Assortment Auctions: A Myersonian Characterization for Markov Chain based Choice Models
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We introduce the assortment auction optimization problem, defined as follows. A seller has a set of substitute products with exogenously-given prices. Each buyer has a ranked list from which she would like to purchase at most one product. The buyers report their lists to the seller, who then allocates products to the buyers using a truthful mechanism, subject to constraints on how many products can be allocated. The seller collects revenues equal to the prices of the products allocated, and would like to design an auction to maximize total revenue, when the buyers' lists are drawn independently from known distributions. If there is a single buyer, then our problem reduces to the assortment optimization problem, which is solved for Markov Chain choice models. We extend this result and compute the optimal assortment auction when each buyer's list distribution arises from its own Markov chain. Moreover, we show that the optimal auction is structurally “Myersonian”, in that each buyer is assigned a virtual valuation based on her list and Markov chain, and then the mechanism maximizes virtual surplus. Since Markov Chain choice models capture valuation distributions, our optimal assortment auction generalizes the classical Myerson's auction. Markov chains also capture the commonly used MNL choice model. We show that without the Markov chain assumption, the optimal assortment auction may be structurally non-Myersonian. Finally, we apply the concept of an assortment auction in online assortment problems. We show that any personalized assortment policy is a special case of a truthful assortment auction, and that moreover, the optimal auction provides a tighter relaxation for online policies than the commonly-used “deterministic LP”. Using this fact, we improve many online assortment policies, and derive the first approximation guarantees that strictly exceed 1-1/e.
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