Abstract
In this study, we introduce and examine the Egalitarian property for some power indices on the class of simple games. This property means that after intersecting a game with a symmetric or anonymous game the difference between the values of two comparable players does not increase. We prove that the Shapley–Shubik index, the absolute Banzhaf index, and the Johnston score satisfy this property. We also give counterexamples for Holler, Deegan–Packel, normalized Banzhaf and Johnston indices. We prove that the Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alonso-Meijide J, Freixas J (2010) A new power index based on minimal winning coalitions without any surplus. Decis Support Syst 49: 70–76
Banzhaf J (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Carreras F, Freixas J (2004) A power analysis of linear games with consensus. Math Soc Sci 48: 207–221
Carreras F, Freixas J (2008) On ordinal equivalence of power measures given by regular semivalues. Math Soc Sci 55: 221–234
Carreras F, Freixas J, Puente M (2003) Semivalues as power indices. Eur J Oper Res 149: 676–687
Coleman J (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman B, Falchikov N (eds) Social Choice. Gordon & Breach, New York, pp 269–300
Deegan J, Packel E (1978) A new index of power for simple n-person games. Int J Game Theory 7: 113–123
Felsenthal D, Machover M (1998) The measurement of voting power. Edward Elgar Publishing Limited, Cheltenham
Felsenthal D, Machover M (2001) The Treaty of Nice and qualified majority voting. Soc Choice Welf 18: 431–464
Freixas J (2010) On ordinal equivalence of the Shapley and Banzhaf values for cooperative games. Int J Game Theory 39: 513–527
Gambarelli G, Owen G (2002) Power in political and business structures. In: Holler M, Kliemt H, Schmidtchen D, Streit M (eds) Power and fairness, Jahrbuch für Neue Politische Ökonomie, vol 20. Tübingen, Mohr-Siebeck, pp 57–68
Hirokawa M, Vlach M (2006) Power analysis of voting by count and account. Kybernetika 42: 483–493
Hirokawa M, Xu P (2005) Small creditors’ power in civil rehablitation—a compound game of a simple majority and a weighted majority. Technical Report, Hosei University, Mimeo.
Holler M (1982) Forming coalitions and measuring voting power. Political Stud 30: 262–271
Holler M (2002) On power and fairness and their relationships: an introductory chapter. In: Holler M, Kliemt H, Schmidtchen D, Streit M (eds) Power and fairness, Jahrbuch für Neue Politische Ökonomie, vol 20. Tübingen, Mohr-Siebeck, pp 1–27
Isbell J (1958) A class of simple games. Duke Math J 25: 423–439
Johnston R (1978) On the measurement of power: some reactions to Laver. Environ Plan A 10: 907–914
Laruelle A, Valenciano F (2002) Inequality among EU citizens in the EU’s council decision procedure. Eur J Political Econ 18: 475–498
Peleg B (1992) Voting by count and account. In: Selten R (eds) Rational interaction. Springer-Verlag, Berlin, pp 307–317
Rae D (1969) Decision-rules and individual values in constitutional choice. Am Political Sci Rev 63: 40–56
Shapley L, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Political Sci Rev 48: 787–792
Taylor A (1995) Mathematics and politics. Strategy, voting, power and proof. Textbooks in mathematical sciences. Springer-Verlag, New York
Taylor A, Zwicker W (1999) Simple games. Desirability relations, trading, pseudoweightings. Princeton University Press, Princeton, NJ
Turnovec F (2002) Evaluation of Commission’s proposal of double simple majority voting rule in the Council of Ministers of the European Union. In: Holler M, Kliemt H, Schmidtchen D, Streit M (eds) Power and fairness, Jahrbuch fur Neue Politische Okonomie vol, 20. Mohr Siebeck, Tufbingen, pp 103–122
Weymark J (1981) Generalized Gini inequality indices. Math Soc Sci 1: 409–430
Acknowledgments
Research partially funded by Grants SGR 2009-1029 of Generalitat de Catalunya and MTM 2009-08037 from the Spanish Science and Innovation Ministry, and by the Polish National Budget Funds 2010–2013 for Science under the Grant N N514 044438. The first author acknowledges the support of the Barcelona Graduate School of Economics and of the Government of Catalonia.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Freixas, J., Marciniak, D. Egalitarian property for power indices. Soc Choice Welf 40, 207–227 (2013). https://doi.org/10.1007/s00355-011-0593-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-011-0593-7