The Complexity of Power-Index Comparison

TL;DR

This paper proves that comparing the power indices of players in weighted voting games is PP-complete for both Shapley-Shubik and Banzhaf indices, and it establishes the complexity of the raw Shapley-Shubik index as #P-complete.

Contribution

It demonstrates the PP-completeness of power index comparison problems and refines the complexity classification of the raw Shapley-Shubik index as #P-complete.

Findings

Power index comparison is PP-complete for both indices.

Raw Shapley-Shubik index is many-one #P-complete.

Raw Shapley-Shubik index is not #P-parsimonious-complete.

Abstract

We study the complexity of the following problem: Given two weighted voting games G' and G'' that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [SS54] and the Banzhaf power index [Ban65,DS79]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also study the complexity of the raw Shapley-Shubik power index. Deng and Papadimitriou [DP94] showed that the raw Shapley-Shubik power index is #P-metric-complete. We strengthen this by showing that the raw Shapley-Shubik power index is many-one complete for #P. And our strengthening cannot possibly be further improved to parsimonious completeness, since we observe that, in contrast with the raw Banzhaf power index, the raw Shapley-Shubik power index is not #P-parsimonious-complete.