Are Gross Substitutes a Substitute for Submodular Valuations

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The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of complement free valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: GS gSS Submodular gSS XOS gSS Subadditive$. The GS class has always been more enigmatic than its counterpart classes, both in its definition and in its relation to the other classes. For example, while it is well understood how closely the Submodular, XOS and Subadditive classes (point-wise) approximate one another, approximability of these classes by GS remained wide open. In particular, the largest gap known between Submodular and GS valuations was some constant ratio smaller than 2. Our main result is the existence of a submodular valuation (one that is also budget additive) that cannot be approximated by GS within a ratio better than $Ømega(łog m/łogłog m), where m is the number of items. En route, we uncover a new symmetrization operation that preserves GS, which may be of independent interest. We show that our main result is tight with respect to budget additive valuations. However, whether GS approximates general submodular valuations within a poly-logarithmic factor remains open, even in the special case of concave of GS valuations (a subclass of Submodular containing budget additive). For concave of Rado valuations (Rado is a significant subclass of GS, containing, e.g., weighted matroid rank functions and OXS), we show approximability by GS within an O(łog2m) factor.

Original languageEnglish
Title of host publicationEC 2021 - Proceedings of the 22nd ACM Conference on Economics and Computation
Number of pages19
ISBN (Electronic)9781450385541
StatePublished - 18 Jul 2021
Event22nd ACM Conference on Economics and Computation, EC 2021 - Virtual, Online, Hungary
Duration: 18 Jul 202123 Jul 2021

Publication series

NameEC 2021 - Proceedings of the 22nd ACM Conference on Economics and Computation


Conference22nd ACM Conference on Economics and Computation, EC 2021
CityVirtual, Online


  • gross substitutes
  • valuation functions

All Science Journal Classification (ASJC) codes

  • Computational Theory and Mathematics
  • Computer Networks and Communications


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