Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games

We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. In an egalitarian objective, the social cost would be the maximal job completion time, i.e. the makespan of the schedule. In an utilitarian objective, the social cost would be the average completion time. Instead of studying one of those objectives, we focus on the more general class of `k-norm (for some parameter k) on job completion times as social cost. This permits to balance overall quality of service and fairness. In this setting, a coordination mechanism is a fixed policy, which specifies how jobs assigned to a same machine will be scheduled. This policy is known to the players and influences therefore their behavior. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the `k-norm social cost. We consider policies with different amount of knowledge about jobs: non-clairvoyant (not depending on the job processing times), strongly-local (where the schedule of machine i depends only on processing times for this machine i and jobs j assigned to i) and local (where the schedule of machine i depends only on processing times for all machines i′ and jobs j assigned to i). The analysis relies on the smooth argument together with adequate inequalities, called smooth inequalities. With this unified framework, we are able to prove the following results. First, we study the inefficiency in `k-norm social costs of a strongly-local policy SPT that schedules the jobs non-preemptively in order of increasing processing times and a non-clairvoyant policy EQUI that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time. We show that the price of anarchy of policy SPT is O(k). We also prove a lower bound of Ω(k/ log k) for all deterministic, non-preemptive, strongly-local and non-waiting policies (non-waiting policies produce schedules without idle times). These results ensure that SPT is close to optimal with respect to the class of `k-norm social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price of anarchy O(2). Second, we consider the makespan (`∞-norm) social cost by making connection within the `k-norm functions. We revisit some local policies and provide simpler, unified proofs from the framework’s point of view. With the highlight of the approach, we derive a local policy Balance. This policy guarantees a price of anarchy of O(logm), which makes it the currently best known policy among the anonymous local policies that always admit a pure Nash equilibrium. ∗PRiSM, Université de Versailles St-Quentin-en-Yvelines, France †LIP6, Université Pierre et Marie Curie, France ‡IBISC, Université Evry Val d’Essonne, France 1 ar X iv :1 20 2. 43 02 v1 [ cs .G T ] 2 0 Fe b 20 12