L S Penrose's limit theorem: Tests by simulation

Abstract

L S Penrose's Limit Theorem–which is implicit in Penrose (1952, p. 72) [Penrose, 1952. On the Objective Study of Crowd Behavior. H. K. Lewis and Co, London, p. 72] and for which he gave no rigorous proof–says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then–under certain conditions–the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover (2004) [Lindner I., Machover M. 2004. L.S. Penrose's limit theorem: proof of some special cases. Mathematical Social Sciences 47, 37–49] prove some special cases of Penrose's Limit Theorem. They give a simple counter-example showing that the theorem does not hold in general even under the conditions assumed by Penrose; but they conjecture, in effect, that under rather general conditions it holds ‘almost always’–that is with probability 1–for large classes of weighted voting games, for various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated with respect to the Penrose–Banzhaf index for a quota of 50% but not for other values; with respect to the Shapley–Shubik index the conjecture is corroborated for all values of the quota (short of 100%).