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.
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
This is a preview of subscription content, log in to check access.
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
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