Elsevier

Journal of Economic Theory

Volume 160, December 2015, Pages 216-242
Journal of Economic Theory

A finite decentralized marriage market with bilateral search

https://doi.org/10.1016/j.jet.2015.09.005Get rights and content

Abstract

I study a model in which a finite number of men and women look for future spouses via random pairwise meetings. The central question is whether equilibrium marriage outcomes are stable matchings when search frictions are small. The answer is they can but need not be. For any stable matching there is an equilibrium leading to it almost surely. However there may also be equilibria leading to an unstable matching almost surely. A restriction to simpler strategies or to markets with aligned preferences rules out such equilibria. However unstable—even Pareto-dominated—matchings may still arise with positive probability under those two restrictions combined. In addition, inefficiency due to delay may remain significant despite vanishing search frictions. Finally, a condition is identified under which all equilibria are outcome equivalent, stable, and efficient.

Introduction

The stable matching is the main solution concept for cooperative two-sided matching problems under nontransferable utility. Many centralized mechanisms are designed to implement stable matchings.1 However, whether outcomes of decentralized two-sided matching markets correspond to stable matchings remains unclear. The present paper addresses this question by considering a decentralized two-sided matching market modeled as a search and matching game. Following Gale and Shapley (1962) I inherit the interpretation that the game represents the situation in which unmarried men and women gather in a marketplace to look for future spouses. The game starts with an initial market à la Gale–Shapley, henceforth referred to as a marriage market, consisting of finitely many men and women with heterogeneous preferences. In every period a meeting between a randomly selected pair of a man and a woman takes place, during which they sequentially decide whether to marry each other. Mutual agreement leads to marriage. Married couples leave the game. Disagreement leads to separation. Separated people continue searching. The game ends when no mutually acceptable pairs of a man and a woman are left. Search is costly due to frictions parametrized as a common discount factor that diminishes the value of a future marriage. A game outcome, reflecting who has married whom and who stays single, corresponds to a matching for the initial market. The central question addressed in the paper is whether matchings that obtain in equilibria are stable matchings for the initial market when search frictions are small. The analysis focuses on a near-frictionless setting in order to test the general conjecture that if in a decentralized market the participants have easy access to each other with low costs then equilibrium outcomes would be in the core of the underlying market.2 The paper shows that the answer to the central question is indeterminate at best and No in general, in contrast to what has been conjectured on this matter.3 First, for any stable matching there is an equilibrium leading to that matching almost surely (Proposition 4.2), that is, every player expects to marry according to the pairing scheme implied by the matching. This result establishes that the set of all stable matchings is contained in the set of all matchings that may arise in equilibria. Then it is shown that the latter set may contain unstable matchings as well: Under certain preference structures there are equilibria leading to an unstable matching almost surely (Example 1). The paper proceeds to propose two conditions, each of which rules out such equilibria: 1. The players do not condition their behavior on the actions during any past failed meeting (Proposition 4.4). 2. The players' preferences satisfy the Sequential Preference Condition, a condition that implies a certain degree of preference alignment (Proposition 4.6). However the two conditions, separately or combined, are not sufficient to rule out equilibria in which unstable matchings arise with positive probability; some of the probable matchings may even be Pareto-dominated (Example 3). Another source of inefficiency is delay: Significant loss of efficiency due to delay may be present in an equilibrium even if search frictions are arbitrarily small (Example 4, Example 5). The paper ends with a uniqueness result that is pro-stability and efficiency: If the players' preferences satisfy a strengthening of the Sequential Preference Condition which implies a stronger degree of alignment, then all equilibria are outcome equivalent, stable, and efficient (Proposition 4.8).

The present paper contributes to the literature on search and matching games in which a marriage market is embedded. The central question of the literature agrees with that of the present paper: Do equilibrium outcomes correspond to stable matchings? An early paper in this literature (Roth and Vande Vate, 1990) studies the steady state of a search and matching game with short-sighted players and concludes that a stable matching obtains almost surely. Later papers consider sophisticated players. McNamara and Collins (1990), Burdett and Coles (1997), Eeckhout (1999), Bloch and Ryder (2000) and Smith (2006) assume that the underlying marriage market admits a unique stable matching that is positively assortative. Their results confirm that equilibrium outcomes retain some extent of assorting. Adachi (2003) and Lauermann and Nöldeke (2014) consider a market with a general preference structure. Adachi (2003) studies a model in which the steady state stock of active players is exogenously maintained and confirms that equilibrium outcomes converge to stable matchings as search frictions vanish. Lauermann and Nöldeke (2014) consider endogenous steady states and finds that all limit outcomes are stable if and only if the underlying market has a unique stable matching.

The model considered in this paper also embeds a marriage market in a search and matching game. In contrast to the previously cited papers, all of which study steady state equilibria in a stationary setting, the present model features a nonstationary search situation. Indeed, the market shrinks as players marry and leave. Moreover, all but Roth and Vande Vate (1990) consider a market with a continuum of nameless players, whereas in the present model the market is finite and the players are identifiable. Nonstationarity and finiteness make the present model qualitatively different from most models considered in the literature. It follows that the set of matchings that may obtain in equilibria of the present model is in general different from that of a stationary and continuum model.

Another related literature investigates models embedding a marriage market in a sequential bargaining game reminiscent of the deferred acceptance protocol in Gale and Shapley (1962). This literature includes Alcalde (1996), Diamantoudi et al. (2015), Pais (2008), Suh and Wen (2008), Niederle and Yariv (2009), Bloch and Diamantoudi (2011), and Haeringer and Wooders (2011).4 Like the present paper, these papers consider a finite marriage market that shrinks as players marry and leave. The difference between models in this literature and those in the search and matching literature, including the present model, is the search technology. A sequential bargaining game models a market with directed search: When it is his or her turn to move, a player can reach and deal with any player of the opposite sex without delay or uncertainty. In contrast, a search and matching game models a market with undirected search: Bilateral meetings are stochastic; one needs patience and luck to encounter a particular person. One common finding among papers with a sequential bargaining model is that some or all stable matchings can be supported in equilibria. Such equilibria bear resemblance to equilibria that lead to a particular stable matching almost surely in the present model, see Proposition 4.2. On the other hand, unstable matchings may also obtain in equilibria of a sequential bargaining game, which is the case in Diamantoudi et al. (2015), Suh and Wen (2008) and Haeringer and Wooders (2011). This common finding is also in accordance with results in the present paper. However, because of random search, the model in the present paper may have equilibria that have no counterpart in a sequential bargaining model. For instance, in a typical sequential bargaining model, an equilibrium in pure strategies leads to one matching deterministically, whereas in this model an equilibrium in pure strategies may lead to several possible matchings, because the players' strategies may depend on which of the multiple probable paths the history has taken. In this respect, nonstationarity has little influence in a sequential bargaining model because a player's expected payoff remains unchanged as the game unfolds, whereas in the present model exogenous uncertainty may drastically change a player's continuation prospect.

A third related literature5 studies whether the Walrasian price can be supported in equilibria of a search and bargaining game in which an exchange economy, instead of a marriage market, is embedded. Papers from this literature and the present paper are united under the theory of non-cooperative foundation of cooperative solution concepts. Indeed, the present model can be seen as the nontransferable utility version of the models considered in Rubinstein and Wolinsky (1990) and Gale and Sabourian (2006).

The layout of the paper is as follows: Section 2 introduces the game. Section 3 sets up an analytic framework. Section 4 provides the analysis. Section 5 concludes. Lengthy proofs and additional examples are found in Appendix A.

Section snippets

The marriage market

There are two disjoint sets of players: the set of men M and the set of women W. A generic man is denoted as m, a woman as w, and a pair of a man and a woman as (m,w). A man might end up marrying some wW or remaining single. All men's preferences over W{s}, where s stands for being single, are represented by u:M×(W{s})R where u(m,) is m's Bernoulli utility function over W{s}. Likewise all women's preferences are represented by v:(M{s})×WR where v(,w) is w's Bernoulli utility function

Equilibria

For most of the analysis the solution concept that will be applied is the subgame perfect equilibrium. In addition I consider two equilibrium selection criteria to accommodate more restrictive information settings.

For history h let g(h) denote the sequence (mt,wt,Rt)t=1:τ(h) where mt and wt are the man and woman who met on date t under h, Rt{marriage,separation} is the result of that meeting, and τ(h) is the date of the last concluded meeting under h. A strategy profile σ satisfies the

Preliminary results

The following lemma collects some useful results for future reference.

Lemma 4.1

For a subgame perfect equilibrium σ let π(x) denote the expected payoff for player x under σ. The following are true for σ:

  • (a)

    π(x)0 for any xMW.

  • (b)

    (m,w) marry with positive probability only if m is acceptable to w.7

Conclusion

This paper studies a search and matching game with a marriage market embedded, and analyzes whether matchings that arise in equilibria are stable when search frictions are small. It is found that this is not the case in general. Unstable matchings may arise for many reasons and under restrictive conditions. Moreover, significant loss of efficiency due to delay may be incurred in equilibria even if search frictions are small. A condition that implies preference alignment in a strong sense

Acknowledgments

I wish to thank the editor and two referees, V. Bhaskar, Tilman Börgers, In-Koo Cho, Scott Kominers, John Quah, Bartley Tablante, and the participants at the 2013 Asian Meeting of the Econometric Society and the 2014 Stony Brook International Conference on Game Theory for enlightening discussions and helpful comments. I am especially grateful to Stephan Lauermann for his encouragement and guidance which made possible the fruition of this paper.

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