  We study the classic bilateral trade setting. Myerson and Satterthwaite show
that there is no Bayesian incentive compatible and budget-balanced mechanism
that obtains the gains from trade of the first-best mechanism. Consider the
random-offerer mechanism: with probability $\frac{1}{2}$ run the
\emph{seller-offering} mechanism, in which the seller offers the buyer a
take-it-or-leave-it price that maximizes the expected profit of the seller, and
with probability $\frac{1}{2}$ run the \emph{buyer-offering} mechanism. Very
recently, Deng, Mao, Sivan, and Wang showed that the gains from trade of the
random-offerer mechanism is at least a constant factor of $\frac 1
{8.23}\approx 0.121$ of the gains from trade of the first best mechanism.
Perhaps a natural conjecture is that the gains-from-trade of the random-offerer
mechanism, which is known to be at least half of the gains-from-trade of the
second-best mechanism, is also at least half of the gains-from-trade of the
first-best mechanism. However, in this note we exhibit distributions such as
the gains-from trade of the random-offerer mechanism is smaller than a
$0.495$-fraction of the gains-from-trade of the first-best mechanism.
