsidering the surplus function Φ ′ xy = max (Φ xy , Φ yx ) which satisfies Assumption 1. Call optimized symmetric problem the prob-lem with surplus Φ ′ xy and population count n x . Then:(i) the optimal surplus in the nonexchangeable roommate problem co-incides with the optimal surplus in the corresponding optimized symmetricproblem, namely W ′P ( n, Φ) = W P (cid:0) n, Φ ′ (cid:1) (ii) the nonexchangeable roommate problem has a stable matching if andonly if the optimized symmetric problem has a stable matching. Given Theorem 7, all results in Sections 3 and 4 hold in the general(nonexchangeable) case. In particular: • Theorem 1 extends to the general case: the social surplus in the room-mate problem with asymmetric surplus Φ xy is approximated by a bi-partite problem with surplus function Φ ′ xy = max (Φ xy , Φ yx ) /
2, ormore formally: W ′P ( n, Φ) ≤ W B (cid:0) n, n, Φ ′ / (cid:1) ≤ W ′P ( n, Φ) + |X | Φ , and as an extension of Proposition 2, equality holds in particular whenthe number of individuals in each types are all even. • Theorem 3 extends as well: there is a stable matching in the roommateproblem with asymmetric surplus Φ xy if and only if there is equalityin the first equality above, that is: W ′P ( n, Φ) = W B (cid:0) n, n, Φ ′ / (cid:1) . • All the asymptotic results in Section 4 hold true: in the asymmetricroommate problem, there is approximate stability and the optimalmatching solves a linear programming problem.17
Conclusion
From a technical perspective, our results are open to various extensions.First, the empirical tools developed in the bipartite setting, especially forthe analysis of the marriage markets (see Choo and Siow, 2006, Chiappori etal., 2017, Fox, 2010, Galichon and Salani´e, 2016, to cite only a few ) can beextended to other contexts where the bipartite constraint is relaxed. Theseinclude law firms or doctor practices, but also team jobs such as pilot/copilot(and more generally team sports), as well as “tickets” in US presidentialelections, marriage markets incorporating single-sex households, and manyothers.To be more specific, assume that the joint surplus ˜Φ ij generated by amatch between two individuals i (of type x ) and j (of type y ) is separable in the sense defined by Chiappori et al., 2017:˜Φ ij = Φ xy + ε iy + η jx ;separability assumes that unobserved heterogeneity terms do not interact inthe formation of joint surplus.If the partnership has symmetric roles, then it is easy to see that Φ xy must be symmetric in ( x, y ), and that η and ε must be the same family ofrandom variables: ˜Φ ij = Φ xy + ε iy + ε jx . Apart from this specific restriction, if there is a stable matching then theresults of Chiappori et al., 2017 apply: there exist U and V such that U xy + V xy ≡ Φ xy and in equilibrium, if i of type x and j of type y matchthen i obtains surplus which stems from the maximization of U xz + ε iz withrespect to z , including zero for the singlehood option in the maximization. Inaddition, symmetry implies that U xy = V yx . This boils down the matchingequilibrium to a series of simple discrete-choice problems. We refer thereader to Galichon and Salani´e, 2017 for a short description of separablemodels, and to Galichon and Salani´e, 2016 for a much more complete studyof identification and estimation.Secondly, while our analysis has been conducted in the discrete case, itwould be interesting to extend our results to the case where there is an in-finite number of agents with a continuum of types. We conjecture that thiscould be done, at some cost in terms of the mathematics required . Thirdly, Graham, 2011 has a good discussion of this burgeoning literature. The relevant tools here come from the theory of optimal transportation, see Vil-lani, 2003 and McCann and Guillen, 2013. For the precise connection between matching
18e conjecture that the same “cloning” technique could be applied to matchesinvolving more than two partners—the multipartite reference, in that case,being the “matching for teams” context studied by Carlier and Ekeland,2010. Moreover, it seems natural to apply this technique when utility isnot transferable. One may think of assigning arbitrarily genders to bothclones of each type, and considering a bipartite stable matching betweenthe two genders. Such a matching will be stable in the roommate matchingframework if the bipartite matching of the cloned populations is symmetric.However, such a symmetric stable bipartite matching of the cloned popula-tion may not exist. Therefore, the usefulness of cloning to restore stabilityin the non-transferable utility version of the roommate problem is an openquestion.Finally, some roommate problems involve extensions to situations wheremore than two partners can form a match; but the two-partner case is a goodplace to start the analysis. Here, we have shown that when the population islarge enough with respect to the number of observable types, the structure ofthe roommate problem is the same as the structure of the bipartite matchingproblem. Most empirical applications of matching models under TU use aframework as in this paper in order to understand, depending on the context,how the sorting on a given matching market depends on age, education orincome, but also height, BMI, marital preferences, etc. . We leave all thisfor future research.On a more substantive front, our conclusions are somewhat mixed. Whileexistence issues may be serious in specific contexts, large markets with adiscrete distribution of skills (or human capital) tend to be largely immunefrom these problems. Specifically, and to put things a bit loosely, two factorsmake partnerships between workers belonging to the same population morelikely to be stable: (i) when individuals can be clustered into a small numberof basic categories (the latter being defined by either a given level of humancapital or a specific combination of skills), and (ii) when different workersbelonging to the same category are “close substitutes” to each other—in thesense that substituting one for the other does not change much the jointsurplus created in any partnership. For instance, we expect that medicalpractices formed by a largish number of doctors with similar specialties models and optimal transportation theory, see Ekeland, 2010, Gretsky et al., 1999 andChiappori et al., 2010. It is also worth mentioning the recent contribution of Ghoussouband Moameni, 2013, which uses the same type of mathematical structure for very differentpurposes. See for instance Choo and Siow, 2006, Chiappori and Oreffice, 2008 Chiappori et al.,2012 among many others. This would also true of firms that rely on a verycharismatic individual for inspiration. The early (1969–84) trajectory ofApple under Steve Jobs may be a case in point. Last but not least, sportteams involving a small number of superstars should exhibit stability issues,especially when several stars are associated within the same team. We arenot aware of any systematic, empirical analysis of these issues; however, itis fair to say that casual empiricism seems to support these predictions.When partnerships are least likely to be stable, firm-specific capital islikely to stabilize a partnership. Regulation may also play a useful role.While we do not pursue this here, one can imagine cases when non-competeor “no poaching” clauses that make mobility more costly could actually bewelfare-improving, if the courts allow them. Becker, 1991, p. 330 alreadycited the ability of homosexual unions to “dissolve without judiciary pro-ceedings, alimony, or child support payments” as one reason why they areless stable than heterosexual unions. This is an interesting topic for furtherresearch.
References
Azevedo, E., Weyl, E., & White, A. (2013). Walrasian equilibrium in large,quasilinear markets.
Theoretical Economics , , 281–290.Balinski, M. (1970). On maximum matchings, minimum coverings, and theirconnections. In H. Kuhn (Ed.), Proc. of the princeton symposium onmathematical programming . Princeton University Press.Becker, G. (1973). A theory of marriage, part i.
Journal of Political Economy , , 813–846.Becker, G. (1974). A theory of marriage, part ii. Journal of Political Econ-omy , , S11–S26.Becker, G. (1991). A treatise on the family . Harvard University Press. Consultancies are an intermediate case: while junior consultants may be relativelyinterchangeable, leadership matters in finding clients and conserving them.
Economic Theory , , 397–418.Chiappori, P.-A., McCann, R., & Nesheim, L. (2010). Hedonic price equilib-ria, stable matching, and optimal transport: Equivalence, topology,and uniqueness. Economic Theory , 1–49.Chiappori, P.-A., & Oreffice, S. (2008). Birth control and female empower-ment: An equilibrium analysis.
Journal of Political Economy , ,113–140.Chiappori, P.-A., Oreffice, S., & Quintana–Domeque, C. (2012). Fatter at-traction: Anthropometric and socioeconomic characteristics in themarriage market. Journal of Political Economy , , 659–695.Chiappori, P.-A., & Salani´e, B. (2016). The econometrics of matching mod-els. Journal of Economic Literature , , 832–861.Chiappori, P.-A., Salani´e, B., & Weiss, Y. (2017). Partner choice and themarital college premium. American Economic Review , , 2107–2169.Choo, E., & Siow, A. (2006). Who marries whom and why. Journal of Po-litical Economy , , 175–201.Chung, K.-S. (2000). On the existence of stable roommate matchings. Gamesand Economic Behavior , , 206–230.Ciscato, E., Galichon, A., & Gouss´e, M. (2015). Like attract like: A structuralcomparison of homogamy across same-sex and different-sex house-holds [mimeo Sciences Po].Ehrlich, I., & Kim, J. (2015). Immigration, human capital formation andendogenous economic growth.
Journal of Human Capital , , 518–563.Ekeland, I. (2010). Notes on optimal transportation. Economic Theory , ,437–459.Fox, J. (2010). Identification in matching games. Quantitative Economics , , 203–254.Gale, D., & Shapley, L. (1962). College admissions and the stability of mar-riage. American Mathematical Monthly , , 9–14.Galichon, A., & Salani´e, B. (2016). Cupid’s invisible hand: Social surplusand identification in matching models [mimeo].Galichon, A., & Salani´e, B. (2017). The econometrics and some propertiesof separable matching models.
American Economic Review Papersand Proceedings , , 251–255.Ghoussoub, N., & Moameni, A. (2013). A self-dual polar factorization forvector fields. Communications on Pure and Applied Mathematics , , 905–933. 21raham, B. (2011). Econometric methods for the analysis of assignmentproblems in the presence of complementarity and social spillovers.In J. Benhabib, A. Bisin, & M. Jackson (Eds.), Handbook of socialeconomics . Elsevier.Gretsky, N., Ostroy, J., & Zame, W. (1999). Perfect competition in thecontinuous assignment model.
Journal of Economic Theory , , 60–118.Gudmunsson, J. (2014). When do stable roommate matchings exist? a re-view. Review of Economic Design , , 151–161.Gusfield, D., & Irving, R. (1989). The stable marriage problem: Structureand algorithms . MIT Press.Irving, R. (1985). An efficient algorithm for the ‘stable roommates’ problem.
Journal of Algorithms , , 577–595.Karlander, J., & Eriksson, K. (2001). Stable outcomes of the roommate gamewith transferable utility. International Journal of Game Theory , ,555–569.Klaus, B., & Nichifor, A. (2010). Consistency in one-sided assignment prob-lems. Social Choice and Welfare , , 415–433.Klaus, B., & Klijn, F. (2010). Smith and rawls share a room: Stability andmedians. Social Choice and Welfare , , 647–667.McCann, R., & Guillen, N. (2013). Five lectures on optimal transportation:Geometry, regularity and applications. In G. Dafni (Ed.), Analysisand geometry of metric measure spaces: Lecture notes of the semi-naire de mathematiques superieures . American Mathematical Soci-ety.Mortensen, D., & Pissarides, C. (1994). Job creation and job destruction inthe theory of unemployment.
Review of Economic Studies , , 397–415.Rodrigues-Neto, A. (2007). Representing roommates’ preferences with sym-metric utilities. Journal of Economic Theory , , 545–550.Shapley, L., & Shubik, M. (1971). The assignment game i: The core. Inter-national Journal of Game Theory , , 111–130.Sotomayor, M. (2005). The roommate problem revisited [mimeo U. S˜ao Paulo].Talman, D., & Yang, Z. (2011). A model of partnership formation.
Journalof Mathematical Economics , , 206–212.Tan, J. (1991). A necessary and sufficient condition for the existence of acomplete stable matching. Journal of Algorithms , , 154–178.Villani, C. (2003). Topics in optimal transportation . American MathematicalSociety. 22
Appendix: Proofs
Our proofs use an auxiliary object: the highest possible surplus for a frac-tional roommate matching, namely W F ( n, Φ) = max µ ∈F ( n ) X x µ xx Φ xx + X x = y µ xy Φ xy . (11)where F ( n ) is the set of fractional (roommate) matchings , which relaxes theintegrality constraint on µ : F ( n ) = (cid:0) µ xy (cid:1) : µ xx + P y = x µ xy ≤ n x µ xy = µ yx µ xy ≥ . (12)The program (11) has no immediate economic interpretation since frac-tional roommate matchings are infeasible in the real world; and while ob-viously W P ( n, Φ) ≤ W F ( n, Φ), the inequality in general is strict. We aregoing to show, however, that the difference between the two programs van-ishes when the population becomes large. Moreover, we will establish a linkbetween (11) and the surplus at the optimum of the associated bipartitematching problem.We start by proving:
Lemma A.1 W F ( n, Φ) = W B ( n, n, Φ / . (13) Moreover, problem (11) has a half-integral solution.
Proof of Lemma A.1.
First consider some fractional roommate matching µ ∈ F ( n ), and define ν xy = µ xy if x = yν xx = 2 µ xx . As a (possibly fractional) bipartite matching, clearly ν ∈ B ( n, n ); and X x µ xx Φ xx + X x = y µ xy Φ xy X x,y ∈X ν xy Φ xy . Now the right-hand side is the aggregate surplus achieved by ν in the bi-partite matching problem with margins ( n, n ) and surplus function Φ /
2. Itfollows that W F ( n, Φ) ≤ W B ( n, n, Φ / . (14)23onversely, let ( ν xy ) maximize aggregate surplus over B ( n, n ) with sur-plus Φ /
2. By symmetry of Φ, ( ν yx ) also is a maximizer; and since (7) is alinear program, ν ′ xy = ν xy + ν yx also maximizes it. Define µ ′ xy = ν ′ xy if x = yµ ′ xx = ν xx . Then 2 µ ′ xx + X y = x µ ′ xy = ν xx + 12 X y = x ( ν xy + ν yx )= 12 ( ν xx + X y = x ν xy )+ 12 ( ν xx + X y = x ν yx ) . Now ν xx + P y = x ν xy ≤ n x by the scarcity constraint of “men” of type x ,and ν xx + P y = x ν yx ≤ n x by the scarcity constraint of “women” of type x .It follows that µ ′ ∈ F ( n ), and X x µ ′ xx Φ xx + X x = y µ ′ xy Φ xy X x,y ∈X ν xy Φ xy . Therefore the values of the two programs coincide.Half-integrality follows from the Birkhoff-von Neumann theorem: therealways exists an integral solution ν of the associated bipartite matchingproblem, and the construction of µ ′ makes it half-integral .Given Lemma A.1, we can now prove Theorem 1. Proof of Theorem 1.
The first inequality simply follows from the fact that P ( n ) ⊂ F ( n ). Let us now show the second inequality. Lemma A.1 provedthat W F ( n, Φ) = W B ( n, n, Φ / µ achieve the maximum in W F ( n, Φ),so that W F ( n, Φ) = X x µ xx Φ xx + X x = y µ xy Φ xy . The half-integrality of the solution of problem (11) also follows from a general theoremof Balinski, 1970; but the proof presented here is self-contained. ⌊ x ⌋ denote the floor rounding of x ; by definition, x < ⌊ x ⌋ + 1, so that W F ( n, Φ) < X x ⌊ µ xx ⌋ Φ xx + X x = y (cid:4) µ xy (cid:5) Φ xy X x Φ xx + X x = y Φ xy . The right-hand side can also be rewritten as X x,y (cid:4) µ xy (cid:5) Φ xy + X x,y Φ xy . But ⌊ µ ⌋ is in B ( n, n ), and is integer by construction; therefore X x,y ∈X (cid:4) µ xy (cid:5) Φ xy ≤ W P ( n, Φ) . Finally, X x,y ∈X Φ xy ≤ |X | Φso that W F ( n, Φ) ≤ W P ( n, Φ) + |X | Φ . A.1 Proof of Proposition 2
Proof.
Let n ′ x = n x . By Lemma A.1, problem W F ( n ′ , Φ) has an half-integral solution µ ′ ; therefore problem W F ( n, Φ) has an integral solution2 µ ′ , which must also solve (7). It follows that W P ( n, Φ) = W F ( n, Φ) . A.2 Proof of Theorem 3
Proof.
By Theorem A.1, Problem (11) coincides with a bipartite match-ing problem between marginal ( n x ) and itself. By well-known results onbipartite matching, there exist vectors ( v x ) and ( w y ) such that v x ≥ , w y ≥ v x + w y ≥ Φ xy µ xy >
0. Setting u x = v x + w x u x ≥ u x + u y ≥ Φ xy and X x ∈X n x u x = X x ∈X µ xx Φ xx + X x = y µ xy Φ xy µ, u ) is stable.Conversely, assume that µ is a stable roommate matching. Then bydefinition, there is a vector ( u x ) such that u x ≥ u x + u y ≥ Φ xy and X x ∈X n x u x = X x ∈X µ xx Φ xx + X x = y µ xy Φ xy . Therefore ( u, A = 0) are Lagrange multipliers for the linear programmingproblem (11), and µ is an optimal solution of (11); finally, µ is integral sinceit is a feasible roommate matching. QED.(i), (ii) and (iii) follow, as there exist integral solutions of (11) if andonly if W P ( n, Φ) = W F ( n, Φ) , and W F ( n, Φ) coincides with W B ( n, n, Φ /
2) from Lemma A.1.
A.3 Proof of Theorem 4
Proof.
For each type x , remove one individual of type x to the populationif n x is odd. The resulting subpopulation differs from the previous one byat most |X | individuals, and there is an even number of individuals of eachtype; hence by Proposition 3.1 there exists a stable matching.Each individual so picked can be compensated with his payoff u x . Since u x ≤ Φ, the total cost of compensating at most one individual of each typeis bounded from above by |X |
Φ. 26 .4 Proof of Proposition 5
Proof.
By Theorem 1, in the large population limitlim k →∞ W P (cid:0) n k , Φ (cid:1) N k = W F ( f, Φ)and Lemma A.1 yields the conclusion.
A.5 Proof of Proposition 6
Proof. (i) The number of individuals to be removed is bounded from aboveby |X | , hence its frequency tends to zero as |X | /N →
0. (ii) follows fromthe fact that W F ( n, Φ) − W P ( n, Φ) N → . A.6 Proof of Theorem 7
Proof. (i) Consider an optimal solution µ xy to W P ( n, Φ ′ ). For any pair x = y such that Φ xy > Φ yx , set π xy = µ xy , and π xy = 0 if Φ xy < Φ yx .If Φ xy = Φ yx , set π xy and π yx arbitrarily nonnegative integers such that π xy + π yx = µ xy ; set π xx = µ xx . Then π is feasible for the optimizedsymmetric problem, and one has X x ∈X µ xx Φ ′ xx + X x = y µ xy Φ ′ xy X x,y ∈X π xy Φ xy so that W P (cid:0) n, Φ ′ (cid:1) ≤ W ′P ( n, Φ) . Conversely, consider π xy an optimal solution to W ′P ( n, Φ). First observethat if Φ xy < Φ yx then π xy = 0; otherwise subtracting one from µ xy andadding one to π yx would lead to an improving feasible solution, contradictingthe optimality of π . Set µ xy = π xy + π yx , x = yµ xx = π xx so that X x ∈X µ xx Φ ′ xx + X x = y µ xy Φ ′ xy X x,y ∈X π xy Φ xy W ′P ( n, Φ) ≤ W P (cid:0) n, Φ ′ (cid:1) . (ii) Assume there is a stable matching π xy in the nonexchangeable room-mate problem. Then if there is a matched pair ( x, y ) in that order, onecannot have Φ yx > Φ xy ; otherwise the coalition ( y, x ) would be blocking.Hence one can define µ xy = π xy + π yx , x = yµ xx = π xx and the matching µ is stable in the optimized symmetric problem. Con-versely, assume that the matching µ is stable in the optimized symmetricproblem. Then it is not hard to see that, defining π from µ as in the firstpart of (i) above, the matching ππ