Autonomous Agents and Multi-Agent Systems

, Volume 20, Issue 2, pp 105–122 | Cite as

Approximating power indices: theoretical and empirical analysis

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


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.


Power indices Power index Coalitional games Shapley value Banzhaf power index Shapley–Shubik power index Power index approximation 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Yoram Bachrach
    • 1
  • Evangelos Markakis
    • 2
    • 3
  • Ezra Resnick
    • 1
  • Ariel D. Procaccia
    • 1
  • Jeffrey S. Rosenschein
    • 1
  • Amin Saberi
    • 4
  1. 1.School of Engineering and Computer ScienceThe Hebrew UniversityJerusalemIsrael
  2. 2.Center for Mathematics and Computer Science (CWI)AmsterdamThe Netherlands
  3. 3.Athens University of Economics and BusinessAthensGreece
  4. 4.Department of Management Science and EngineeringStanford UniversityPalo AltoUSA

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