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Abstract
Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.
Notes
Notes
1. Recent estimates for the population are taken from http://epp.eurostat.ec.europa.eu.
2. In Citation11 the authors call this problem the voting game design problem.
3. We would like to remark that the counts for weighted voting games with 6 ≤ n ≤ 8 voters are wrongly stated in Citation10, but the methods should work.
4. Additionally, some computational tricks and heuristics were used to reduce the number of linear programs.
5. Here one may also read the coalition vectors as integers written in their binary expansion and use the ordinary ordering ≤ of integers.
6. We may also add inequalities to ensure that all numbers of swings for a given voter have the same parity.