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Smoothed and average-case approximation ratios of mechanisms: beyond the worst-case analysis

Smoothed and average-case approximation ratios of mechanisms: beyond the worst-case analysis
Smoothed and average-case approximation ratios of mechanisms: beyond the worst-case analysis
The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves.
These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.
1-15
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Deng, Xiaotie
772c0705-a735-43dc-8988-f5c527572574
Gao, Yansong
203d3e45-b3e2-4e34-8632-c10ac44d6ccd
Zhang, Jie
6bad4e75-40e0-4ea3-866d-58c8018b225a
Deng, Xiaotie
772c0705-a735-43dc-8988-f5c527572574
Gao, Yansong
203d3e45-b3e2-4e34-8632-c10ac44d6ccd
Zhang, Jie
6bad4e75-40e0-4ea3-866d-58c8018b225a

Deng, Xiaotie, Gao, Yansong and Zhang, Jie (2017) Smoothed and average-case approximation ratios of mechanisms: beyond the worst-case analysis. In 42nd International Symposium on Mathematical Foundations of Computer Science. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 15 pp, pp. 1-15. (doi:10.4230/LIPIcs.MFCS.2017.16).

Record type: Conference or Workshop Item (Paper)

Abstract

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves.
These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.

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Accepted/In Press date: 12 June 2017
e-pub ahead of print date: 22 November 2017
Organisations: Agents, Interactions & Complexity

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Local EPrints ID: 411697
URI: https://eprints.soton.ac.uk/id/eprint/411697
PURE UUID: ff4e0ae3-f667-49bf-b46e-5e2f296680e0
ORCID for Jie Zhang: ORCID iD orcid.org/0000-0003-1380-9952

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Date deposited: 22 Jun 2017 16:31
Last modified: 06 Jun 2018 12:14

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Contributors

Author: Xiaotie Deng
Author: Yansong Gao
Author: Jie Zhang ORCID iD

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