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 Scarf's Lemma. Then 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. Finally we describe the interpretation of these results for stable matching problems, and in particular, for the hospitals/residents problem with couples.