Selfish Load Balancing and Atomic Congestion Games

Abstract

Abstract. We revisit a classical load balancing problem in the modern context of decentralized systems and self-interested clients. In particular, there is a set of clients, each of whom must choose a server from a permissible set. Each client has a unit-length job and selfishly wants to minimize its own latency (job completion time). A server's latency is inversely proportional to its speed, but it grows linearly with (or, more generally, as the pth power of) the number of clients matched to it. This interaction is naturally modeled as an atomic congestion game, which we call selfish load balancing. We analyze the Nash equilibria of this game and prove nearly tight bounds on the price of anarchy (worst-case ratio between a Nash solution and the social optimum). In particular, for linear latency functions, we show that if the server speeds are relatively bounded and the number of clients is large compared with the number of servers, then every Nash assignment approaches social optimum. Without any assumptions on the number of clients, servers, and server speeds, the price of anarchy is at most 2.5. If all servers have the same speed, then the price of anarchy further improves to 1 + 2/ √ 3 ≈ 2.15. We also exhibit a lower bound of 2.01. Our proof techniques can also be adapted for the coordinated load balancing problem under L 2 norm, where it slightly improves the best previously known upper bound on the competitive ratio of a simple greedy scheme.

The Sequential Price of Anarchy for Atomic Congestion Games

“…This model, and variants thereof, are also known as load balancing games and, with respect to the quality of equilibria, have a vast literature, e.g. [10,4,15].…”

“…Constraints (11), (12), and (13) make sure that there exists at least one subgame perfect action for each player. Constraints (14), (15), and (16) make sure no player can improve from any subgame perfect action; when action c is subgame perfect when players 1, 2 choose actions a, b respectively, then cost 3 (abc) ≤ cost 3 (abc ) for any action c . When action b is subgame perfect for player 2 when player 1 chooses action a, then cost 2 (ab) ≤ cost 2 (ab ) for all actions b .…”

Graphical Congestion Games

“…By defining an elegant potential function he showed that they always possess (and always converge to) pure Nash equilibria [34,35]. Since then, different variations of the original model have been proposed [32,33,40] and, in the last decade, they have come across the analysis of the Computer Science community with the purpose of characterizing the complexity of computing their pure Nash equilibria [14] and evaluating their suboptimality [3,9,13,16,[38][39][40] in terms of price of stability [1] and anarchy [30].…”

Competitive Online Multicommodity Routing

“…They proved an upper bound on the competitive ratio of 3 + 2 √ 2. In the same context, Suri, Toth, and Zhou [26] and Caragiannis et al [8] studied Nash solutions for every released job and showed that the resulting online algorithm outperforms the greedy strategy of [3]. Their setting, however, is restricted to m parallel arcs and all released jobs have to be assigned to exactly one machine (arc).…”