Two Values for Transferable Utility Games with Coalition and Graph Structure

In this paper we introduce and characterize two new values for transferable utility games with graph restricted communication and a priori unions. Both values are obtained by applying the Shapley value to an associated TU-game. The graph-partition restricted TU-game is obtained by taking the Myerson graph restricted game and of that the Kamijo partition restricted game. In this game the dividend of any coalition that is neither a subset of a union nor a union of unions is zero. The partition-graph restricted TU-game is obtained by taking the partition restricted game and of that the graph restricted game. In this game the dividend of any coalition that is not connected in the graph is zero. We apply the values to an economic example in which the players in a union represent the cities in a country and the graph represents a network of natural gas pipelines between the cities.