11institutetext: University of Michigan, Ann Arbor
11email<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>11email:
<EMAIL_ADDRESS>
# Bayesian Persuasion in Sequential Trials

Shih-Tang Su Supported in part by NSF grant ECCS 2038416 and MCubed 3.0.11
Vijay G. Subramanian Supported in part by NSF grants ECCS 2038416, CCF
2008130, and CNS 1955777.11 Grant Schoenebeck Supported in part by NSF grants
CCF 2007256 and CAREER 1452915.11

###### Abstract

We consider a Bayesian persuasion 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, 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 arbitrary nodes 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.

###### Keywords:

Information design Bayesian persuasion Signaling games.

##  1 Introduction

Information design studies how informed agents (senders) persuade uninformed
agents (receivers) to take specific actions by influencing the uninformed
agents’ beliefs via information disclosure in a game. The canonical Kamenica-
Gentzkow model [16] is one where the sender can commit to an information
disclosure policy (signaling strategy) before learning the true state. Once
the state is realized, a corresponding (randomized) signal is sent to the
receiver. Then, the receiver takes an action, which results in payoffs for
both the sender and the receiver. Senders in information design problems only
need to manipulate the receivers’ beliefs with properly chosen signals. The
manipulated beliefs will create the right incentives for the receiver to
spontaneously take specific actions that benefit the sender (in expectation).
In (classical) mechanism design, however, the story is different: the designer
is unaware of the agents’ private information, and the agents communicate
their private information to the designer, who then has to provide incentives
via (monetary) transfers or other means. The flexibility afforded by
information design that allows the sender to benefit from information
disclosure without implementing utility-transfer mechanisms has led to greater
applicability of the methodology: various models and theories can be found in
survey papers such as [3] and [15].

Our work is motivated by many real-world problems where persuasion schemes are
applicable, but the sender is constrained in the choice of signals available
for information design. Specifically, we are interested in problems that are
naturally modeled via multi-phase trials where the interim outcomes determine
the subsequent experiments. Further, we insist that some of the experiments
are given in an exogenous manner. This feature imposes restrictions on the
sender’s signaling space, and without it, we would have a classical Bayesian
persuasion problem with an enlarged signal space. Our goal is to study the
impact of such constraints on the optimal signaling scheme, and in particular,
to contrast it with the optimal signaling schemes in classical Bayesian
persuasion.

The following motivating example describes a possible real-world scenario.

######  Example 1 (Motivating example - Acquiring funds from a venture capital
firm)

We consider a scenario where a start-up is seeking funds from a venture
capital firm. The process for this will typically involve multiple rounds of
negotiation and evaluation: some of these will be demonstrations of the start-
up’s core business idea, and the others will be assessments by the venture
capital firm following their own screening procedures. The start-up will have
to follow the venture capital firm’s screening procedures but chooses its
product demonstrations. Based on these stipulations, the start-up needs to
design its demonstrations to maximize its chance of getting funded.

Figure 1: Example of a negotiation process – a startup vs. a venture capital
firm.

In the example above, the start-up (sender) has to generate an information
disclosure scheme to get the desired funds from the venture capital firm
(receiver). Then the screening procedures set by the venture capital firm are
analogous to our determined experiments, and the demonstrations carried out by
the start-up are the (sender) designed experiments. For example, in Figure 1
we present one plausible interaction where the start-up company designs
demonstrations A, B, and C (circles in the figure) and the venture capital
firm has pre-determined screening examinations W, X, Y, and Z (rectangles in
the figure). Whereas we have illustrated this example via a balanced tree, if
we have an unbalanced tree owing to the receiver deciding in the middle, we
can modify it to a balanced tree by adding the required number of dummy
stages.

The sender’s reduced flexibility on her signaling strategies under some
predetermined experiments with arbitrary positions and informativeness
differentiates our work from the growing literature on dynamic information
design. Our model considers a problem with the following features: a static
state space, a sequential information disclosure environment, and a signaling
space restricted by some exogenous constraints whose harshness may depend on
the proposed singling schemes. Models with a static state space, an
unrestricted signal space but a variety of sequential information disclosure
environments have been studied to capture features in different real-world
problems: e.g., with multiple senders [12, 19], with costly communication [14,
22], allowing for sequential decision making [10], or with partial commitment
[1, 22]. Models with dynamic states and sequential information disclosure
environments are usually studied under an informed sender with the knowledge
of dynamically changing state(s); a variety of works in this category lie in
state change detection [9, 11] or routing games [21]. Although several works
[7, 13, 18] also consider constrained signaling schemes, these works either
consider the signal space to be smaller than the action space [13, 7] or
consider a noisy signaling environment [18]. Models with exogenous information
[17, 5, 4], can be viewed as sequential information disclosure problems with
exogenous determined experiments placed in determined phases. The sequential
information disclosure in our model, which actually enlarges the signal space,
makes our work different from above works. To keep the focus of the paper on
sequential trials, we discuss the broader literature on constrained senders,
algorithmic information design, and works related to the receiver’s experiment
design111See Section 5 in [23] for details. in our online version [23].

According to the motivating example illustrated in Figure 1, the persuasion
problem considers a sequence of experiments where experiments further along in
the tree depend on the outcomes of previous phases. The experiment to be run
in each phase is either exogenously determined or chosen by the sender. In the
game, the sender chooses designed experiments with knowledge of the prior, the
determined experiments, and the receiver’s utility function, but before the
state of the world is realized. After the sender commits to the experiments
(i.e., the signaling strategy), the state of the world is realized, and a
specific sequence of experiments is conducted based on the realization of the
underlying random variables. The receiver then takes an action depending on
the entire sequence of outcomes. The prior, the sender’s and receiver’s
utility functions, the determined experiments, and the designed experiments
(after the sender finalizes them) are assumed to be common knowledge. We study
this problem for binary states of the world, first for two-phase binary-
outcome trials, and then generalized to multi-phase binary-outcome trials. We
then generalize to non-binary experiments (still with an underlying binary
state space). In the online version [23] we add games with an additional stage
where the receiver moves before the sender to decide some or all of the
determined experiments, perhaps with some constraints.

Contributions: The main contributions of this work are:

  1. 1.

To the best of our knowledge, within the multi-phase Bayesian persuasion
framework, we are the first to study the design of sender’s signaling schemes
with exogeneously determined experiments in arbitrary positions. Our results
highlight the difference between “exogenously determined experiments” and
“exogenously given information” in the dynamic information design, where the
former gives greater flexibility and allows for more heterogeneity.

  2. 2.

We explicitly solve the optimal signaling scheme in two-phase trials.
Moreover, using structural insights gained from two-phase trials, we present a
dynamic programming algorithm to derive the optimal signaling in general
multi-phase trials via backward iteration.

  3. 3.

We analyze the impact of constraints on the sender via the determined
experiments by contrasting the performance with the classical Bayesian
persuasion setting and when using classical Bayesian persuasion optimal
signaling schemes when the sender is constrained. As a part of this, we
provide a sufficient condition for when a sequential trial is equivalent to
classical Bayesian persuasion with a potentially enlarged signal space.

##  2 Problem Formulation

There are two agents, a sender (Alice) and a receiver (Bob), participating in
the game. We assume binary states of the world,
$\Theta=\\{\theta_{1},\theta_{2}\\}$, with a prior belief
$p\coloneqq\mathbb{P}(\theta_{1})$ known to both agents. The receiver has to
take an action $\Phi\in\\{\phi_{1},\phi_{2}\\}$ which can be thought of as a
prediction of the true state. We assume that the receiver’s utility is given
by $u_{r}(\phi_{i},\theta_{j})=1_{\\{i=j\\}}$ for all $i,j\in\\{1,2\\}$. To
preclude discussions on trivial cases and to simplify the analysis, we assume
that the sender always prefers the action $\phi_{1}$, and her utility is
assumed to be $u_{s}(\phi_{1},\theta_{i})=1$ and
$u_{s}(\phi_{2},\theta_{i})=0$ for all $i\in\\{1,2\\}$.

Before the receiver takes his action, a trial consisting of multiple phases
will be run, and the outcome in each phase will be revealed to him. In each
phase, one experiment will be conducted, which is chosen according to the
outcomes in earlier phases. Hence, the experiment outcomes in earlier phases
not only affect the interim belief but also influence the possible (sequence
of) experiments that will be conducted afterward. In the most sender-friendly
setup where the sender can choose any experiment in each phase without any
constraints, the problem is equivalent to the classical Bayesian persuasion
problem with an enlarged signal space. However, when some experiments are pre-
determined conditional on a set of outcomes, the sender must take these
constraints into account to design her optimal signaling structure.

To present our results on the influence of multiple phases on the sender’s
signaling strategy, we start with a model of two-phase trials with binary-
outcome experiments in the rest of this section. We then analyze the optimal
signaling strategy of this model in Section 3. After that, we will introduce
the general model of multiple-phase trials with binary-outcome experiments and
propose a systematic approach to analyze the optimal signaling structure in
Section 4.

###  2.1 Model of Binary-Outcome Experiments in Two-Phase Trials

There are two phases in the trial: phase I and phase II. Unlike in the
classical Bayesian persuasion problem, our goal is for the sender to not have
the ability to choose the experiments to be conducted in both phases of the
trial. We will start by assuming that the sender can choose any binary-outcome
experiment in phase I, but both the phase-II experiments (corresponding to the
possible outcomes in phase I) are determined. Formally, in phase I, there is a
binary-outcome experiment with two possible outcomes
$\omega_{1}\in\Omega_{1}=\\{\omega_{A},\omega_{B}\\}$ and each outcome
corresponds to a determined binary-outcome experiment, $E_{A}$ or $E_{B}$,
which will be conducted in phase II, respectively. The sender can design the
experiment in phase I via choosing a probability pair
$(p_{1},p_{2})\in[0,1]^{2}$, where $p_{i}=\mathbb{P}(\omega_{A}|\theta_{i})$.
Once the probability pair $(p_{1},p_{2})$ is chosen, the interim belief of the
true state $\mathbb{P}(\theta_{1}|\omega_{1})$ can be calculated (while
respecting the prior) as follows:

$\displaystyle\mathbb{P}(\theta_{1}|\omega_{A})=\frac{pp_{1}}{pp_{1}+(1-p)p_{2}}\text{
and
}\mathbb{P}(\theta_{1}|\omega_{B})=\frac{p(1-p_{1})}{p(1-p_{1})+(1-p)(1-p_{2})}.$
(1)

On the other hand, the phase-II experiments are given in an exogenous manner
beyond the sender’s control. In phase II, one of the binary-outcome
experiments, $E\in\\{E_{A},E_{B}\\}$ will be conducted according to the
outcome, $\omega_{A}$ or $\omega_{B}$ of the phase-I experiment. If
$\omega_{A}$ is realized, then experiment $E_{A}$ will be conducted in phase
II; if $\omega_{B}$ is realized, the experiment $E_{B}$ will be conducted in
phase II. Similarly, we can denote the possible outcomes
$\omega_{2}\in\Omega_{2X}=\\{\omega_{XP},\omega_{XF}\\}$ when the experiment
$E_{X}$ is conducted, where notation $P,F$ can be interpreted as passing or
failing the experiment. Likewise, the phase-II experiments can be represented
by two probability pairs $E_{1}=(q_{A1},q_{A2})$ and $E_{2}=(q_{B1},q_{B2})$,
where $q_{Xi}$ denotes the probability that the outcome $\omega_{XP}$ is
realized conditional on the experiment $E_{X}$ and the state $\theta_{i}$,
i.e., $q_{Xi}=\mathbb{P}(\omega_{XP}|\theta_{i},E_{X})$.

In real-world problems, regulations, physical constraints, and natural limits
are usually known to both the sender and the receiver before the game starts.
Hence, we assume that the possible experiments $E_{1},E_{2}$ that will be
conducted in phase II are common knowledge. Given the pairs
$(q_{A1},q_{A2}),(q_{B1},q_{B2})$, the sender’s objective is to maximize her
expected utility by manipulating the posterior belief (of state $\theta_{1}$)
in each possible outcome of phase II. However, since the phase II experiments
are predetermined, the sender can only indirectly manipulate the posterior
belief by designing the probability pair $(p_{1},p_{2})$ of the phase-I
experiment. As the sender prefers the action $\phi_{1}$ irrespective of the
true state, her objective is to select an optimal probability pair
$(p_{1},p_{2})$ to maximize the total probability that the receiver is willing
to take action $\phi_{1}$. Recalling the receiver’s utility function discussed
above, the receiver’s objective is to maximize the probability of the
scenarios where the action index matches the state index. Thus, the receiver
will take action $\phi_{1}$ if the posterior belief
$\mathbb{P}(\theta_{i}|\omega)\geq\frac{1}{2}$ and take action $\phi_{2}$
otherwise. After taking the receiver’s objective into the account, the
sender’s optimization problem can be formulated as below:

$\displaystyle\max_{p_{1},p_{2}}\sum_{\omega_{2}\in\\{\omega_{AP},\omega_{AF},\omega_{BP},\omega_{BF}\\}}\mathbb{P}(\phi_{1},\omega_{2})$
(2) s.t.
$\displaystyle\Big{(}\mathbb{P}\big{(}\theta_{1}|\omega_{AY},q_{A1},q_{A2},p_{1},p_{2}\big{)}-\tfrac{1}{2}\Big{)}\Big{(}\mathbb{P}(\phi_{1},\omega_{AY})-\tfrac{1}{2}\Big{)}\geq
0~{}\forall~{}Y\in\\{P,F\\},$
$\displaystyle\Big{(}\mathbb{P}\big{(}\theta_{1}|\omega_{BY},q_{B1},q_{B2},p_{1},p_{2}\big{)}-\tfrac{1}{2}\Big{)}\Big{(}\mathbb{P}(\phi_{1},\omega_{BY})-\tfrac{1}{2}\Big{)}\geq
0~{}\forall~{}Y\in\\{P,F\\},$
$\displaystyle\mathbb{P}(\omega_{AP})=pp_{1}q_{A1}+(1-p)p_{2}q_{A2},$
$\displaystyle\mathbb{P}(\omega_{BP})=p(1-p_{1})q_{B1}+(1-p)(1-p_{2})q_{B2},$
$\displaystyle\mathbb{P}(\omega_{AP})+\mathbb{P}(\omega_{AF})=pp_{1}+(1-p)p_{2},$
$\displaystyle\mathbb{P}(\omega_{BP})+\mathbb{P}(\omega_{BF})=p(1-p_{1})+(1-p)(1-p_{2}),$
$\displaystyle\mathbb{P}(\phi_{1},\omega_{2})\in[0,1]~{}\forall\omega_{2}\in\\{\omega_{AP},\omega_{AF},\omega_{BP},\omega_{BF}\\},\qquad
p_{1},p_{2}\in[0,1].$

In the sender’s optimization problem (2), the first two inequalities are
constraints of incentive-compatibility (IC) that preclude the receiver’s
deviation. The IC constraints can be satisfied when both terms in the brackets
are positive or negative. That is to say; the sender can only persuade the
receiver to take action $\phi_{1}$/$\phi_{2}$ when the posterior belief (of
$\theta_{1}$) is above/below $0.5$. While we have written the IC constraints
in a nonlinear form for compact presentation, in reality they’re linear
constraints. The next four equations are constraints that make the sender’s
commitment (signaling strategy) Bayes plausible222A commitment is Bayes-
plausible [16] if the expected posterior probability of each state equals its
prior probability, i.e.,
$\sum_{\omega\in\Omega}\mathbb{P}(\omega)\mathbb{P}(\theta_{i}|\omega)=\mathbb{P}(\theta_{i})$..
Hence, there are $4$ IC constraints and $4$ Bayes-plausible constraints in the
optimization problem for a two-phase trial. However, in an $N$-phase trial,
both the number of IC constraints and the number of Bayes-plausible
constraints will expand to $2^{N}$ each. Although the linear programming (LP)
approach can solve this optimization problem, solving this LP problem in large
Bayesian persuasion problems can be computationally hard [8].Hence, instead of
solving this optimization problem via an LP, we aim to leverage structural
insights discovered in the problem to derive the sender’s optimal signaling
structure.

We end this section by emphasizing that this model is the only non-trivial
two-phase trial configuration when determined and designed experiments
coexist. In other configurations such that some of the phase-II experiments
can be designed by sender, the model can be reduced to a corresponding single-
phase trial in the sense that the single-phase trial will yield the same
payoffs for both sender and receiver when they play optimally. (Note that the
reduced model may have a different prior if the experiment in phase-I is
determined).

##  3 Binary-outcome Experiments in Two-phase Trials

In this section the sender’s optimization problem presented in (2) Section
2.1, is solved starting with the simplest non-trivial case. There are only two
phases in the trial studied here, and from this we will develop more insight
into how different types of experiments (determined versus sender-designed)
influence the optimal signaling strategy of the sender. To be more specific,
we will analyze how two determined experiments (in phase II) and one sender-
designed experiment (in phase I) will impact the sender’s optimal signaling
strategy. Before we present the general case, we discuss a subset class of
two-phase trials that are similar to single-phase trials. In this class of
two-phase trials, in one of the phase-II experiments, called a trivial
experiment, the outcome distribution is independent of the true state. Trivial
experiments [2], also called (Blackwell) non-informative experiments in some
literature, are frequently used as benchmarks to compare the agents’ expected
utility change under different signaling schemes/mechanisms, e.g., [22, 20,
21]. This two-phase model with a trivial experiment tries to capture real-
world problems with one actual (and costly) experiment, e.g., clinical trials,
venture capital investments, or space missions. Since the experiment is
costly, a screening procedure is provided to decide whether it is worth
conducting the experiment. We will then analyze the optimal signaling strategy
in the general scenario, where both experiments in phase II are non-trivial.

###  3.1 Experiments with screenings

We start by analyzing the sender’s optimal strategy (signaling structure) in a
simple scenario where there is one non-trivial experiment conducted in phase
II. The sender’s authority on choosing the probability pair $(p_{1},p_{2})$
controls the screening process. To avoid any ambiguity, we first define what a
trivial experiment is.

###### Definition 1

An experiment $E$ is trivial if the distribution of its outcomes
$\triangle_{E}$ is independent of the state of the world:
$\triangle_{E}=\triangle_{E|\theta_{i}}$ for all $\theta_{i}\in\Theta$.

When a trivial experiment (in phase II) is conducted, the posterior belief of
the state stays the same as the interim belief derived in (1). When there
exists a trivial experiment in the two phase-II trial options, then Lemma 1
states that the sender and the receiver’s expected utility under the optimal
signaling strategy is the same as in the (single-phase) classical Bayesian
persuasion problem.

###### Lemma 1

When the state space is binary, both sender and receiver’s expected utilities
are the same in the following two Bayesian persuasion schemes under each
scheme’s optimal signaling strategy:

  1. 1.

Bayesian persuasion in a single-phase trial,

  2. 2.

Bayesian persuasion in a two-phase trial with a sender-designed phase-I
experiment and a trivial experiment in phase II.

In the single-trial classical Bayesian persuasion setting, the optimal
signaling strategy only mixes the two possible states in one outcome (e.g.,
when the prosecutor claims the suspect is guilty). On the other outcome, the
sender reveals the true state with probability one (e.g., when the prosecutor
says the suspect is innocent). When there is a trivial experiment in phase II,
the other experiment (supposing that it will be conducted at outcome
$\omega_{B}$) will be rendered defunct by the sender’s choice of experiments
in phase I. This phenomenon occurs because the sender can always choose to
reveal the true state when the non-trivial experiment is to be conducted,
i.e., by setting $\mathbb{P}(\theta_{1}|E_{B})=1$ or
$\mathbb{P}(\theta_{2}|E_{B})=1$; and the classical Bayesian persuasion
strategy can be replicated. In essence, having a trivial experiment in the
phase-II trial does not constrain the sender.

###  3.2 Assumptions and induced strategies

Next we detail the optimal signaling strategy in our two-phase trial setting
with general binary-outcome experiments. To aid in the presentation and to
avoid repetition, we make two assumptions without loss of generality and
introduce several explanatory concepts before the analysis.

###### Lemma 2

We can make the following two assumptions WLOG.

  1. 1.

The probability of passing a phase-II experiment under $\theta_{1}$ is greater
than or equal to the probability under $\theta_{2}$, i.e., $q_{A1}\geq q_{A2}$
and $q_{B1}\geq q_{B2}$.

  2. 2.

When the true state is $\theta_{1}$, the experiment conducted when outcome
$\omega_{A}$ occurs is more informative333In terms of the Blackwell
informativeness from [6]. than the experiment conducted when outcome
$\omega_{B}$ occurs, i.e., $q_{A1}\geq q_{B1}$.

The sender’s strategy consists of the following: choice of phase-I experiment
parameters $(p_{1},p_{2})$ and the persuasion strategies in phase-II for each
outcome of the phase-I experiment. To understand better the choices available
to the sender and her reasoning in determining her best strategy, we will
study the possible persuasion strategies in phase-II; these will be called
induced strategies to distinguish them from the entire strategy. Given the
assumptions above on phase-II experiments, it’ll turn out we can directly rule
out one class of induced strategies from the sender’s consideration. The other
set of induced strategies will need careful assessment that we present next.

###### Claim

When the inequalities of the assumption $q_{A1}\geq q_{A2}$ and $q_{B1}\geq
q_{B2}$ in Lemma 2 are strict, for any phase-II experiment
$E_{X}\in\\{E_{A},E_{B}\\}$), taking action $\phi_{1}$ when $E_{X}$ fails but
taking action $\phi_{2}$ when $E_{X}$ passes is not an incentive compatible
strategy for the receiver for any interim belief
$~{}\mathbb{P}(\theta_{1}|E_{X})\in[0,1]$.

The above claim can be verified by comparing the posterior belief
$\mathbb{P}(\theta_{1}|\omega)$ of each possible outcome
$\omega\in\\{\omega_{XP},\omega_{XF}\\}$ and the receiver’s corresponding best
response. Therefore, upon the outcome of a phase-I experiment being revealed
(to be either $\omega_{A}$ or $\omega_{B}$), the sender only has three
different classes of “induced strategies” by which to persuade the receiver in
phase II:
($\alpha_{X}$) Suggest action $\phi_{1}$ only when the phase-II experiment
outcome is a pass;
($\beta_{X}$) Suggest action $\phi_{1}$ no matter the result of phase-II
experiment; and
($\gamma_{X}$) Suggest action $\phi_{2}$ irrespective of the result of phase-
II experiment, which is equivalent to not persuading the receiver to take the
sender-preferred action.

Given these three classes of induced strategies and the freedom to choose
different induced strategies based on the phase-I experiment’s outcome, the
sender can use any combination of these $3^{2}$ choices to form a set of
strategies $\mathcal{S}$. To simplify the representation, we use
$(c_{A},d_{B})$, $c,d\in\\{\alpha,\beta,\gamma\\}$ to represent a “type of
strategy” of the sender. Note that to specify a strategy $S$ within the set of
strategies, i.e., $S\in\mathcal{S}$, the probability pair $(p_{1},p_{2})$ has
to be determined first. Before we analyze the different strategies, we discuss
the relationship between the given phase-II experiments, induced strategies,
and the incentive-compatibility requirements from the sender’s side (to avoid
profitable deviations by the receiver). To avoid ambiguity, hereafter, when we
mention incentive compatibility/incentive compatible requirements/IC
strategies, we mean the condition/requirements/strategies of a sender’s
commitment satisfying the following statement: for every possible realized
signal under this commitment, the receiver taking the sender-suggested action
is incentive-compatible.

###  3.3 Constraints given by phase-II experiments

By her choice of the experiment in phase I, the sender decides how to split
the prior into the interim beliefs for the two experiments available in phase-
II. The resulting interim-beliefs then lead to certain induced strategies at
stage-II being applicable, i.e., incentive compatible (for the receiver). In
other words, the probability pair $(p_{1},p_{2})$ must make each (applied)
induced strategy yield the maximum utility for the receiver. These
requirements constrain the sender’s choice of $(p_{1},p_{2})$, and the sender
needs to account for the (reduced) choice while deciding the split of the
prior. Table 1 summarizes the impact in terms of the parameters of the phase-I
experiment via primary requirements on $(p_{1},p_{2})$ driven by the incentive
compatibility while using each class of induced strategies. Hereafter, when we
use IC requirements without additional specification, we mean primary IC
requirements. From the entries in the table, it is clear that the phase-II
experiments (indirectly) limit the sender’s strategy selection where this
limitation arises due the receiver’s IC requirements for each induced strategy
(when that induced strategy is used).

Table 1: IC requirements of the sender’s commitment based on the induced strategy Induced strategy | Primary IC requirement | Induced strategy | Primary IC requirement
---|---|---|---
$\alpha_{A}$ | $p_{1}\geq\frac{1-p}{p}\frac{q_{A2}}{q_{A1}}p_{2}$ | $\alpha_{B}$ | $1-p_{1}\geq\frac{1-p}{p}\frac{q_{B2}}{q_{B1}}(1-p_{2})$
$\beta_{A}$ | $p_{1}\geq\frac{1-p}{p}\frac{1-q_{A2}}{1-q_{A1}}p_{2}$ | $\beta_{B}$ | $1-p_{1}\geq\frac{1-p}{p}\frac{1-q_{B2}}{1-q_{B1}}(1-p_{2})$
$\gamma_{A}$ | $p_{1}\leq\frac{1-p}{p}\frac{q_{A2}}{q_{A1}}p_{2}$ | $\gamma_{B}$ | $1-p_{1}\leq\frac{1-p}{p}\frac{q_{B2}}{q_{B1}}(1-p_{2})$

With this in mind, the sender’s experiment design in phase I, essentially, is
to select between different combinations of these induced strategies such that
each induced strategy satisfies the constraint listed in Table 1. Hence, we
next seek to understand how these IC constraints collectively determine the
sender’s strategy selection. To answer this, we first discuss the relationship
between induced strategies, IC requirements, and the sender’s expected
utility.

From the sender’s perspective, each induced strategy and its corresponding
signals provide a path to persuade (or dissuade) the receiver to take action
$\phi_{1}$. Since the sender’s objective is to maximize the probability that
action $\phi_{1}$ is taken, she would like to use the “most efficient”444The
efficiency of a strategy is defined as
$\frac{\mathbb{P}(\phi_{1}|\text{interim belief, the induced strategy
used})}{\mathbb{P}(\theta_{1}|\text{interim belief, the induced strategy
used})}$. pair of induced strategies to persuade the receiver555When the prior
falls in the region where the optimal signaling strategy is non-trivial.. To
better understand the “efficiency” of induced strategies, we evaluate each
induced strategy under a given phase-II experiment $E_{X}$:
$\bullet$ $\alpha_{X}$ strategy: To persuade receiver to take action
$\phi_{1}$ via this induced strategy, the sender needs to ensure that
$\mathbb{P}(\theta_{1}|\omega_{XP})\geq\frac{1}{2}$. Hence, the interim belief
$\mathbb{P}(\theta_{1}|E_{X})$ must satisfy
$\mathbb{P}(\theta_{1}|E_{X})\in\left[\frac{q_{X2}}{q_{X1}+q_{X2}},\frac{1-q_{X2}}{2-q_{X1}-q_{X2}}\right]$,
otherwise a commitment using $\alpha_{X}$ induced strategy will never be
incentive-compatible. From the sender’s perspective, the most efficient
strategy to persuade the receiver using $\alpha_{X}$ induced strategy is to
design the phase-I experiment such that
$\mathbb{P}(\theta_{1}|E_{X})=\frac{q_{X2}}{q_{X1}+q_{X2}}$. At this interim
belief, the sender experiences a relative expected utility $2q_{X1}$ (with
respect to the prior). When
$\mathbb{P}(\theta_{1}|E_{X})\in\left(\tfrac{q_{X2}}{q_{X1}+q_{X2}},\frac{1-q_{X2}}{2-q_{X1}-q_{X2}}\right)$,
the sender’s marginal expected utility when the interim belief increases is
$q_{X1}-q_{X2}$.
$\bullet$ $\beta_{X}$ strategy: To persuade receiver to take action $\phi_{1}$
with this induced strategy the sender needs to ensure that both inequalities
$\mathbb{P}(\theta_{1}|\omega_{XP})\geq\frac{1}{2}$ and
$\mathbb{P}(\theta_{1}|\omega_{XF})\geq\frac{1}{2}$ hold. Given the assumption
in Lemma 2, namely $q_{X1}\geq q_{X2}$, the only constraint that can be tight
is $\mathbb{P}(\theta_{1}|\omega_{XF})\geq\frac{1}{2}$. Hence, IC commitments
using $\beta_{X}$ induced strategy exist only when the interim belief
$\mathbb{P}(\theta_{1}|E_{X})\geq\frac{1-q_{X2}}{2-q_{X1}-q_{X2}}$. From the
sender’s perspective, the most efficient strategy to persuade the receiver
using a $\beta_{X}$ induced strategy is to design the phase-I experiment such
that $\mathbb{P}(\theta_{1}|E_{X})=\frac{1-q_{X2}}{2-q_{X1}-q_{X2}}$ with the
resulting relative expected utility $1+\frac{1-q_{X1}}{1-q_{X2}}$. Unlike an
$\alpha_{X}$ induced strategy where the sender still gets a positive utility
gain when the interim belief increases, for a $\beta_{X}$ induced strategy the
sender’s marginal expected utility gain when the interim belief increases is
$0$ when $\mathbb{P}(\theta_{1}|E_{X})>\frac{1-q_{X2}}{2-q_{X1}-q_{X2}}$.
$\bullet$ $\gamma_{X}$ strategy: Since the sender suggests the receiver to
take action $\phi_{2}$ in this strategy, the sender’s expected utility is $0$
when using this induced strategy.

According to the discussion above, it is clear that the sender will not use
the set of strategies corresponding to $(\gamma_{A},\gamma_{B})$ unless the
prior $p=0$. Besides, we know that different induced strategies provide
different relative expected utility to the sender. When induced strategies are
used in the most efficient manner, the relative expected utility under a
$\alpha_{X}$ induced strategy is at most $2q_{X1}$, and the average expected
utility under a $\beta_{X}$ induced strategy is at most
$1+\tfrac{1-q_{X1}}{1-q_{X2}}$.

Since these two values capture the best scenario that the sender can achieve
by tailoring the interim belief under the given experiment, we define this
pair of ratios, $(2q_{X1},1+\tfrac{1-q_{X1}}{1-q_{X2}})$ as a function of (the
given) experiment, denoted by $PerP(E_{X})$; this pair is called the
persuasion potential.

###### Definition 2

Given an experiment $E_{X}=(q_{X1},q_{X2})$, the persuasion potential of this
experiment, $PerP(E_{X})$, is the pair
$\Big{(}2q_{X1},1+\frac{1-q_{X1}}{1-q_{X2}}\Big{)}$.

To provide some insights on the importance and use of the persuasion potential
we preview Corollary 1. Corollary 1 states that the sender only uses induced
strategies in the most efficient manner, i.e.,
$\mathbb{P}(\theta_{1}|E_{X})=\tfrac{q_{X2}}{q_{X1}+q_{X2}}$ when an induced
strategy $\alpha_{X}$ is used and
$\mathbb{P}(\theta_{1}|E_{X})=\tfrac{1-q_{X2}}{2-q_{X1}-q_{X2}}$ when an
induced strategy $\beta_{X}$ is used. Thus, the persuasion potential can
simplify the sender’s search for the optimal signaling strategy. When a
particular induced strategy is used in the most efficient manner described in
the above parameter, the interim belief is now determined. Therefore, the
sender does not need to search for the optimal signaling strategy from the
whole set of IC strategies but only needs to search from a small number of
strategies that generate the particular interim beliefs.

###  3.4 Persuasion ratio and the optimal signaling structure

Since the sender wants to maximize the total probability of action $\phi_{1}$,
she needs to compare different sets of strategies formed by different pairs of
induced strategies. To compare each set of strategies, we introduce the
persuasion ratio of a set of strategies for a given value of the prior.

###### Definition 3

Given a set of incentive-compatible strategies $\mathcal{S}$, e.g.,
$\mathcal{S}=(c_{A},d_{B})$ with $c,d\in\\{\alpha,\beta,\gamma\\}$ which
satisfying IC requirements, the persuasion ratio of the set of strategies
$\mathcal{S}$ is the maximum total probability of action $\phi_{1}$ is taken
(under a strategy within the set) divided by the prior $p$,
$PR(\mathcal{S},p)=\max_{S\in\mathcal{S}}\frac{\mathbb{P}(\phi_{1}|S)}{p}$.

Careful readers may notice that if we multiply the persuasion ratio with the
prior, the value will be the (maximum) expected utility the sender can achieve
from the given set of strategies. Since the sender’s expected utility is
monotone increasing in the prior $p$ regardless of which set of strategies the
sender adopts, the persuasion ratio under a given prior can be viewed as the
relative utility gain this set of strategies can offer to the sender. Hence,
given a specific prior, if a set of strategies has a higher persuasion ratio
with respect to another set of strategies, the sender should use a strategy in
the former instead of the latter.

According to this discussion, we can draw a persuasion ratio curve for each
set of strategies as the prior is varied in $[0,1]$. Abusing notation, we
represent the persuasion ratio curve by $PR(\mathcal{S})$. It may appear that
an optimization needs to be carried out for each value of the prior. However,
structural insights presented in the following two lemmas considerably
simplify the analysis. Properties presented in Lemma 3 narrow down the space
where the sender needs to search for the optimal signaling strategy. This
allows us to depict persuasion ratio curves $PR(S)$ for some basic strategies.
On top of that, Lemma 4 provides a systematic approach to derive persuasion
ratio curves for all types of strategies.

###### Lemma 3

Given a type of strategy $\mathcal{S}$ and a prior $p$, there exists a
(sender’s) optimal strategy $S\in\mathcal{S}$ which satisfies one of the
following two conditions:

  1. 1.

At least one IC requirement of the constituent induced strategies is tight;

  2. 2.

There is a signal that will be sent with probability $1$ under $S$.

Before discussing the simplifications that Lemma 3 yields in terms of the key
properties for solving the problem, we give an intuitive outline of the proof
of Lemma 3. When the IC requirements of the two induced strategies are not
tight and both signals are sent with non-zero probability in a strategy $S$,
the sender can increase her expected utility by slightly raising the
probability of the signal with a higher persuasion ratio (and adjust the
probability of the other signal to respect the prior) to form a strategy
$S_{+}$. The sender can keep doing this ‘slight’ modification of her
strategies until either one of the IC conditions is satisfied or the signal is
sent with probability one.

Given this lemma, the persuasion ratio curve of the following types of
strategies: $(\alpha_{A},\gamma_{B})$, $(\beta_{A},\gamma_{B})$,
$(\gamma_{A},\alpha_{B})$, $(\gamma_{A},\beta_{B})$ can be determined
immediately since the IC requirement can never be tight for the $\gamma$ class
induced strategy. For the remaining four types of strategies: $(c_{A},d_{B})$,
$c,d\in\\{\alpha,\beta\\}$, the following lemma aids in solving for the
strategy meeting the persuasion ratio without a point-wise calculation. As a
preview of result of Lemma 5, once we derive the persuasion ratio curve for
each type of strategies via Lemma 4, we can immediate identify the optimal
signaling strategy by overlaying those curves in one figure.

###### Lemma 4

Given the persuasion ratio curves of the types of strategies
$(c_{A},\gamma_{B})$ and $(\gamma_{A},d_{B})$, denoted by
$PR((c_{A},\gamma_{B}),p)$ and $PR((\gamma_{A},d_{B}),p)$, respectively, the
persuasion ratio curve of the set of incentive compatible strategies
$(c_{A},d_{B})$, denoted by $PR((c_{A},d_{B}),p)$, is the generalized concave
hull 666If $PR((c_{A},\gamma_{B}),\cdot)$ and $PR((\gamma_{A},d_{B}),\cdot)$
were the same function $f(\cdot)$, then this would be its convex hull. of the
functions $PR((c_{A},\gamma_{B}),p)$ and $PR((\gamma_{A},d_{B}),p)$ when
$\mathcal{S}_{(c_{A},d_{B})}(p)\neq\emptyset$:

$PR((c_{A},d_{B}),p)=\max_{\begin{subarray}{c}x,u,v\in[0,1],\\\
xu+(1-x)v=p\end{subarray}}xPR((c_{A},\gamma_{B}),u)+(1-x)PR((\gamma_{A},d_{B}),v)),\vspace{-8pt}$
(3)

where $\mathcal{S}_{(c_{A},d_{B})}(p)$ is the set of IC strategies of
$(c_{A},d_{B})$ at $p$.

The proof of Lemma 4 uses the structure of the sender’s expected utility when
no $\gamma$ induced strategy is used in the types of strategies employed. The
sender’s expected utility function under $(c_{A},d_{B})$ at prior $p$ can be
represented as a linear combination of her utility function under
$(c_{A},\gamma_{B})$ at prior $u$ and her utility function under
$(\gamma_{A},d_{B})$ at prior $v$, then the optimization problem of solving
the optimal phase-I experiment parameters $(p_{1},p_{2})$ can be transformed
to the maximization problem in the statement of Lemma 4. As we have discussed
the means to determine each type of strategy’s persuasion ratio curve, Lemma 5
illustrates the persuasion ratio curve of the optimal signaling strategy.

###### Lemma 5

The persuasion ratio curve of the optimal signaling strategy $PR^{*}(p)$ is
the upper envelope of the different types of strategies’ persuasion curves.
Further, the optimal signaling strategy (under a given prior) is the strategy
that reaches the frontier of the persuasion ratio curve (at that prior).

Since a higher persuasion ratio indicates a higher (sender’s) expected utility
for every given prior, the sender will choose the upper envelope of the
persuasion curves of the different types of strategies. Because the set
$\mathcal{S}_{(c_{A},d_{B})}(p)$ could be empty for some prior values with a
corresponding persuasion ratio $PR(c_{A},d_{B}),p)=0$, the main effort in
proving Lemma 5 is to show the existence of an incentive-compatible commitment
on the frontier of the persuasion ratio curve at every possible prior.
Finally, once the persuasion ratio curve of the optimal signaling strategy is
determined, we can immediately infer an optimal signaling strategy $S^{*}$
under a specific prior.

For a two-phase trial or to solve the last two phases of a trial with more
than two phases studied in Section 4, the following corollary can further
simplify the sender’s optimization procedure.

###### Corollary 1

Let $\Pi^{*}(p)$ represents the optimal signaling strategy at prior $p$. If
$\mathbb{P}(\phi_{1}|\Pi^{*}(p))<1$, then the following two statements are
true:

  1. 1.

When $\alpha_{X}$ is used in $\Pi^{*}(p)$, the interim belief is
$\mathbb{P}_{\Pi^{*}(p)}(\theta_{1}|E_{X})=\tfrac{q_{X2}}{q_{x1}+q_{X2}}$.

  2. 2.

When $\beta_{X}$ is used in the optimal signaling strategy $\Pi^{*}(p)$, the
interim belief is
$\mathbb{P}_{\Pi^{*}(p)}(\theta_{1}|E_{X})=\frac{1-q_{X2}}{2-q_{x1}-q_{X2}}$.

With Corollary 1, the Equation (3) in Lemma 4 reduces to a linear equation.
Hence, the comparison in Lemma 5 and the computation in Lemma 4 can be reduced
to a comparison of the (unique) corresponding IC strategies (if one exists)
under the interim belief listed in Corollary1 for different types of
strategies.

###  3.5 Comparison with classical Bayesian persuasion strategies

Given the optimal signaling strategy derived in Lemma 5, one natural follow-up
question is the quantification of the sender’s utility improvement obtained by
adopting the optimal signaling strategy in comparison to using strategies
structurally similar to the optimal strategies in classical Bayesian
persuasion for a binary state of the world. Owing the page limit, we directly
define a class of strategies structurally similar to the classical Bayesian
persuasion strategy below and provide the justification in our online version
[23].

###### Definition 4

With binary states of the world, a (binary-state) Bayesian persuasion (BBP)
strategy is a strategy that “mixes two possible states in one signal and
reveals the true state on the other signal”.

Given the model defined in Section 2.1, a BBP strategy is forced to use at
least one $\gamma_{X}$ induced strategy777Because using either $\alpha_{X}$,
$\beta_{X}$ requires a mixture of two possible states in one signal.. Given a
fixed type of strategy, e.g., $(\alpha_{A},\gamma_{B})$, an optimal BBP
strategy using this type of strategy can be solved by the concavification
approach after the calculation of the sender’s expected utility under interim
beliefs. After solving the optimal BBP strategy of a given strategy type via
concavification respectively, the optimal BBP strategy is the strategy in the
set of
$\\{(\alpha_{A},\gamma_{B}),(\beta_{A},\gamma_{B}),(\gamma_{A},\alpha_{B}),(\gamma_{A},\beta_{B})\\}$
which yields the highest expected utility for the sender. Figure 2 plots the
sender’s expected utility for the optimal signaling strategy and the optimal
BBP strategy under a given pair of phase-II experiments:
$(q_{A1},q_{A2})=(0.8,0.2),(q_{B1},q_{B2})=(0.7,0.3)$. The blue line in Figure
2 is the benchmark of the sender’s maximum expected utility in a single-phase
scenario where the sender chooses the experiment. Note this would also be the
optimal performance if one of phase-II trials were changed to a trivial
experiment. As we can see, the sender’s expected utility is lowered owing to
the determined phase-II experiments. For low-priors, the optimal signaling
strategy derived in Section 3.4 and the optimal BBP strategy give the sender
the same expected utility888Because the optimal signaling strategy in the low-
prior region is $(\alpha_{A},\gamma_{B})$ here.. However, as the prior
increases, a utility gap between the optimal signaling strategy derived in
Section 3.4 and the optimal BBP strategy appears and then increases until the
receiver will take $\phi_{1}$ with probability one. The utility gap starts
when the optimal signaling strategy uses strategies $(\alpha_{A},\beta_{B})$
or $(\alpha_{B},\beta_{A})$ which are not considered in BBP strategies.

Figure 2: Sender’s utility under different problem settings and strategies

##  4 Binary-outcome Experiments in Multi-phase trials

This section generalizes the structural results in Section 3 to multi-phase
trials. First, we generalize the model in Section 2.1 to multi-phase trials
and then propose a dynamic programming algorithm to solve for the optimal
signaling strategy. The state for the dynamic program will be the interim
belief on the state of the world that results at any node in the extensive-
form delineation of the problem. As the belief at each level is determined
based on the actions in earlier stages (if any), in the backward iteration
procedure, we will determine the optimal choice of experiments by the sender
(if there is a choice) for any possible interim belief. In this dynamic
programming, there is only a terminal reward that arises from the receiver’s
action based on the outcome of the final trial and based on the receiver’s
resulting posterior beliefs.

###  4.1 Model of binary-outcome experiments in multi-phase trials

There are $N$ phases in a trial where one binary-outcome experiment will be
conducted in each phase. However, as in the two-phase-trial settings, the
specific experiment conducted in each phase is determined by the earlier
phases’ outcomes. Therefore, we can model an $N$-phase trial by a height-$N$
binary tree where each leaf node represents an outcome revealed to the
receiver, and each non-leaf node represents an experiment. With this binary
tree, experiment $E_{i,j}$ represents the $j^{th}$ experiment to be conducted
at level $i$. When $j$ is odd, an experiment $E_{i,j}$ will be conducted only
if the experiment $E_{i-1,(j+1)/2}$ is conducted and passed. Similarly, when
$j$ is even, an experiment $E_{i,j}$ will be conducted only if the experiment
$E_{i-1,j/2}$ is conducted but it fails. In this binary tree, some
experiments, e.g, $E_{i,j}$, are determined. However, some experiments, e.g.,
$E_{k,l}$, can be designed by the sender; all the parameters are chosen before
any experiment is conducted and are common knowledge. In such experiments, the
sender can choose a probability pair, e.g., $(p_{kl1},p_{kl2})\in[0,1]^{2}$.
In contrast to the model defined in Section 2.1, here determined experiments
and sender-designed experiments can be at any level of the tree with the
placement arbitrary but carried out before the sender receives her signals. In
other words, unlike the model in Section 2.1, a sender may be able to design
an experiment at final level (phase $N$) owing to a determined experiment
outcome at phase $N-1$.

In this model, the prior, the experiments that are determined (their location
on the tree and their parameters), and the sender-designed experiments (their
location on the tree) are common knowledge. The sender has to design all her
experiments simultaneously and before the game starts (when the state is
realized); the designed experiments’ parameters are then revealed to the
receiver (again before the game starts). Given the experiments designed by the
sender and the realized outcome of a sequence of experiments, the receiver
will take an action to guess the true state of the world. For simplicity of
analysis, we keep the sender and the receiver’s utilities the same as in
Section 2. Then, the sender’s objective is to jointly design the set of
experiments that she has the flexibility to choose to maximize her expected
utility, which is nothing but the probability of the receiver taking action
$\phi_{1}$. Before proceeding, we point out that this model can be easily
generalized to unbalanced binary trees straightforwardly by adding dummy nodes
with determined trivial experiments defined in Section 3.1 to construct an
equivalent balanced binary tree.

###  4.2 Determined versus sender-designed experiments

Given the model, the sender can manipulate the phase-$K$ interim belief only
when designing an experiment at phase $K-1$. If an experiment at phase $K-1$
is determined, then the phase-$K$ interim belief is a function of interim
belief at phase $K-1$. Therefore, figuring out how these two types of
experiments, determined and sender-designed experiments, will influence every
given phase’s interim belief is the key to solving for the optimal signaling
strategy. We start by noting that if the posterior belief at a leaf node is
given, then the receiver’s action is determined - he will take the action with
the highest posterior probability unless there is a tie, in which case he is
indifferent and will follow the sender’s recommendation. Therefore, we can use
backward iteration and the principle of optimality to determine the optimal
signaling. We start by considering the last phase’s experiments when the
sender can design them.

####  4.2.1 Experiments at phase $N$

Recall the result we have discussed in Section 3.2, a determined experiment in
the last phase (phase-II in Section 3.2) limits the sender’s strategy choice
to one of three induced strategies. Besides, the best scenario that the sender
can achieve via using these induced strategies (without violating the IC
requirement) is captured by the persuasion potential of the determined
experiment. However, when there is a sender-designed experiment at phase $N$
and the interim belief999See the corresponding footnote in [23] for the
discussion of the interim belief $\tilde{p}>\frac{1}{2}$.
$\tilde{p}\leq\frac{1}{2}$, the sender can always design an experiment which
makes two states equally likely when this experiment passes and reveal the
less-preferred state (by sender) when it fails. If we cast this sender-
designed experiment in terms of a determined experiment, the sender-designed
experiment will have a persuasion potential $(2,2)$101010See the corresponding
footnote in [23] for the detailed derivation.. Thus, no matter the type of
experiment at phase $N$, we can capture the sender’s optimal set of induced
strategies via a persuasion potential.

####  4.2.2 Experiments in phase $N-1$

In the second-last phase, results in Section 3.4 describe a sender-designed
experiment’s role in the optimal signaling strategy: pick the strategy on the
frontier of all persuasion-ratio curves. However, if the experiment is
determined in the second-last phase, an additional constraint on the interim
belief between the second-last phase and the last phase is enforced. That is
to say, the set of (feasible) strategies will shrink. Fortunately, after
enforcing the constraints, the process of searching for the optimal signaling
strategy under a determined experiment is the same as the sender-designed
experiment, i.e., pick the strategy in the frontier of all persuasion ratio
curves. Therefore, at each possible branch of phase $N-1$, we can plot an
optimal persuasion ratio curve capturing the sender’s optimal signaling
strategy at phase $N-1$ and phase $N$.

Input: The set of determined experiments $\mathbf{E_{D}}$, the binary tree
structure

Output: The optimal persuasion ratio curve

1. For each experiment at phase $N$, $E_{N,i}~{}i\in\\{1,...,2^{N}\\}$, solve its persuasion potential $Prep(E_{N,i})$

2. For each experiment at phase $N-1$, $E_{N-1,i}~{}i\in\\{1,...,2^{N-1}\\}$, find the optimal persuasion ratio curve using $(Prep(E_{N,2i-1}),Prep(E_{N,2i}))$.

3. K=N-2

4. while  _$K >0$_ do

For each experiment at phase $K$, $E_{K,i}~{}i\in\\{1,...,2^{K}\\}$, find the
optimal persuasion ratio curve using equation (4) or Claim 2

K=K-1

end while

5. Return the optimal persuasion ratio curve at phase $1$

ALGORITHM 1 Dynamic programming approach for multi-phase trials

####  4.2.3 Experiments in earlier phases

Now we consider experiments in earlier phases. When we have a determined
experiment in phase-$K$, e.g., $E_{k,i}=(q_{Ki1},q_{Ki2})$, and we have solved
the optimal persuasion ratio curves of its succeeding phase ($(K+1)$), i.e.,
$PR^{*}_{K+1,2i-1}(p)$ and $PR^{*}_{K+1,2i}(p)$, then the optimal persuasion
ratio curve at this determined phase-$K$ experiment $E_{k,i}$ is just a linear
combination of $PR^{*}_{K+1,2i-1}(p)$ and $PR^{*}_{K+1,2i}(p)$ can be written
as follows:

$\displaystyle PR^{*}_{K,i}(p)$ $\displaystyle=$
$\displaystyle(pq_{Ki1}+(1-p)q_{Ki2})PR^{*}_{K+1,2i-1}\Big{(}\tfrac{pq_{Ki1}}{pq_{Ki1}+(1-p)q_{Ki2}}\Big{)}+(p(1-q_{Ki1})$
(4) $\displaystyle+$
$\displaystyle(1-p)(1-q_{Ki2}))PR^{*}_{K+1,2i}\Big{(}\tfrac{p(1-q_{Ki1})}{p(1-q_{Ki1})+(1-p)(1-q_{Ki2})}\Big{)}\vspace{-6pt}$

For a sender-designed experiment at phase K, e.g., $E_{K,j}$, if we have
already solved the optimal persuasion ratio curves of its succeeding
phase-$(K+1)$ experiments $E_{K+1,2j-1}$ and $E_{K+1,2j}$, the sender’s best
design at $E_{K,j}$ is to find a linear combination of $PR^{*}_{K+1,2j-1}(p)$
and $PR^{*}_{K+1,2j}(p)$ which yield the highest persuasion ratio for every
phase-$K$ interim belief $p$. Since the persuasion ratio curve is monotone
decreasing in the belief, the optimal persuasion ratio curve can be
constructed similar to Lemma 4 as shown in Corollary 2.

###### Corollary 2

Given two persuasion ratio curves at phase $K+1$, $PR^{*}_{K+1,2j-1}(p)$ and
$PR^{*}_{K+1,2j}(p)$, the optimized persuasion ratio curve $PR^{*}_{K,j}(p)$
at phase $K$ is the maximum convex combination of $PR^{*}_{K+1,2j-1}(p)$ and
$PR^{*}_{K+1,2j}(p)$, i.e.,

$\displaystyle
PR^{*}_{K,j}(p)=\max_{x,u,v\in[0,1],xu+(1-x)v=p}xPR^{*}_{K+1,2j-1}(u)+(1-x)PR^{*}_{K+1,2j}(v)$
(5)

####  4.2.4 Non-binary Outcome Experiments

When the experiments have non-binary outcomes, the same approach derived above
works with an increased number of phases (if complexity is not an issue).

For general non-binary experiments, see the proof of Lemma 6 in [23] for a
detailed construction from non-binary to binary experiments.

###### Lemma 6

Given a non-binary experiment
$E=\\{q_{1,1},...,q_{1,n};q_{2,1},...,q_{2,n}\\}$, we can replace it by
$\lceil\log_{2}n\rceil$ levels of binary outcome experiments.

###  4.3 Multi-phase model and classical Bayesian persuasion

At the end of this section, we mention a class of special multi-phase trials
where the sender’s expected utility under the optimal signaling strategy is
equivalent to utility obtained from a single-phase Bayesian persuasion model.
Inspired by the two-phase example with a trivial experiment in 1, the sender
can implement a signaling strategy similar to single-phase Bayesian persuasion
when there exists a trivial experiment in the last phase and she can design
experiments in earlier phases. When the sender can design all earlier phases,
she can voluntarily reduce the signal space in effect via designing the
experiment $E_{i,j}$ to be $E_{i,j}=(1,1)$ or $E_{i,j}=(0,0)$, i.e., a non-
informative experiment. By doing this, the sender can reduce the multi-phase
trial to an equivalent two-phase trial model and then a straightforward
extension of Lemma 1 will hold when there exists a trivial experiment in the
last phase. The following Lemma 7 further generalizes the class of multi-phase
models where the sender has the same expected utility as a single-phase
Bayesian persuasion problem with a necessary pruning process defined in
Definition 5. Owing to the page limit, explanations and some preliminary
analysis about the robustness of signaling strategies under small
perturbations from trials satisfying Lemma 7 is only available in our online
version [23].

###### Definition 5

Given an N-phase trial model $M$, a pruned N-phase trial model $Prun(M)$ is a
model which recursively replaces every subtree of $M$ by a revealing
experiment $E_{\theta}=(1,0)$ if the subtree satisfies the following
condition, starting from the leaves: the (sub)root of this subtree has a
trivial (determined) experiment $E_{X}$ with at least one of its succeeding
experiments non-trivial (but determined).

Note that the pruned tree will potentially be unbalanced.

###### Lemma 7

Given an $N$-phase trial $M$ with binary-outcome experiments, if there exists
a pruned N-phase trial model $Prun(M)$ such that the following two conditions
hold, then the sender’s expected utility is given by an equivalent single-
phase Bayesian persuasion model.

  1. 1.

For every non-trivial determined experiment, its sibling is either a trivial
or a sender-designed experiment.

  2. 2.

There exists a least one sender-designed experiment in each (from root to
leaf) experiment sequence of $Prun(M)$ .

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