  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.
