The Exact Approximation Ratio of the Optimal Fixed-Price Mechanism in Bilateral Trade

Abstract

In bilateral trade, a seller owns an item that a buyer may value more, and their values are drawn independently from known distributions. A fixed-price mechanism posts a single price and trades exactly when both agents accept it. How much of the first-best expected welfare can the optimal fixed price guarantee? A sequence of analytic and computer-assisted results narrowed the worst-case ratio to the interval $[0.7292,0.73805]$, but its exact value remained open. We determine this ratio exactly: it equals $0.7380243357\dots$, given in closed form through the unique root of an explicit one-dimensional equation. The guarantee holds for every pair of independent nonnegative distributions with finite means, including atomic and unbounded ones, and an explicit family of instances shows that no larger constant is possible. The proof reduces the problem, through put and call transforms of the two distributions and a worst-case choice of seller, to a one-dimensional control problem for the buyer. A logarithmic change of variables makes this problem strictly convex, so its first-order conditions certify a global optimum. The optimizer switches exactly once from an interior arc to the boundary and can be integrated in closed form. Realizing it yields a bounded seller and a bounded buyer together with a vanishing buyer atom at a diverging value. The worst case is approached by this family but attained by no instance.