The Instability of Instability∗

Democratic simple majority voting is perhaps the most widely used method of group decision making in our time. Standard 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 classic 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 discrete proposal space, and others. One consequence of this work is to render precise and rigorous, the solution proposed by Tullock [64, 65] 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 non-centeredness of a population. As a by-product of the analysis, we establish the statistical consistency of the sample yolk radius.