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

Abstract

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 of 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 (the value of the 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 uniformly related machines. The results are expressed in terms of s, which is the maximum speed ratio between any two machines. For uniformly related machines, we prove that the pos is unbounded for s > 2, and the poa is unbounded for s ≥ 2. For the remaining cases we show that while the poa grows to infinity as s tends to 2, the pos is at most 2 for any s ≤ 2.

Maximizing the minimum load: The cost of selfishness

“…Similarly to the results of [11], one can argue that for any number of uniformly related machines these measures are unbounded.…”

“…This is very different from the results in the situation of the makespan minimization social goal, where the poa is finite [5,15]. A study of the cases where the speed ratio is no larger than 2 can be found in [11,20,30].…”

“…It is shown in [11] 1 that for this more general variant, even the pos is unbounded already for any number of machines m ≥ 2 with a maximum speed ratio larger than 2, and the poa is unbounded for a speed ratio of at least 2. This is very different from the results in the situation of the makespan minimization social goal, where the poa is finite [5,15].…”

“…These measures are unbounded already for two uniformly related machines [11], and therefore we focus on identical machines. We show that the pos is 1, and derive tight bounds on the pure poa for m ≤ 7 and on the overall pure poa, showing that its value is exactly 1.7.…”

Inefficiency of the Nash equilibrium for selfish machine covering on two hierarchical uniform machines

“…Chen et al [4] improve the results for the case with identical machines, where the PoS is exactly 1 and the overall PoA is exactly 1.7 for the case with m identical machines. Epstein et al [9] analyze the PoA and PoS for a special case of m uniform machines. Epstein et al [8] and Tan et al [19] completely solve the problem of measuring the inefficiency of equilibria on two uniform machines (see Fig.…”