Abstract
Decision rules for Yes–No voting systems are placed in a probabilistic framework. Selfdual and permutationally invariant distributions are introduced. Under such distributions, the mean success margin of the majority rule and of the unanimity rule are shown to bound the mean success margin of all other decision rules. For bloc decision rules in the Penrose/Banzhaf model, a product formula for the voters’ influence probabilities is derived. Other indices and the Shapley/Shubik model are also discussed.
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References
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Colomer JM, McLean I (1998) Electing popes: approval balloting and qualified-majority rule. J Interdiscip Hist 29: 1–22
Coleman JS (1971) Control of collectivities and the power of a collectivity to act. In: Lieberman D (eds) Social choice. Gordon and Breach, New York, pp 269–300
Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4: 131–140
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4: 99–131
Felsenthal DS, Machover M (1998) The measurement of voting power—theory and practice, problems and paradoxes. Edward Elgar, Chaltenham
Felsenthal DS, Machover M (2002) Annexations and alliances: when are blocs advantageous a priori?. Social Choice Welf 19: 295–312
Käufl A, Ruff O, Pukelsheim F (2010) Abstentions in the German Bundesrat and ternary decision rules (forthcoming)
Kirsch W, Langner J (2009) Power indices and minimal winning coalitions. Social Choice Welf 34: 33–46
Laruelle A, Valenciano F (2004) On the meaning of Owen-Banzhaf coalitional value in voting situations. Theory Decis 56: 113–123
Laruelle A, Valenciano F (2005) Assessing success and decisiveness in voting situations. Social Choice Welf 24: 171–197
Leutgäb P, Pukelsheim F (2009) List apparentements in local elections: a lottery. Homo Oecon 26: 489–500
Owen G (1971) Political games. Naval Res Logist Q 18: 345–355
Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109: 53–57
Shapley LS (1962) Simple games: an outline of the descriptive theory. Behav Sci 7: 59–66
Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48: 787–792
Straffin PD (1977a) Homogeneity, independence, and power indices. Public Choice 30: 107–118
Straffin PD (1977b) Majority rule and general decision rules. Theory Decis 8: 351–360
Straffin PD (1978) Probability models for power indices. In: Ordeshook PD (eds) Game theory and political sience. New York University Press, New York, pp 477–510
Straffin PD (1988) The Shapley–Shubik and Banzhaf power indices as probabilities. In: Roth AE (eds) The Shapley value—essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge , pp 59–66
Von Neumann J, Morgenstern O (1944) Game theory and economic behavior. Princeton University Press, Princeton
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Ruff, O., Pukelsheim, F. A probabilistic synopsis of binary decision rules. Soc Choice Welf 35, 501–516 (2010). https://doi.org/10.1007/s00355-010-0450-0
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DOI: https://doi.org/10.1007/s00355-010-0450-0