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Am empirical comparison of the performance of classical power indices
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Leech, Dennis (2001) Am empirical comparison of the performance of classical power indices. Working Paper. Coventry: University of Warwick, Department of Economics. (Warwick economic research papers).

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Abstract
Power indices are general measures of the relative voting power of individual members of a voting body. They are useful in helping understand and design voting bodies particularly those which employ weighted voting, in which different members having different numbers of votes. It is well known that in such bodies a member's voting power, in the sense of their capacity to affect the outcomes of votes called, rarely corresponds to the actual number of votes allocated to him. Many voting bodies for which this is an important consideration exist: examples include international organisations (notably the World Bank, the IMF, the European Union), the US presidential Electoral College and corporations in which votes are proportionate to stockholdings. Two classical power indices dominate the literature: the ShapleyShubik index and the Banzhaf index (also known by other names). Both are based on the idea that a member's power depends on the relative number of times they can change a coalition from losing to winning by joining it and adding their vote. They may be defined in probabilistic terms as the probability of being able to swing the result of a vote, where all possible outcomes are taken as equiprobable. The indices differ however in the way they count voting coalitions. In probabilistic terms they use different coalition models and therefore differ in precisely what is meant by equiprobable outcomes. The indices have been used in a number of empirical applications but their relative performance has remained an open question for many years, a factor, which has hindered the wider acceptance of the approach. Where both the indices have been used for the same case, they have often given different results, sometimes substantially so, and theoretical studies of their properties have not been conclusive. There is therefore a need for comparative testing of their relative performance in practical contexts. Very little work of this type has been done however for a number of reasons: lack of independent indicators of power in actual voting bodies with which to compare them, difficulties in obtaining consistent data on a voting body over time with sufficient variation in the disposition of votes among members of actual legislatures and the lack of independent criteria against which the results of the indices may be judged. It has also been hampered to some extent by lack of easily available algorithms for computing the indices in large games. This paper assesses the indices against a set of reasonable criteria in terms of shareholder voting power and the control of the corporation in a large cross section of British companies. Each company is a separate voting body and there is much variation in the distribution of voting shares among them. Moreover reasonable criteria exist against which to judge the indices. New algorithms for the ShapleyShubik and Banzhaf indices are applied to detailed data on beneficial ownership of 444 large UK companies without majority control. Because some of the data is missing, both finite and oceanic games of shareholder voting are studied to overcome this problem. The results, judged against these criteria, are unfavorable to the ShapleyShubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocks vary.
Item Type:  Working or Discussion Paper (Working Paper) 

Subjects:  H Social Sciences > HB Economic Theory J Political Science > JC Political theory 
Divisions:  Faculty of Social Sciences > Economics 
Library of Congress Subject Headings (LCSH):  Power (Social sciences), Game theory, Group theory, Voting reseach 
Series Name:  Warwick economic research papers 
Publisher:  University of Warwick, Department of Economics 
Place of Publication:  Coventry 
Date:  October 2001 
Number:  No.563 
Number of Pages:  37 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
Description:  Original version June 2000; this revision October 2001 
Adapted As:  Leech, D. (2001). Am empirical comparison of the performance of classical power indices. Political Studies, 50(1), pp. 122. 
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URI:  http://wrap.warwick.ac.uk/id/eprint/1614 
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