Statistical Mechanics of Voting

Decision procedures aggregating the preferences of multiple agents can produce cycles and hence outcomes which have been described heuristically as “chaotic.” We make this description precise by constructing an explicit dynamical system from the agents' preferences and a voting rule. The dynamics form a one-dimensional statistical mechanics model; this suggests the use of the topological entropy to quantify the complexity of the system. We compute the expected complexity of a voting rule and the degree of cohesion/diversity among agents using random matrix models—ensembles of statistical mechanics models—in some representative cases.