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 $\epsilon \ge 1/(m+3)$.

翻译：在霍特林空间竞争模型中，单位质量的选民分布于区间$[0,1]$（其位置对应政治倾向），$m$个候选人各自选择该区间内的不同位置作为策略。每位选民投票给距离最近的候选人，候选人通过调整策略最大化得票数。已知当候选人超过两人时，该模型可能不存在均衡态，但此前尚未明确能达到何种程度的近似均衡。本研究探讨该模型中的近似均衡——若策略配置中任何候选人的得票增量不超过$\epsilon$，则称为（可加性）$\epsilon$均衡，并给出可实现的近似值$\epsilon$的紧致或近紧致界。对于3个候选人的情形，我们发现无论选民如何分布，均有$\epsilon \ge 1/12$。这揭示了令人惊奇的结论：无论选民分布与候选人策略配置如何，总有至少$1/12$的选票被“闲置”。进一步研究表明，最坏情况下存在选民分布使得$\epsilon \ge 1/6$，且该界是紧致的——即总能通过计算得到$1/6$近似均衡。随后我们分析$m$个候选人的一般情形，证明随着$m$增大，体系趋近于精确均衡：通过多项式时间算法总能实现$1/(m+1)$近似均衡。通过构造满足$\epsilon \ge 1/(m+3)$的选民分布，我们证明该界具有渐近紧致性。