Spanning connectivity games

TL;DR

This paper investigates the computational complexity of voting power indices in the spanning connectivity game, revealing which indices are #P-complete or polynomial-time computable, and establishing links between different indices' computational difficulties.

Contribution

It provides complexity classifications for computing various voting power indices in the spanning connectivity game, including new polynomial-time algorithms and hardness results.

Findings

Banzhaf and Shapley-Shubik indices are #P-complete to compute in SCGs.

Holler and Deegan-Packel indices can be computed in polynomial time.

Polynomial-time algorithms exist for Banzhaf indices in graphs with bounded treewidth.

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 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.