Mathematical Social Sciences 39 2000 119–132 www.elsevier.nl locate econbase
Existence of stable outcomes and the lattice property for a unified matching market
Marilda Sotomayor
˜ ˜
FEA-USP-Department of Economics , Universidade de Sao Paulo, 908 Butanta SP,
05508-900 CEP Sao Paulo, Brazil Received 1 December 1997; received in revised form 1 June 1998; accepted 1 February 1999
Abstract
This paper establishes the existence of stable matchings, the lattice property of the core and the existence of optimal stable payoffs for each side of the market. We do this in a matching market
consisting of a ‘mixed economy’ in which some firms compete by means of salary and others have no flexibility over terms of appointment. Common proofs are used, which unify the traditional
discrete and continuous cases, namely the marriage and assignment models.
2000 Elsevier Science B.V. All rights reserved.
Keywords : Matching; Stable matching; Core
JEL classification : C78; D78
1. Introduction
There is now a large literature on matching, both theoretical and empirical, concerned with stable matchings and mechanisms which achieve them. An unusual feature of the
theoretical literature is that quite similar results have been established for both discrete and continuous models, initially with fundamentally dissimilar proofs, and it has turned
out to be difficult to unify these models. This divide goes back to Gale and Shapley 1962 and Shapley and Shubik 1972 which show the non-emptiness of the set of
stable matchings for discrete and continuous models, respectively, using combinatorial arguments in one case and linear programming arguments in the other. These same
families of arguments can then be extended to show that the sets of stable matchings in
Tel.: 155-11-211-8706; fax: 155-11-814-3814. E-mail address
: marildasusp.br M. Sotomayor 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
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each kind of model share many common properties such as a lattice structure in terms of the common preferences of agents on one side of the market, with corresponding optimal
outcomes for each side of the market at which agents on that side will have no incentive to misrepresent their preferences etc. see Roth and Sotomayor, 1990, for a review of the
literature.
The empirical importance of both kinds of models comes from the variety of labor markets. For example, new law school graduates may enter the market for associate
positions in private law firms, which compete with each other in terms of salary, or they may seek employment as law clerks to federal circuit court judges, which are civil
service positions with predetermined fixed salaries. Traditionally the former kind of market has been modeled as an Assignment Game, in which salary may be negotiated
and may vary continuously on the set of real numbers, while the latter has been modeled as a Marriage Market. In this market salary is modeled as part of the job description and
it is one of the factors that determine the preferences that workers have over firms.
One line of work which went some way towards unifying these disparate models involved creating linear programming formulations of stable matchings in discrete
markets for example, Vande Vate, 1989; Rothblum, 1992; Roth et al., 1993. Those papers revealed a good deal of surprising algebraic structure in the set of stable
matchings of discrete markets, but the large differences between the linear programming formulations for the discrete case and the continuous case only emphasized that similar
results were being obtained for very different reasons in the two cases.
Another approach has been to look at models which generalize both the Marriage Model and the Assignment Model, and try to obtain the common results for the two in
the more general model. This was the approach taken in Roth and Sotomayor 1996. In contrast with previous treatments, this paper derives the parallel conclusions for the two
sets of models in the same way from the same assumptions. It shows the lattice property of the core, the existence of optimal stable outcomes and some comparative statics
results.
However, that paper does not provide any existence theorem of stable outcomes. Furthermore, in the model of Roth and Sotomayor 1996 a given agent either is in the
discrete market or is in the continuous one. This depends on the set of allowable salaries. For empirical purposes, the problem with such a model is of course that all individuals
have their choices restricted to only one of the markets, although some of them may wish to enter both kinds of markets simultaneously. In the real world these markets are
really parts of a single market.
Kaneko 1982 provides a quite general and complex model that generalizes the Assignment Game of Shapley and Shubik. He observes that the Marriage model is a
special case of this model. He proves the non-emptiness of the core but not the lattice result or related structural properties.
Eriksson and Karlander 1997 also provides a single market in which the two kinds of agents can trade. However, they prove the existence of stable outcomes and the lattice
property of the core for a particular case of this model, which does not include the continuous market as it was proposed by Shapley and Shubik.
In the present paper, following the idea of Eriksson and Karlander 1997, we unify the two kinds of markets by presenting a single market that includes the discrete and
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continuous markets. The interest of this model is that it mimics real life to the effect that an agent may trade in both markets simultaneously.
For this market we propose to fill some existing gaps. We first prove the non- emptiness of the set of stable outcomes. Gale and Shapley 1962 uses an algorithm to
prove the existence of stable outcomes for the marriage market; Shapley and Shubik 1972 uses the Duality Theorem to prove the same result for the continuous Assignment
Game; Demange et al. 1986 proves the existence of stable outcomes for the Assignment Game with an integer payoff matrix by using a mechanism where objects
are auctioned simultaneously.
An old challenge in the literature is to find a simple proof of the existence result for both models, using similar arguments. The proof of Kaneko 1982, which uses the
technique of balanced games, is very complicated because of the complexity of its model. The recent proof of Eriksson and Karlander 1997 does not hold for all possible
assignment games. It uses a version of the proof of Demange et al. 1986 which only applies when the payoff matrices have integer entries.
In this paper we use combinatorial arguments. The main feature of our proof is that it is simpler and shorter than the proofs of the other authors. Moreover, it holds for both
models without any restriction. When restricted to the Marriage Model it coincides with the existence proof of Sotomayor 1996. When restricted to the Assignment Game it is
also very short and it can be used as an alternative proof of the non-emptiness of the core of this model.
Next we prove that the set of stable payoffs is endowed with the structure of a complete lattice. Roughly speaking, if A and B are two stable outcomes then it might be
that some workers resp. firms will get more income under A than under B and others will get more under B than under A. The lattice theorem implies that there is a stable
outcome C, which gives each worker resp. firm the larger of the two amounts and also one, D, which gives each of them the smaller amount. This is of interest because,
together with the compactness of the set of stable payoffs, it shows that among all stable payoffs there is one and only one which is worker optimal resp. firm optimal,
meaning that all workers resp. firms get as much income under it as under any other stable payoff.
Eriksson and Karlander 1997 proves the lattice result under a very restrictive assumption. It assumes strict preferences for the Marriage Market and unicity of the
optimal matching for the Assignment Game. John Conway, in Knuth 1976, requires strictness of the preferences to prove the lattice property for the Marriage Market. Our
assumption is that the strong core and the weak core coincide. In particular our result holds for the Marriage Market when preferences are strict. It always holds for the
Assignment Market independently of the number of optimal matchings. In the marriage model, the assumption of strict preferences causes those two sets to coincide, while in
the continuous models the two sets coincide because agents have continuous preferences and prices can be adjusted continuously.
Finally, as a corollary of the lattice property and its completeness, we show that, when the strong core and the weak core coincide, the optimal stable payoff for each side
exists. This paper is organized as follows: Section 2 describes the generalized model. Section
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3 proves the existence of stable outcomes. Section 4 shows that the core is a complete lattice and that it has a unique optimal stable payoff for each side of the market. Two
illustrative examples are presented at the end of this section.
2. The mathematical model
