Inefficiency of multiplicative approximate Nash equilibrium for scheduling games

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 \(\theta \ge 0\).