Scheduling Games with Machine-Dependent Priority Lists

TL;DR

This paper studies scheduling games on related machines with machine-dependent priority lists, proving NP-hardness of equilibrium existence, characterizing special cases, and analyzing the efficiency and computation of Nash equilibria.

Contribution

It introduces a comprehensive analysis of scheduling games with machine-specific priorities, including complexity results, equilibrium existence conditions, algorithms, and inefficiency bounds.

Findings

NP-hardness of deciding pure Nash equilibrium existence

Existence of algorithms for certain classes of instances

Bounded inefficiency of Nash equilibria in specific cases

Abstract

We consider a scheduling game on parallel related machines, in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We prove that it is NP-hard to decide if a pure Nash equilibrium exists and characterize four classes of instances in which a pure Nash equilibrium is guaranteed to exist. For each of these classes, we give an algorithm that computes a Nash equilibrium, we prove that best-response dynamics converge to a Nash equilibrium, and we bound the inefficiency of Nash equilibria with respect to the makespan of the schedule and the sum of completion times. In addition, we show that although a pure Nash equilibrium is guaranteed to exist in instances with identical machines, it is NP-hard to approximate the best Nash equilibrium with respect to both objectives.