Location of Repository

Computation of power indices

By Dennis Leech


Power indices are a useful tool for studying voting systems in which different members have different numbers of votes. Many international organisations are organised in this way in order to accommodate differences in size of countries, including the system of QMV in the European Union Council of Ministers. A power index is a measure of power based on the idea that a member’s power is his ability to swing a decision by changing the way his vote is cast.\ud This paper addresses the problem of the computation of the two most widely used power indices, the so-called classical power indices, the Banzhaf index (and also the related Coleman indices) and the Shapley-Shubik index. It discusses the various methods that have been proposed in the literature: Direct Enumeration, Monte Carlo Simulation, Generating Functions, Multilinear Extensions Approximation, the Modified MLE Approximation Method. The advantages and disadvantages of the algorithms are discussed including computational complexity. It also describes methods for so called “oceanic games”.\ud The paper also discusses the so-called “inverse problem” of finding what the weights should be given the desired power indices. The method is potentially useful as providing a basis for designing a voting system with a given desired distribution of power among the members, for example, to reflect differences in population or financial contributions. Examples are given from the International Monetary Fund, shareholder voting and the European Union Council

Topics: HM
Publisher: University of Warwick, Department of Economics
Year: 2002
OAI identifier: oai:wrap.warwick.ac.uk:1542

Suggested articles



  1. (1954). A Method for Evaluating the Distribution of Power in a doi
  2. (1999). A Priori Power Measures and the Institutions of the European Union”, doi
  3. (2002). An Empirical Comparison of the Performance of Classical Power Indices," doi
  4. (1983). and H.S.Wilf doi
  5. Computing Power Indices for Large Voting Games”, doi
  6. (1971). Control of Collectivities and the Power of a Collectivity to Act," in B.Lieberman (ed), Social Choice,
  7. (1992). Empirical Analysis of the Distribution of a priori Voting Power: Some Results for the British Labour Party Conference and Electoral College", doi
  8. Evaluation of a Presidential Election Game”, doi
  9. (1995). Game Theory,(3rd Edition)
  10. (1998). Is the Allocation of Voting Power among EU States Fair?”, doi
  11. (1979). Mathematical Properties of the Banzhaf Value," doi
  12. (1998). Measurement of Voting Power, Cheltenham, Edward Elgar.Computation of Power Indices -------------------, doi
  13. Multilinear Extensions and the Banzhaf Value," doi
  14. (2000). of Power Indices Some Web Links There are some good websites about power indices. These are a few interesting and useful sites. An interesting on-line power indices calculator by Kazuo Morota and Yashuaki Oisho,
  15. (1994). Power and Stability in Politics," chapter 32 of Aumann, Robert doi
  16. (1981). Power, Voting and Voting Power, doi
  17. (2001). Shareholder Voting Power and Corporate Governance: a Study of Large British Companies,"
  18. (1998). The Bicameral Postulates and Indices of a Priori Voting Power," Theory and Decision,
  19. (1946). The Elementary Statistics of Majority Voting," doi
  20. (1968). The Optimum Addition of Points to Quadrature Formulae,” doi
  21. (1982). The Problem of the Right Distribution of Voting Power", doi
  22. (1988). The Relationship between Shareholding Concentration and Shareholder Voting Power in British Companies: a Study of the Application of Power Indices for Simple Games," doi
  23. (2001). The Treaty of Nice and doi
  24. (1962). Values of Large Games VI: Evaluating the Electoral College Exactly, RM-3158, The Rand Corporation ,
  25. (1988). Voting Games, Power Indices and Presidential Elections,"
  26. (1994). Voting Power in the EC Decision Making and the Consequences of Two Different Enlargements," doi
  27. (2002). Voting Power in the Governance of the International Monetary Fund", Annals of Operations Research, Special Issue on Game Practice (Guest-Editors:
  28. (1965). Weighted Voting Doesn’t Work: A Mathematical Analysis”,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.