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

Abstract

The inverse equilibrium bidding strategies $\{ v_i(b) \}_{i=1}^n$ 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 ($n\gg1$), 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 $\{ v_i^\prime(b) \}_{i=1}^n$ but not for $\{ v_i(b) \}_{i=1}^n$. (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 $\{ v_i(b) \}_{i=1}^n$. The accuracy of the boundary-layer approximation is confirmed numerically, for both moderate and large values of $n$.