Computation of the Shapley Value of Minimum Cost Spanning Tree Games: #P-Hardness and Polynomial Cases

We show that computing the Shapley value of minimum cost spanning tree games is #P-hard even if the cost functions are restricted to be {0, 1}-valued. The proof is by a reduction from counting the number of minimum 2-terminal vertex cuts of an undirected graph, which is #P-complete. We also investigate minimum cost spanning tree games whose Shapley values can be computed in polynomial time. We show that if the cost function of the given network is a subtree distance, which is a generalization of a tree metric, then the Shapley value of the associated minimum cost spanning tree game can be computed in O(n4) time, where n is the number of players.