We explore brokerage between traders in an online learning framework. At any round $t$, two traders meet to exchange an asset, provided the exchange is mutually beneficial. The broker proposes a trading price, and each trader tries to sell their asset or buy the asset from the other party, depending on whether the price is higher or lower than their private valuations. A trade happens if one trader is willing to sell and the other is willing to buy at the proposed price. Previous work provided guidance to a broker aiming at enhancing traders' total earnings by maximizing the gain from trade, defined as the sum of the traders' net utilities after each interaction. In contrast, we investigate how the broker should behave to maximize the trading volume, i.e., the total number of trades. We model the traders' valuations as an i.i.d. process with an unknown distribution. If the traders' valuations are revealed after each interaction (full-feedback), and the traders' valuations cumulative distribution function (cdf) is continuous, we provide an algorithm achieving logarithmic regret and show its optimality up to constant factors. If only their willingness to sell or buy at the proposed price is revealed after each interaction ($2$-bit feedback), we provide an algorithm achieving poly-logarithmic regret when the traders' valuations cdf is Lipschitz and show that this rate is near-optimal. We complement our results by analyzing the implications of dropping the regularity assumptions on the unknown traders' valuations cdf. If we drop the continuous cdf assumption, the regret rate degrades to $\Theta(\sqrt{T})$ in the full-feedback case, where $T$ is the time horizon. If we drop the Lipschitz cdf assumption, learning becomes impossible in the $2$-bit feedback case.

翻译：本文研究在线学习框架下交易者之间的经纪行为。在任意轮次$t$中，若交易对双方均有利，则两名交易者会面交换资产。经纪人提出交易价格，每位交易者根据价格是否高于或低于其私有估值，尝试出售自身资产或从对方购买资产。当一方愿意以提议价格出售且另一方愿意购买时，交易即告达成。先前研究为旨在通过最大化交易收益（定义为每次交互后交易者净效用之和）来提升交易者总收益的经纪人提供了指导。与之相反，本文探究经纪人应如何行动以实现交易量（即交易总次数）的最大化。我们将交易者估值建模为具有未知分布的独立同分布过程。若交易者估值在每次交互后完全披露（全反馈），且其估值累积分布函数连续，我们提出一种实现对数后悔度的算法，并证明该算法在常数因子范围内具有最优性。若每次交互后仅披露交易者在提议价格下的买卖意愿（2比特反馈），当交易者估值累积分布函数满足Lipschitz条件时，我们提出一种实现多对数后悔度的算法，并证明该速率近乎最优。我们通过分析放弃对未知交易者估值累积分布函数的正则性假设所产生的影响来完善研究结论。若放弃累积分布函数连续性假设，全反馈情况下的后悔度将退化至$\Theta(\sqrt{T})$，其中$T$为时间跨度。若放弃累积分布函数Lipschitz条件假设，2比特反馈情况下的学习将不可实现。