Fixed-Price Approximations in Bilateral Trade

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}[(...