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 design of 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, in: Proceedings of the 31st annual ACM symposium on theory of computing (STOC), 1999). One version of this problem, which includes a verification component, is studied by Koutsoupias (Theory Comput Syst 54(3):375–387, 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 by Koutsoupias (Theory Comput Syst 54(3):375–387, 2014) is upper bounded by a constant. This positive result asymptotically separates the average-case ratio from the worst-case ratio. It indicates that the optimal mechanism devised for a worst-case guarantee works well on average.
Highlights
The field of Algorithmic Mechanism Design [19, 20, 22] focuses on optimization problems where the input is provided by self-interested agents that participate in the mechanism by reporting their private information
The approximation ratio is the canonical measure for evaluating the performance of a truthful mechanism towards this goal
Deng et al [7] and Gao and Zhang [9] propose an alternative measure, the average-case approximation ratio, that compares the performance of the truthful mechanism against the optimal mechanism, averaged over all possible inputs when they follow a specific distribution
Summary
The field of Algorithmic Mechanism Design [19, 20, 22] focuses on optimization problems where the input is provided by self-interested agents that participate in the mechanism by reporting their private information. If it turns out to be a large value, one can hardly be certain about the mechanism’s performance as it may perform well on most inputs and perform poorly on only a few inputs To address this issue, Deng et al [7] and Gao and Zhang [9] propose an alternative measure, the average-case approximation ratio, that compares the performance of the truthful mechanism against the optimal mechanism, averaged over all possible inputs when they follow a specific distribution. The machines (alternatively, speaking of agents in game-theoretical settings) are rational and want to minimize their execution time They may achieve this by misreporting their processing times to the mechanism. It would be interesting to understand how well the optimal mechanism given in [11] performs on average when the instances are chosen from a certain distribution
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