Computation of power indices

Abstract

Power indices are a useful tool for studying voting systems in which different members have different numbers of votes. Many international organisations are organised in this way in order to accommodate differences in size of countries, including the system of QMV in the European Union Council of Ministers. A power index is a measure of power based on the idea that a member’s power is his ability to swing a decision by changing the way his vote is cast. This paper addresses the problem of the computation of the two most widely used power indices, the so-called classical power indices, the Banzhaf index (and also the related Coleman indices) and the Shapley-Shubik index. It discusses the various methods that have been proposed in the literature: Direct Enumeration, Monte Carlo Simulation, Generating Functions, Multilinear Extensions Approximation, the Modified MLE Approximation Method. The advantages and disadvantages of the algorithms are discussed including computational complexity. It also describes methods for so called “oceanic games”. The paper also discusses the so-called “inverse problem” of finding what the weights should be given the desired power indices. The method is potentially useful as providing a basis for designing a voting system with a given desired distribution of power among the members, for example, to reflect differences in population or financial contributions. Examples are given from the International Monetary Fund, shareholder voting and the European Union Council