A tighter welfare guarantee for first-price auctions

Abstract

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.