The cost of selfishness for maximizing the minimum load on uniformly related machines

Consider the following scheduling game. A set of jobs, each controlled by a selfish agent, are to be assigned to m uniformly related machines. The cost of a job is defined as the total load on the machine that its job is assigned to. A job is interested in minimizing its cost, while the social objective is maximizing the minimum load (cover) over the machines. This goal is different from the regular makespan minimization goal, which was extensively studied in a game theoretic context. We study the price of anarchy (POA) and the price of stability (POS) for unrelated machines. We consider use the parameter smax, which is the maximum speed ratio between any two machines. We show that on related machines, POS is unbounded for smax > 2, and the POA is unbounded for smax ≥ 2. We study the remaining cases and show that while the POA tends to grows to infinity as smax tends to 2, the POS is at most 2 for any smax ≤ 2. Finally, we analyze the POA and POS form = 2.