The stable fixtures problem with payments

Abstract

We generalize two well-known game-theoretic models by introducing multiple partners matching games, defined 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 payoff is a mapping p: N ×N → ℝ with p(i, j) + p(j, i) = w(ij) if ij ∈ M and p(i, j) = p(j, i) = 0 if ij ∉ M. The pair (M,p) is called a solution. A pair of players i, j with ij ∈ E\M 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 off. A solution is stable if it has no blocking pairs. We give a polynomial-time algorithm that either finds 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 different ways and initiate a study on the core of the corresponding cooperative game, where coalitions of any size may be formed.