Smooth inequalities and equilibrium inefficiency in scheduling games
Abstract
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 the paper, we focus on a general class of ℓk -norm (for parameter k) on job completion times as social cost, that permits to balance overall quality of service and fairness. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the ℓk-norm social cost. We consider strongly-local and local policies (the policies with different amount of knowledge about jobs). First, we study the inefficiency in ℓk-norm social costs of a strongly-local policy SPT that schedules the jobs non-preemptively in order of increasing processing times. We show that the price of anarchy of policy SPT is O(kk+1/k) and this bound is optimal (up to a constant) for all deterministic, non-preemptive, strongly-local and non-waiting policies (non-waiting policies produce schedules without idle times). Second, we consider the makespan (ℓ∞-norm) social cost by making connection within the ℓk-norm functions. We present 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.
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