First price auctions in the asymmetric N bidder case

I consider the first price auction when the bidders' valuations may be diff erently distributed. I show that every Bayesian equilibrium is an 'essentia lly' pure equilibrium formed by bid functions whose inverses are solutions of a system of differential equations with boundary conditions. I then prov e the existence of an equilibrium. I prove its uniqueness when the valuatio n distributions have a mass point at the lower extremity of the support. I give sufficient conditions for uniqueness when every valuation distribution is one of two atomless distributions. I establish inequalities between equ ilibrium strategies when relations of stochastic dominance exist between va luation distributions.