CHARACTERIZATION AND COMPUTATION OF NASH-EQUILIBRIA FOR AUCTIONS WITH INCOMPLETE INFORMATION

We consider a continuous sealed-bid auction model for an indivisible object with two bidders and incomplete information on both sides where the bidders' evaluations are assumed to be independently distributed on some real intervals. The price the winner (the highest bidder) has to pay is a given convex combination of the highest and the second highest (lowest) bid. It is shown that, for all but the second highest bid-price auction, all equilibrium-strategies are continuously differentiable and strictly monotonically increasing, and moreover, that the set of Nash-equilibria is completely characterized by a boundary value problem for a system of singular differential equations. In the case of symmetric data (independently and identically distributed true values) and for a particular class of asymmetric distributions (including uniform distributions), the boundary value problem is solved explicitly and uniquely.