# An Improved Welfare Guarantee for First Price Auctions

## TL;DR

This paper improves the theoretical welfare guarantee for first price auctions in Bayesian settings, showing they achieve at least 74.3% of the optimal welfare, surpassing previous bounds and using a novel analysis approach.

## Contribution

It provides a tighter welfare guarantee for first price auctions by employing a new analysis that leverages agents' independent value distributions, moving beyond smoothness techniques.

## Key findings

- Welfare of first price auctions is at least 0.743 of optimal in Bayes-Nash equilibrium.
- Previous bound was approximately 0.63, now improved to 0.743.
- Empirical examples show actual welfare can reach about 0.869 of optimal.

## 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 $1-1/e \approx .63$, derived in Syrgkanis and Tardos (2013) using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example (from Hartline et al. (2014)), the first price auction achieves a $\approx .869$-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the $1-1/e \approx .63$ 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.