Encyclopedia of Law and Economics

2019 Edition
| Editors: Alain Marciano, Giovanni Battista Ramello

Power Indices

  • Dennis LeechEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7753-2_375


Many voting bodies are constituted on a principle of accountability whereby a member’s influence is intended to be a reflection of a measure of size such as financial contribution or population. Examples include the joint stock company, the US Electoral College, the IMF executive and the EU council of ministers.

Power indices are a tool for addressing the (often ignored) problem inherent in this: that the constitution defines the voting rules and not the effective voting powers they imply. Indices due to Penrose-Banzhaf and Shapley-Shubik are often used. That they ignore preferences is often cited as a limitation with regard to positivistic analysis of existing constitutions but an advantage when they are used to address the normative problem of designing voting rules. Power indices can reveal hidden properties of voting rules that are not obvious at a superficial level. The issue of how to construct behavioral power indices that do take account of preferences remains an important research dimension. Power indices can also help us understand multi-tiered governance structures such as federal constitutions or corporate networks, an area where there is need for further research.

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  1. Banzhaf JF III (1964) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317Google Scholar
  2. Braham M, Holler MJ (2005) The impossibility of a preference-based power index. J Theor Polit 17(1):137–157CrossRefGoogle Scholar
  3. Coleman JS (1971) Control of collectivities and the power of a collectivity to act. Soc Choice:269–300Google Scholar
  4. Crama Y, Leruth L (2007) Control and voting power in corporate networks: concepts and computational aspects. Eur J Oper Res 178(3):879–893CrossRefGoogle Scholar
  5. Fara R, Leech D, Salles M (2014) Voting power and procedures, SpringerGoogle Scholar
  6. Felsenthal D, Machover M (1998) The measurement of voting power. Edward Elgar, CheltenhamCrossRefGoogle Scholar
  7. Holler MJ, Nurmi H (2013) Power, voting, and voting power: 30 years after. Springer, BerlinCrossRefGoogle Scholar
  8. Leech D (2002) An empirical comparison of the performance of classical power indices. Political Stud 50(1):1–22CrossRefGoogle Scholar
  9. Leech D (2003) Computing power indices for large voting games. Manag Sci 49:831–838CrossRefGoogle Scholar
  10. Leech D (2013) Power indices in large voting bodies. Public Choice 155(1–2):61–79CrossRefGoogle Scholar
  11. Leech D, Leech R Algorithms for voting power indices: www.ecaae.ac.uk
  12. Lindner I, Machover M (2004) LS Penrose’s limit theorem: proof of some special cases. Math Soc Sci 47(1):37–49CrossRefGoogle Scholar
  13. Miller NR (2013) A priori voting power and the US Electoral College. In: Holler MJ, Nurmi H (eds) Power, voting, and voting power: 30 years after. Springer, Berlin, pp 411–442CrossRefGoogle Scholar
  14. Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57CrossRefGoogle Scholar
  15. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48(9):787–792CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of WarwickCoventryUK