The Stable Fixtures Problem with Payments

We generalize two well-known game-theoretic models by introducing multiple partners matching games, de ned by a graph G = (N;E), with an integer vertex capacity function b and an edge weighting w. The set N consists of a number of players that are to form a set M E of 2-player coalitions ij with value w(ij), such that each player i is in at most b(i) coalitions. A payo is a mapping p : N N ! R with p(i; j) + p(j; i) = w(ij) if ij 2 M and p(i; j) = p(j; i) = 0 if ij =2 M. The pair (M; p) is called a solution. A pair of players i; j with ij 2 E nM blocks a solution (M; p) if i; j can form, possibly only after withdrawing from one of their existing 2-player coalitions, a new 2-player coalition in which they are mutually better o . A solution is stable if it has no blocking pairs. We give a polynomial-time algorithm that either nds that no stable solution exists, or obtains a stable solution. Previously this result was only known for multiple partners assignment games, which correspond to the case where G is bipartite (Sotomayor, 1992) and for the case where b 1 (Biro et al., 2012). We also characterize the set of stable solutions of a multiple partners matching game in two di erent ways and perform a study on the core of the corresponding cooperative game, where coalitions of any size may be formed. In particular we show that the standard relation between the existence of a stable solution and the non-emptiness of the core, which holds in the other models with payments, is no longer valid for our (most general) model.