糖心TV

Abstract. I investigate what we know about voting power indices when there are 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 (eg the EUCM, US Electoral College). This is true for both the Banzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (that has been dubbed the Penrose limit theorem). On the other hand, we have the results from the older literature  from cooperative game theory, due to Shapley and his collaborators. This shows that, where there are 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 - so called oceanic games - the powers of the voters with finite weight tend to limiting values in general not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies, eg. the IMF/WB boards, corporate shareholder control. We give a new precise statement of the Penrose Limit theorem and show that convergence occurs in the limit, not as the number of players goes to infinity but  as the Laakso-Taagepera index of political fragmentation (the political science name for the numbers-equivalent corresponding to the Hirschman-Herfindahl index) increases without  limit. I claim that this new version reconciles the different theoretical and empirical results that have been found for large voting games.