Spanning connectivity games

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values and Shapley-Shubik indices is #P-complete for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, computing the Shapley value is #P-complete for simple games represented by the set of minimal winning coalitions, Threshold Network Flow Games, Vertex Connectivity Games and Coalitional Skill Games.

[1]  Haris Aziz,et al.  Complexity of comparison of influence of players in simple games , 2008, ArXiv.

[2]  S RosenscheinJeffrey,et al.  Power in threshold network flow games , 2009 .

[3]  Herbert Hamers,et al.  Operations research games: A survey , 2001 .

[4]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[5]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[6]  Evangelos Markakis,et al.  Approximating power indices , 2008, AAMAS.

[7]  J. Deegan,et al.  A new index of power for simplen-person games , 1978 .

[8]  Ely Porat,et al.  Power and stability in connectivity games , 2008, AAMAS.

[9]  Jeffrey S. Rosenschein,et al.  Computing the Banzhaf power index in network flow games , 2007, AAMAS '07.

[10]  Forming Coalitions and Measuring Voting Power , 1982 .

[11]  Michael Maschler,et al.  Spanning network games , 1998, Int. J. Game Theory.

[12]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..

[13]  Stef Tijs,et al.  Monotonic games are spanning network games , 1993 .

[14]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

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

[16]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[17]  B. Bollobás.  Surveys in Combinatorics , 1979 .

[18]  Ulrik Brandes,et al.  Network Analysis: Methodological Foundations , 2010 .

[19]  Michael O. Ball,et al.  Computational Complexity of Network Reliability Analysis: An Overview , 1986, IEEE Transactions on Reliability.

[20]  Imma Curiel,et al.  Cooperative Game Theory and Applications: Cooperative Games Arising from Combinatorial Optimization Problems , 1997 .

[21]  Jeffrey S. Rosenschein,et al.  Power in threshold network flow games , 2009, Autonomous Agents and Multi-Agent Systems.

[22]  Yves Crama,et al.  Control and voting power in corporate networks: Concepts and computational aspects , 2007, Eur. J. Oper. Res..

[23]  Piotr Faliszewski,et al.  The complexity of power-index comparison , 2008, Theor. Comput. Sci..

[24]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[25]  Jeffrey S. Rosenschein,et al.  Coalitional skill games , 2008, AAMAS.