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There is no consensus on the properties of voting power indices when there is a large number of voters in a weighted-voting body. On the one hand, in some real-world cases that have been studied the power indices have been found to be nearly proportional to the weights (e.g., the EUCM, US Electoral College); this is true for both the Penrose-Banzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (known subsequently as the Penrose limit theorem, that has been shown to hold under certain conditions). On the other hand, we have the older literature from cooperative game theory, due to Shapley and his collaborators, showing that, where there is a finite number of voters whose weights remain constant in relative terms, and where the quota remains constant in relative terms, while the total number of voters increases without limit, the powers of the voters with finite weight tend to limiting values that are, in general, not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies (e.g., the IMF/WB boards, corporate shareholder meetings). This paper proposes a restatement of the Penrose limit theorem and shows that, in general, both the 'classical' power indices converge in the limit to proportionality with weights as the Laakso-Taagepera index of political fragmentation increases. This new version reconciles the different theoretical and empirical results that have been found for large voting bodies. Adapted from the source document.

Power indices are general measures of the relative a priori voting power of individual members of a voting body. They are useful for both positive and normative analysis of voting bodies particularly those using weighted voting. This paper applies new algorithms for computing the rival Shapley-Shubik and Banzhaf indices for large voting bodies to shareholder voting power in a cross section of British companies. Each company is a separate voting body and there is much variation in ownership between them resulting in different power structures. Because the data are incomplete, both finite and 'oceanic' games of shareholder voting are analysed. The indices are appraised, using reasonable criteria, from the literature on corporate control. The results are unfavourable to the Shapley-Shubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocs vary.

This paper examines the system of Qualified Majority Voting, used by the Council of the European Union, from the perspective of enlargement of the EU. It uses an approach based on power indices due to Penrose (1946), Banzhaf (1965), & Coleman (1971) to make two analyses: (1) the question of the voting power of member countries from the point of view of fairness & (2) the question of how the threshold number of votes required for QMV should be determined. It studies two scenarios for change from 2005 onwards envisaged by the Nice Treaty: (1) no enlargement, the EU comprising 15 member countries; & (2) full enlargement to 27 members by the accession of all the present 12 candidates. The proposal is made that fair weights be determined algorithmically as a technical or routine matter as the membership changes. The analysis of how the threshold affects power shows the trade-offs that countries face between their blocking power & the power of the Council to act. The main findings are: (A) that the weights laid down in the Nice Treaty are close to being fair, the only significant discrepancies being the underrepresentation of Germany & Romania, & the overrepresentation of Spain & Poland; (B) the threshold required for a decision is set too high for the Council to be an effective decision making body. 5 Tables, 7 Figures, 28 References. Adapted from the source document.

Abstract. This paper provides an empirical analysis using power indices derived by regarding the Conference and Electoral College as weighted majority games. It is shown that both main power indices, the Shapley‐Shubik and the Banzhaf, may be computed with reasonable accuracy by an approximation method based on probabilistic‐voting assumption. The results show voting power to be more concentrated than voting weight, that of some large trade unions being enhanced at the expense of smaller affiliated organisations and constituency parties (and, in the Electoral College, MPs). The results also provide insights into the comparative properties of the two classical power indices.

Es werden die a priori Verteilung von Stimmrechten und der tatsächliche Einfluß von Stimmen bei parteiinternen Wahlgängen der britischen Labour Party untersucht. Anhand von "power-indices", dem Shaphey-Shubik-Index und dem Banzaf-Index wird der Einfluß der angeschlossenen Gewerkschaften auf Parteitagen der Labour Party und des Wahlmännergremiums zur Wahl des Parteivorsitzenden untersucht. (AuD-Pls)