The Core Matchings of Markets with Transfers

We characterize the structure of the set of core matchings of an assignment game (a two-sided market with transfers). Such a set satisfies a property we call consistency. Consistency of a set of matchings states that, for any matching ν, if, for each agent i there exists a matching μ in the set for which μ (i) = ν (i), then ν is in the set. A set of matchings satisfies consistency if and only if there is an assignment game for which all elements of the set maximize the surplus. We also identify conditions under which we can assume the assignment game has nonnegative values.