Bilateral trade models one of the most fundamental economic interactions: the intermediation between two strategic agents, a seller and a buyer, willing to trade a good. We consider the learning version of the problem, where the goal is to learn a mechanism from a sampled dataset of agents' valuations to maximize either profit or economic efficiency. While known learning algorithms are characterized by high sensitivity to the input dataset, we specifically study this problem through the lens of differential privacy, ensuring that each data point does not significantly affect the probability of learning any specific mechanism. For our results, we adopt the PAC-learning framework: with high probability, the learning algorithm should output a mechanism that is at most an additive $α$ away from optimal, in a $\varepsilon$-differentially private way. As a first result, we show that differential privacy and (near)-optimality are not achievable for general distributions. Surprisingly, assuming that the distribution underlying the agents' valuations is $σ$-smooth, we recover nearly optimal sample-complexity bounds for both economic efficiency and profit. For profit, we show how to construct in polynomial time an $α$-optimal and $\varepsilon$-differentially private mechanism using $\tildeΘ(\frac{1}{σ\varepsilonα^2})$ samples. For efficiency, measured by the gain from trade, we achieve the same result using $\tildeΘ(\frac{1}{\varepsilonα}+\frac{1}{α^2})$ samples. Notably, these bounds are essentially tight in the precision parameter $α$, since achieving $α$-optimality (ignoring differential privacy) requires at least $\frac{1}{α^2}$ samples.

翻译：双边贸易建模了最基本的经济交互之一：两个战略主体（卖家和买家）在中间环节中进行商品交易。我们考虑该问题的学习版本，其目标是从代理估值的采样数据集中学习一个机制，以最大化利润或经济效率。由于已知的学习算法对输入数据集高度敏感，我们特意通过差分隐私的视角研究这一问题，确保每个数据点不会显著影响学习任何特定机制的概率。在结果中，我们采用PAC学习框架：以高概率，学习算法应以ε-差分隐私方式输出一个机制，该机制与最优机制之间的差异至多为附加项α。首先，我们证明对于一般分布，差分隐私与（近）最优性无法同时实现。令人惊讶的是，假设代理估值的分布是σ-光滑的，我们恢复了经济效率和利润的近最优样本复杂度界限。对于利润，我们展示了如何使用$\tildeΘ(\frac{1}{σ\varepsilonα^2})$个样本在多项式时间内构造一个α-最优且ε-差分隐私的机制。对于以贸易收益衡量的效率，我们使用$\tildeΘ(\frac{1}{\varepsilonα}+\frac{1}{α^2})$个样本实现了相同结果。值得注意的是，这些界限在精度参数α上本质上是紧的，因为实现α-最优性（忽略差分隐私）至少需要$\frac{1}{α^2}$个样本。