  We study correlated equilibria and coarse equilibria of simple first-price
single-item auctions in the simplest auction model of full information. Nash
equilibria are known to always yield full efficiency and a revenue that is at
least the second-highest value. We prove that the same is true for all
correlated equilibria, even those in which agents overbid -- i.e., bid above
their values.
  Coarse equilibria, in contrast, may yield lower efficiency and revenue. We
show that the revenue can be as low as 26% of the second-highest value in a
coarse equilibrium, even if agents are assumed not to overbid, and this is
tight. We also show that when players do not overbid, the worst-case bound on
social welfare at coarse equilibrium improves from 63% of the highest value to
81%, and this bound is tight as well.
