Sequential Persuasion Using Limited Experiments

Bayesian persuasion and its derived information design problem has been one of the main research agendas in the economics and computation literature over the past decade. However, when attempting to apply its model and theory, one is often limited by the fact that the sender can only implement very restricted information structures. Moreover, in this case, the sender can possibly achieve higher expected utility by performing a sequence of feasible experiments, where the choice of each experiment depends on the outcomes of all previous experiments. Indeed, it has been well observed that real life persuasions often take place in rounds during which the sender exhibits experiments/arguments sequentially. We study the sender's expected utility maximization using finite and infinite sequences of experiments. For infinite sequences of experiments, we characterize the supremum of the sender's expected utility using a function that generalizes the concave closure definition in the standard Bayesian persuasion problem. With this characterization, we first study a special case where the sender can use feasible experiments to achieve the optimal expected utility of the standard Bayesian persuasion without feasibility constraints, which is a trivial utility upper bound, and establish structural findings about the sender's optimal sequential design in this case. Then we derive conditions under which the sender's optimal sequential design exists; when an optimal sequential design exists, there exists an optimal design that is Markovian, i.e., the choice of the next experiment only depends on the receiver's current belief.