We study a Bayesian persuasion problem with externalities. In this model, a principal sends signals to inform multiple agents about the state of the world. Simultaneously, due to the existence of externalities in the agents' utilities, the principal also acts as a correlation device to correlate the agents' actions. We consider the setting where the agents are categorized into a small number of types. Agents of the same type share identical utility functions and are treated equitably in the utility functions of both other agents and the principal. We study the problem of computing optimal signaling strategies for the principal, under three different types of signaling channels: public, private, and semi-private. Our results include revelation-principle-style characterizations of optimal signaling strategies, linear programming formulations, and analysis of in/tractability of the optimization problems. It is demonstrated that when the maximum number of deviating agents is bounded by a constant, our LP-based formulations compute optimal signaling strategies in polynomial time. Otherwise, the problems are NP-hard.

翻译：我们研究了一个具有外部性的贝叶斯劝说问题。在此模型中，委托人向多个智能体发送信号以告知其世界状态。同时，由于智能体效用函数中存在外部性，委托人还充当了关联设备以协调智能体的行动。我们考虑智能体被划分为少量类型的情形。同类型智能体具有相同的效用函数，并且在其他智能体及委托人的效用函数中受到平等对待。我们研究了在三种不同信号通道（公开、私有和半私有）下为委托人计算最优信号策略的问题。我们的结果包括：最优信号策略的显示原理式刻画、线性规划建模，以及对优化问题可解性/难解性的分析。研究表明，当最大偏离智能体数量受常数限制时，我们基于线性规划的模型可在多项式时间内计算出最优信号策略；否则，该问题是NP难的。