The geometry of voting power: Weighted voting and hyper-ellipsoids

Abstract

Abstract. In cases where legislators represent districts that vary in population, the design of fair legislative voting rules requires an understanding of how the number of votes cast by a legislator is related to a measure of her influence over collective decisions. We provide three new characterizations of weighted voting, each based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyperellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose-Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights.

Representation-compatible power indices

“…The same counter-example also applies for the Banzhaf index, since in this game the two power vectors coincide. Houy and Zwicker (2014) characterize the set of representations that is compatible with the Banzhaf index in a general weighted game.…”

“…Two recent theoretical studies provide conditions for which the weight distribution and the power distribution coincide. These results are available for the Banzhaf index by Houy and Zwicker (2014) and the nucleolus by Kurz, Napel and Nohn (2014). An exception is the recently introduced Minimum Sum Representation Index (MSRI) index by Freixas and Kaniovski (2014), which is specifically designed to fulfill proportionality.…”

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

“…The same article provided a new sufficient condition for exact coincidence of nucleolus and weights, which future research can presumably weaken. Coincidence of power and weights has also been studied recently by Houy and Zwicker (2014) for the PBI. Analogous findings for the SSI remain to be developed.…”

The Average Representation - A Cornucopia of Power Indices?

“…The theoretical literature shows that, in general, we cannot, although there are particular cases when it may be possible. Recently [Houy & Zwicker, 2014] have characterized a class of weighted majority games, which admit a representation using their respective Banzhaf distribution. For the nucleolus, we known that (q(x ⋆ , v), x ⋆ ) is a representation of a constant-sum weighted majority game, where x ⋆ (v) denotes the nucleolus of v, and q(x ⋆ , v) denotes the corresponding maximum excess [Maschler et al, 2013, Theorem 20.52].…”