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 $M\subseteq E$ 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 $p: N \times N \rightarrow {\mathbb R}$ with $p(i,j)+p(j,i)=w(ij)$ if $ij\in M$ and $p(i,j)=p(j,i)=0$ if $ij\notin M$. The pair (M, p) is called a solution. A pair of players i, j with $ij\in E\setminus M$ 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 $b\equiv 1$ (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.

P. Biró—Supported by the Hungarian Academy of Sciences under Momentum Programme LD-004/2010, by the Hungarian Scientific Research Fund - OTKA (no. K108673), and by János Bolyai Research Scholarship of the Hungarian Academy of Sciences.

D. Paulusma—Supported by EPSRC Grant EP/K025090/1.