On the Structure and Complexity of Worst-Case Equilibria

We study an intensively studied resource allocation game introduced by Koutsoupias and Papadimitriou where n weighted jobs are allocated to m identical machines. It was conjectured by Gairing et al. that the fully mixed Nash equilibrium is the worst Nash equilibrium for this game w. r. t. the expected maximum load over all machines. The known algorithms for approximating the so-called “price of anarchy” rely on this conjecture. We present a counter-example to the conjecture showing that fully mixed equilibria cannot be used to approximate the price of anarchy within reasonable factors. In addition, we present an algorithm that constructs so-called concentrated equilibria that approximate the worst-case Nash equilibrium within constant factors.