Asymmetric First-Price Auctions: A Dynamical Systems Approach

28 Pages Posted: 14 Apr 2010

See all articles by Gadi Fibich

Gadi Fibich

Tel Aviv University - School of Mathematical Sciences

Nir Gavish

Technion-Israel Institute of Technology - Faculty of Mathematics

Date Written: January 4, 2010

Abstract

We introduce a new approach for analysis and numerical simulations of asymmetric first-price auctions, which is based on dynamical systems. We apply this approach to asymmetric auctions in which players’ valuations are power-law distributed, i.e., Fi(v) = v^{α_i} , i = 1, · · · , n. We utilize the dynamical-systems formulation to provide a simple proof of the existence and uniqueness of the equilibrium strategies in the case of two types of players. In the case of n types of players, the singularity point of the original system corresponds to a saddle point with n − 1 admissible directions of the dynamical system. This relation enables us to use forward solutions in the analysis and in the numerical simulations, in contrast with previous analytic and numerical studies which used backward solutions. The dynamical systems approach provides an intuitive explanation for why the standard backward-shooting method for computing the equilibrium strategies is inherently unstable, and enables us to devise a stable forward-shooting method. In particular, in the case of two types of players this method is extremely simple, as it does not require any shooting.

Keywords: asymmetric auctions, first-price auctions, dynamical systems, numerical methods

JEL Classification: C63, D44, C72, D82

Suggested Citation

Fibich, Gadi and Gavish, Nir, Asymmetric First-Price Auctions: A Dynamical Systems Approach (January 4, 2010). Available at SSRN: https://ssrn.com/abstract=1589726 or http://dx.doi.org/10.2139/ssrn.1589726

Gadi Fibich (Contact Author)

Tel Aviv University - School of Mathematical Sciences ( email )

Tel Aviv 69978
Israel

Nir Gavish

Technion-Israel Institute of Technology - Faculty of Mathematics ( email )

Haifa 32000
Israel
97248294181 (Phone)

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