Nash equilibrium structure of Cox process Hotelling games

TL;DR

This paper analyzes a multi-player game involving Poisson point processes on Polish spaces, characterizing the existence and uniqueness of Nash equilibria based on players' intensity functions and their proportionality.

Contribution

It provides a characterization of Nash equilibria in Cox process Hotelling games, establishing conditions for existence and proving uniqueness when equilibria exist.

Findings

Nash equilibria are unique and proportional in pure strategies when they exist.

Examples where Nash equilibria do not exist are provided.

The paper highlights open problems regarding the criteria for equilibrium existence.

Abstract

We study an N-player game where a pure action of each player is to select a non-negative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of its open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and comprised of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.