The minimum sum representation as an index of voting power

Abstract

We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley-Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

Representation-Compatible Power Indices

“…The BZI and the SSI do not show the paradox in this example, although examples are known in which the BZI is liable to the bloc paradox. Also, the MSRI of Freixas and Kaniovski (2014) displays the bloc paradox in this example. 37;25,20,17,15,9,6,2,1].…”

“…study the frequency of the occurrence of the donation paradox in weighted games with a small number of players, providing examples for the Banzhaf and Johnston indices. The Shapley-Shubik index is immune to both the bloc and donation paradoxes Freixas and Kaniovski (2014). provide an example, which also shows that the MSR index is liable to the donation paradox.…”

“…Unfortunately, the minimum sum representations are not unique for n ≥ 8 (Kurz 2012). Nevertheless, one can take a convex combination of all such minimum sum representations, which yields, after a normalization, the MSR index recently introduced in Freixas and Kaniovski (2014). 8 One motivation for minimizing weights is to minimizing the cost of political representation by minimizing the number of representatives.…”

Bounds for the Diameter of the Weight Polytope

“…Representation compatibility of ϕ for [q; w] is automatically satisfied for the modified nucleolus (modiclus) [20], minimum sum representation index [6] or one of the power indices based on averaged representations [8] for all weighted games and for the Penrose-Banzhaf index for all spherically separable simple games [7]. The theorem also applies to the bargaining model for weighted games analyzed in [17], cf.…”

Mostly Sunny: A Forecast of Tomorrow's Power Index Research

“…For a 27member assembly, as considered by Le , its computation is an almost insurmountable obstacle for non-experts. So we see a future for more easy-to-use software, especially for the computation of technically more demand-ing constructs (as, e.g., also the minimum sum representation index recently introduced by Freixas and Kaniovski, 2014). For power analysis based on convex policy spaces, algorithmic considerations are still in their infancy.…”