A Pseudo-Polynomial Algorithm for Computing Power Indices in Graph-Restricted Weighted Voting Games

Weighted voting games allow for studying the distribution of power between agents in situations of collective decision making. While the conventional version of these games assumes that any agent is always ready to cooperate with all others, recently, more involved models have been proposed, where cooperation is subject to restrictions. Following Myerson [1977], such restrictions are typically represented by a graph that expresses available communication links among agents. In this paper, we study the time complexity of computing two well-known power indices - the Shapley-Shubik index and the Banzhaf index – in the graph-restricted weighted voting games. We show that both are #P-complete and propose a dedicated dynamic-programming algorithm that runs in pseudo-polynomial time for graphs with the bounded treewidth.