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.

[1]  Daniel Granot,et al.  Minimum cost spanning tree games , 1981, Math. Program..

[2]  Jeroen Kuipers,et al.  Note Computing the nucleolus of min-cost spanning tree games is NP-hard , 1998, Int. J. Game Theory.

[3]  H. Hirai.  Characterization of the Distance between Subtrees of a Tree by the Associated Tight Span , 2006 .

[4]  Daniel Granot,et al.  The Relationship Between Convex Games and Minimum Cost Spanning Tree Games: A Case for Permutationally Convex Games , 1982 .

[5]  Daniel Granot,et al.  On the core and nucleolus of minimum cost spanning tree games , 1984, Math. Program..

[6]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[7]  Eyal Winter.  Chapter 53 The shapley value , 2002 .

[8]  Kazutoshi Ando,et al.  REDUCTION OF ULTRAMETRIC MINIMUM COST SPANNING TREE GAMES TO COST ALLOCATION GAMES ON ROOTED TREES , 2010 .

[9]  J. S. Mateo.  The Shapley Value , 2012 .

[10]  C. Colbourn,et al.  Computing 2-terminal reliability for radio-broadcast networks , 1989 .

[11]  Vincent Feltkamp,et al.  On the irreducible core and the equal remaining obligations rule of minimum cost spanning extension problems , 1994 .

[12]  Sándor P. Fekete,et al.  On the complexity of testing membership in the core of min-cost spanning tree games , 1997, Int. J. Game Theory.

[13]  A. Roth.  The Shapley value , 2005, Game Theory.

[14]  Henk Norde,et al.  Minimum cost spanning tree games and population monotonic allocation schemes , 2004, Eur. J. Oper. Res..

[15]  L. Shapley.  A Value for n-person Games , 1988 .

[16]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[17]  Rahul Savani,et al.  Power Indices in Spanning Connectivity Games , 2009, AAIM.

[18]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[19]  Anirban Kar,et al.  Cost monotonicity, consistency and minimum cost spanning tree games , 2004, Games Econ. Behav..

[20]  Anirban Kar,et al.  Axiomatization of the Shapley Value on Minimum Cost Spanning Tree Games , 2002, Games Econ. Behav..

[21]  A. Belianin,et al.  A Game-Theoretic Approach , 2001 .

[22]  C. G. Bird,et al.  On cost allocation for a spanning tree: A game theoretic approach , 1976, Networks.

[23]  Roland Bacher,et al.  Determinants of matrices related to the Pascal triangle , 2002 .

[24]  Gustavo Bergantiños,et al.  The optimistic TU game in minimum cost spanning tree problems , 2007, Int. J. Game Theory.

[25]  Daniel J. Kleitman,et al.  Cost allocation for a spanning tree , 1973, Networks.

[26]  Hervé Moulin,et al.  Sharing a minimal cost spanning tree: Beyond the Folk solution , 2010, Games Econ. Behav..

[27]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .