A Simple Proof of the Uniqueness of the Equilibrium in First Price Auctions

There exists an explicit formula for the equilibrium strategies of the first price auction only in the symmetric case where the bidders’ valuations are distributed identically. The first price auction is considerably more difficult to investigate in the general case where the valuations may be distributed differently. Authors have searched to gain insights by computing numerical estimates of the equilibria for some valuations. Once they have found an equilibrium, the uniqueness of the equilibrium would spare these researchers the effort to look for other equilibria. Among the analytical results already available in the literature, there is a simple proof of the uniqueness when the lowest acceptable bid is a mass point of all valuation distributions, except possibly one. Under the assumption of strict log-concavity at the lowest acceptable bid, this paper provides a simple proof of the uniqueness in the general case where they may be 2 or more atomless valuation distributions