SIAM Journal on Computing, Ahead of Print.
Persuasion, defined as the act of exploiting an informational advantage in order to influence the decisions of others, is ubiquitous. Indeed, persuasive communication has been estimated to account for almost a third of all economic activity in the U.S. This paper examines persuasion through a computational lens, focusing on what is perhaps the most basic and fundamental model in this space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow [Am. Econ. Rev., 101 (2011), pp. 2590--2615]. Here there are two players, a sender and a receiver. The receiver must take one of a number of actions with an a priori unknown payoff, and the sender has access to additional information regarding the payoffs of the various actions for both players. The sender can commit to revealing a noisy signal regarding the realization of the payoffs of various actions, and would like to do so to maximize her own payoff in expectation assuming that the receiver rationally acts to maximize his own payoff. When the payoffs of various actions follow a joint distribution (the common prior), the sender's problem is nontrivial, and its computational complexity depends on the representation of this prior. We examine the sender's optimization task in three of the most natural input models for this problem, and essentially pin down its computational complexity in each. When the payoff distributions of the different actions are independently and identically distributed (i.i.d.) and given explicitly, we exhibit a polynomial-time (exact) algorithmic solution, and a “simple” $(1-1/e)$-approximation algorithm. Our optimal scheme for the i.i.d. setting involves an analogy to auction theory, and makes use of Border's characterization of the space of reduced-forms for single-item auctions. When action payoffs are independent but nonidentical with marginal distributions given explicitly, we show that it is \#P-hard to compute the optimal expected sender utility. In doing so, we rule out a generalized Border's theorem, in the sense of Gopalan, Nisan, and Roughgarden [Public projects, boolean functions, and the borders of Border's theorem, in Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC '15, ACM, New York, 2015, p. 395], for this setting. Finally, we consider a general (possibly correlated) joint distribution of action payoffs presented by a black box sampling oracle, and exhibit a fully polynomial-time approximation scheme (FPTAS) with a bicriteria guarantee. Our FPTAS is based on Monte Carlo sampling, and its analysis relies on the principle of deferred decisions. Moreover, we show that this result is the best possible in the black-box model for information-theoretic reasons.

中文翻译：

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贝叶斯算法说服力

《 SIAM计算杂志》，预印本。
说服力无所不在，它被定义为利用信息优势来影响他人决策的行为。确实，说服性传播已被估计占美国所有经济活动的近三分之一。本文通过计算视角来考察说服力，着眼于该领域最基本和最基本的模型：著名的贝叶斯说服模型。 Kamenica和Gentzkow [Am。经济。Rev. 101（2011），pp。2590--2615]。这里有两个播放器，一个发送者和一个接收者。接收方必须采取先验未知收益的多种动作之一，发送者可以访问有关两个参与者的各种动作的收益的其他信息。发送者可以承诺揭示有关实现各种动作的收益的嘈杂信号，并希望这样做，以最大化其自身的收益，并假设接收者合理地采取行动以最大化其自身的收益。当各种动作的收益遵循联合分布（共同先验）时，发送方的问题就不那么重要了，其计算复杂度取决于先验的表示形式。我们针对此问题在三种最自然的输入模型中检查了发件人的优化任务，并从根本上确定了每种模型中的发送者计算复杂度。当不同动作的收益分布独立且相同地分布（iid）并明确给出时，我们将展示多项式时间（精确）算法解和“简单” $（1-1 / e）$逼近算法。我们为iid设置的最佳方案涉及拍卖理论，并利用Border对单项拍卖的简化形式空间的刻画。当行动收益是独立的，但是在明确给出边际分布的情况下是不相同的时，我们证明很难计算出最佳预期发送者效用。这样，我们排除了Gopalan，Nisan和Roughgarden [公共项目，布尔函数以及Border定理的边界，在第16届ACM经济与计算会议论文集，EC' 15，ACM，纽约，2015年，第15页。395]，用于此设置。最后，我们考虑由黑匣子抽样预言家提供的行动收益的一般（可能是相关的）联合分布，并展示具有双标准保证的完全多项式时间近似方案（FPTAS）。我们的FPTAS基于蒙特卡洛采样，其分析依赖于延迟决策的原理。此外，由于信息论的原因，我们证明了这个结果在黑盒模型中是最好的。