Equilibria in Large Position-Optimization Games

2026-02-16

AI Summary

This study investigates the existence, structure, and convergence behavior of Nash equilibria in large-scale symmetric location optimization games, encompassing classical settings such as Hotelling games and prediction contests. By formulating a general model in which players select locations in a metric space to maximize the probability mass of covered targets, and leveraging symmetry assumptions together with asymptotic analysis, the work demonstrates that as the number of players tends to infinity, both pure-strategy and symmetric mixed-strategy Nash equilibria concentrate on a finite set of pseudo-targets and converge to a limiting distribution induced by the underlying target distribution. The analysis unifies and extends several classical results, provides explicit bounds on convergence rates, and establishes the existence and extremal properties of equilibria in the large-population regime.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• 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
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

We propose a general class of symmetric games called position-optimization games. Given a probability distribution $Q$ over a set of targets $\mathcal{Y}$, the $n$ players each choose a position in a space $\mathcal{X}$. A player's utility is the $Q$-mass of targets they are closest to under some proximity measure, with ties broken evenly. Our model captures Hotelling games and forecasting competitions, among other applications. We show that for sufficiently large $n$, both pure and symmetric mixed Nash equilibria exist, and moreover are extreme: all players play on a finite set of pseudo-targets $\mathcal{X}^* \subseteq \mathcal{X}$. We further show that both pure and symmetric mixed equilibria converge to the distribution $P$ on $\mathcal{X}^*$ induced by $Q$, and bound the convergence rate in $n$. The generality of our model allows us to extend and strengthen previous work in Hotelling games, and prove entirely new results in forecasting competitions and other applications.

Problem

• position-optimization games
• Nash equilibrium
• large games
• Hotelling games
• forecasting competitions

Innovation

• position-optimization games
• Nash equilibrium
• convergence rate
• pseudo-targets
• symmetric games