The menu complexity of "one-and-a-half-dimensional" mechanism design

Raghuvansh R. Saxena, Ariel Schvartzman, S. Matthew Weinberg

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Scopus citations

Abstract

We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-halfdimensional" setting where a single bidder has both a value and a deadline for receiving an item [FGKK16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN13]. We show the following when the bidder has n possible deadlines: Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity ≥ 2n-1. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK16].Fully polynomial menu complexity is neces-sary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1-n)-approximation to the optimal revenue with menu complexity O(n3=2 q minfn;ln(vmax)g) = O(n2/n), where vmax denotes the largest value in the support of integral distributions. There exist instances where any mechanism guaranteeing a multiplicative (1- O(1=n2))-approximation to the optimal revenue requires menu complexity (n2). Our main technique is the polygon approximation of concave functions [Rot92], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW17] on the menu complexity of optimal auctions for a budget-constrained buyer.

Original languageEnglish (US)
Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
EditorsArtur Czumaj
PublisherAssociation for Computing Machinery
Pages2026-2035
Number of pages10
ISBN (Electronic)9781611975031
DOIs
StatePublished - 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
Duration: Jan 7 2018Jan 10 2018

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
CountryUnited States
CityNew Orleans
Period1/7/181/10/18

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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