Nearly Tight Bounds on Approximate Equilibria in Spatial Competition on the Line

In Hotelling's model of spatial competition, a unit mass of voters is distributed in the interval $[0,1]$ (with their location corresponding to their political persuasion), and each of $m$ candidates selects as a strategy his distinct position in this interval. Each voter votes for the nearest candidate, and candidates choose their strategy to maximize their votes. It is known that if there are more than two candidates, equilibria may not exist in this model. It was unknown, however, how close to an equilibrium one could get. Our work studies approximate equilibria in this model, where a strategy profile is an (additive) $\epsilon$-equilibria if no candidate can increase their votes by $\epsilon$, and provides tight or nearly-tight bounds on the approximation $\epsilon$ achievable. We show that for 3 candidates, for any distribution of the voters, $\epsilon \ge 1/12$. Thus, somewhat surprisingly, for any distribution of the voters and any strategy profile of the candidates, at least $1/12$th of the total votes is always left ``on the table.'' Extending this, we show that in the worst case, there exist voter distributions for which $\epsilon \ge 1/6$, and this is tight: one can always compute a $1/6$-approximate equilibria. We then study the general case of $m$ candidates, and show that as $m$ grows large, we get closer to an exact equilibrium: one can always obtain an $1/(m+1)$-approximate equilibria in polynomial time. We show this bound is asymptotically tight, by giving voter distributions for which $\epsilon \ge 1/(m+3)$.