We consider a scheduling game on parallel related machines, in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We prove that it is NP-hard to decide if a pure Nash equilibrium exists and characterize four classes of instances in which a pure Nash equilibrium is guaranteed to exist. For each of these classes, we give an algorithm that computes a Nash equilibrium, we prove that best-response dynamics converge to a Nash equilibrium, and we bound the inefficiency of Nash equilibria with respect to the makespan of the schedule and the sum of completion times. In addition, we show that although a pure Nash equilibrium is guaranteed to exist in instances with identical machines, it is NP-hard to approximate the best Nash equilibrium with respect to both objectives.

翻译：我们考虑并行相关机器上的调度博弈，其中作业通过选择处理机器来最小化其完成时间。每台机器使用独立的优先级列表来决定其上作业的处理顺序。我们证明判定纯纳什均衡是否存在是NP-难问题，并刻画了保证存在纯纳什均衡的四类实例。针对每一类实例，我们给出计算纳什均衡的算法，证明最优响应动态收敛至纳什均衡，并分别从调度完工时间和完成时间总和两个维度界定了纳什均衡效率损失的上界。此外，我们证明尽管在相同机器实例中纯纳什均衡必然存在，但针对这两个目标函数近似求解最优纳什均衡仍是NP-难的。