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 differ in many dimensions. This paper partially solves the problem in an auction setting with characteristics stochastically independent across bidders. The solution applies to the multidimensional versions of incentive contracts (Laffont and Tirole (1987) and Che (1993)) and nonlinear pricing (Armstrong (1996)). First, the paper proves that the multidimensionality requires that an optimal auction exclude a positive measure of bidders. Consequently, a standard auction without a reserve price or entrance fee is not optimal. Second, the paper obtains an explicit formula for optimal mechanisms, adopting the assumption of multiplicative separability from Armstrong (1996). Our optimal mechanism is almost equivalent to a Vickrey auction with a reserve price, except that the bids are ranked by an optimal scoring rule, which assigns scores to the multidimensional bids. This ``scoring-rule auction'' is optimal among all mechanisms if incentive compatibility constraints are non-binding (guaranteed by a hazard-rate assumption), and it is optimal among a smaller class of mechanisms if the constraints are binding. Our solution implies that an optimizing seller would induce downward distortion of a bid's nonmonetary provisions from the first-best configuration. Applied to multidimensional nonlinear pricing, our solution yields an explicit optimal pricing function.