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

Gadi Fibich, Nir Gavish

Research output: Contribution to journalArticlepeer-review

Abstract

The inverse equilibrium bidding strategies {vi(b)}i=1n in a first-price auction with n asymmetric bidders, where vi 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 {vi(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 {vi(b)}i=1n. The accuracy of the boundary-layer approximation is confirmed numerically, for both moderate and large values of n.

Original languageEnglish
Pages (from-to)229-251
Number of pages23
JournalSIAM Journal on Applied Mathematics
Volume75
Issue number1
DOIs
StatePublished - 2015

Keywords

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

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