Equilibrium Computation in the Hotelling-Downs Model of Spatial Competition

TL;DR

This work tackles the computational problem of finding equilibria in the Hotelling-Downs line model of spatial competition. It develops three algorithms to compute equilibria across different combinations of discrete and continuous voter and candidate placements, delivering exact equilibria when candidate positions are discrete and approximate equilibria (within a factor of 4ε) when both locations are continuous. A key contribution is showing how to bound the bit complexity of equilibria, proving that if equilibria exist, there is one with rational coordinates and bounded denominators, which enables grid-based reductions to discrete dynamic programs. The results bridge computation and economic theory in spatial elections, offering practical methods for equilibrium computation and revealing structural properties like the necessity of non-midpoint positions and inherent bit-size limitations. These advances pave the way for computing equilibria in extended spatial models and graph-based variants, with potential applications in political economics and computational social choice.

Abstract

The Hotelling-Downs model is a natural and appealing model for understanding strategic positioning by candidates in elections. In this model, voters are distributed on a line, representing their ideological position on an issue. Each candidate then chooses as a strategy a position on the line to maximize her vote share. Each voter votes for the nearest candidate, closest to their ideological position. This sets up a game between the candidates, and we study pure Nash equilibria in this game. The model and its variants are an important tool in political economics, and are studied widely in computational social choice as well. Despite the interest and practical relevance, most prior work focuses on the existence and properties of pure Nash equilibria in this model, ignoring computational issues. Our work gives algorithms for computing pure Nash equilibria in the basic model. We give three algorithms, depending on whether the distribution of voters is continuous or discrete, and similarly, whether the possible candidate positions are continuous or discrete. In each case, our algorithms return either an exact equilibrium or one arbitrarily close to exact, assuming existence. We believe our work will be useful, and may prompt interest, in computing equilibria in the wide variety of extensions of the basic model as well.

Figures (8)

• Figure 1: Distribution of voters and candidates for the Example \ref{['ex:violation']}. Candidates are indicated by black triangles and $\epsilon=\frac{1}{5}.$
• Figure 2: An equilibrium with five candidates and 24 voters. The red circles indicate the voters' locations, with the numbers below each circle representing the number of voters at that location. The blue numbers correspond to the lengths of the respective line segments
• Figure 3: An instance with eight candidates and ten equidistant voters with consecutive distance $1$, showing an equilibrium for the candidates.
• Figure 4: An instance with 3 candidates and 30 voters. The red circles indicate the voters' locations, with the numbers below each circle representing the number of voters at that location. The blue numbers correspond to the lengths of the respective line segments.
• Figure 5: An equilibrium with 5 candidates and 24 voters. The red circles indicate the voters' locations, with the numbers below each circle representing the number of voters at that location. The blue numbers correspond to the lengths of the respective line segments. At any equilibrium, the middle candidate must remain between the two inner dashed lines.

Theorems & Definitions (81)

• Definition 1: $\epsilon$-equilibrium
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Proposition 3