We examine fixed-price mechanisms in bilateral trade through the lens of regret minimization. Our main results are twofold. (i) For independent values, a near-optimal $\widetildeΘ(T^{2/3})$ tight bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback. (ii) For correlated/adversarial values, a near-optimal $Ω(T^{3/4})$ lower bound for $\textsf{Global Budget Balance}$ fixed-price mechanisms with two-bit/one-bit feedback, which improves the best known $Ω(T^{5/7})$ lower bound obtained in the work [BCCF24] and, up to polylogarithmic factors, matches the $\widetilde{\mathcal{O}}(T^{3 / 4})$ upper bound obtained in the same work. Our work in combination with the previous works [CCCFL24mor, CCCFL24jmlr, AFF24, BCCF24] (essentially) gives a thorough understanding of regret minimization for fixed-price bilateral trade. En route, we have developed two technical ingredients that might be of independent interest: (i) A novel algorithmic paradigm, called $\textit{fractal elimination}$, to address one-bit feedback and independent values. (ii) A new $\textit{lower-bound construction}$ with novel proof techniques, to address the $\textsf{Global Budget Balance}$ constraint and correlated values.

翻译：我们通过遗憾最小化的视角研究了双边贸易中的固定价格机制。主要结果包括两方面：（i）对于独立取值情形，在具有两位/一位反馈的全局预算平衡固定价格机制下，得到了一个近乎最优的$\widetildeΘ(T^{2/3})$紧界；（ii）对于相关/对抗取值情形，在具有两位/一位反馈的全局预算平衡固定价格机制下，得到了一个近乎最优的$Ω(T^{3/4})$下界，该结果改进了文献[BCCF24]中已知的最佳$Ω(T^{5/7})$下界，并在多对数因子范围内匹配了同一文献中得到的$\widetilde{\mathcal{O}}(T^{3/4})$上界。我们的工作与前期研究[CCCFL24mor, CCCFL24jmlr, AFF24, BCCF24]相结合，（实质上）提供了对固定价格双边贸易中遗憾最小化的全面理解。在此过程中，我们开发了两个可能具有独立价值的技术要素：（i）一种新颖的算法范式，称为**分形消除法**，用于处理一位反馈和独立取值情形；（ii）一种新的**下界构造**方法，结合创新的证明技术，用于处理全局预算平衡约束和相关取值情形。