The polynomial-time hierarchy and sparse oracles
Journal of the ACM (JACM)
NP-completeness of some problems concerning voting games
International Journal of Game Theory
Probabilistic polynomial time is closed under parity reductions
Information Processing Letters
On the complexity of cooperative solution concepts
Mathematics of Operations Research
The Complexity of Planar Counting Problems
SIAM Journal on Computing
NP-completeness for calculating power indices of weighted majority games
Theoretical Computer Science
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A heuristic technique for multi-agent planning
Annals of Mathematics and Artificial Intelligence
A linear approximation method for the Shapley value
Artificial Intelligence
An anytime approximation method for the inverse Shapley value problem
Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems - Volume 2
Note: The complexity of power-index comparison
Theoretical Computer Science
Boolean combinations of weighted voting games
Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems - Volume 1
Approximating power indices: theoretical and empirical analysis
Autonomous Agents and Multi-Agent Systems
Enumeration and exact design of weighted voting games
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1 - Volume 1
Using complexity to protect elections
Communications of the ACM
Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
False-name manipulations in weighted voting games
Journal of Artificial Intelligence Research
The Shapley value as a function of the quota in weighted voting games
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
Complementary cooperation, minimal winning coalitions, and power indices
Theoretical Computer Science
On random quotas and proportional representation in weighted voting games
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
Sharing rewards in cooperative connectivity games
Journal of Artificial Intelligence Research
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Weighted voting games provide a simple model of decision-making in human societies and multi-agent systems. Such games are described by a set of players, a list of players@? weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in a weighted voting game is traditionally identified with her Shapley-Shubik index or her Banzhaf index, two classic power measures that reflect the player@?s marginal contribution under different coalition formation scenarios. In this paper, we investigate by how much one can change a player@?s power, as measured by these indices, by modifying the quota. We give tight bounds on the changes in the individual player@?s power that can result from a change in quota. We then describe an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces her power (as measured by both indices) to 0. We also study how the choice of quota can affect the relative power of the players. Finally, we investigate scenarios where one@?s choice in setting the quota is constrained. We show that optimally choosing between two values of the quota is complete for the complexity class PP, which is believed to be significantly more powerful than NP. On the other hand, we empirically demonstrate that even small changes in quota can have a significant effect on a player@?s power.