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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

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.

Shahar Dobzinski and Ariel Shaulker. Rigidity in Mechanism Design and Its Applications. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 44:1-44:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dobzinski_et_al:LIPIcs.ITCS.2023.44, author = {Dobzinski, Shahar and Shaulker, Ariel}, title = {{Rigidity in Mechanism Design and Its Applications}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {44:1--44:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.44}, URN = {urn:nbn:de:0030-drops-175479}, doi = {10.4230/LIPIcs.ITCS.2023.44}, annote = {Keywords: Revenue Maximization, Auctions} }

Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

A rapidly growing literature on lying in behavioral economics and psychology shows that individuals often do not lie even when lying maximizes their utility. In this work, we attempt to incorporate these findings into the theory of mechanism design.
We consider players that have a preference for truth-telling and will only lie if their benefit from lying is sufficiently larger than the loss of the others. To accommodate such players, we introduce α-moral mechanisms, in which the gain of a player from misreporting his true value, comparing to truth-telling, is at most α times the loss that the others incur due to misreporting. Note that a 0-moral mechanism is a truthful mechanism.
We develop a theory of moral mechanisms in the canonical setting of single-item auctions within the "reasonable" range of α, 0 ≤ α ≤ 1. We identify similarities and disparities to the standard theory of truthful mechanisms. In particular, we show that the allocation function does not uniquely determine the payments and is unlikely to admit a simple characterization. In contrast, recall that monotonicity characterizes the allocation function of truthful mechanisms.
Our main technical effort is invested in determining whether the auctioneer can exploit the preference for truth-telling of the players to extract more revenue comparing to truthful mechanisms. We show that the auctioneer can indeed extract more revenue when the values of the players are correlated, even when there are only two players. However, we show that truthful mechanisms are revenue-maximizing even among moral ones when the values of the players are independently drawn from certain identical distributions (e.g., the uniform and exponential distributions).
A by-product of our proof that optimal moral mechanisms are truthful is an alternative proof to Myerson’s optimal truthful mechanism characterization in the settings that we consider. We flesh out this approach by providing an alternative proof that does not involve moral mechanisms to Myerson’s characterization of optimal truthful mechanisms to all settings in which the values are independently drawn from regular distributions (not necessarily identical).

Shahar Dobzinski and Sigal Oren. Mechanism Design with Moral Bidders. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dobzinski_et_al:LIPIcs.ITCS.2022.55, author = {Dobzinski, Shahar and Oren, Sigal}, title = {{Mechanism Design with Moral Bidders}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {55:1--55:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.55}, URN = {urn:nbn:de:0030-drops-156513}, doi = {10.4230/LIPIcs.ITCS.2022.55}, annote = {Keywords: Mechanism Design, Cognitive Biases, Revenue Maximization} }