Abstract

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