The Price of Anarchy on Uniformly Related Machines Revisited

Recent interest in Nash equilibria led to a study of the price of anarchy (POA) and the strong price of anarchy (SPOA) for scheduling problems. The two measures express the worst case ratio between the cost of an equilibrium (a pure Nash equilibrium, and a strong equilibrium, respectively) to the cost of a social optimum. The atomic players are the jobs, and the delay of a job is the completion time of the machine running it, also called the load of this machine. The social goal is to minimize the maximum delay of any job, while the selfish goal of each job is to minimize its own delay, that is, the delay of the machine running it. We consider scheduling on uniformly related machines. While previous studies either consider identical speed machines or an arbitrary number of speeds, focusing on the number of machines as a parameter, we consider the situation in which the number of different speeds is small. We reveal a linear dependence between the number of speeds and the POA. For a set of machine of at most p speeds, the POA turns out to be exactly p + 1. The growth of the POA for large numbers of related machines is therefore a direct result of the large number of potential speeds. We further consider a well known structure of processors, where all machines are of the same speed except for one possibly faster machine. We investigate the POA as a function of both the speed ratio between the fastest machine and the number of slow machines.