An improved bound for the price of anarchy for related machine scheduling

In this paper, we introduce an improved upper bound for the efficiency of Nash equilibria in utilitarian scheduling games on related machines. The machines have varying speeds and adhere to the shortest processing time first policy. The goal of each job is to minimize its completion time, while the social objective is to minimize the sum of completion times. Our main finding establishes an upper bound of 2-1/(4m-2) on the price of anarchy for the general case of m machines. We improve this bound to 3/2 for the case of two machines, and to 2-1/(2 m) for the general case of m machines when the machines have divisible speeds, i.e., if the speed of each machine is divisible by the speed of any slower machine.