On Infinite Separations Between Simple and Optimal Mechanisms
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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 𝒟. It is known that there exist priors 𝒟 such that simple mechanisms – those with bounded menu complexity – extract an arbitrarily small fraction of the optimal revenue. This paper considers the opposite direction: given a correlated distribution 𝒟 witnessing an infinite separation between simple and optimal mechanisms, what can be said about 𝒟? Previous work provides a framework for constructing such 𝒟: it takes as input a sequence of k-dimensional vectors satisfying some geometric property, and produces a 𝒟 witnessing an infinite gap. Our first main result establishes that this framework is without loss: every 𝒟 witnessing an infinite separation could have resulted from this framework. Even earlier work provided a more streamlined framework. Our second main result establishes that this restrictive framework is not tight. That is, we provide an instance 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.
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