Safely Learning Optimal Auctions: A Testable Learning Framework for Mechanism Design

When can the distributional assumptions of theorems and learning algorithms be trusted? Inspired by this question, Rubinfeld and Vasilyan (2023) initiated the study of testable learning. In this schema, we always learn one of the following two things: either we have achieved the desired accuracy regardless of whether the distributional assumptions are satisfied, or the input distribution does not satisfy the original distributional assumptions. Motivated by the challenge of relying on strong distributional assumptions in many theorems in mechanism design, we develop a testable learning framework for mechanism design. Traditional models in mechanism design assume that value distributions satisfy some notion of regularity. Unfortunately, testing regularity is not possible in the original testable learning framework as we show. To bypass this impossibility, we propose a regularized version of the testable learning framework. Under this framework, we always learn one of the following two things: either we achieve high revenue compared to the best possible revenue of any regular distribution close to the input distribution, or the input distribution does not satisfy regularity. We then use this framework to provide: 1) a tester-learner pair for revenue optimal mechanisms, 2) a tester for whether the fundamental Bulow-Klemperer Theorem (Bulow and Klemperer 1996) is applicable to a given dataset, and 3) a tester to confirm the existence of an anonymous reserve price that results in the anonymous price auction securing a constant fraction of the optimal revenue.

Many results in auction design assume that you know the exact shape of the bidders’ value distributions. These assumptions are often not met in the real world and cannot be checked in practice. This raises a key question: when can the distributional assumptions of theorems and learning algorithms be trusted? A framework called Testable Learning has been proposed to answer this question by either delivering the promised performance of the algorithm or warning you that the assumptions of the algorithm fail. In our paper, we adapt this framework to auction design, where the critical distributional assumption is called “regularity”. Under our new framework, we develop an algorithm that takes in an arbitrary distribution and either secures revenue that is close to the best possible for any nearby regular distribution or signals that the dataset is irregular. We also demonstrate how our algorithm can be utilized to verify when other important theorems in auction design, which rely on regularity, can be applied to data.