Price of Anarchy of Multi-Stage Machine Scheduling Games

In this paper, we extend the discussion of the price of anarchy of machine scheduling games to a multi-stage machine setting. The multi-stage setting arises naturally in manufacturing pipelines and distributed computing workflows, when each job must traverse a fixed sequence of processing stages. While the classical makespan price of anarchy of $2 - \frac{1}{m}$ has been established for sequential scheduling on identical machines, the efficiency loss in multi-stage scheduling has, to the best of our knowledge, not been previously analyzed. We assume that each task follows a greedy strategy and gets assigned to the least-loaded machine upon arrival at each stage. Notably, we observe that in multi-stage environments, greedy behavior generally does not coincide with a subgame perfect Nash equilibrium. We continue with analyzing the equilibrium under greedy choices, since it is logical for modeling selfish agents with limited computational power, and may also model a central scheduler performing the common least-load scheduling heuristics. Under this model, we first show that in single-stage scheduling, greedy choice again yields an exact price of anarchy of $2 - \frac{1}{m}$. In multi-stage scheduling, we show that the completion time from one stage to the next increases by at most two times the maximum job execution time. Using this relationship, we derived the price of anarchy of multistage scheduling under greedy choices to lie within $[2 - \frac{1}{m}, 3 - \frac{1}{m}]$, where $m$ denote the maximum number of machines in one stage.