The polynomial-time hierarchy and sparse oracles
Journal of the ACM (JACM)
The complexity of optimization problems
Journal of Computer and System Sciences - Structure in Complexity Theory Conference, June 2-5, 1986
NP-completeness of some problems concerning voting games
International Journal of Game Theory
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Introduction to the theory of complexity
Introduction to the theory of complexity
On the complexity of cooperative solution concepts
Mathematics of Operations Research
The Complexity of Planar Counting Problems
SIAM Journal on Computing
Power balance and apportionment algorithms for the United States Congress
Journal of Experimental Algorithmics (JEA)
Rank aggregation methods for the Web
Proceedings of the 10th international conference on World Wide Web
NP-completeness for calculating power indices of weighted majority games
Theoretical Computer Science
On some central problems in computational complexity.
On some central problems in computational complexity.
Divide and conquer: false-name manipulations in weighted voting games
Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems - Volume 2
Power Indices in Spanning Connectivity Games
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
Survey: The consequences of eliminating NP solutions
Computer Science Review
Proof systems and transformation games
Annals of Mathematics and Artificial Intelligence
Computing cooperative solution concepts in coalitional skill games
Artificial Intelligence
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We study the complexity of the following problem: Given two weighted voting games G驴 and G驴驴 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 [16] 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 [5], who showed that the Shapley-Shubik power index is #P-metric-complete.