Approximate Equilibria and Ball Fusion

Abstract

We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. W e study the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum maximum latency had global regulation been available. For the case of uniform links it is known that all Nash equilibria belong to this class of strategies. We a r e i n terested in bounding the coordination ratio (or price of anarchy) of these strategies de ned as the worst-case ratio of the maximum (over all links) expected latency over the optimum maximum latency. The load balancing aspect of the problem immediately implies a lower bound ; ln m ln ln m of the coordination ratio. We g i v e a tight ( u p t o a m ultiplicative c o n s t a n t) upper bound. To show the upper bound, we analyze a variant o f t h e classical balls and bins problem, in which balls with arbitrary weights are placed into bins according to arbitrary probability distributions. At the heart of our approach is a new probabilistic tool that we c a l l

Network uncertainty in selfish routing

“…The problem of computing pure Nash equilibria was studied for congestion games in [3] and for weighted congestion games in [6]. The KP-Model [13] and its Nash Equilibria were studied extensively in the last years; see, for example, [2,4,6,12,15,16] and [5] for a survey. Feldmann et al [4] and Gairing et al [7] propose algorithms to transform any user strategy to a Nash equilibrium without increasing the maximum congestion.…”

“…Subsequently, tight bounds were proposed for it in [2,12] for identical links, in [2] for related links, and in [1] for congestion games with linear latency functions.…”

Algorithmic Game Theory and Applications

“…The price of anarchy for this game has been shown to be (log m/log log m) if either the users or the links are identical [18,59], and (log m/log log log m) for weighted users and uniformly related links [18]. On the contrary, [17] showed that the price of anarchy is far worse and can be even unbounded for arbitrary latency functions.…”

Coordination Mechanisms

“…When the users act selfishly at a Nash equilibrium the resulting allocation may be suboptimal. The price of anarchy, that is, the worst-case ratio of the maximum latency at a Nash equilibrium over the optimal allocation can be as high as Θ(log m/ log log m) [25,14,26]. The question is "How can we improve the price of anarchy?…”