A Unifying Model for Matching Situations

We present a unifying framework for transferable utility coalitional games that are derived from a non-negative matrix in which every entry represents the value obtained by combining the corresponding row and column. We assume that every row and every column is associated with a player, and that every player is associated with at most one row and at most one column. The instances arising from this framework are called matching games, and they encompass assignment games and permutation games as two polar cases. We show that the core of a matching game is always nonempty by proving that the set of matching games coincides with the set of permutation games. Then we focus on two separate problems. First, we exploit the wide range of situations comprised in our framework to investigate the relationship between matching games with different player sets but defined by the same underlying matrix. We show that the core is not only immune to the merging of a row player and a column player, but also to the reverse anipulation, i.e., to the splitting of a player into a row player and a column player. Other common solution concepts fail to be either merging-proof or splitting-proof in general. Second, we focus on permutation games only and we analyze the set of all matrices that define permutation games with the same core. In contrast to assignment games, we show that there can be multiple matrices whose entries cannot be raised without modifying the core of the corresponding permutation game and that, for small instances, every such matrix defines an exact game.