On Pure and (approximate) Strong Equilibria of Facility Location Games

TL;DR

This paper analyzes the efficiency of stable and coalitionally resilient equilibria in Facility Location games, providing bounds on social cost losses and showing existence of approximate strong equilibria with tight bounds in various network settings.

Contribution

It offers the first comprehensive analysis of Price of Stability and Price of Anarchy for strong equilibria in Facility Location games, including bounds and existence results.

Findings

Constant bounds for PoS in unweighted metric networks.

Existence of $e$-approximate strong equilibria.

Upper bounds on SPoA related to total agent weight.

Abstract

We study social cost losses in Facility Location games, where $n$ selfish agents install facilities over a network and connect to them, so as to forward their local demand (expressed by a non-negative weight per agent). Agents using the same facility share fairly its installation cost, but every agent pays individually a (weighted) connection cost to the chosen location. We study the Price of Stability (PoS) of pure Nash equilibria and the Price of Anarchy of strong equilibria (SPoA), that generalize pure equilibria by being resilient to coalitional deviations. A special case of recently studied network design games, Facility Location merits separate study as a classic model with numerous applications and individual characteristics: our analysis for unweighted agents on metric networks reveals constant upper and lower bounds for the PoS, while an $O(\ln n)$ upper bound implied by previous work is tight for non-metric networks. Strong equilibria do not always exist, even for the unweighted metric case. We show that $e$-approximate strong equilibria exist ($e=2.718...$). The SPoA is generally upper bounded by $O(\ln W)$ ($W$ is the sum of agents' weights), which becomes tight $\Theta(\ln n)$ for unweighted agents. For the unweighted metric case we prove a constant upper bound. We point out several challenging open questions that arise.