我们研究了在卖方和买方的估值具有上下文性时的重复双边交易。更具体地说，代理的估值由一个上下文向量与两个未知的d维向量的内积给出——一个用于买方，一个用于卖方。在每个时间步t，学习者会接收到一个上下文并发布两个价格，一个给卖方，一个给买方，如果双方都接受自己的价格，交易就会发生。我们研究了这个问题的两个目标，即交易收益和利润，并证明了在一种令人惊讶的强基准下具有无遗憾性：最佳全知动态策略。在自然场景中，当学习者分别观察到代理是否接受价格——即所谓的“两位反馈”时，我们设计了实现交易收益的O(d log d)遗憾和利润最大化O(d log log T + d log d)遗憾的算法。这两种结果都是紧致的，除了log(d)因子外，并实现了每一步预算平衡，意味着学习者永远不会产生负利润。在信息较少的“一位反馈”模型中，学习者只观察到交易是否发生。对于这种场景，我们证明了紧致的两位反馈遗憾范围仍然可以实现，但代价是允许学习者可能产生一个顺序O(d log d)的小负利润，这明显与时间范围无关。作为最终的结果，我们研究了一位反馈和每一步预算平衡的结合。在那里，我们设计了一个交易收益算法，其遗憾与时间范围无关，但指数级依赖于维度d。对于利润最大化，我们维持这个指数依赖于维度，其乘以一个log T因子。
英文摘要：
We study repeated bilateral trade when the valuations of the sellers and the buyers are contextual. More precisely, the agents' valuations are given by the inner product of a context vector with two unknown $d$-dimensional vectors -- one for the buyers and one for the sellers.
At each time step $t$, the learner receives a context and posts two prices, one for the seller and one for the buyer, and the trade happens if both agents accept their price. We study two objectives for this problem, gain from trade and profit, proving no-regret with respect to a surprisingly strong benchmark: the best omniscient dynamic strategy.
In the natural scenario where the learner observes \emph{separately} whether the agents accept their price -- the so-called \emph{two-bit} feedback -- we design algorithms that achieve $O(d\log d)$ regret for gain from trade, and $O(d \log\log T + d\log d)$ regret for profit maximization. Both results are tight, up to the $\log(d)$ factor, and implement per-step budget balance, meaning that the learner never incurs negative profit.
In the less informative \emph{one-bit} feedback model, the learner only observes whether a trade happens or not. For this scenario, we show that the tight two-bit regret regimes are still attainable, at the cost of allowing the learner to possibly incur a small negative profit of order $O(d\log d)$, which is notably independent of the time horizon. As a final set of results, we investigate the combination of one-bit feedback and per-step budget balance. There, we design an algorithm for gain from trade that suffers regret independent of the time horizon, but \emph{exponential} in the dimension $d$. For profit maximization, we maintain this exponential dependence on the dimension, which gets multiplied by a $\log T$ factor.
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université Paul Sabatier（保罗萨巴蒂尔大学）；Politecnico di Milano（米兰理工大学）；Università degli Studi di Milano（米兰大学）；University of Ottawa（渥太华大学）
Tight Regret Bounds for Fixed-Price Bilateral Trade
Shanghai Jiao Tong University（上海交通大学）；Huawei’s Taylor Lab（华为泰勒实验室）；Shanghai University of Finance and Economics, Laboratory of Interdisciplinary Research of Computation and Economics (SUFE)（上海金融学院，计算与经济学交叉研究实验室（SUFE））
Bilateral Trade: A Regret Minimization Perspective
Università degli Studi di Milano（米兰大学）；Toulouse School of Economics (TSE)（图卢兹经济学院）；Artificial and Natural Intelligence Toulouse Institute (ANITI)（图卢兹人工与自然智能研究所）；Istituto Italiano di Tecnologia（意大利理工学院）；Sapienza Università di Roma（罗马一大（罗马智慧大学））
Repeated Bilateral Trade Against a Smoothed Adversary
Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）；University of Ottawa（渥太华大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；Sapienza Università di Roma（罗马大学）
Hong Kong University of Science and Technology（香港科技大学）；Renmin University of China（中国人民大学）；University at Buffalo（布法罗大学）；Xi’an Jiaotong University（西安交通大学）
Training Language Models for Bilateral Trade with Private Information
Department of Economics, Yale University（耶鲁大学经济学系）；Yale School of Management, Yale University（耶鲁大学管理学院）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）；Department of Industrial Engineering, Tsinghua University（清华大学工业工程系）
The Exact Welfare Guarantee of Fixed-Price Bilateral Trade
Adrasteia Labs and UIUC（阿德拉西亚实验室和伊利诺伊大学厄巴纳-香槟分校）；Zhongnan University of Economics and Law（中南财经政法大学）；Tsinghua University（清华大学）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Boston College（波士顿学院）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
A Parametric Contextual Online Learning Theory of Brokerage
Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université Paul Sabatier（保罗·萨巴蒂埃大学）；University of Ottawa（渥太华大学）；Politecnico di Milano（米兰理工大学）；Università degli Studi di Milano（米兰大学）
