Power Indices in Spanning Connectivity Games

  • Haris Aziz
  • Oded Lachish
  • Mike Paterson
  • Rahul Savani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5564)


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 is #P-complete and computing Shapley-Shubik indices or values is NP-hard 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. This answers (positively) an open question of whether computing Shapley-Shubik indices for a simple game represented by the set of minimal winning coalitions is NP-hard.


Network connectivity coalitional games Banzhaf index Shapley-Shubik index 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Haris Aziz
    • 1
  • Oded Lachish
    • 1
  • Mike Paterson
    • 1
  • Rahul Savani
    • 1
  1. 1.Computer Science DepartmentUniversity of WarwickCoventryUK

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