The Power of Simple Mechanisms: A Tight $e$-Approximation for Gains from Trade in Matching Markets

Abstract

We study how well simple mechanisms approximate gains from trade (GFT) in two-sided matching markets with independent buyer values and seller costs, where feasible outcomes form an arbitrary downward-closed family of matchings. This model includes bilateral trade and double auctions as special cases. We focus on the Generalized Random-Offerer (GRO) mechanism, which is an equal mixture of the Generalized Sellers-Offering Mechanism (GSOM) and the Generalized Buyers-Offering Mechanism (GBOM). We determine GRO's exact worst-case approximation ratio with respect to first-best GFT, showing that it is $e$. This improves the previous $3.15$ approximation guarantee (Babaioff et al. STOC 2026) for the same mechanism. In bilateral trade, the result implies a $1/e$ guarantee for the random-offerer mechanism, improving the previous $1/\pi$ bound (Jo 2026). Our analysis uses a two-dimensional inequality in quantile space to compare first-best GFT with optimal one-sided auction profits. For each potential trade, we identify the region of buyer values and seller costs for which the first-best allocation selects that trade, and then randomly shrink this region in quantile space, yielding posted-price rules whose expected profits can be evaluated exactly. We establish tightness of the $e$ approximation by constructing markets with regular type distributions, pairwise disjoint trading edges, and a single knapsack constraint. On these instances, GRO's expected GFT approaches a $1/e$ fraction of first-best GFT, therefore ruling out any better guarantee even under these restrictions.