Abstract: This paper studies scheduling games with potential penalties on the move of jobs. There are a set of machines and a set of jobs. Each job could choose one machine and be processed by the chosen one. Jobs that try to change their original choice will incur additional costs, which is proportional to its processing time with proportional parameter $\delta \geqslant 0$. A schedule is a $\delta$-NE if no job has the incentive to change its choice unilaterally. The $\delta$-NE is a generation of Nash equilibrium, and its inefficiency can be measured by the $\delta$-PoA, which is also a generalization of the Price of Anarchy. For the game with the social cost of minimizing the makespan, the exact $\delta$-PoA for any number of machines and any $\delta \geqslant 0$ is obtained. For the game with the social cost of maximizing the minimum machine load, an upper bound on the $\delta$-PoA is presented, and tight bounds are given for $2 \leqslant m \leqslant 11$ and any $\delta \geqslant 0$, where $m$ is the number of machines.

Key words: Scheduling game, Price of Anarchy, Nash equilibrium, Parallel machine