  We study the performance of the Empirical Revenue Maximizing (ERM) mechanism
in a single-item, single-seller, single-buyer setting. We assume the buyer's
valuation is drawn from a regular distribution $F$ and that the seller has
access to {\em two} independently drawn samples from $F$. By solving a family
of mixed-integer linear programs (MILPs), the ERM mechanism is proven to
guarantee at least $.5914$ times the optimal revenue in expectation. Using
solutions to these MILPs, we also show that the worst-case efficiency of the
ERM mechanism is at most $.61035$ times the optimal revenue. These guarantees
improve upon the best known lower and upper bounds of $.558$ and $.642$,
respectively, of [Daskalakis & Zampetakis, '20].
