About This Course

This course begins where Game Theory leaves off. There, information was something a player had: a type drawn by nature, a private cost, a value for an object. Players with private information chose what to do, and the others drew inferences from it. Here information becomes something a party chooses. A school chooses how to grade, a rating agency how many grades to use, a platform what to tell its drivers about demand, a regulator how hard to make a stress test. The question is what such a choice can achieve, and the answer, developed over ten chapters, is the theory of Bayesian persuasion and information design.

The reader is assumed to have finished the Game Theory course, in particular Bayes–Nash equilibrium (its Chapter 7), signaling games and perfect Bayesian equilibrium (its Chapter 8), and mechanisms and the revelation principle (its Chapter 9). Conditional probability, convex functions, and the duality theorem of linear programming are used freely; the library’s Probability Theory and Continuous Optimization courses contain everything needed. Measure-theoretic probability appears only where a chapter works with a continuum of states, and those passages say what they import.

The course has four parts. The first two chapters make beliefs the object of study: every experiment is summarized by the distribution of posterior beliefs it produces, the only restriction on that distribution is that it averages to the prior, and information is valuable to a decision maker because the value of a belief is convex. The next three chapters ask what an informed party can communicate when it cannot commit to how it reports: free and unverifiable talk, verifiable evidence that can be withheld, and communication through a mediator. Chapters 6 to 8 give the informed party full commitment to an experiment and solve the resulting persuasion problem by concavification, by linear programming and duality, and, when only the posterior mean matters, through the convex order. The last two chapters move to many receivers, where the relevant object is the Bayes correlated equilibrium, and to the weaker forms of commitment and the richer receivers that current research studies.

Cheap talk comes before persuasion on purpose. Persuasion is easiest to appreciate against what an informed party can achieve without commitment, and the comparison between the two, made concrete at the end of Chapter 3, is the thread that ties the communication models together.

By the end the reader should be able to take an unfamiliar sender–receiver or designer–players environment, decide which model of communication fits it, formulate the design problem, solve it, prove that the proposed experiment is optimal, and say which assumption drives the answer. Each chapter closes with what the original papers add and where the research frontier lies, so that the reader can move on to the literature directly.

Every chapter ends with exercises that run from computation to proof, and every chapter includes at least one exercise that asks the reader to find the error in a plausible argument, because in this subject the tempting arguments about information are often wrong in instructive ways. Applications are drawn from canonical examples, industrial organization, and operations and platforms. The companion volume solves many exercises in full and gives hints or answers for the rest. Every numerical claim in the worked examples was recomputed by machine; the reader should still verify each one by hand, and should attempt each exercise before reading its solution.