Abstract
This paper discusses a numerical scheme for computing the Banzhaf swing probability when votes are neither equiprobable nor independent. Examples indicate a substantial bias in the Banzhaf measure of voting power if neither assumption is met. The analytical part derives the exact magnitude of the bias due to the common probability of an affirmative vote deviating from one half and due to common correlation in unweighted simple-majority games. The former bias is polynomial, the latter is linear. A modified square-root rule for two-tier voting systems that takes into account both the homogeneity and the size of constituencies is also provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banzhaf JF (1965). Weighted voting does not work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Barbera S and Jackson MO (2006). On the weights of nations: assigning voting weights in a heterogeneous union. J Polit Econ 114: 317–339
Berg S (1985). Paradox of voting under an urn-model: the effect of homogeneity. Pub Choice 47: 377–387
Berg S (1993). Condorcet’s jury theorem, dependency among jurors. Social Choice and Welfare 10: 87–95
Boland PJ (1989). Majority systems and the Condorcet jury theorem. Statistician 38: 181–189
Braham M and Holler M (2005). The impossibility of a preference-based power index. J Theor Polit 17: 137–157
Braham M and Holler M (2005). Power and preferences again. A reply to Napel and Widgrén. J Theor Polit 17: 389–395
Chamberlain G and Rothschild M (1981). A note on the probability of casting a decisive vote. J Econ Theory 25: 152–162
Everitt BS (1992). The analysis of contingency tables. Chapman & Hall, London
Felsenthal DS, Machover M (1998) The measurement of voting power: theory and practice, problems and paradoxes. Edward Elgar
Garrett G and Tsebelis G (2001). Even more reasons to resist the temptation of power indices in the European Union. J Theor Polit 13: 99–105
Gelman A, Katz JN and Bafumi J (2004). Standard voting power indices don’t work: an empirical analysis. Br J Polit Sci 34: 657–674
Gelman A, Katz JN and Tuerlinckx F (2002). The mathematics and statistics of voting power. Stat Sci 17: 420–435
Good I and Mayer L (1975). Estimating the efficacy of a vote. Behav Sci 20: 25–33
Grofman B (1981). Fair apportionment and the Banzhaf index. Am Math Mont 88: 1–5
Hall AR (2005). Generalized method of moments. Advanced texts in econometrics. Oxford University Press, Oxford
Kaniovski YM and Pflug GC (2007). Risk assessment for credit portfolios: a coupled Markov chain model. J Bank Finance 31: 2303–2323
Kirman A, Widgrén M (1995) European economic decision-making policy: progress or paralysis? Econ Policy 10:423–460
Leech D and Leech R (2006). Voting power and voting blocs. Pub Choice 127: 285–303
Maaser N and Napel S (2007). Equal representation in two-tier voting systems. Soc Choice Welf 28: 401–420
Merrill~III S and Grofman B (1999). A unified theory of voting: directional and proximity spatial models. Cambridge University Press, Cambridge
Napel S and Widgrén M (2004). Power measurement as sensitivity analysis: a unified approach. J Theor Polit 16: 517–538
Napel S and Widgrén M (2005). The possibility of a preference-based power index. J Theor Polit 17: 377–387
Owen G (1972). Multilinear extensions of games. Manage Sci 18: 64–79
Page SE (2006). Path dependence. Q J Polit Sci 1: 87–115
Penrose LS (1946). The elementary statistics of majority voting. J R Stat Soc 109: 53–57
Shapley LS and Shubik M (1954). A method of evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792
Steunenberg B, Schmidtchen D and Koboldt C (1999). Strategic power in the European Union: evaluating the distribution of power in policy games. J Theor Polit 11: 339–366
Straffin PD (1977). Homogeneity, independence and power indices. Pub Choice 30: 107–118
Straffin PD (1994). Power and stability in politics. In: Aumann, RJ and Hart, S (eds) Handbook of game theory, Vol 2, chap 32, pp. Elsevier, Amsterdam
Tsebelis G (1995). Decision making in political systems: veto players in presidentialism, parliamentarism, multicameralism and multipartyism. Br J Polit Sci 25: 289–325
Widgrén M (1995). Probabilistic voting power in the EU Council: the cases of trade policy and social regulation. Scand J Econ 97: 345–356
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kaniovski, S. The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent. Soc Choice Welfare 31, 281–300 (2008). https://doi.org/10.1007/s00355-007-0280-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-007-0280-x