Introduction to algorithms
NP-completeness of some problems concerning voting games
International Journal of Game Theory
On the complexity of cooperative solution concepts
Mathematics of Operations Research
NP-completeness for calculating power indices of weighted majority games
Theoretical Computer Science
Computing Power Indices for Large Voting Games
Management Science
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
A randomized method for the shapley value for the voting game
Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems
Computing the Banzhaf power index in network flow games
Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
A linear approximation method for the Shapley value
Artificial Intelligence
Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems - Volume 2
Proof systems and transformation games
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Fingerprinting ratings for collaborative filtering: theoretical and empirical analysis
SPIRE'10 Proceedings of the 17th international conference on String processing and information retrieval
Collusion in VCG path procurement auctions
WINE'10 Proceedings of the 6th international conference on Internet and network economics
False-name manipulations in weighted voting games
Journal of Artificial Intelligence Research
Manipulating the quota in weighted voting games
Artificial Intelligence
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
Crowd IQ: aggregating opinions to boost performance
Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 1
The inverse shapley value problem
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Automated analysis of weighted voting games
Proceedings of the 13th International Conference on Electronic Commerce
Crowd IQ: measuring the intelligence of crowdsourcing platforms
Proceedings of the 3rd Annual ACM Web Science Conference
Proof systems and transformation games
Annals of Mathematics and Artificial Intelligence
Efficient computation of shapley values for demand response programs
Proceedings of the fourth international conference on Future energy systems
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
A heuristic approximation method for the Banzhaf index for voting games
Multiagent and Grid Systems
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Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much "real power" a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley---Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required.