Location Games on Networks: Existence and Efficiency of Equilibria

TL;DR

This paper analyzes location games on networks, proving the existence of pure Nash equilibria for many retailers and comparing their efficiency to optimal arrangements using price of anarchy and stability metrics.

Contribution

It establishes the existence of pure Nash equilibria in location games on networks with many retailers and quantifies their efficiency relative to optimal solutions.

Findings

Pure Nash equilibria exist when the number of retailers is large enough.

Asymptotically, the price of anarchy is bounded by two.

The price of stability approaches one as the number of retailers increases.

Abstract

We consider a game where a finite number of retailers choose a location, given that their potential consumers are distributed on a network. Retailers do not compete on price but only on location, therefore each consumer shops at the closest store. We show that when the number of retailers is large enough, the game admits a pure Nash equilibrium and we construct it. We then compare the equilibrium cost borne by the consumers with the cost that could be achieved if the retailers followed the dictate of a benevolent planner. We perform this comparison in term of the price of anarchy, i.e., the ratio of the worst equilibrium cost and the optimal cost, and the price of stability, i.e., the ratio of the best equilibrium cost and the optimal cost. We show that, asymptotically in the number of retailers, these ratios are bounded by two and one, respectively.