Power Indices in Spanning Connectivity Games

Abstract

Abstract. 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 ShapleyShubik 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.

Power Indices and the Measurement of Control in Corporate Structures

“…When dealing with large databases, as in Crama et al [2003], Crama and Leruth [2007], Aminadav et al [2011], questions of algorithmic efficiency become of paramount importance. Much literature has been devoted to the computation of power indices of simple games and of special classes of games, such as weighted majority games; see, e.g., Bilbao [2000], Matsui and Matsui [2000], Klinz and Woeginger [2005], Aziz et al [2009], Bachrach et al [2010], Crama and Hammer [2011]. Various methodologies have been proposed, based on dynamic programming, generating functions, Monte-Carlo simulation, multilinear extensions, and so on.…”

Computation of the Shapley value of minimum cost spanning tree games: #P-hardness and polynomial cases

“…(Proof) We reduce #MINIMUM s-t VERTEX CUT to MCSTG-SHAPLEY, where we shall use the proof technique used in [3]. Let G = (N, E) be an undirected graph with s, t ∈ N being non-adjacent.…”

Graph-based Coalitional Games - An Analysis via Characteristics

“…The reasons why we have selected this type of coalitional games are the following: Graph-based games are particularly interesting to solve network-based problems of all sorts, which often occur in operations research. They offer a manifold of graph-based characteristics that can be analysed, and several graph-based games have already been treated in the literature over the last three years [2,5,6,12].…”

“…Over the last few years a series of papers [1,2,3,5,6,12,15,21,24] has been published that analyse the computational complexity of solution concepts applied to different types of coalitional games, as well as the complexity to Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.…”