First Price Auctions in the Asymmetric N Bidder Case

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Abstract
The first price auction is the auction procedure awarding the item to the highest bidder at the price equal to his bid. Much attention has been devoted to the two bidder case or to the symmetric case where the bidders' valuations are identically and independently distributed. We consider the general case where the valuations' distributions may be different. Furthermore, we allow an arbitrary number of bidders as well as mixed strategies. We show that every Bayesian equilibrium is an " essentially " pure equilibrium formed by bid functions whose inverses are solutions of a system of differential equations with boundary conditions. We then prove the existence of a Bayesian equilibrium. We prove the uniqueness of the equilibrium when the valuation distributions have a mass point at the lower extremity of the support. When every bidder's valuation distribution is one of two atomless distributions, we give assumptions under which the equilibrium is unique. The n-tuples of distributions that result from symmetric settings after some bidders have colluded satisfy these assumptions. We establish inequalities between equilibrium strategies when relations of stochastic dominance exist between valuation distributions.
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