False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

TL;DR

This paper proves that determining beneficial false-name splitting or merging in weighted voting games is PP-hard, providing exact complexity bounds and resolving open questions about their computational difficulty.

Contribution

It establishes the PP-completeness of beneficial merging and splitting problems for key power indices, advancing understanding of their computational complexity.

Findings

Beneficial merging and splitting are PP-hard.

Matching upper bounds are provided for these problems.

Results imply these problems are unlikely to be in NP unless the polynomial hierarchy collapses.

Abstract

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.