We study repeated bilateral trade from a fairness perspective. At each round, a fresh seller-buyer pair arrives, and the platform posts a price before observing the traders' valuations. Trade occurs only if both agents accept the price. Rather than maximizing only the gain from trade, we consider platforms that seek balanced divisions of the generated surplus. We show that natural fairness desiderata lead to a one-parameter Rawls-to-Nash family of fair-gain objectives, obtained by aggregating the seller's and buyer's net gains through nonpositive Hölder means. Unlike the standard gain-from-trade objective and the Rawlsian fair-gain objective studied in prior work, our proposed objectives induce a new statistical structure in which expected rewards are recovered from threshold feedback through a two-dimensional singular-kernel integral identity. This leads to a nonstandard pure-exploration problem whose natural estimators are rectangular double sums with row-column dependence and singular weights. Assuming independent i.i.d. seller and buyer valuation sequences with arbitrary unknown marginals, we characterize the optimal learning rates for the whole Rawls-to-Nash family of fair-gain objectives, giving matching fixed-confidence sample-complexity and regret bounds up to polylogarithmic factors.

翻译：我们从公平性视角研究重复双边交易。每一轮中，新到达的卖方-买方对出现，平台在观察交易者估值前发布价格。仅当双方均接受价格时交易才会发生。我们考虑的平台不仅追求交易收益最大化，还寻求对产生的盈余进行平衡分配。研究表明，自然的公平性需求引出了一族由非正赫尔德均值聚合卖方与买方净收益得到的原始-纳什公平收益目标（单参数族）。与标准交易收益目标及先前工作中研究的罗尔斯公平收益目标不同，我们所提出的目标诱导出一种全新统计结构——通过阈值反馈恢复预期奖励时，需借助二维奇异核积分恒等式。这引出一个非标准纯探索问题，其自然估计量为具有行列依赖性与奇异权重的矩形双重和。在假设卖方与买方估值序列满足独立同分布且边际分布完全未知的条件下，我们刻画了整个原始-纳什公平收益目标族的最优学习速率，给出了匹配至多对数因子级别的固定置信度样本复杂度与遗憾界。