Despite the complexity of the first price auction in the general asymmetric case, analytical results have started to emerge in the literature. Authors have also searched to gain insights by computing numerical estimates of the equilibria for some probability distributions of the valuations. This...

I consider the first price auction when the bidders' valuations may be differently distributed. I 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. I then...

We add a resale stage to standard auctions with two bidders. Bids are either kept secret or made public. Either the auction winner or the auction loser chooses the resale price. We characterize an infinity of equilibria of the second-price auction and a unique equilibrium of the first-price...

We deviate from the symmetric case of the independent private value model by allowing the bidders’ value distributions, which depend on parameters, to be slightly different. We show that previous results about the equality to the first-order in the parameters between revenues from the...

We study the first price auction game with an arbitrary number of bidders when the bidders' valuations are independent from each other. In technical words, we work within the "independent private value model." We show that if the supports of the valuation probability distributions have the same...

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...

If there is asymmetry among the bidders taking part in a first price auction, the equilibrium is not expost efficient. We consider a simple model of first price auction among two bidders where resale always occurs in case of inefficiency. We obtain mathematical formulas for the equilibrium...

We prove that, around the symmetric case, where the values are identically distributed, the equilibrium of the first price auction is jointly differentiable with respect to general bidder-specific parameters of the value distributions. We show that the revenue equivalence between the first-price...