On Infinite Separations Between Simple and Optimal Mechanisms

Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track

Bibtex Paper Supplemental


Alexandros Psomas, Ariel Schvartzman Cohenca, S. Weinberg


We consider a revenue-maximizing seller with $k$ heterogeneous items for sale to a single additive buyer, whose values are drawn from a known, possibly correlated prior $\mathcal{D}$. It is known that there exist priors $\mathcal{D}$ such that simple mechanisms --- those with bounded menu complexity --- extract an arbitrarily small fraction of the optimal revenue~(Briest et al. 2015, Hart and Nisan 2019). This paper considers the opposite direction: given a correlated distribution $\mathcal{D}$ witnessing an infinite separation between simple and optimal mechanisms, what can be said about $\mathcal{D}$?\citet{hart2019selling} provides a framework for constructing such $\mathcal{D}$: it takes as input a sequence of $k$-dimensional vectors satisfying some geometric property, and produces a $\mathcal{D}$ witnessing an infinite gap. Our first main result establishes that this framework is without loss: every $\mathcal{D}$ witnessing an infinite separation could have resulted from this framework. An earlier version of their work provided a more streamlined framework (Hart and Nisan 2013). Our second main result establishes that this restrictive framework is not tight. That is, we provide an instance $\mathcal{D}$ witnessing an infinite gap, but which provably could not have resulted from the restrictive framework. As a corollary, we discover a new kind of mechanism which can witness these infinite separations on instances where the previous ``aligned'' mechanisms do not.