## 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 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=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 boundarylayer 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.

Original language | English |
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Pages (from-to) | 229-251 |

Number of pages | 23 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 75 |

Issue number | 1 |

DOIs | |

State | Published - 2015 |

## Keywords

- Asymmetric auctions
- Backward shooting
- Boundary value problems
- Boundary-layer theory
- First-price auctions
- Simulations
- Singular perturbations