The Sample Complexity of Up-to-$\varepsilon$ Multi-Dimensional Revenue Maximization

TL;DR

This paper investigates how many samples are needed to learn near-optimal revenue-maximizing auctions in complex multi-dimensional settings with multiple bidders, providing a polynomial sample complexity bound and extending previous results.

Contribution

It introduces a nonparametric approach that achieves polynomial sample complexity for learning approximately optimal auctions in general multi-dimensional settings, including non-subadditive valuations.

Findings

Polynomial sample complexity for learning near-optimal auctions.

Extension to general multi-dimensional valuation settings.

Exact incentive compatibility in single-bidder or single-good cases.

Abstract

We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of $n$ additive bidders whose values for $m$ heterogeneous items are drawn independently. For any such instance and any $\varepsilon>0$, we show that it is possible to learn an $\varepsilon$-Bayesian Incentive Compatible auction whose expected revenue is within $\varepsilon$ of the optimal $\varepsilon$-BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally, and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive, and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well-understood, our corollary for this case extends slightly the state-of-the-art.