Are Gross Substitutes a Substitute for Submodular Valuations?

TL;DR

This paper investigates the relationship between gross substitutes and submodular valuations, revealing limitations in approximating submodular functions with GS and introducing new symmetrization techniques.

Contribution

It proves that certain submodular valuations cannot be well-approximated by GS, and introduces a novel symmetrization operation that preserves GS properties.

Findings

Submodular valuations can be hard to approximate by GS within a logarithmic factor.

A new symmetrization operation that preserves GS is introduced.

Approximation bounds are established for a broader class called concave of Rado valuations.

Abstract

The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of {\em complement free} valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: $GS \subsetneq Submodular \subsetneq XOS \subsetneq Subadditive$. 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. 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 $\Omega(\frac{\log m}{\log\log m})$, 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. Additionally, for a class of submodular functions that we refer to as {\em concave of Rado} valuations (this class greatly extends budget additive valuations), we show approximability by GS within an $O(\log^2m)$ factor.