Modeling large electorates with Fourier series, with applications to Nash equilibria in proximity and directional models of spatial competition

In this paper we introduce harmonic analysis (Fourier series) as a tool for characterizing the existence of Nash equilibria in two-dimensional spatial majority rule voting games with large electorates. We apply our methods both to traditional proximity models and to directional models. In the latter voters exhibit preferences over directions rather than over alternatives, per se. A directional equilibrium can be characterized as a Condorcet direction, in analogy to the Condorcet (majority) winner in the usual voting models, i.e., a direction which is preferred by a majority to (or at least is not beaten by) any other direction. We provide a parallel treatment of the total median condition for equilibrium under proximity voting and equilibrium conditions for directional voting that shows that the former result is in terms of a strict equality (a knife-edge result very unlikely to hold) while the latter is in terms of an inequality which is relatively easy to satisfy. For the Matthews [3] directional model and a variant of the Rabinowitz and Macdonald [7] directional model, we present a sufficiency condition for the existence of a Condorcet directional vector in terms of the odd-numbered components of the Fourier series representing the density distribution of the voter points. We interpret our theoretical results by looking at real-world voter distributions and direction fields among voter points derived from U.S. and Norwegian survey data. The authors are indebted to Dorothy Green and Chau Tran for library assistance. The paper was written while the first author was a Visiting Scholar in the Department of Biostatistics at The University of Washington. The data sets used in the study were made available by the Inter-university Consortium for Political and Social Research (ICPSR) and by the Norwegian Social Science Data Services (NSD). Bernt Aardal and Henry Valen were the Principal Investigators of the Norwegian Election Study of 1989 and the Norwegian Bureau of Statistics collected the data. Neither the ICPSR, the NSD, the Norwegian Bureau of Statistics, nor the principal investigators of either study are responsible for the analysis or interpretation presented here.