Optimal Equilibrium Locations for Proximity Spatial Theory of Voting

Dr. S. Gikiri Thuo

Morgan State University

March 5, 2003
202 Madison Hall
3:00 PM (Coffee and Cookies at 2:30)

Abstract

We analyze the proximity spatial models for at-large elections. Using a cumulative voting heuristic, we identify conditions necessary for a symmetric Nash equilibrium to exist when voters' ideal points have a standard normal distribution. We first place two candidates away from the median voter, on the opposite sides, and investigate conditions necessary for equilibrium to exist when a third candidate is within an epsilon neighborhood of either of the other two candidates. Next we investigate equilibrium conditions when the third candidate is allowed to assume positions anywhere on R, the single predictive dimension space. From the first analysis we develop an equation whose solutions provide the only possibility for a symmetric Nash equilibrium to exist. Using optimization techniques, we approximate those solutions by a nice elementary function whose properties we know. From the second analysis, we derive mathematical models, which we analyze to establish intervals of beta for which a symmetric Nash equilibrium exists. We then extend our analysis to a more plausible distribution for the cumulative voting method, the "double normal" distribution. The problem model we developed in this case is complex and hence necessitates analysis via simulation. Through our simulation model we illustrate the electoral potential of cumulative voting to yield fair representation when plurality voting does not.