Random Preferences, Acyclicity, and the Threshold of Rationality

Abstract

Draw an agent's preferences over $n$ alternatives at random as follows: independently, for each pair of distinct alternatives, with probability $1-p$ the agent is indifferent between the alternatives in the pair, and with probability $p$ the agent is equally likely to prefer one alternative over the other. The agent's preferences are rational if they are transitive or, equivalently, if there are no preference cycles. I show that rationality exhibits a threshold behavior: if $p$ is asymptotically smaller than $n^{-2}$ then the agent's preferences are rational with high probability, and if $p$ is asymptotically larger than $n^{-2}$ then the agent's preferences are not rational with high probability. The main result of this paper embeds some existing results on the probability of preference cycles and of Condorcet cycles.