Computing Power Indices in Weighted Majority Games with Formal Power Series

2025-11-19

AI Summary

For weighted majority games, computing the Banzhaf and Shapley–Shubik power indices becomes computationally prohibitive when the number of players (n) and the quota (q) are large. This paper introduces, for the first time, formal power series techniques to this domain, proposing a fast pseudopolynomial algorithm based on generating functions and divide-and-conquer convolution. The method achieves optimal time complexities of (O(n + q log q)) for the Banzhaf index and (O(nq log q)) for the Shapley–Shubik index—substantially improving upon existing dynamic programming and enumeration approaches. Under the condition (q = 2^{o(n)}), the algorithm enables efficient batch computation of power indices for all players. It thus bridges theoretical optimality with practical scalability, offering both asymptotic efficiency and empirical feasibility for large-scale weighted voting systems.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

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 $O(n+qlog (q))$ 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 $O(nqlog (q))$ time. Our algorithms are faster than existing algorithms when $q=2^{o(n)}$. Our algorithms exploit efficient computation techniques for formal power series.

Problem

• Developing fast algorithms to compute Banzhaf power indices efficiently
• Calculating Shapley-Shubik indices using pseudo-polynomial-time methods
• Improving computational efficiency for weighted majority games analysis

Innovation

• Fast pseudo-polynomial algorithms for power indices
• Banzhaf index computed in O(n+q log(q)) time
• Shapley-Shubik index computed in O(nq log(q)) time