**Asymmetric auctions**

Gadi Fibich, Tel Aviv University

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.