Marriage, honesty, and stability

Many centralized two-sided markets form a matching between participants by running a stable marriage algorithm. It is a well-known fact that no matching mechanism based on a stable marriage algorithm can guarantee truthfulness as a dominant strategy for participants. However, as we will show in this paper, in a probabilistic setting where the preference lists of one side of the market are composed of only a constant (independent of the the size of the market) number of entries, each drawn from an arbitrary distribution, the number of participants that have more than one stable partner is vanishingly small. This proves (and generalizes) a conjecture of Roth and Peranson [23]. As a corollary of this result, we show that, with high probability, the truthful strategy is the best response for a given player when the other players are truthful. We also analyze equilibria of the deferred acceptance stable marriage game. We show that the game with complete information has an equilibrium in which a (1 - o(1)) fraction of the strategies are truthful in expectation. In the more realistic setting of a game of incomplete information, we will show that the set of truthful strategies form a (1 + o(1))-approximate Bayesian-Nash equilibrium. Our results have implications in many practical settings and were inspired by the work of Roth and Peranson [23] on the National Residency Matching Program.

[1]  Lawrence Bodin,et al.  High Tech for a Higher Authority: The Placement of Graduating Rabbis from Hebrew Union College - Jewish Institute of Religion , 2003, Interfaces.

[2]  Boris G. Pittel,et al.  The Average Number of Stable Matchings , 1989, SIAM J. Discret. Math..

[3]  Noam Nisan,et al.  Algorithmic Mechanism Design , 2001, Games Econ. Behav..

[4]  Rajeev Motwani,et al.  Stable husbands , 1990, SODA '90.

[5]  Alvin E. Roth,et al.  Sorority Rush as a Two-Sided Matching Mechanism , 1991 .

[6]  Boris G. Pittel.  On likely solutions of a stable matching problem , 1992, SODA '92.

[7]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[8]  Alvin E. Roth,et al.  Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , 1990 .

[9]  A. Roth.  The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics , 2002 .

[10]  David A. Freedman,et al.  Machiavelli and the Gale-Shapley Algorithm , 1981 .

[11]  A. Roth.  The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory , 1984, Journal of Political Economy.

[12]  Noam Nisan,et al.  Algorithmic mechanism design (extended abstract) , 1999, STOC '99.

[13]  Chung-Piaw Teo,et al.  Gale-Shapley Stable Marriage Problem Revisited: Strategic Issues and Applications , 1999, IPCO.

[14]  Alvin E. Roth,et al.  The Economics of Matching: Stability and Incentives , 1982, Math. Oper. Res..

[15]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[16]  Joan Feigenbaum,et al.  A BGP-based mechanism for lowest-cost routing , 2002, PODC '02.

[17]  Éva Tardos,et al.  An approximate truthful mechanism for combinatorial auctions with single parameter agents , 2003, SODA '03.

[18]  A. Roth,et al.  The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design , 1999, The American economic review.

[19]  G. Brightwell.  THE STABLE MARRIAGE PROBLEM: STRUCTURE AND ALGORITHMS (Foundations of Computing) , 1991 .

[20]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[21]  L. B. Wilson,et al.  Stable marriage assignment for unequal sets , 1970 .

[22]  A. Roth.  The National Residency Matching Program as a labor market. , 1996, Journal of the American Medical Association (JAMA).

[23]  D. A. Sprott.  Urn Models and Their Application—An Approach to Modern Discrete Probability Theory , 1978 .