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
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
Quantifying inefficiency in cost-sharing mechanisms
Journal of the ACM (JACM)
False name manipulations in weighted voting games: splitting, merging and annexation
Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems - Volume 1
Power Indices in Spanning Connectivity Games
AAIM '09 Proceedings of the 5th International Conference on Algorithmic Aspects in Information and Management
Variable Influences in Conjunctive Normal Forms
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
Manipulating the quota in weighted voting games
AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 1
Approximating power indices: theoretical and empirical analysis
Autonomous Agents and Multi-Agent Systems
Honor among thieves: collusion in multi-unit auctions
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1 - Volume 1
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1 - Volume 1
Internet economics: the use of Shapley value for ISP settlement
IEEE/ACM Transactions on Networking (TON)
On cooperative settlement between content, transit, and eyeball internet service providers
IEEE/ACM Transactions on Networking (TON)
Efficient computation of the shapley value for game-theoretic network centrality
Journal of Artificial Intelligence Research
Computing cooperative solution concepts in coalitional skill games
Artificial Intelligence
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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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 also 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.