In machine scheduling, a set of
jobs must be scheduled on a set of
machines. Each job
incurs a processing time of
and the goal is to schedule jobs so as to minimize some global objective function, such as the maximum makespan of the schedule considered in this paper. Often in practice, each job is controlled by an independent selfish agent who chooses to schedule his job on machine which minimizes the (expected) completion time of his job. This scenario can be formalized as a game in which the players are job owners; the strategies are machines; and the disutility to each player in a strategy profile is the completion time of his job in the corresponding schedule (a player’s objective is to minimize his disutility). The equilibria of these games may result in larger-than-optimal overall makespan. The ratio of the worst-case equilibrium makespan to the optimal makespan is called the
price of anarchy
of the game. In this paper, we design and analyze scheduling policies, or
, for machines which aim to minimize the price of anarchy (restricted to pure Nash equilibria) of the corresponding game. We study coordination mechanisms for four classes of multiprocessor machine scheduling problems and derive upper and lower bounds for the price of anarchy of these mechanisms. For several of the proposed mechanisms, we also are able to prove that the system converges to a pure Nash equilibrium in a linear number of rounds. Finally, we note that our results are applicable to several practical problems arising in networking.