Matching games: the least core and the nucleolus

A matching game is a cooperative game defined by a graph $G=(V,E)$. The player set is $V$ and the value of a coalition $S \subseteq V$ is defined as the size of a maximum matching in the subgraph induced by $S$. We show that the nucleolus of such games can be computed efficiently. The result is based on an alternative characterization of the least core which may be of independent interest. The general case of weighted matching games remains unsolved. \u

Two-sided Matching, Who Marries Whom? And what Happens upon Divorce?

Conventional two-sided matching game is a one-period game. In this note, we contribute to the existing literature by examining a multi-period two-sided matching problem allowing for the possibility of a divorce. We assume that the matching game is played repeatedly and the payoff matrix changes over time. It is shown that the rule of divorce will affect the equilibrium of a marriage game. An empirical implication of our result is that a country with a well-developed financial market will have a better marital outcome as compared to a less-developed country.

The Eeckhout Condition and the Subgame Perfect Implementation of Stable Matching

We investigate an extensive form sequential matching game of perfect information. We show that the subgame perfect equilibrium of the sequential matching game leads to the unique stable matching when the Eeckhout Condition (2000) for existence of a unique stable matching holds, regardless of the sequence of agents. This result does not extend to preferences that violate the Eeckhout Condition, even if there is a unique stable matching.Matching; unique stable matching; subgame perfect equilibrium

Social Norms and Choice: A Weak Folk Theorem for Repeated Matching Games

A folk theorem for repeated matching games is established that holds if the stage game is not a pure coordination game. It holds independent of population size and for all matching rules-including rules that depend on players choices or the history of play. This paper also establishes an equilibrium condition and using this discovers two differences between the equilibria of repeated matching games and standard repeated games. Trigger strategies are not equilibria and there is no simple optimal penal code.

Shapley Meets Shapley

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page