The Instability of Instability

Abstract

Democratic simple majority voting is perhaps the most widely used method of group decision making in our time. Current theory, based on instability theorems, predicts that a group employing this method will almost always fail to reach a stable conclusion. There is one case that the classical instability theorems do not treat: an even number of voters in 2 dimensions. We resolve this remaining case, proving that instability occurs with probability converging rapidly to 1 as the population increases. But empirical observations do not support the gloomy predictions of the instability theorems. We show that the instability theorems are themselves unstable in the following sense: if the model of voter behavior is altered however slightly to incorporate any of several plausible characteristics of decision-making, then the instability theorems do not hold and in fact the probability of stability converges to 1 as the population increases, when the population is sampled from a centered distribution. The assumptions considered are: a cost of change; bounded rationality; perceptual thresholds; a cost of uncertainty; a discrete proposal space, and others. One consequence of this work is to render precise and rigorous the solution proposed by Tullock (63,64) and refined by Arrow (2) to the impossibility problem. The stability results all hold for arbitrary dimension, and generalize to establish a tradeoff between the characteristics and the degree of noncenteredness of a population. As a by-product of the analysis, we establish the statistical consistency of the sample yolk radius