Spatial Analysis of Cumulative Voting with Dynamical System Simulation
by
Duane A. Cooper
Dept. of Mathematics, Morehouse College, Atlanta GA 30314

In this talk, we examine a the development of a limited spatial model of the election method of cumulative voting, after which we explore how the model can be formulated and analyzed as a dynamical system We analyze a one-dimensional proximity spatial model on which three candidates vie for two seats with voters uniformly distributed on (0, 1). Under this heuristic, a voter plumps his votes, awarding both to the nearest candidate, if that candidate is much closer to the voter than the second nearest candidate (i.e., if the distance d(v, c1) < 1/ß·d(v, c2) with ß > 1) or if the second nearest candidate is not much closer to the voter than the third candidate (i.e., if d(v, c2) > 1/ß·d(v, c3) ). Otherwise, the voter splits his votes one apiece for the two nearest candidates. We prove that two candidates cannot be defeated at equilibrium positions of 1/4 + (1/2)/(ß2+1) and 3/4 – (1/2)/(ß2+1) and that the equilibrium state exists if and only if ß > 1 + sqrt(2) . Knowing the conditions for and location of the equilibria, we will analyze the model as a dynamical system to determine whether and when the candidates indeed gravitate towards the equilibrium positions.

Date received: February 24, 2003