  We consider the bilateral trade problem, in which two agents trade a single
indivisible item. It is known that the only dominant-strategy truthful
mechanism is the fixed-price mechanism: given commonly known distributions of
the buyer's value $B$ and the seller's value $S$, a price $p$ is offered to
both agents and trade occurs if $S \leq p \leq B$. The objective is to maximize
either expected welfare $\mathbb{E}[S + (B-S) \mathbf{1}_{S \leq p \leq B}]$ or
expected gains from trade $\mathbb{E}[(B-S) \mathbf{1}_{S \leq p \leq B}]$.
  We improve the approximation ratios for several welfare maximization variants
of this problem. When the agents' distributions are identical, we show that the
optimal approximation ratio for welfare is $\frac{2+\sqrt{2}}{4}$. With just
one prior sample from the common distribution, we show that a
$3/4$-approximation to welfare is achievable. When agents' distributions are
not required to be identical, we show that a previously best-known
$(1-1/e)$-approximation can be strictly improved, but $1-1/e$ is optimal if
only the seller's distribution is known.
