Harsanyi Power Solutions for Graph-restricted Games

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A solution for TU-games assigns a set of payoff distributions (possibly empty or consisting of a unique element) to every TU-game. Harsanyi solutions are solutions that are based on distributing dividends. In this paper we consider games with limited communication structure in which the edges or links of an undirected graph on the set of players represent binary communication links between the players such that players can cooperate if and only if they are connected. For such games we discuss Harsanyi solutions whose dividend shares are based on power measures for nodes in corresponding communication graphs. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games (i.e. the class of TU-games) and equals the position value on the class of cycle-free graph games. Another example is the Harsanyi power solution that is based on the equal power measure, which turns out to be the Myerson value. Various applications of our results are provided.