Coordination Mechanisms on Unrelated Machines with Arbitrary Priority Lists.

In job-scheduling games, each job is a selfish player that selects a machine to minimize its own completion time. Coordination mechanisms are employed to reduce the inefficiency of equilibria that result from such decentralized decision-making. This paper contributes to the extensive body of research on coordination mechanisms by investigating their application to unrelated parallel machines, where each machine may use its own scheduling policy to determine the processing order of assigned jobs. Since pure Nash equilibria (NE) are not guaranteed to exist in this setting, we identify and characterize several classes of instances-motivated by real-world applications-in which a NE is guaranteed to exist. For each such class, we design an algorithm to compute a NE, prove the convergence of best-response dynamics, and analyze the inefficiency of equilibria with respect to the makespan. In addition, we study two fundamental problems: (1) computing a NE schedule with low makespan, and (2) selecting, given a matrix of processing times, machine-specific scheduling policies that guarantee the existence of a NE with low makespan. For both problems, we establish computational hardness results.