# The Average Tree Solution for Cooperative Games with Communication Structure

Working paper

Issue number:

RM/08/026

Publisher:

Maastricht University

Year:

2008

We study cooperative games with communication structure, represented by an undirected graph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class of games. Given the graph structure we deﬁne a collection of spanning trees, where each spanning tree speciﬁes a particular way by which players communicate and determines a payoﬀ vector of marginal contributions of all the players. The average tree solution is deﬁned to be the average of all these payoﬀ vectors. It is shown that if a game has a complete communication structure, then the proposed solution coincides with the Shapley value, and that if the game has a cycle-free communication structure, it is the solution proposed by Herings, van der Laan and Talman (2008). We introduce the notion of link-convexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is even weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than supper-additivity.