Fractional solutions for capacitated NTU-games, with applications to stable matchings

In this paper we investigate some new applications of Scarf's Lemma. First, we introduce the notion of fractional core for NTU-games, which is always nonempty by the Lemma. Stable allocation is a general solution concept for games where both the players and their possible cooperations can have capacities. We show that the problem of finding a stable allocation, given a finitely generated NTU-game with capacities, is always solvable by a variant of Scarf's Lemma. Then we describe the interpretation of these results for matching games. Finally we consider an even more general setting where players' contributions in a joint activity may be different. We show that a stable allocation can be found by the Scarf algorithm in this case as well, and we demonstrate the usage of this method for the hospitals resident problem with couples. This problem is relevant in many practical applications, such as NRMP (National Resident Matching Program).