# Generalized Coleman-Shapley indices and total-power monotonicity

- 6 Downloads

## Abstract

We introduce a new axiom for power indices, which requires the total (additively aggregated) power of the voters to be nondecreasing in response to an expansion of the set of winning coalitions; the total power is thereby reflecting an increase in the collective power that such an expansion creates. It is shown that total-power monotonic indices that satisfy the standard semivalue axioms are probabilistic mixtures of generalized Coleman-Shapley indices, where the latter concept extends, and is inspired by, the notion introduced in Casajus and Huettner (Public choice, forthcoming, 2019). Generalized Coleman-Shapley indices are based on a version of the random-order pivotality that is behind the Shapley-Shubik index, combined with an assumption of random participation by players.

## Keywords

Simple games Voting power Shapley-Shubik index Banzhaf index Coleman-Shapley index Semivalues Power of collectivity to act Total-power monotonicity axiom Probabilistic mixtures## JEL classification numbers

C71 D72## Notes

## References

- Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343Google Scholar
- Banzhaf JF (1966) Multi-member electoral districts—Do they violate the “One Man, One Vote” principle. Yale Law J 75:1309–1338CrossRefGoogle Scholar
- Banzhaf JF (1968) One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College. Vilanova Law Review 13:304–332Google Scholar
- Brams SJ (2013) Game Theory and Politics, Dover Books on Mathematics. Dover Publications, MineolaGoogle Scholar
- Casajus A (2012) Amalgamating Players, Symmetry, and the Banzhaf Value. Int J Game Theory 41:497–515CrossRefGoogle Scholar
- Casajus A, Huettner F (2018) Decomposition of solutions and the shapley value. Games Econ Behav 108:37–48CrossRefGoogle Scholar
- Casajus A, Huettner F (2019) The Coleman-Shapley-index: being decisive within the coalition of the interested. Public Choice, forthcomingGoogle Scholar
- Coleman JS (1971) Control of collectives and the power of a collectivity to act. In: Lieberman Bernhardt (ed) Social choice. Gordon and Breach, New York, pp 192–225Google Scholar
- Dubey P (1975) On the uniqueness of the shapley value. Int J Game Theory 4:131–139CrossRefGoogle Scholar
- Dubey P, Neyman A, Weber RJ (1981) Value theory without efficiency. Math Oper Res 6:122–128CrossRefGoogle Scholar
- Dubey P, Einy E, Haimanko O (2005) Compound voting and the Banzhaf index. Games Econo Behav 51:20–30CrossRefGoogle Scholar
- Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4:99–131CrossRefGoogle Scholar
- Einy E (1987) Semivalues of simple games. Math Oper Res 12:185–192CrossRefGoogle Scholar
- Felsenthal DS, Machover M (1998) The measurement of voting power: theory and practice, problems and paradoxes. Edward Elgar Publishers, LondonCrossRefGoogle Scholar
- Haimanko O (2018) The axiom of equivalence to individual power and the Banzhaf index. Games Econ Behav 108:391–400CrossRefGoogle Scholar
- Hart S, Mas-Colell (1996) Bargaining and Value. Econometrica 64:357–380CrossRefGoogle Scholar
- Lehrer E (1988) Axiomatization of the Banzhaf value. Int J Game Theory 17:89–99CrossRefGoogle Scholar
- Nowak AS (1997) On an axiomatization of the Banzhaf value without the additivity axiom. Int J Game Theory 26:137–141CrossRefGoogle Scholar
- Owen G (1968) A note on the shapley value. Manag Sci 14:731–732CrossRefGoogle Scholar
- Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57CrossRefGoogle Scholar
- Shapley LS (1953) A value for \(n\)-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of Games II (annals of mathematical studies 28). Princeton University Press, PrincetonGoogle Scholar
- Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792CrossRefGoogle Scholar
- Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shapley value: essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 101–121CrossRefGoogle Scholar