Rigidity in Mechanism Design and Its Applications

Shahar Dobzinski, Ariel Shaulker

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We introduce the notion of rigidity in auction design and use it to analyze some fundamental aspects of mechanism design. We focus on the setting of a single-item auction where the values of the bidders are drawn from some (possibly correlated) distribution F. Let f be the allocation function of an optimal mechanism for F. Informally, S is (linearly) rigid in F if for every mechanism M with an allocation function f where f and f agree on the allocation of at most x-fraction of the instances of S, it holds that the expected revenue of M is at most an x fraction of the optimal revenue. We start with using rigidity to explain the singular success of Cremer and McLean's auction assuming interim individual rationality. Recall that the revenue of Cremer and McLean's auction is the optimal welfare if the distribution obeys a certain “full rank” conditions, but no analogous constructions are known if this condition does not hold. We show that the allocation function of the Cremer and McLean auction has logarithmic (in the size of the support) Kolmogorov complexity, whereas we use rigidity to show that there exist distributions that do not obey the full rank condition for which the allocation function of every mechanism that provides a constant approximation is almost linear. We further investigate rigidity assuming different notions of individual rationality. Assuming ex-post individual rationality, if there exists a rigid set then the structure of the optimal mechanism is relatively simple: the player with the highest value “usually” wins the item and contributes most of the revenue. In contrast, assuming interim individual rationality, there are distributions with a rigid set S where the optimal mechanism has no obvious allocation pattern (in the sense that its Kolmogorov complexity is high). Since the existence of rigid sets essentially implies that the hands of the designer are tied, our results help explain why we have little hope of developing good, simple and generic approximation mechanisms in the interim individual rationality world.

Original languageEnglish
Title of host publication14th Innovations in Theoretical Computer Science Conference, ITCS 2023
EditorsYael Tauman Kalai
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772631
DOIs
StatePublished - 1 Jan 2023
Event14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States
Duration: 10 Jan 202313 Jan 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume251
ISSN (Print)1868-8969

Conference

Conference14th Innovations in Theoretical Computer Science Conference, ITCS 2023
Country/TerritoryUnited States
CityCambridge
Period10/01/2313/01/23

All Science Journal Classification (ASJC) codes

  • Software

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