“Near Optimality of Second Price Mechanisms in a Class of Asymmetric Auctions,” V. Mares & J. Swinkels (2010)

Auctions are often asymmetric, in that some bidders are preferred by the seller/buyer. For instance, government procurement often wishes to give a “bonus” to minority-headed firms, and farm produce purchasers might wish to penalize growers who are shipping from a distance. The optimal mechanism in asymmetric auctions, however, is often quite complicated and hence difficult to implement. Mares and Swinkels show that the “virtual surplus” function common in auction theory has a tight link with a concept called rho-concavity which measures the degree of concavity of a function at a given point. They then propose a simple mechanism – choose a scale q in (0,1] and give a winning bidder a bonus of qD(i), where D(i) is a measure of how much the seller “prefers” agent i. Depending on what properties the distribution of bidder costs has, the simple mechanism can retain a surprisingly high percentage of the seller surplus compared to the optimal mechanism; for instance, if the density of costs c is monotonically increasing, then the simple mechanism generates at least 75% of the buyer surplus that the optimal mechanism generates vis-a-vis a nondiscriminatory auction. That is, the simple mechanism will always beat a symmetric auction given some very mild assumptions, and this result holds even when the auctioneer does not know the true cost distribution, but rather only knows it lies in some class. This suggests that auction designers may want to consider asymmetric auctions, since though they seem difficult to implement, their simple analogues are still vast improvements on optimal mechanisms that treat all bidders the same.

http://www.kellogg.northwestern.edu/faculty/mares/htm/personal/A2.pdf

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