Optimal Auction in a Multidimensional World
A long-standing unsolved problem, often arising from auctions with multidimensional bids, is how to design seller-optimal auctions when bidders' private characteristics ("types") differ in many dimensions. This paper solves the problem, assuming bidder-types stochastically independent across bidders. First, it proves that in any optimal auction, with positive probability, the object is not sold. Thus, a standard auction without a reserve price is not optimal. Second, and more importantly, the paper obtains an explicit formula for optimal auctions in a class of environments. The optimal mechanism is almost equivalent to a Vickrey auction with reserve price, except that the bids are ranked by an optimal scoring rule, which assigns scores to the multidimensional bids. When the hazard rate of a statistic of bidder-types is monotone, this auction is optimal among all "scoring mechanisms," where a winner chooses a multidimensional payment bundle subject to a type-specific rule. Our optimal auction implies that an optimizing seller would not evaluate bid by her own preferences; instead, she would induce downward distortion of nonmonetary provisions from the first-best configuration. Applied to multidimensional nonlinear pricing, our design of optimal auction yields an explicit optimal pricing function.
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