Computing Power Indices in Weighted Majority Games with Formal Power Series

TL;DR

This paper introduces efficient pseudo-polynomial algorithms for calculating power indices in weighted majority games, significantly improving computational speed for large player sets by leveraging formal power series techniques.

Contribution

The paper presents novel fast algorithms for computing Banzhaf and Shapley--Shubik indices in weighted majority games, outperforming existing methods when the quota is subexponential in the number of players.

Findings

Banzhaf index computed in O(n+q log q) time.

Shapley--Shubik index computed in O(nq log q) time.

Algorithms outperform previous methods for quotas q=2^{o(n)}.

Abstract

In this paper, we propose fast pseudo-polynomial-time algorithms for computing power indices in weighted majority games. We show that we can compute the Banzhaf index for all players in $O(n+q\log (q))$ time, where $n$ is the number of players and $q$ is a given quota. Moreover, we prove that the Shapley--Shubik index for all players can be computed in $O(nq\log (q))$ time. Our algorithms are faster than existing algorithms when $q=2^{o(n)}$. Our algorithms exploit efficient computation techniques for formal power series.