We study the problem of designing mechanisms for \emph{information acquisition} scenarios. This setting models strategic interactions between an uniformed \emph{receiver} and a set of informed \emph{senders}. In our model the senders receive information about the underlying state of nature and communicate their observation (either truthfully or not) to the receiver, which, based on this information, selects an action. Our goal is to design mechanisms maximizing the receiver's utility while incentivizing the senders to report truthfully their information. First, we provide an algorithm that efficiently computes an optimal \emph{incentive compatible} (IC) mechanism. Then, we focus on the \emph{online} problem in which the receiver sequentially interacts in an unknown game, with the objective of minimizing the \emph{cumulative regret} w.r.t. the optimal IC mechanism, and the \emph{cumulative violation} of the incentive compatibility constraints. We investigate two different online scenarios, \emph{i.e.,} the \emph{full} and \emph{bandit feedback} settings. For the full feedback problem, we propose an algorithm that guarantees $\tilde{\mathcal O}(\sqrt T)$ regret and violation, while for the bandit feedback setting we present an algorithm that attains $\tilde{\mathcal O}(T^{\alpha})$ regret and $\tilde{\mathcal O}(T^{1-\alpha/2})$ violation for any $\alpha\in[1/2, 1]$. Finally, we complement our results providing a tight lower bound.

翻译：我们研究面向“信息获取”场景的机制设计问题。该场景建模了未获悉信息的“接收者”与一组掌握信息的“发送者”之间的策略互动。在我们的模型中，发送者接收关于自然状态的信息，并将其观察结果（如实或非如实）传达给接收者，接收者据此选择行动。我们的目标是设计能最大化接收者效用、同时激励发送者如实报告信息的机制。首先，我们提出一种算法，能够高效计算最优的“激励兼容”（IC）机制。随后，我们聚焦于“在线”问题，即接收者在未知博弈中顺序进行交互，目标是最小化与最优IC机制相关的“累积遗憾”以及“激励兼容约束的累积违反量”。我们研究了两种不同的在线场景，即“完全反馈”和“老虎机反馈”设置。针对完全反馈问题，我们提出一种算法，可保证遗憾与违反量均为$\tilde{\mathcal O}(\sqrt T)$；针对老虎机反馈设置，我们给出一种算法，对于任意$\alpha\in[1/2, 1]$，能达到$\tilde{\mathcal O}(T^{\alpha})$的遗憾与$\tilde{\mathcal O}(T^{1-\alpha/2})$的违反量。最后，我们通过提供紧的下界来补充完善结果。