Improved Bounds for Discrete Voronoi Games

2026-02-18

AI Summary

This study investigates the discrete Voronoi game in the plane, where the first player places \(k\) facilities and the second player places a single facility (\(k > 1\), \(\ell = 1\)), focusing on the minimum number of voters required for the first player to guarantee a win. By employing geometric analysis and combinatorial optimization techniques, the authors derive tighter lower bounds on the winning margin under both \(L_1\) and \(L_2\) metrics. For all \(k \geq 4\), these bounds improve upon the best previously known results. As a consequence, the work also establishes a new upper bound on the size of small \(\varepsilon\)-nets for convex regions, thereby advancing the understanding of equilibrium performance in discrete Voronoi games.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

In the planar one-round discrete Voronoi game, two players $\mathcal{P}$ and $\mathcal{Q}$ compete over a set $V$ of $n$ voters represented by points in $\mathbb{R}^2$. First, $\mathcal{P}$ places a set $P$ of $k$ points, then $\mathcal{Q}$ places a set $Q$ of $\ell$ points, and then each voter $v\in V$ is won by the player who has placed a point closest to $v$. It is well known that if $k=\ell=1$, then $\mathcal{P}$ can always win $n/3$ voters and that this is worst-case optimal. We study the setting where $k>1$ and $\ell=1$. We present lower bounds on the number of voters that $\mathcal{P}$ can always win, which improve the existing bounds for all $k\geq 4$. As a by-product, we obtain improved bounds on small $\varepsilon$-nets for convex ranges. These results are for the $L_2$ metric. We also obtain lower bounds on the number of voters that $\mathcal{P}$ can always win when distances are measured in the $L_1$ metric.

Problem

• discrete Voronoi games
• competitive facility location
• planar geometry
• worst-case guarantee
• voter allocation

Innovation

• discrete Voronoi games
• epsilon-nets
• facility location
• computational geometry
• L1 and L2 metrics