Bayesian persuasion involves designing mechanisms to influence a rational agent's actions. Emir Kamenica and Matthew Gentzkow analyze when a sender can persuade a receiver to act in the sender's interest. They introduce a persuasion mechanism, a game between sender and receiver defined by an information structure and message technology. The sender can only influence the receiver's beliefs, not their payoffs. The paper derives necessary and sufficient conditions for the existence of a persuasion mechanism that strictly benefits the sender and characterizes the optimal mechanism. The authors show that the sender benefits from persuasion if the receiver does not take the sender's preferred action by default and if the receiver's action is constant in a neighborhood of the prior belief. The optimal mechanism depends on the concavity or convexity of the sender's payoff function. If the sender's payoff is concave in the receiver's beliefs, no disclosure is optimal; if it is convex, full disclosure is optimal. The paper also generalizes these results to various scenarios, including cases where payoffs depend only on the expected state. It demonstrates that the value of an optimal mechanism can be determined by analyzing the concave closure of the sender's expected utility function. The results have applications in advertising, legal proceedings, and political campaigns, among other areas. The authors conclude that the optimal mechanism can be constructed by finding a Bayes-plausible distribution of posteriors that maximizes the sender's expected utility.Bayesian persuasion involves designing mechanisms to influence a rational agent's actions. Emir Kamenica and Matthew Gentzkow analyze when a sender can persuade a receiver to act in the sender's interest. They introduce a persuasion mechanism, a game between sender and receiver defined by an information structure and message technology. The sender can only influence the receiver's beliefs, not their payoffs. The paper derives necessary and sufficient conditions for the existence of a persuasion mechanism that strictly benefits the sender and characterizes the optimal mechanism. The authors show that the sender benefits from persuasion if the receiver does not take the sender's preferred action by default and if the receiver's action is constant in a neighborhood of the prior belief. The optimal mechanism depends on the concavity or convexity of the sender's payoff function. If the sender's payoff is concave in the receiver's beliefs, no disclosure is optimal; if it is convex, full disclosure is optimal. The paper also generalizes these results to various scenarios, including cases where payoffs depend only on the expected state. It demonstrates that the value of an optimal mechanism can be determined by analyzing the concave closure of the sender's expected utility function. The results have applications in advertising, legal proceedings, and political campaigns, among other areas. The authors conclude that the optimal mechanism can be constructed by finding a Bayes-plausible distribution of posteriors that maximizes the sender's expected utility.