Average-case Approximation Ratio of Scheduling without Payments

TL;DR

This paper analyzes the average-case approximation ratio of a scheduling mechanism without payments, showing it is bounded by a constant under certain probabilistic assumptions, contrasting with its worst-case ratio.

Contribution

It introduces an average-case analysis for a scheduling mechanism, demonstrating a constant upper bound on the approximation ratio under i.i.d. cost distributions, unlike the worst-case bound.

Findings

Average-case ratio is bounded by a constant under i.i.d. costs.

Contrasts average-case performance with worst-case ratio of (n+1)/2.

Mechanism performs well on average despite poor worst-case performance.

Abstract

Apart from the principles and methodologies inherited from Economics and Game Theory, the studies in Algorithmic Mechanism Design typically employ the worst-case analysis and approximation schemes of Theoretical Computer Science. For instance, the approximation ratio, which is the canonical measure of evaluating how well an incentive-compatible mechanism approximately optimizes the objective, is defined in the worst-case sense. It compares the performance of the optimal mechanism against the performance of a truthful mechanism, for all possible inputs. In this paper, we take the average-case analysis approach, and tackle one of the primary motivating problems in Algorithmic Mechanism Design -- the scheduling problem [Nisan and Ronen 1999]. One version of this problem which includes a verification component is studied by [Koutsoupias 2014]. It was shown that the problem has a tight approximation ratio bound of (n+1)/2 for the single-task setting, where n is the number of machines. We show, however, when the costs of the machines to executing the task follow any independent and identical distribution, the average-case approximation ratio of the mechanism given in [Koutsoupias 2014] is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio, and indicates that the optimal mechanism for the problem actually works well on average, although in the worst-case the expected cost of the mechanism is Theta(n) times that of the optimal cost.