The paper provides a complete characterization of Nash equilibria for games in which n candidates choose a strategy in the form of a platform, each from a circle of feasible platforms, with the aim of maximizing the stretch of the circle from where the candidate's platform will receive support from the voters. Using this characterization, it is shown that if the sum of all players' payoffs is 1, the Nash equilibrium payoff of each player in an arbitrary Nash equilibrium must be restricted to the interval [1/2(n − 1), 2/(n + 1)]. This implies that in an election with four candidates, a candidate who is attracting less than one-sixth of the voters can do better by changing his or her strategy.