# Generalized Coleman-Shapley indices and total-power monotonicity

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## 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

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