When Are Welfare Guarantees Robust?

TL;DR

This paper rigorously examines the robustness of welfare guarantees when bidders' valuations only approximately satisfy the gross substitutes condition, revealing limitations and conditions under which positive results still hold.

Contribution

It provides a formal analysis of the limitations of welfare guarantees under approximate gross substitutes valuations and offers positive results under additional structural assumptions.

Findings

Optimal social welfare cannot be approximated within a subpolynomial factor for approximate valuations.

Demand oracles cannot be simulated efficiently for approximate valuations.

Certain algorithms degrade gracefully under specific structural assumptions.

Abstract

Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders' valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values, that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS, and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations.