Annals of Operations Research

, Volume 264, Issue 1–2, pp 235–265 | Cite as

Representation-compatible power indices

  • Serguei Kaniovski
  • Sascha Kurz
Original Paper


This paper studies power indices based on average representations of a weighted game. If restricted to account for the lack of power of null voters, average representations become coherent measures of voting power, with power distributions being proportional to the distribution of weights in the average representation. This makes these indices representation-compatible, a property not fulfilled by classical power indices. In this paper we introduce two computationally cheaper alternatives to the existing representation-compatible power indices, and study the properties of a family that now comprises four measures.


First average representation Power index Proportionality between weights and power 

Mathematics Subject Classification

91A12 91A80 



We are greatly indebted to three anonymous reviewers, whose critique considerably improved the quality of this paper. We would also like to thank the participants of the 27th European Conference on Operational Research in 2015 and the annual conference of the German Operations Research Society (GOR) in 2015 for their comments and suggestions.


  1. Ansolabehere, S., Snyder, J. M., Strauss, A. B., & Ting, M. M. (2005). Voting weights and formateur advantages in the formation of coalition governments. American Journal of Political Science, 49, 550–563.CrossRefGoogle Scholar
  2. Aziz, H., Paterson, M., & Leech, D. (2007). Efficient algorithm for designing weighted voting games. Working Paper of the University of Warwick.Google Scholar
  3. Baldoni, V., Berline, N., De Loera, J. A., Dutra, B., Köppe, M., Moreinis, S., et al. (2014). A user’s guide for LattE integrale v1.7.2.
  4. Bilbao, J. M., Fernández, J. R., Losada, A. J., & López, J. J. (2000). Generating functions for computing power indices efficiently. TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, 8, 191–213.CrossRefGoogle Scholar
  5. Brams, S. J. (1990). Negotiation games: Applying game theory to bargaining and arbitration. Abingdon: Routledge.CrossRefGoogle Scholar
  6. Carreras, F., & Freixas, J. (1996). Complete simple games. Mathematical Social Sciences, 32, 139–155.CrossRefGoogle Scholar
  7. Chalkiadakis, G., Elkind, E., & Wooldridge, M. (2011). Computational aspects of cooperative game theory (Synthesis lectures on artificial intelligence and machine learning). San Rafael: Morgan & Claypool Publishers.Google Scholar
  8. Crama, Y., & Hammer, P. L. (2011). Boolean functions: Theory, algorithms, and applications. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  9. De Loera, J. A., Dutra, B., Köppe, M., Moreinis, S., Pinto, G., & Wu, J. (2013). Software for exact integration of polynomials over polyhedra. Computational Geometry: Theory and Applications, 46, 232–252.CrossRefGoogle Scholar
  10. Dubey, P., & Shapley, L. S. (1979). Mathematical properties of the Banzhaf power index. Mathematics of Operations Research, 4, 99–131.CrossRefGoogle Scholar
  11. Einy, E., & Lehrer, E. (1989). Regular simple games. International Journal of Game Theory, 18, 195–207.CrossRefGoogle Scholar
  12. Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: Theory and practice, problems and paradoxes. Cheltenham: Edward Elgar.CrossRefGoogle Scholar
  13. Felsenthal, D. S., & Machover, M. (2004). A priori voting power: What is it all about? Political Studies Review, 2, 1–23.CrossRefGoogle Scholar
  14. Felsenthal, D. S., & Machover, M. (2005). Voting power measurement: A story of misreinvention. Social Choice and Welfare, 25, 485–506.CrossRefGoogle Scholar
  15. Freixas, J., & Gambarelli, G. (1997). Common internal properties among power indices. Control and Cybernetics, 26, 591–604.Google Scholar
  16. Freixas, J., & Kaniovski, S. (2014). The minimum sum representation as an index of voting power. European Journal of Operational Research, 233, 739–748.CrossRefGoogle Scholar
  17. Freixas, J., & Kurz, S. (2014). On minimum integer representations of weighted games. Mathematical Social Sciences, 67, 9–22.CrossRefGoogle Scholar
  18. Freixas, J., & Molinero, X. (2009). On the existence of a minimum integer representation for weighted voting systems. Annals of Operations Research, 166, 243–260.CrossRefGoogle Scholar
  19. Freixas, J., & Molinero, X. (2010). Detection of paradoxes of power indices for simple games. In L. A. Petrosjan & N. A. Zenkevich (Eds.), Contributions to Game Theory and Management (pp. 82–90). Saint Petersburg: Graduate School of Management, Saint Petersburg State University.Google Scholar
  20. Freixas, J., & Pons, M. (2010). Hierarchies achievable in simple games. Theory and Decision, 68, 393–404.CrossRefGoogle Scholar
  21. Houy, N., & Zwicker, W. S. (2014). The geometry of voting power: Weighted voting and hyper-ellipsoids. Games and Economic Behavior, 84, 7–16.CrossRefGoogle Scholar
  22. Isbell, J. R. (1956). A class of majority games. Quarterly Journal of Mechanics and Applied Mathematics, 7, 183–187.Google Scholar
  23. Kaniovski, S., & Kurz, S. (2015). The average representation—A cornucopia of power indices? Homo Oeconomicus, 32, 169–182.Google Scholar
  24. Kurz, S. (2012). On minimum sum representations for weighted voting games. Annals of Operations Research, 196, 361–369.CrossRefGoogle Scholar
  25. Kurz, S., Napel, S., & Nohn, A. (2014). The nucleolus of large majority games. Economics Letters, 123, 139–143.CrossRefGoogle Scholar
  26. Taylor, A. D., & Pacelli, A. M. (2008). Mathematics and politics: Strategy, voting, power, and proof. Berlin: Springer.CrossRefGoogle Scholar
  27. Taylor, A. D., & Zwicker, W. S. (1999). Simple games: Desirability relations, trading, and pseudoweightings. Princeton, NJ: Princeton University Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Austrian Institute of Economic Research (WIFO)ViennaAustria
  2. 2.University of BayreuthBayreuthGermany

Personalised recommendations