Myerson’s seminal characterization of the revenue-optimal auction for a single item remains a cornerstone of mechanism design. However, generalizing this framework to multi-item settings has proven exceptionally challenging. Even under restrictive assumptions, closed-form characterizations of optimal mechanisms are rare and are largely confined to the single-agent case, departing from the two-item setting only when prior distributions are uniformly distributed. In this work, we build upon the bi-valued setting introduced by Yao (EC 2017), where each item’s value has support 2 and lies in \(\{a,b\}\). Yao’s result provides the only known closed-form optimal mechanism for multiple agents. We extend this line of work along three natural axes, establishing the first closed-form optimal mechanisms in each of the following settings: (i) \(n\) i.i.d. agents and \(m\) i.i.d. items, (ii) \(n\) non-i.i.d. agents and two i.i.d. items, and (iii) \(n\) i.i.d. agents and two non-i.i.d. items. Our results lie at the limit of what is considered possible, since even with a single agent and \(m\) bi-valued non-i.i.d. items, finding the optimal mechanism is #P-Hard. We finally generalize the discrete analog of a result from Daskalakis et al. (Econometrica 2017), showing that for a single agent with \(m\) items drawn from arbitrary (non-identical) discrete distributions, grand bundling is optimal when all item values are sufficiently large. We further show that for any continuous product distribution, grand bundling achieves \(\mathsf{OPT} - \epsilon\) revenue for large enough values.

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).