Approximate Equilibria and Ball Fusion

We consider selfish routing over a network consisting of m parallel links through which $n$ selfish users route their traffic trying to minimize their own expected latency. We 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 are interested in bounding the coordination ratio (or price of anarchy) of these strategies defined 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 Ω(ln m ln ln m) of the coordination ratio. We give a tight (up to a multiplicative constant) upper bound. To show the upper bound, we analyze a variant of the 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 call ball fusion; this tool is used to reduce the variant of the problem where balls bear weights to the classical version (with no weights). Ball fusion applies to more general settings such as links with arbitrary capacities and other latency functions.

Authors and Affiliations

1. Department of Computer Science, University of California at Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1361, USA and Department of Informatics, University of Athens, 15771 Athens, Greece

   Elias Koutsoupias

2. Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus , Cyprus

   Marios Mavronicolas

3. Department of Computer Engineering and Informatics, University of Patras, 265 00 Patras and Computer Technology Institute, 261 10 Patras, Greece

   Paul Spirakis

Corresponding authors

Correspondence to Elias Koutsoupias, Marios Mavronicolas or Paul Spirakis.

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