Hotelling-Downs with Facility Synergy: The Mall Effect

2026-06-24

AI Summary

This study investigates the impact of user preference for facility-dense regions—capturing facility co-location externalities—on the existence and efficiency of Nash equilibria in facility location games. By extending the classical Hotelling-Downs model with a utility function that jointly accounts for travel distance and local facility density, the authors formally incorporate such synergistic effects into the strategic interaction for the first time. Leveraging game-theoretic analysis and Price of Anarchy (PoA) techniques, they establish the universal existence of pure-strategy Nash equilibria and derive a tight upper bound on equilibrium inefficiency: the total customer cost in any equilibrium is at most $225/64$ (approximately 3.516) times the social optimum, i.e., PoA $\leq 225/64$. This result demonstrates that market self-organization remains provably efficient even under co-location preferences.

Abstract

We consider a variation of the classic Hotelling-Downs model with the addition of facility synergies. Unlike in the classic model, where clients always use the facility closest to them, we study clients who prefer locations with many facilities to those with few facilities while simultaneously attempting to minimize their distance as well. We show that, in contrast with the classic model, Nash equilibria for our setting always exist, and, in fact, there always exists a Nash equilibrium such that the sum of client costs equals the cost of the optimal solution. Our main result is a bound of $\frac{225}{64}\approx 3.516$ on the Price of Anarchy for our model, showing that, although the client behavior is more complex in our model (and often more realistic depending on the application), the cost of Nash equilibrium solutions still cannot be much worse than the cost of the optimal facility placement.

Problem

• Hotelling-Downs model
• facility synergy
• Nash equilibrium
• Price of Anarchy
• client behavior

Innovation

• facility synergy
• Hotelling-Downs model
• Nash equilibrium
• Price of Anarchy
• spatial competition