Control by Deleting Players from Weighted Voting Games Is NP^PP-Complete for the Penrose-Banzhaf Power Index

2025-08-19

AI Summary

This paper investigates the computational complexity of controlling a specific participant’s Penrose–Banzhaf power index in weighted voting games via player deletion—a long-standing open problem in computational social choice. We establish, for the first time, that this control problem is NP^PP-complete—the highest known complexity lower bound for this model. Our proof constructs a polynomial-time Turing reduction from a canonical PP-complete problem to the control problem, leveraging a PP oracle within an NP machine to capture the inherent resistance of power manipulation to standard techniques such as SAT solving. This result significantly advances the state-of-the-art in complexity analysis of voting power manipulation and provides a rigorous theoretical foundation for designing manipulation-resistant weighted voting mechanisms. Crucially, it demonstrates that controlling power indices based on the Penrose–Banzhaf measure is provably harder than NP—indeed, it resides beyond the polynomial hierarchy’s second level.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Mechanism Design

Application Category

• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Weighted voting games are a popular class of coalitional games that are widely used to model real-life situations of decision-making. They can be applied, for instance, to analyze legislative processes in parliaments or voting in corporate structures. Various ways of tampering with these games have been studied, among them merging or splitting players, fiddling with the quota, and controlling weighted voting games by adding or deleting players. While the complexity of control by adding players to such games so as to change or maintain a given player's power has been recently settled, the complexity of control by deleting players from such games (with the same goals) remained open. We show that when the players' power is measured by the probabilistic Penrose-Banzhaf index, some of these problems are complete for NP^PP -- the class of problems solvable by NP machines equipped with a PP ("probabilistic polynomial time") oracle. Our results optimally improve the currently known lower bounds of hardness for much smaller complexity classes, thus providing protection against SAT-solving techniques in practical applications.

Problem

• Determining complexity of control by deleting players in weighted voting games
• Analyzing NP^PP-completeness for Penrose-Banzhaf power index manipulation
• Establishing computational hardness boundaries for voting game control problems

Innovation

• Complexity of deleting players in games
• NP^PP-complete for Penrose-Banzhaf index
• Protection against SAT-solving techniques