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.