We study online bilateral trade, where a learner facilitates repeated exchanges between a buyer and a seller to maximize the Gain From Trade (GFT), i.e., the social welfare. In doing so, the learner must guarantee not to subsidize the market. This constraint is usually imposed per round through Weak Budget Balance (WBB). Despite that, Bernasconi et al. [2024] show that a Global Budget Balance (GBB) constraint on the profit -- enforced over the entire time horizon -- can improve the GFT by a multiplicative factor of two. While this might appear to be a marginal relaxation, this implies that all existing WBB-focused algorithms suffer linear regret when measured against the GBB optimum. In this work, we provide the first algorithm to achieve sublinear regret against the GBB benchmark in stochastic environments under one-bit feedback. In particular, we show that when the joint distribution of valuations has a bounded density, our algorithm achieves $\widetilde{\mathcal{O}}(T^{3/4})$ regret. Our result shows that there is no separation between the one-dimensional problem of learning the optimal WBB price and the two-dimensional problem of learning the optimal GBB distribution over pairs of prices.

翻译：我们研究在线双边贸易问题，其中学习者通过促进买卖双方之间的重复交易来最大化贸易收益，即社会福利。在此过程中，学习者必须保证不向市场提供补贴。这一约束通常通过每轮执行的弱预算平衡条件来实现。尽管如此，Bernasconi等人[2024]的研究表明，对整体时间跨度上的利润施加全局预算平衡约束，可以将贸易收益提升至两倍乘数。尽管这看似是一种边际放松，但意味着所有现有专注于弱预算平衡的算法，在相对于全局预算平衡最优解进行衡量时，都会遭受线性遗憾。在本工作中，我们提出了首个在单比特反馈的随机环境中，能够针对全局预算平衡基准实现次线性遗憾的算法。具体而言，我们证明当估值联合分布具有有界密度时，我们的算法可实现$\widetilde{\mathcal{O}}(T^{3/4})$的遗憾。我们的结果表明，学习最优弱预算平衡价格的一维问题与学习最优全局预算平衡价格对分布的二维问题之间不存在分离性。