  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.
