Values of Graph-Restricted Games

We consider the problem of modifying n-person games so as to take account of the difficulties imposed by lack of communications, and the opportunities this might accord to intermediaries. In this model, the members of a finite set are simultaneously players in a game and vertices of a graph. A combination of these two structures gives rise to a new, modified game in which the only effective coalitions are those corresponding to connected partial graphs. We study the relationship between the power indices of the original game and the restricted game; for the special case where the graph is a tree, this relationship is especially easy to analyze. Several examples are studied in detail.