Asymmetric auctions
Gadi Fibich, Tel Aviv University

Abstract: 
In a first-price auction, the highest bidder wins the object and pays his bid.  When all bidders are symmetric, the equilibrium strategy can be calculated explicitly. In practice, however, bidders are typically asymmetric.  In that case, the asymmetric equilibrium strategies are the solutions of a non-standard, nonlinear boundary-value problem. This boundary-value problem is hard to analyze, or even to solve numerically.  As a result, very little is known about asymmetric first-price auctions, and more generally about asymmetric auctions.  For example, in the symmetric case, the Revenue Equivalence Theorem implies that all auction mechanisms are revenue equivalent. In contrast, revenue ranking of asymmetric auctions is an open problem.

In this talk I will first use a perturbation approach to derive an asymptotic Revenue Equivalence Theorem for asymmetric auctions. Then, I will show that asymmetric first-price auctions can be analyzed using dynamical systems. This approach will reveal the ``nature of non-uniqueness" of the model, and lead to an existence and uniqueness proof. I will then show that the standard numerical method for this problem is inherently unstable, and suggest an alternative numerical method, which is stable.  

This is joint work with Nir Gavish, Arieh Gavious and Aner Sela.