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

Abstract

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. For unweighted agents on metric networks we prove upper and lower bounds on P oS, while an O(ln n) upper bound implied by previous work is tight for nonmetric networks. We also prove a constant upper bound for the SPoA of metric networks when strong equilibria exist. For the weighted game on general networks we prove existence of e-approximate (e = 2.718 . . .) strong equilibria and an upper bound of O(ln W ) on SPoA (W is the sum of agents' weights), which becomes tight Θ(ln n) for unweighted agents.

Improved Bounds for Facility Location Games with Fair Cost Allocation

“…The agent tries to minimize his individual cost; it consists of the (weighted) distance of i from v and a fair share of the facility installation cost at v. Facility installation cost at v is shared evenly among all agents having chosen v, or proportionally to their demand weight in case of weighted agents. In this paper we study pure Nash and strong equilibria (PNE and SE) of the described facility location game, and improve or extend previous results [1,2]. We prove a super-polynomial lower bound on the complexity of the Iterative Best Response algorithm for finding PNE, and analyze a polynomial-time algorithm for finding approximate PNE on metric networks.…”

“…. ) was shown in [2], even for the non-metric weighted case. Nguyen Kim [12] showed that PNE do not always exist for metric facility location games with weighted agents.…”

“…The upper bounding is with reference to the socially optimum cost; this ratio is known as the Price of Anarchy [3]. A previous result [2] provided an upper bound only in case of existence of exact SE.…”

“…The notion of SE is due to Aumann [11]. Metric Facility Location with unweighted players is the only case of the model of [1], in which tight constant bounds are known for the Price of Stability [2]. Also, existence of e-approximate SE (e = 2.718 .…”

“…. factor shown in [2] for general Facility Location games. Finally, we develop an analysis for the Price of Anarchy of α-approximate SE for metric Facility Location games with unweighted agents, and show that it is SP oA = O(α ln α).…”