If the value cumulative distribution functions are log-concave at the highest lower extremity of their supports of the first-price auction in the asymmetric indepent private values model.

We show that Zheng (2002)’s optimal mechanism in the presence of resale can be interpreted as an equilibrium of an ascending-price auction and, in the two-bidder case, as an equilibrium with a no-regret property of the English and second-price auctions. We also show that it can be extended...

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...

An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two-step functions that depend on a single parameter. We characterize the symmetric Bayesian equilibrium and prove its existence and...

In the presence of resale and under more general assumptions than Zhengʼs (2002), implementation of Myersonʼs (1981) optimal allocation with two bidders is achieved through the second-price auction. In a special class of asymmetric n-bidder models, it is achieved through the English auction.

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...