Intransitivities in multidimensional spatial voting: period three implies chaos

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.

[1]  D. Black The theory of committees and elections , 1959 .

[2]  Charles R. Plott,et al.  A Notion of Equilibrium and Its Possibility Under Majority Rule , 1967 .

[3]  R. McKelvey Intransitivities in multidimensional voting models and some implications for agenda control , 1976 .

[4]  Gerald H. Kramer,et al.  A dynamical model of political equilibrium , 1977 .

[5]  N. Schofield,et al.  Instability of Simple Dynamic Games , 1978 .

[6]  Charles R. Plott,et al.  Committee Decisions under Majority Rule: An Experimental Study , 1978, American Political Science Review.

[7]  Linda B. Cohen,et al.  Cyclic sets in multidimensional voting models , 1979 .

[8]  R. McKelvey General Conditions for Global Intransitivities in Formal Voting Models , 1979 .

[9]  K. Shepsle Institutional Arrangements and Equilibrium in Multidimensional Voting Models , 1979 .

[10]  Steven A. Matthews,et al.  Constrained Plott Equilibria, Directional Equilibria and Global Cycling Sets , 1980 .

[11]  N. Schofield Generic Instability of Majority Rule , 1983 .

[12]  Edward W. Packel,et al.  Limiting distributions for continuous state Markov voting models , 1984 .

[13]  Norman Schofield,et al.  Social equilibrium and cycles on compact sets , 1984 .

[14]  R. Devaney An Introduction to Chaotic Dynamical Systems , 1990 .

[15]  B. Grofman,et al.  The Core and the Stability of Group Choice in Spatial Voting Games , 1988, American Political Science Review.

[16]  Donald G. Saari,et al.  A dictionary for voting paradoxes , 1989 .

[17]  Donald G. Saari,et al.  The Borda dictionary , 1990 .

[18]  D. Saari Erratic behavior in economic models , 1991 .

[19]  Donald G. Saari Symmetry extensions of “neutrality” I: Advantage to the condorcet loser , 1992 .