The celebrated Myerson--Satterthwaite theorem shows that in bilateral trade, no mechanism can be simultaneously fully efficient, Bayesian incentive compatible (BIC), and budget balanced (BB). This naturally raises the question of how closely the gains from trade (GFT) achievable by a BIC and BB mechanism can approximate the first-best (fully efficient) benchmark. The optimal BIC and BB mechanism is typically complex and highly distribution-dependent, making it difficult to characterize directly. Consequently, much of the literature analyzes simpler mechanisms such as the Random-Offerer (RO) mechanism and establishes constant-factor guarantees relative to the first-best GFT. An important open question concerns the worst-case performance of the RO mechanism relative to first-best (FB) efficiency. While it was originally hypothesized that the approximation ratio $\frac{\text{GFT}_{\text{FB}}}{\text{GFT}_{\text{RO}}}$ is bounded by $2$, recent work provided counterexamples to this conjecture: Cai et al. proved that the ratio can be strictly larger than $2$, and Babaioff et al. exhibited an explicit example with ratio approximately $2.02$. In this work, we employ AlphaEvolve, an AI-guided evolutionary search framework, to explore the space of value distributions. We identify a new worst-case instance that yields an improved lower bound of $\frac{\text{GFT}_{\text{FB}}}{\text{GFT}_{\text{RO}}} \ge \textbf{2.0749}$. This establishes a new lower bound on the worst-case performance of the Random-Offerer mechanism, demonstrating a wider efficiency gap than previously known.

翻译：著名的Myerson–Satterthwaite定理表明，在双边贸易中，不存在能够同时实现完全效率、贝叶斯激励相容（BIC）和预算平衡（BB）的机制。这自然引出一个问题：一个满足BIC和BB的机制所能实现的贸易收益（GFT）能在多大程度上逼近最优（完全效率）基准。最优的BIC和BB机制通常结构复杂且高度依赖于价值分布，难以直接刻画。因此，大量文献转而分析更简单的机制，如随机报价者（RO）机制，并建立了相对于最优GFT的常数倍保证。一个重要的开放性问题涉及RO机制相对于最优（FB）效率的最坏情况性能。尽管最初猜想近似比$\frac{\text{GFT}_{\text{FB}}}{\text{GFT}_{\text{RO}}}$的上界为$2$，但近期研究给出了该猜想的反例：Cai等人证明该比值可以严格大于$2$，Babaioff等人则给出了一个比值约为$2.02$的显式实例。在本工作中，我们采用AI引导的进化搜索框架AlphaEvolve来探索价值分布空间。我们发现了一个新的最坏情况实例，其给出了改进的下界$\frac{\text{GFT}_{\text{FB}}}{\text{GFT}_{\text{RO}}} \ge \textbf{2.0749}$。这为随机报价者机制的最坏情况性能确立了一个新的下界，揭示了比以往所知更显著的效率差距。