  We study the problem of designing a two-sided market (double auction) to
maximize the gains from trade (social welfare) under the constraints of
(dominant-strategy) incentive compatibility and budget-balance. Our goal is to
do so for an unknown distribution from which we are given a polynomial number
of samples. Our first result is a general impossibility for the case of
correlated distributions of values even between just one seller and two buyers,
in contrast to the case of one seller and one buyer (bilateral trade) where
this is possible. Our second result is an efficient learning algorithm for one
seller and two buyers in the case of independent distributions which is based
on a novel algorithm for computing optimal mechanisms for finitely supported
and explicitly given independent distributions. Both results rely heavily on
characterizations of (dominant-strategy) incentive compatible mechanisms that
are strongly budget-balanced.
