Large Asymmetric First-Price Auctions - A Boundary-Layer Approach

The inverse equilibrium bidding strategies {v i (b)} n i=1 in a first-price auction with n asymmetric bidders, where v i is the value of bidder i and b is the bid, are solutions of a system of n first-order ordinary differential equations, with 2n boundary conditions and a free boundary on the right. In this study we show that when the number of bidders is large (n 1), this problem has a boundary-layer structure with several nonstandard features: (1) The small parameter does not multiply the highest-order derivative. (2) The number of equations goes to infinity as the small parameter goes to zero. (3) The boundary-layer structure is for the derivatives {v i (b)} n i=1 but not for {v i (b)} n i=1. (4) In the boundary-layer region, the solution is the sum of an outer solution in the original variable and an inner solution in the rescaled boundary-layer variable. Using boundary-layer theory, we compute an O(1/n 3) uniform approximation for {v i (b)} n i=1. The accuracy of the boundary-layer approximation is confirmed numerically, for both moderate and large values of n. 1. Introduction. Auction is an important economic mechanism, which is central to the modern economy. For example, in 2013 the US treasury auctioned securities in a total sum of 7.9 trillion dollars. Google makes most of its profits by selling sponsored links via online auctions. The first systematic analysis of auctions was done in 1961 by Vickrey [15]. Since then, auctions have been the subject of an intense study. In this study we analyze a boundary value problem that arises in the study of asymmetric first-price private-value auctions, in which n risk-neutral bidders compete for a single object. Each of the bidders submits his bid in a closed envelope; the highest bidder wins the object and pays his bid, while all other bidders pay nothing. In this case, the inverse equilibrium bidding strategies {v i (b)}