On the Robustness of Mechanism Design under Total Variation Distance

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 $\mathcal{O}$, extending a recent result of Brustle et al. ([BCD20], 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. ([BGLT19], 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 ([DK19], EC 2019) as a special case; (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. ([PSCW22], 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 ([CO21], EC 2021).