  We consider a Bayesian persuasion or information design problem where the
sender tries to persuade the receiver to take a particular action via a
sequence of signals. This we model by considering multi-phase trials with
different experiments conducted based on the outcomes of prior experiments. In
contrast to most of the literature, we consider the problem with constraints on
signals imposed on the sender. This we achieve by fixing some of the
experiments in an exogenous manner; these are called determined experiments.
This modeling helps us understand real-world situations where this occurs:
e.g., multi-phase drug trials where the FDA determines some of the experiments,
funding of a startup by a venture capital firm, start-up acquisition by big
firms where late-stage assessments are determined by the potential acquirer,
multi-round job interviews where the candidates signal initially by presenting
their qualifications but the rest of the screening procedures are determined by
the interviewer. The non-determined experiments (signals) in the multi-phase
trial are to be chosen by the sender in order to persuade the receiver best.
With a binary state of the world, we start by deriving the optimal signaling
policy in the only non-trivial configuration of a two-phase trial with
binary-outcome experiments. We then generalize to multi-phase trials with
binary-outcome experiments where the determined experiments can be placed at
any chosen node in the trial tree. Here we present a dynamic programming
algorithm to derive the optimal signaling policy that uses the two-phase trial
solution's structural insights. We also contrast the optimal signaling policy
structure with classical Bayesian persuasion strategies to highlight the impact
of the signaling constraints on the sender.
