Intransitivities in multidimensional spatial voting: period three implies chaos

Abstract

It is well known that multidimensional spatial voting can involve intransitivity and cycles, resulting in outcomes anywhere in the policy space. These results are typically referred to as the “chaos theorems”. In this paper, I show that the connection between non-equilibrium spatial voting and “chaos” is not merely semantic, but is theoretic. Using symbolic dynamics, I show that if a three-cycle intransitivity among social choices exists, then cycles of all lengths greater than three are possible. This result is then used to establish the three conditions of sensitive dependence on initial conditions, topological transitivity, and dense periodic points, demonstrating the formal connection between multidimensional spatial voting and chaotic nonlinear dynamics.