TY - JOUR
T1 - Large asymmetric first-price auctions - A boundary-layer approach
AU - Fibich, Gadi
AU - Gavish, Nir
N1 - Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.
PY - 2015
Y1 - 2015
N2 - 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.
AB - 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.
KW - Asymmetric auctions
KW - Backward shooting
KW - Boundary value problems
KW - Boundary-layer theory
KW - First-price auctions
KW - Simulations
KW - Singular perturbations
UR - http://www.scopus.com/inward/record.url?scp=84923859487&partnerID=8YFLogxK
U2 - 10.1137/140968811
DO - 10.1137/140968811
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AN - SCOPUS:84923859487
SN - 0036-1399
VL - 75
SP - 229
EP - 251
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 1
ER -