Large asymmetric first-price auctions - A boundary-layer approach

The inverse equilibrium bidding strategies {v_i(b)}_i=1ⁿ 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 蠑 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)}_i=1ⁿ but not for {v_i(b)}_i=1ⁿ. (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 boundarylayer theory, we compute an O(1/n³) uniform approximation for {v_i(b)}_i=1ⁿ. The accuracy of the boundary-layer approximation is confirmed numerically, for both moderate and large values of n.