This paper studies the inefficiency of multiplicative approximate Nash Equilibrium for scheduling games. There is a set of machines and a set of jobs. Each job could choose one machine and be processed by the chosen one. A schedule is a \(\theta \)-NE if no player has the incentive to deviate so that it decreases its cost by a factor larger than \(1+\theta \). The \(\theta \)-NE is a generation of Nash Equilibrium and its inefficiency can be measured by the \(\theta \)-PoA, which is also a generalization of the Price of Anarchy. For the game with the social cost of minimizing the makespan, the exact \(\theta \)-PoA for any number of machines and any \(\theta \ge 0\) is obtained. For the game with the social cost of maximizing the minimum machine load, we present upper and lower bounds on the \(\theta \)-PoA. Tight bounds are provided for cases where the number of machines is between 2 and 7 and for any \(\theta \ge 0\).

本文研究了乘法近似纳什均衡在安排比赛方面的低效率。有一组机器和一组作业。每个作业可以选择一台机器并由所选机器进行处理。如果没有玩家有动机偏离，以至于它以大于 \（1+\theta \） 的系数降低其成本，那么赛程就是 \（\theta \）-NE。\（\theta \）-NE 是纳什均衡的一代，它的低效率可以用 \（\theta \）-PoA 来衡量，这也是无政府状态价格的推广。对于具有最小化 makespan 的社会成本的游戏，可以获得任意数量的机器和任何 \（\theta \ge 0\） 的精确 \（\theta \） -PoA。对于具有最大化最小机器负载的社会成本的游戏，我们提出了 \（\theta \）-PoA 的上限和下限。对于机器数量在 2 到 7 之间以及任何 \（\theta \ge 0\） 的情况，都提供了严格的边界。