On Robustness to $k$-wise Independence of Optimal Bayesian Mechanisms

TL;DR

The paper studies how robust revenue-optimal Bayesian single-item auctions are to relaxations of mutual independence among bidders. It shows Myerson’s mechanism is not pairwise-robust but is 3-wise robust, highlighting a separation between pairwise and higher-order independence in nonidentical marginals, while Myerson’s auction under symmetric i.i.d. marginals and second-price with anonymous reserve (AR) exhibit stronger pairwise-robustness: AR is pairwise-robust across general marginals, and in the symmetric case, Myerson’s mechanism attains a constant-factor robustness bound of about 2.63. The authors develop a mixture of constructive counterexamples and intricate probabilistic-analytic arguments (including ex-ante relaxations and tail/core decompositions) to establish tight bounds; they also compare AR and Myerson across identical and nonidentical marginals, revealing practical guidance for robust mechanism design under limited correlation assumptions. Overall, the work clarifies when simple robust auctions can outperform revenue-optimized mechanisms under weaker independence, informing design choices in environments with uncertain correlation structures. The results implicate broader applicability of correlation-robust thinking in auction design and provide concrete constants capturing the price of correlation uncertainty.

Abstract

This paper reexamines the classic problem of revenue maximization in single-item auctions with $n$ buyers under the lens of the robust optimization framework. The celebrated Myerson's mechanism is the format that maximizes the seller's revenue under the prior distribution, which is mutually independent across all $n$ buyers. As argued in a recent line of work (Caragiannis et al. 22), (Dughmi et al. 24), mutual independence is a strong assumption that is extremely hard to verify statistically, thus it is important to relax the assumption. While optimal under mutual independent prior, we find that Myerson's mechanism may lose almost all of its revenue when the independence assumption is relaxed to pairwise independence, i.e., Myerson's mechanism is not pairwise-robust. The mechanism regains robustness when the prior is assumed to be 3-wise independent. In contrast, we show that second-price auctions with anonymous reserve, including optimal auctions under i.i.d. priors, lose at most a constant fraction of their revenues on any regular pairwise independent prior. Our findings draw a comprehensive picture of robustness to $k$-wise independence in single-item auction settings.

Figures (2)

• Figure 1: Curves in the proof of Theorem \ref{['thm: myersoniid']}
• Figure 2: Curves in the proof of Lemma \ref{['lem: reg_quantile']}

Theorems & Definitions (48)

• Definition 1
• Definition 2: $k$-wise-robustness
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Claim 3.1
• proof
• Lemma 4.1: caragiannis2022relaxing
• Lemma 4.2: boros1989closed