Approximating power indices: Theoretical and empirical analysis

Yoram Bachrach*, Evangelos Markakis, Ezra Resnick, Ariel D. Procaccia, Jeffrey S. Rosenschein, Amin Saberi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


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.

Original languageAmerican English
Pages (from-to)105-122
Number of pages18
JournalAutonomous Agents and Multi-Agent Systems
Issue number2
StatePublished - Mar 2010

Bibliographical note

Funding Information:
Acknowledgments The second author would like to thank Krzysztof Apt for introducing him to the concepts of power indices and for providing useful pointers to the literature. We would also like to thank Elchanan Mossel for valuable discussions. This work was partially supported by Israel Science Foundation Grant #898/05.


  • Banzhaf power index
  • Coalitional games
  • Power index
  • Power index approximation
  • Power indices
  • Shapley value
  • Shapley-Shubik power index


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