Optimal Auctions with Restricted Allocations

Abstract

We study the problem of designing optimal auctions under restrictions on the set of permissible allocations. In addition to allowing us to restrict to deterministic mechanisms, we can also indirectly model non-additive valuations. We prove a strong duality result, extending a result due to Daskalakis et al. [2015], that guarantees the existence of a certificate of optimality for optimal restricted mechanisms. As a corollary of our result, we provide a new characterization of the set of allocations that the optimal mechanism may actually use. To illustrate our result we find and certify optimal mechanisms for four settings where previous frameworks do not apply, and provide new economic intuition about some of the tools that have previously been used to find optimal mechanisms.

Bounding the menu-size of approximately optimal auctions via optimal-transport duality

“…We note that both Pavlov (2011) and Kash and Frongillo (2016) conjecture that the conclusion of Theorem 6 holds under more general conditions than the McAfee-McMillan hazard condition. An affirmation of (either of) these conjectures would, by the above proof, immediately imply that the conclusion of Theorem 4 holds under the same generalized assumptions.…”

“…We now present only the minimal amount of detail from the extensive analysis of that is required to follow our proof; the interested reader is referred to their paper or to the excellent survey of Daskalakis (2015), whose notation we follow, for the full details that lie beyond the scope of this paper. (See also Giannakopoulos and Koutsoupias (2014) for a slightly different duality approach, and Kash and Frongillo (2016) for an extension.) In their analysis, identify a signed Radon measure 5 µ on [0, 1] 2 5 To understand our proof there is no need to be familiar with the general definition of a signed Radon measure.…”

On optimal mechanisms in the two-item single-buyer unit-demand setting

“…In Section 2, we first formulate an optimization problem under the unit-demand setting. We next compute its dual using a result in [25], and solve it for three representative examples of (c, b 1 , b 2 ). The main purpose behind these examples is to bring out the variety in structure, and therefore the difficulty in guessing and computing, the dual measure for more general settings.…”

“…(i) We identify the dual to the problem of optimal auction in the restricted unit-demand setting, using a result in [25] 1 . We then argue that the computation of the dual measure in the unit-demand setting using the approach of optimal transport in [18] 1.2], and show that the optimal dual variable differs significantly with variation in c, thus making it hard to discover the correct dual measure.…”

“…[25, Cor. 4.1] Let u * and γ * be feasible for the aforementioned primal (4) and dual (5) problems, respectively.…”