On the quality and complexity of pareto equilibria in the job scheduling game

In the well-known scheduling game, a set of jobs controlled by selfish players wishes each to minimize the load of the machine on which it is executed, while the social goal is to minimize the makespan, that is, the maximum load of any machine. We consider this problem on the three most common machines models, identical machines, uniformly related machines and unrelated machines, with respect to both weak and strict Pareto optimal Nash equilibria. These are kinds of equilibria which are stable not only in the sense that no player can improve its cost by changing its strategy unilaterally, but in addition, there is no alternative choice of strategies for the entire set of players where no player increases its cost, and at least one player reduces its cost (in the case of strict Pareto optimality), or where all players reduce their costs (in the case of weak Pareto optimality). We give a complete classification of the social quality of such solutions with respect to an optimal solution, that is, we find the Price of Anarchy of such schedules as a function of the number of machines, m. In addition, we give a full classification of the recognition complexity of such schedules.