Maximizing the minimum load: The cost of selfishness

Abstract

a b s t r a c tWe consider a scheduling problem on m machines, where each job is controlled by a selfish agent. Each agent is only interested in minimizing its own cost, defined as the total load of the machine that its job is assigned to. We consider the objective of maximizing the minimum load (the value of the cover) over the machines. Unlike the regular makespan minimization problem, which was extensively studied in a game-theoretic context, this problem has not been considered in this setting before.We study the price of anarchy (poa) and the price of stability (pos). 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. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, we show that the mixed poa grows exponentially with m for this problem.In addition, we consider a similar setting of selfish jobs with a different objective of minimizing the maximum ratio between the loads of any pair of machines in the schedule.We show that under this objective the pos is 1 and the pure poa is 2, for any m ≥ 2.

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

“…Epstein et al [8] first study the inefficiency of equilibria of machine covering games on uniform and identical 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.…”

Symmetry exploitation for Online Machine Covering with Bounded Migration

“…Chen et al [3] proved tight bounds for the approximability of jump-optimal solutions. Their result is stated in a game theoretical framework, where jump-optimal solutions are equivalent to pure Nash equilibria for the Machine Covering game (see for example [19]).…”

“…Regarding local search applied to load balancing problems, many neighborhoods have been studied such as Jump, Swap, Push and Lexicographical Jump in the context of makespan minimization on related machines [17], makespan minimization on restricted parallel machines [15], and also multi-exchange neighborhoods for makespan minimization on identical parallel machines [8]. For the case of machine covering, Chen et al [3] study the Jump neighborhood in a game-theoretical context, proving that every locally optimal solution is 1.7-approximate and that this factor is tight.…”

Symmetry Exploitation for Online Machine Covering with Bounded Migration

“…Regarding local search applied to load balancing problems, many neighborhoods have been studied such as Jump, Swap, Push and Lexicographical Jump in the context of makespan minimization on related machines [18], makespan minimization on restricted related machines [16], and also multi-exchange neighborhoods for makespan minimization on identical parallel machines [8]. For the case of machine covering, Chen et al [3] study the Jump neighborhood in a game-theoretical context, proving that every locally optimal solution is 1.7-approximate and that this factor is tight.…”

“…Chen et al [3] proved tight bounds for the approximability of jump-optimal solutions. Their result is stated in a game theoretical framework, where jump-optimal solutions are equivalent to pure Nash equilibria for the Machine Covering game (see for example [20]).…”