Representation-Compatible Power Indices

This paper studies power indices based on average representations of a weighted game. If restricted to account for the lack of power of dummy voters, average representations become coherent measures of voting power, with power distributions being proportional to the distribution of weights in the average representation. This makes these indices representation-compatible, a property not fulfilled by classical power indices. Average representations can be tailored to reveal the equivalence classes of voters defined by the Isbell desirability relation, which leads to a pair of new power indices that ascribes equal power to all members of an equivalence class.Comment: 28 pages, 1 figure, and 11 table

Approximating power by weights

Determining the power distribution of the members of a shareholder meeting or a legislative committee is a well-known problem for many applications. In some cases it turns out that power is nearly proportional to relative voting weights, which is very beneficial for both theoretical considerations and practical computations with many members. We present quantitative approximation results with precise error bounds for several power indices as well as impossibility results for such approximations between power and weights.Comment: 23 pages, 1 table, 1 figur

Which criteria qualify power indices for applications? : A comment to "The story of the poor Public Good index"

We discuss possible criteria that may qualify or disqualify power indices for applications. Instead of providing final answers we merely ask questions that are relevant from our point of view and summarize some material from the literature.Comment: 6 pages; typos correcte

Bounds for the diameter of the weight polytope

A weighted game or a threshold function in general admits different weighted representations even if the sum of non-negative weights is fixed to one. Here we study bounds for the diameter of the corresponding weight polytope. It turns out that the diameter can be upper bounded in terms of the maximum weight and the quota or threshold. We apply those results to approximation results between power distributions, given by power indices, and weights.Comment: 16 pages; typos corrected; arXiv admin note: text overlap with arXiv:1802.0049