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

Abstract

Abstract. 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).

Matching couples with Scarf’s algorithm

“…Then we present the integral stable allocation problem with contributions by Biró and Fleiner [3]. Finally we describe the connections between some variants of these two problems.…”

“…The exact description of the Scarf algorithm can be found in [18] and its extension for ISAC is described in [3]. Here we only want to highlight some important facts of the Scarf algorithm (and its extension).…”

“…Biró and Fleiner [3] showed that every stable allocation problem with contributions has a solution that can be obtained by the extended Scarf algorithm. Furthermore, if we have no capacities on the edges then the existence of a stable allocation is guaranteed by the original Scarf lemma, and one stable allocation can be obtained by the original Scarf algorithm.…”

“…Aharoni and Fleiner [2] used this algorithm to find stable fractional matchings for problems where the underlying graph is not necessarily bipartite. Biró and Fleiner [3] generalised this result by showing that Scarf's algorithm can be extended to find stable allocations for NTU-games where both the agents and their cooperations can have capacities, and even when agents have different contributions in a co-operation performed. We shall note though that the Scarf algorithm is not known to be polynomial.…”

“…Furthermore, we show that the HRC problem (where couples may apply for a pair of positions in the same hospital) can be transformed into an integral stable allocation problem with contributions (ISAC) and with no edge capacities, where the stable solutions for the latter problem are stable solutions for the former. Biró and Fleiner [3] demonstrated that the original Scarf algorithm always returns a stable allocation for both the ISA and ISAC problems with no edge capacities, and whenever it outputs an integer solution it solves the problem. Therefore, the Scarf algorithm can be used as a heuristic to solve all of these problems, ISA, ISAC and HRC as well.…”