  We consider a model of bilateral trade with private values. The value of the
buyer and the cost of the seller are jointly distributed. The true joint
distribution is unknown to the designer, however, the marginal distributions of
the value and the cost are known to the designer. The designer wants to find a
trading mechanism that is robustly Bayesian incentive compatible, robustly
individually rational, budget-balanced and maximizes the expected gains from
trade over all such mechanisms. We refer to such a mechanism as an optimal
robust mechanism. We establish equivalence between Bayesian incentive
compatible mechanisms (BIC) and dominant strategy mechanisms (DSIC). We
characterise the worst distribution for a given mechanism and use this
characterisation to find an optimal robust mechanism. We show that there is an
optimal robust mechanism that is deterministic (posted-price), dominant
strategy incentive compatible, and ex-post individually rational. We also
derive an explicit expression of the posted-price of such an optimal robust
mechanism. We also show the equivalence between the efficiency gains from the
optimal robust mechanism (max-min problem) and guaranteed efficiency gains if
the designer could choose the mechanism after observing the true joint
distribution (min-max problem).
