We study the problem of designing mechanisms when agents' valuation functions are drawn from unknown and correlated prior distributions. In particular, we are given a prior distribution $\D$, and we are interested in designing a (truthful) mechanism that has good performance for all ``true distributions'' that are close to $\D$ in Total Variation (TV) distance. We show that DSIC and BIC mechanisms in this setting are strongly robust with respect to TV distance, for any bounded objective function $\Ocal$, extending a recent result of Brustle et al. (\cite{Brustle2020}, EC 2020). At the heart of our result is a fundamental duality property of total variation distance. As direct applications of our result, we (i) demonstrate how to find approximately revenue-optimal and approximately BIC mechanisms for weakly dependent prior distributions; (ii) show how to find correlation-robust mechanisms when only ``noisy'' versions of marginals are accessible, extending recent results of Bei et. al. (\cite{bei2019correlation}, SODA 2019); (iii) prove that prophet-inequality type guarantees are preserved for correlated priors, recovering a variant of a result of D{\"u}tting and Kesselheim (\cite{Dutting19}, EC 2019); (iv) give a new necessary condition for a correlated distribution to witness an infinite separation in revenue between simple and optimal mechanisms, complementing recent results of Psomas et al. (\cite{psomas2022infinite}, NeurIPS 2022); (v) give a new condition for simple mechanisms to approximate revenue-optimal mechanisms for the case of a single agent whose type is drawn from a correlated distribution that can be captured by a Markov Random Field, complementing recent results of Cai and Oikonomou (\cite{Cai21}, EC 2021).

翻译：我们研究当代理人估值函数来自未知且相关的先验分布时的机制设计问题。具体而言，给定一个先验分布 $\D$，我们旨在设计一个（诚实的）机制，使其对于所有与 $\D$ 在总变差（TV）距离上接近的“真实分布”均具有良好性能。我们证明，在此设定下，对于任意有界目标函数 $\Ocal$，DSIC 机制和 BIC 机制关于 TV 距离具有强鲁棒性，这一结果推广了 Brustle 等人（\cite{Brustle2020}，EC 2020）的最新成果。我们结论的核心在于总变差距离的一个基本对偶性质。作为我们结论的直接应用，我们：（i）展示了如何为弱相关先验分布找到近似收益最优且近似 BIC 的机制；（ii）展示了当仅能获取边际分布的“含噪”版本时如何找到相关性鲁棒机制，推广了 Bei 等人（\cite{bei2019correlation}，SODA 2019）的最新结果；（iii）证明了先知不等式类型的保证对于相关先验分布仍然成立，恢复并扩展了 D{\"u}tting 和 Kesselheim（\cite{Dutting19}，EC 2019）的一个结果变体；（iv）为相关分布能够见证简单机制与最优机制之间收益无限分离提供了一种新的必要条件，补充了 Psomas 等人（\cite{psomas2022infinite}，NeurIPS 2022）的最新结果；（v）针对单一代理人类型由可被马尔可夫随机场捕获的相关分布抽取的情形，给出了简单机制近似收益最优机制的新条件，补充了 Cai 和 Oikonomou（\cite{Cai21}，EC 2021）的最新结果。