Price of Anarchy for Machines Models

This entry considers a selfish routing model formally introduced by Koutsoupias and Papadimitriou [10], in which the goal is to route the traffic on parallel links with linear latency functions. One can describe this model as a scheduling problem with m independent machines with speeds s1; : : : ; sm and n independent tasks with weights w1; : : : ;wn. The goal is to allocate the tasks to the machines to minimize the maximum load of the links in the system. It is assumed that all tasks are assigned by noncooperative agents. The set of pure strategies for task i is the set f1; : : : ; mg, and a mixed strategy is a distribution on this set. Given a combination .j1; : : : ; jn/ 2 f1; : : : ; mgn of pure strategies, one for each task, the cost for task i is P jkDji wk sji , which is the time needed for machine ji chosen by task i to complete all tasks allocated to that machine. Similarly, for a combination of pure strategies .j1; : : : ; jn/ 2 f1; : : : ; mgn, the load of machine j is defined as PjkDj wk sj . Given n tasks of length w1; : : : ;wn and m machines with the speeds s1; : : : ; sm, let opt denote the social optimum, that is, the minimum cost over all combinations of pure strategies: