The Complexity of Power-Index Comparison
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  and the Banzhaf power index [3,6]. 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 show that, unlike the Banzhaf power index, the Shapley-Shubik power index is not #P-parsimonious-complete. This finding sets a hard limit on the possible strengthenings of a result of Deng and Papadimitriou , who showed that the Shapley-Shubik power index is #P-metric-complete.
KeywordsWeighted voting games power indices computational complexity
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