Stretch in bottleneck games

In bottleneck congestion games the social cost is the worst congestion (bottleneck) on any resource, and each player selfishly minimizes the worst resource congestion in its strategy. We examine the price of anarchy with respect to the stretch which is a measure of variation in the resource utilization in the strategy sets of the players. The stretch is particularly important in routing problems since it compares the chosen path lengths with the respective shortest path lengths. We show that the price of anarchy in general bottleneck games is bounded by O(sm), where s is the stretch and m is the total number of resources. In linear bottleneck games, where the resource latencies are linearly proportional to the players' workloads, the price of anarchy is improved to O√sm). These bounds are asymptotically tight. For constant stretch linear games we obtain a Θ(√m) improvement over the previously best known bound.