Bayesian Nash Equilibrium in First Price Auction with Discrete Value Distribution

First price auction is widely used in government contract and industrial auctions. We study Bayesian Nash Equilibrium (BNE) in first price auction with discrete value distribution.We constructively prove the existence of BNE in first price auction and provide an algorithm to compute BNE at the same time. Moreover, we prove that BNE is unique. Some previous results of equilibrium in continuous value distribution do not suit for discrete value distribution. We cannot prove the uniqueness result in discrete case by using the uniqueness property in continuous case either. Furthermore, unlike the continuous case, we do not have the computation error when solving ordinary differential equations. Experiments show that our algorithm for discrete distribution is faster and more accurate than applying previous method on the continuous distribution which approximates the very discrete distribution. The results in this paper are derived in the asymmetric independent private values model, which assumes that the distributions of buyers’ valuations are common knowledge.

• Algorithms and Theory

• Economics and Electronic Commerce