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.
We consider computational aspects of a game theoretic approach to network reliability. Consider a network where failure of one node may disrupt communication between two other nodes. We model this network as a simple coalitional game, called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. We show that power indices, which express an agent's ability to affect the outcome of the vertex connectivity game, can be used to identify significant possible points of failure in the communication network, and can thus be used to increase network reliability. We show that in general graphs, calculating the Banzhaf power index is #P-complete, but suggest a polynomial algorithm for calculating this index in trees. We also show a polynomial algorithm for computing the core of a CG, which allows a stable division of payments to coalition agents. We study the complexity of the following problem: Given two weighted voting games G? and G?? that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [16] and the Banzhaf power index [3,6]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also show that, unlike the Banzhaf power index, the Shapley-Shubik power index is not #P-parsimonious-complete. This finding sets a hard limit on the possible strengthenings of a result of Deng and Papadimitriou [5], who showed that the Shapley-Shubik power index is #P-metric-complete. Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. The Banzhaf power index is one of the most popular; it is also well-defined for any simple coalitional game. In this paper, we examine the computational complexity of calculating the Banzhaf power index within a particular multiagent domain, a network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow of a given size from a source vertex to a target vertex. The relative power of each edge agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We show that calculating the Banzhaf power index of each agent in this network flow domain is #P-complete. We also show that for some restricted network flow domains there exists a polynomial algorithm to calculate agents' Banzhaf power indices. This paper proposes to rely on power indices to measure the amount of control held by individual shareholders in corporate networks. The value of the indices is determined by a complex voting game viewed as the composition of interlocked weighted majority games; the compound game reflects the structure of shareholdings. The paper describes an integrated algorithmic approach which allows to deal efficiently with the complexity of computing power indices in shareholding networks, irrespective of their size or structure. In particular, the approach explicitly accounts for the presence of float and of cyclic shareholding relationships. It has been successfully applied to the analysis of real-world financial networks.