Computing Power Indices in Weighted Majority Games with Formal Power Series
Naonori Kakimura, Yoshihiko Terai
TL;DR
The paper addresses the computational challenge of power indices in weighted majority games by introducing fast pseudo-polynomial-time algorithms that leverage formal power series and FFT-based coefficient extraction. It achieves $O(n+q\log(q))$ time for computing all Banzhaf indices and $O(nq\log(q))$ time for all Shapley--Shubik indices, with improved performance when $q=2^{o(n)}$, thereby surpassing prior approaches in these regimes. A two-variable formal power series framework is developed to extend the Shapley--Shubik computation to $O(n+\tilde{n}q\log q)$ time, where $\tilde{n}=\min\{n,q\}$, enabling efficient handling of the combinatorial structure via $f_p=\prod_{\ell\neq p}(1+yx^{w_\ell})$ and $\hat{f}=\prod_p(1+yx^{w_p})$. The work highlights practical implications for political and computational settings by reducing run times and providing a pathway to exact results through modular computations and CRT when needed. Overall, the approach demonstrates how generating functions and FPS arithmetic can yield substantial speedups for classical voting-power problems.
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.