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

TL;DR

This work analyzes additive ε-equilibria in Hotelling-style spatial competition on the unit interval, addressing the instability caused by the nonexistence of exact equilibria when $m\ge3$. It establishes a universal lower bound $ε\ge 1/12$ for three candidates and shows a worst-case $ε\ge 1/6$ with a matching $1/6$-approximation algorithm, while for general $m$ players it achieves $ε\le 1/(m+1)$ with a $1/(m+3)$ lower bound in the worst case; a variant with co-located candidates yields $ε\ge 1/7$. The results are supported by polynomial-time procedures leveraging Eval and Cut queries, analogous to Robertson–Webb cake-cutting queries, and by constructions that prove tightness in both the 3-candidate and asymptotic $m$-candidate regimes. Overall, the paper provides a quantitative, nearly tight picture of how close one can get to equilibrium in spatial competition, with stability improving as the number of candidates grows.

Abstract

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) $ε$-equilibria if no candidate can increase their votes by $ε$, and provides tight or nearly-tight bounds on the approximation $ε$ achievable. We show that for 3 candidates, for any distribution of the voters, $ε\ge 1/12$. 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 $ε\ge 1/6$, 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 $ε\ge 1/(m+3)$.

Figures (8)

• Figure 1: Figure showing bounds on the votes, as shown by Claim \ref{['mid']} and Claim \ref{['extreme']}. Candidates are shown by blue circles. The dashed lines are the mid-points of $x_1,x_2$ and $x_2, x_3$.
• Figure 2: Distribution of voters and candidates for the lower bound of $1/12$ in Theorem \ref{['112ex']}. Candidates are indicated by blue circles.
• Figure 3: Figure showing the votes, as shown by Theorem \ref{['thm:ub-1/6']}. Blue circles show candidates. The dashed lines are the mid-points of $x_1,x_2$ and $x_2, x_3$.
• Figure 4: Change in votes for candidate $k$, if he shifts from $x_k$ to $x_k'$.
• Figure 5: Bounds on the utility of candidate $k+1$ in Theorem \ref{['thm:general-lb']}.

Theorems & Definitions (34)

• Definition 1: $\epsilon$-equilibrium
• Theorem 1
• Claim 1
• proof
• Claim 2
• proof
• Claim 3
• proof : of Theorem \ref{['lb112']}
• Theorem 2
• proof