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arXiv · 2609.35888

The Exact Welfare Guarantee of Fixed-Price Bilateral Trade

Tingyi Lin·Yichen Shi·Ke Dong·Jiazhuo Li·Shawn Yu·Huanxi Zhang
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Abstract

A seller and a buyer with independent private values can trade only at a posted price. We determine the worst case of this mechanism exactly: the best posted price always guarantees a $β_*=0.738024...$ fraction of first-best welfare, where $β_*$ is the root of an explicit equation, the worst-case buyer is unique up to scaling, and no pair of distributions attains the worst case. This closes the gap $[0.7292,0.73805]$ left by a line of work from SODA 2016 through two STOC 2023 papers and AAAI 2026. The proof is an explicit certificate of optimality: after one change of variables, the gap between the optimal value and that of any buyer distribution is a sum of nonnegative integrals, as in a linear-programming dual, and equality identifies the worst-case shape. The certificate also gives the complete tradeoff between gains from trade and the seller's initial welfare: when first-best gains from trade are a fraction $κ$ of initial seller welfare, the exact worst-case fraction $ρ(κ)$ of gains obtained by the best price satisfies $ρ(κ)\sim2/\log(1/κ)$ as $κ\to0$. The worst-case buyer has two constant segments joined by an explicit nonexponential curve, approached through a vanishing atom whose value tends to infinity. The same constant is the exact guarantee of dominant-strategy mechanisms with individual rationality and strong budget balance in every realization. For two units with increasing submodular valuations, an explicit finite instance has ratio below $0.7290804$, so multi-unit trade is strictly harder than single-unit trade. For any bounded ordered buyer pair we characterize the least common-price mass that guarantees a prescribed ratio against every ordered seller pair; a bound on this one-sided functional would determine the exact two-unit constant. Within an explicit family the unique minimum is $0.729080...$, conjectured to be optimal.

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Tingyi Lin, Yichen Shi, Ke Dong, Jiazhuo Li, Shawn Yu, Huanxi Zhang. 2026-09-27. The Exact Welfare Guarantee of Fixed-Price Bilateral Trade. https://arxiv.org/abs/2609.35888

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