The Stable Fixtures Problem with Payments

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 vector 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 a given multiple partners matching game has no stable solution, or obtains a stable solution for it. We characterize the set of stable solutions of a multiple partners matching game in two different ways and show how this leads to simple proofs for a number of known results of Sotomayor (1992,1999,2007) for multiple partners ssignment games and to generalizations of some of these results to multiple partners matching games. We also 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the existence of a stable solution and the non-emptiness of the core, which holds in the other models with payments, is no longer valid for our (most general) model. We also prove that the problem of deciding if an allocation belongs to the core jumps from being polynomial-time solvable for $b\leq 2$ to NP-complete for $b\equiv 3$.