Abstract
A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley–Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley–Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.
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References
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Billot A, Thisse JF (2005) How to share when context matters: The Möbius value as a generalized solution for cooperative games. J Math Econ 41(8): 1007–1029
Deegan J, Packel EW (1978) A new index of power for simple n-person games. Int J Game Theory 7(2): 113–123
Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4(3): 131–139
Dubey P, Weber RJ, Neyman A (1981) Value theory without efficiency. Math Oper Res 6(1): 122–128
Grabisch M, Marichal JL, Roubens M (2000) Equivalent representations of set functions. Math Oper Res 25(2): 157–178
Harsányi JC (1963) A simplified bargaining model of the n-person cooperative game. Int Econ Rev 4: 194–220
Holler MJ, Packel EW (1983) Power, luck and the right index. J Econ (Z Natlökonomie) 43(1): 21–29
Kirsch W, Langner J (2009) Power indices and minimal winning coalitions. Soc Choice Welf 34(1): 33–46
Kóczy LÁ (2012) Beyond lisbon: demographic trends and voting power in the European Union Council of Ministers. Math Soc Sci 63(2): 152–158
Laruelle A, Valenciano F (2001) Shapley–Shubik and Banzhaf indices revisited. Math Operat Res 26(1): 89–104
Lehrer E (1988) An axiomatization of the banzhaf value. Int J Game Theory 17(2): 89–99
Owen G (1986) Values of graph-restricted games. SIAM J Algebr Discret Methods 7(2): 210
Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109(1): 53–57
Rota GC (1964) On the foundations of combinatorial theory I. Theory Möbius Functions. Z Wahrscheinlichkeitstheorie und Verwandte Gebiete 2(4): 340–368
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker A (eds) Contributions to the theory of games. Princeton University Press, pp 307–317
Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Political Sci Rev 48(3): 787–792
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Lange, F., Kóczy, L.Á. Power indices expressed in terms of minimal winning coalitions. Soc Choice Welf 41, 281–292 (2013). https://doi.org/10.1007/s00355-012-0685-z
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DOI: https://doi.org/10.1007/s00355-012-0685-z