The Inverse Problem for Power Distributions in Committees

Several power indices have been introduced in the literature in order to measure the influence of individual committee members on the aggregated decision. Here we ask the inverse question and aim to design voting rules for a committee such that a given desired power distribution is met as closely as possible. We present an exact algorithm for a large class of different power indices based on integer linear programming. With respect to negative approximation results we generalize the approach of Alon and Edelman who studied power distributions for the Banzhaf index, where most of the power is concentrated on few coordinates. It turned out that each Banzhaf vector of an n-member committee that is near to such a desired power distribution, has to be also near to the Banzhaf vector of a k-member committee. We show that such Alon-Edelman type results are possible for other power indices like e.g. the Public Good index or the Coleman index to prevent actions, while they are principally impossible for e.g. the Johnston index