# Reputation-based Persuasion Platforms111The work by Moshe Tennenholtz was
supported by funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement 740435). Itai Arieli acknowledges support from the Ministry of
Science and Technology (grant agreement 19400214).

Itai Arieli, Omer Madmon, Moshe Tennenholtz Faculty of Data and Decision
Sciences, Technion - Israel Institute of Technology,
<EMAIL_ADDRESS>of Data and Decision Sciences, Technion - Israel
Institute of Technology<EMAIL_ADDRESS>of Data and
Decision Sciences, Technion - Israel Institute of Technology,
<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce a two-stage Bayesian persuasion model in which a
third-party platform controls the information available to the sender about
users’ preferences. We aim to characterize the optimal information disclosure
policy of the platform, which maximizes average user utility, under the
assumption that the sender also follows its own optimal policy. We show that
this problem can be reduced to a model of market segmentation, in which
probabilities are mapped into valuations. We then introduce a repeated
variation of the persuasion platform problem in which myopic users arrive
sequentially. In this setting, the platform controls the sender’s information
about users and maintains a reputation for the sender, punishing it if it
fails to act truthfully on a certain subset of signals. We provide a
characterization of the optimal platform policy in the reputation-based
setting, which is then used to simplify the optimization problem of the
platform.

##  1 Introduction

Bayesian persuasion refers to a situation where a sender (such as a seller)
attempts to influence the decision of a receiver (such as a buyer) by
presenting them with information. One classic example is a seller trying to
sell a product of uncertain quality to a buyer. The seller may have some
aggregate knowledge about the preferences and likelihood of purchasing the
product for a group of buyers, but may not be able to distinguish among
individual buyers. In contrast, other parties, such as online selling
platforms like Amazon and eBay, may have access to more specific information
about each buyer and their likelihood of purchasing the product. These
platforms can use this information to reveal relevant characteristics of
individual buyers to the seller, improving the efficiency of the persuasion
process. Our work aims to study this double information disclosure mechanism
in a persuasion setting.

In particular, we study this model under the assumption that the receiver (or
the utility of the receiver) is drawn from a given commonly known distribution
and in parallel independently the quality of the sender’s product is
determined. The quality of the product is known only to the sender and the
user’s utility is known only to the platform. In the first stage, the platform
reveals information to the sender about the receiver and thereafter the
receiver reveals information to the sender as a function of the information it
received from the platform. We study this problem under the assumption that
the platform tries to maximize the expected utility of the receivers.

Unlike the standard Bayesian persuasion setting, in our model, we have a two-
stage process of information revelation. We study this model in two different
settings: a one-shot setting and a repeated setting. In the repeated setting,
which is inspired by reputation considerations, reciprocity, and punishments
play a significant role.

We consider the standard product adoption setting with a binary set of states
of the world (e.g., high-quality product and low-quality product) and a binary
set of receiver actions (e.g., buy and do not buy the product). The sender
only knows a prior distribution of user types, and the platform knows the
realized user type. After the sender receives information from the platform it
forms a posterior on the user type. Then, the state of the world (product
quality) is drawn from a commonly known prior distribution, and is observed
only by the sender. The sender then recommends according to the committed
persuasion strategy and the receiver forms a posterior probability on the
product quality and decides whether to buy or not. Lastly, the sender and
receiver obtain their payoffs based on the receiver’s action and the product
quality. 222Note that the terms ’receiver’ and ’user’ are used
interchangeably.

Our problem, then, is to find the subgame perfect equilibrium that is optimal
for the platform, i.e., an information disclosure policy for the platform that
maximizes the users’ utility. We start with analyzing the one-shot setting. In
this setting, we demonstrate how our model can be reduced to the seemingly
unrelated market segmentation model proposed by Bergemann et al. [BBM15]. In
their model, a monopolistic producer determines an optimal price of a product
in a given market, which is a distribution over consumers’ valuations.
Bergemann et al. provide an efficient algorithm for finding a segmentation of
the initial market into multiple markets, such that the consumers’ utility is
maximized assuming the producer determines an optimal price at each segment
separately. In their terminology, segmentation is essentially a distribution
over markets, such that its expectation is the initial market. The reduction
from the persuasion platform problem to the market segmentation problem is
done by mapping the persuasion thresholds of the users (i.e., the
probabilities of lying to users regarding the product quality) into valuations
in the market segmentation problem.

Next, we introduce a repeated variation of the persuasion platform problem, in
which myopic users arrive sequentially. The one-shot interaction between the
platform, the sender, and the users is similar to before, except now the
platform can not only control the sender’s user information, but it also
maintains a reputation state of the sender. That is, the platform requires the
sender to act truthfully on some signals and punishes the sender if it does
not respect the requirement by permanently suspending the sender from the
market and transmitting it from a high reputation to a low reputation, for all
subsequent time periods.

At the low reputation state, the sender obtains a known, constant utility that
is lower than the monopolistic sender utility, i.e., the utility of the sender
when it is provided with no user information from the platform and relies only
on the prior user distribution. This punishment utility comes from an outside
option, which is less desirable for the sender than the case in which the
platform provides no user information at all.

The lower the utility of the low reputation state is, the larger the
punishment power the platform has, meaning it can request a more truthful
behavior while still being incentive-compatible from the perspective of the
sender.

For the repeated setting, we provide a characterization of an optimal platform
policy. This policy relies on the fact that in the high reputation state, it
is optimal for the platform to adopt the segmentation which is similar to the
one it provided in the one-shot case.

Our contribution is threefold: First, we define a novel model of persuasion
platforms, which captures, for example, the strategic nature of recommendation
systems and aims to provide a mechanism to increase users’ welfare on average.
Second, we show a reduction from the one-shot persuasion platform to the
market segmentation model of Bergemann et al. [BBM15]. This reduction enables
solving the persuasion platform problem in efficient runtime. Finally, we
propose the reputation-based persuasion platform model and provide a
characterization of the optimal platform policy. The characterization makes
use of the algorithm of Bergemann et al. [BBM15], which is used also to solve
the one-shot case.

The paper is organized as follows: in section 3 we introduce and study the
one-shot persuasion platform problem. In section 3.1 we present the persuasion
platform model and the problem of finding an optimal platform policy, in
section 3.2 we discuss the market segmentation model [BBM15], and in section
3.3 we show the reduction from the persuasion platforms problem to the market
segmentation problem. Section 4 then discusses the repeated, reputation-based
variation of the persuasion platforms problem: in section 4.1 we introduce the
model, and section 4.2 deals with the characterization of an optimal platform
policy. The proof of the main Theorem appears in section 4.3. We then conclude
in section 5.

##  2 Related Work

There is a rich literature on markets with asymmetric information. Akerlof
introduced the very first model [Ake70], which introduced the market for
“lemons”, in which sellers have private information regarding the product
quality. More recent models of information disclosure include the cheap talk
[CS82, FR96] and Bayesian persuasion [KG11]. In our model, we rely on the
Bayesian persuasion framework, which differs from cheap talk by the fact that
the sender has commitment abilities in addition to its private information.
Our model describes a two-level persuasion process, in which a third-party
platform controls the information the sender has about the users.

There is also literature on multi-level, sequential Bayesian persuasion, e.g.
[ABS22, LN21, Mah22]. In contrast to these works, in our model the information
that is revealed to the sender is on the user type, whereas the information
that is revealed by the sender is on the product quality. In addition, we also
extend the sequential persuasion setting to a repeated, reputation-based
setting, in which sequential persuasion games are being played for an infinite
number of time periods with myopic users, and characterize the optimal
platform policy in terms of both information revelation and punishment
strategy.

Signaling schemes are well-studied in various economic settings. For instance,
the authors of [EFG+14] have studied optimal signaling schemes in the context
of a second-price auction with probabilistic goods, where optimality is
measured with respect to the auctioneer’s revenue. In contrast, our problem
deals with maximizing the users’ average utility. The authors of [KMP14] have
also studied a repeated dynamic persuasion model, in which the sender is
aiming to persuade the receivers towards exploration in order to maximize
social welfare. Notice that while they considered a single level of
persuasion, in our model we study a two-level persuasion: from the platform to
the sender and from the sender to the receiver.

We rely on the market segmentation model of Bergemann et. al. [BBM15] and
utilize their algorithm to solve our one-shot persuasion platforms problem and
characterize an optimal platform policy in the reputation-based persuasion
platforms problem.

In our reputation-based setting the platform also punishes the sender once it
is caught lying to a user it should not have lied to according to the
platform’s request. The role of reputation in repeated games with incomplete
information is well studied, e.g. at [KW82, CMS04]. In our reputation-based
setting, we exploit a similar notion of reputation in an information design
setting. In particular, [Tad16] studied the role of reputation in online
platform markets, and highlighted empirical evidence for the fact that
reputation systems often have a positive effect on online markets, in terms of
both stability and efficiency. Related theoretical models of reputation and
information design are considered by [MPS22, BQ20]. They used reputation in a
repeated setting in a model that relaxes the commitment power assumption of
the sender.

Other work considered dynamic Bayesian Persuasion models, in which a sender
and a receiver interact repeatedly, and the state of the world evolves
according to a Markovian law [LS21, RSV17]. Since in our model the state of
the world is drawn independently every time period, our problem has a
stationary optimal solution. The authors of [WZF+22] considered a different
Markovian setting, in which an informed sender is willing to persuade a stream
of myopic receivers to take actions that maximize its cumulative utilities.
While they focus on the sender’s optimal persuasion strategy, in this work we
aim to characterize an optimal platform policy that maximizes the average
receivers’ utility.

In [IS22] the authors deal with the receiver’s welfare by presenting a sender-
receiver Bayesian model in which the sender can request additional information
from the receiver. While we are also interested in the receiver’s welfare, we
take a mechanism design perspective rather than allowing additional
communication from the receiver to the sender.

##  3 Persuasion Platforms

###  3.1 The Model

We denote by
$\Theta=\\{\theta_{1},...\theta_{n}\\}\subset\operatorname*{\mathbb{R}}_{+}$
the set of users’ types. We assume $0<\theta_{1}<...<\theta_{n}$.
$x^{*}\in\Delta(\Theta)$ is the prior user distribution.
$\Omega=\\{\omega_{0},\omega_{1}\\}$ is the set of states of the world, i.e.
product qualities. We think of $\omega_{0}$ as the state of the world in which
the product is of low quality, and $\omega_{1}$ is the state of the world in
which the product is of high quality. $\mu\in\Delta(\Omega)$ is the prior
distribution over product qualities. We identify $\mu=P_{\mu}(\omega_{1})$.
Let $A=\\{a_{0},a_{1}\\}$ be the set of user actions (and sender
recommendations), corresponding to buying ($a_{1}$) and not buying ($a_{0}$)
the product. We denote the user’s action by $a$ and the sender’s
recommendation by $\tilde{a}$.

A strategy of the platform is a signaling policy $\sigma:\Theta\to\Delta(S)$
for some abstract finite signal realizations space $S$. The platform’s
strategy can be thought of as a set of conditional distributions over $S$
(conditioned on the user type). Alternatively, a strategy of the platform is a
Bayes-plausible distribution of distributions
$\sigma\in\Delta(\Delta(\Theta))$. That is, $|supp(\sigma)|<\infty$ and
$\operatorname*{\mathbb{E}}_{x\sim\sigma}[x]=x^{*}$. We denote the set of all
Bayes-plausible distributions by $\Sigma$.

A strategy of the sender is a recommendation policy conditioned on the signal
realization and the state of the world, i.e. $p:\Omega\times S\to\Delta(A)$.
We denote $p(\omega,s)=P_{p}(\tilde{a}=a_{1}|\omega,s)$ for any $s\in
S,\omega\in\Omega$. It is a well-known result that the optimal sender strategy
always satisfies $p(\omega_{1},s)=1$ for any $s\in S$. Therefore, the sender’s
strategy can be solely characterized by its recommendation policy conditioned
on the low-quality product, i.e. $p:S\to\Delta(A)$, where
$p(s)=P_{p}(\tilde{a}=a_{1}|\omega_{0},s)$, and
$P_{p}(\tilde{a}=a_{1}|\omega_{1},s)=1$. Note that the domain of $p$ can be
defined as $\Delta(\Theta)$ instead of $S$, as each signal realization induces
a posterior probability over the user types.

The sender has a utility function of $u_{S}(a_{0})=0,u_{S}(a_{1})=1$, and a
user of type $\theta$ has a utility function of
$u_{R}^{\theta}(a_{0},\omega_{0})=u_{R}^{\theta}(a_{0},\omega_{1})=0,u_{R}^{\theta}(a_{1},\omega_{0})=-1,u_{R}^{\theta}(a_{1},\omega_{1})=\theta$.
The platform’s utility is the users’ average utility (w.r.t the distribution
of user distributions induced by its signaling policy $\sigma$). The
interaction between the three entities (platform, sender, and user) is then
defined as follows:

  1. 1.

The platform commits to a strategy $\sigma$.

  2. 2.

The sender observes $\sigma$, and commits to its own strategy $p$.

  3. 3.

A user $\theta\sim x^{*}$ is drawn, and its type is visible only to the
platform.

  4. 4.

The platform sends a signal $s\in S$ to the sender, according to the committed
policy $\sigma$. That is, $s\sim\sigma(\theta)$.

  5. 5.

The state of the world $\omega\sim\mu$ is drawn, and it is visible to the
sender only.

  6. 6.

If $\omega=\omega_{1}$, the sender sends a recommendation $\tilde{a}=a_{1}$ to
the user. Otherwise, the sender sends a recommendation according to the
committed policy $p$. That is, $\tilde{a}\sim p(s)$.

  7. 7.

The user observes $\tilde{a}$ and plays $a=a_{1}$ if and only if
$\operatorname*{\mathbb{E}}_{\omega\sim\tilde{\mu}}[u_{R}^{\theta}(a_{1},\omega)]\geq\operatorname*{\mathbb{E}}_{\omega\sim\tilde{\mu}}[u_{R}^{\theta}(a_{0},\omega)]$,
where $\tilde{\mu}\in\Delta(\Omega)$ is the posterior computed by the user
based on $\mu$ and $\tilde{a}$ (otherwise it plays $a=a_{0}$). Simple algebra
reveals that the condition holds for a user of type $\theta$ if and only if
$p(s)\leq\frac{\mu}{1-\mu}\theta$. We refer to the quota
$\frac{\mu}{1-\mu}\theta$ as the persuasion threshold of a $\theta$-type user,
and denote it by $\tau_{\theta}$.

Figure 1 visualizes the timing of the Platform, Sender, and User interaction.
A user of type $\theta$ then plays according to the best-response mapping:

$BR_{\theta}(\tilde{a},p)=\begin{cases}\tilde{a}&p\leq\tau_{\theta}\\\
a_{0}&p>\tau_{\theta}\end{cases}$

where $\tilde{a}$ is the received recommendation and
$p=P(\tilde{a}=a_{1}|\omega_{0})$; The sender is maximizing:

$U_{S}(\sigma,p)=\operatorname*{\mathbb{E}}_{x\sim\sigma}[U_{S}(p;x)]=\operatorname*{\mathbb{E}}_{x\sim\sigma}\operatorname*{\mathbb{E}}_{\theta\sim
x}\operatorname*{\mathbb{E}}_{\omega\sim\mu}\operatorname*{\mathbb{E}}_{\tilde{a}\sim
p(\omega,x)}[u_{S}(BR_{\theta}(\tilde{a},p(\omega,x)))]$

and the platform is maximizing:

$U_{P}(\sigma,p)=\operatorname*{\mathbb{E}}_{x\sim\sigma}[U_{P}(p;x)]=\operatorname*{\mathbb{E}}_{x\sim\sigma}\operatorname*{\mathbb{E}}_{\theta\sim
x}[U_{R}^{\theta}(p;x)]=\operatorname*{\mathbb{E}}_{x\sim\sigma}\operatorname*{\mathbb{E}}_{\theta\sim
x}\operatorname*{\mathbb{E}}_{\omega\sim\mu}\operatorname*{\mathbb{E}}_{\tilde{a}\sim
p(\omega,x)}[u_{R}^{\theta}(BR_{\theta}(\tilde{a},p(\omega,x)),\omega))]$

Note that we use the notation $U_{S}(p;x)$ for the sender’s expected utility
from playing $p$ conditional on the user posterior distribution $x$, and
denote by $U_{S}(\sigma,p)$ the expected utility when the platform provides
user information according to policy $\sigma$, which is taking the expectation
over all $x\sim\sigma$ (and similarly we use $U_{P}(p;x)$ and
$U_{P}(\sigma,p)$, to denote the expected utility of the platform).
$U_{R}^{\theta}(p;x)$ is the expected utility of a $\theta-$type user when it
responds optimally, as a function of the sender’s strategy $p$ and the user
distribution $x$.

The following lemma provides a closed form of the sender and platform
utilities (the proof is omitted for brevity):

######  Lemma 1.

For every pair of strategies $(\sigma,p)$, the sender and platform utilities
are:

$U_{S}(\sigma,p)=\operatorname*{\mathbb{E}}_{x\sim\sigma}[U_{S}(p;x)]=\sum_{x\in
supp(\sigma)}\sigma(x)\sum_{j=1}^{n}x_{j}\operatorname*{\mathbb{1}}_{p(x)\leq\tau_{\theta_{j}}}(\mu+(1-\mu)p(x))$
$U_{P}(\sigma,p)=\operatorname*{\mathbb{E}}_{x\sim\sigma}[U_{P}(p;x)]=\sum_{x\in
supp(\sigma)}\sigma(x)\sum_{j=1}^{n}x_{j}\operatorname*{\mathbb{1}}_{p(x)\leq\tau_{\theta_{j}}}(\mu\theta_{j}-(1-\mu)p(x))$

where $\operatorname*{\mathbb{1}}_{a\leq b}=1$ if $a\leq b$ and $0$ otherwise.

Note that it is straight-forward to see that any optimal sender strategy
$p^{*}$ must satisfy that for every $x\in\Delta(\Theta)$,
$p^{*}(x)\in\\{\tau_{\theta}\\}_{\theta\in\Theta}$.

PlatformSenderUser$\theta\sim
x^{*}$$\omega\sim\mu$$s\sim\sigma(\theta)$$\tilde{a}\sim p(\omega,s)$ Figure
1: The interaction between the Platform, the Sender, and the User.

###  3.2 Market Segmentation

We now describe a model for market segmentation, presented by [BBM15]. In this
model a monopolist producer sells a good to a continuum of consumers, where
each consumer demands exactly one unit of the good. We denote by
$V=\\{v_{1},...v_{n}\\}$ the set of consumer valuations for the good. We
assume $0<v_{1}<...<v_{n}$. A market $\pi\in\Pi=\Delta(V)$ is a distribution
over the possible valuations.

In a given market $\pi$, the demand for the good at any price in the interval
$(v_{k-1},v_{k}]$ is $\sum_{j=k}^{n}\pi_{j}$ (with the convention that
$v_{0}=0$). A price $v_{k}$ is said to be optimal (for the producer) in a
market $\pi$ if for all $i\in[n]$: $v_{k}\sum_{j=k}^{n}\pi_{j}\geq
v_{i}\sum_{j=i}^{n}\pi_{j}$. We denote by $\Pi_{k}$ the set of all markets
where price $v_{k}$ is optimal. We hold a given initial (aggregate) market
denoted by $\pi^{*}\in\Pi$. We denote by $v^{*}=v_{i^{*}}$ the optimal uniform
price for the initial market $\pi^{*}$. Thus, $\pi^{*}\in\Pi^{*}=\Pi_{i^{*}}$.

Segmentation is a division of the aggregate market into different markets.
Thus, a segmentation $\sigma$ is a finite-support distribution over markets,
with the interpretation that $\sigma(\pi)$ is the proportion of the population
in the market $\pi$. Thus, the set of possible segmentations is:

$\Sigma=\\{\sigma\in\Delta(\Pi)|\sum_{\pi\in
supp(\sigma)}\sigma(\pi)\pi=\pi^{*},|supp(\sigma)|<\infty\\}$.

A pricing rule for a segmentation $\sigma$ specifies a price for each segment
(market in the support of $\sigma$). Formally, a pricing rule for a
segmentation $\sigma$ is a function $\phi:supp(\sigma)\to V$. A pricing rule
$\phi$ is optimal if for each segment $\pi\in supp(\sigma)$, $\phi(\pi)=v_{k}$
implies $\pi\in\Pi_{k}$. That is, the charged price in every market $\pi$ must
be optimal for the producer.

We denote the utility of a consumer with valuation $v_{j}$ when charged price
is $v$ by $W_{j}(v)=\operatorname*{\mathbb{1}}_{v\leq v_{j}}(v_{j}-v)$, and
the utility of a producer charging price $v$ in the market $\pi$ by
$W_{S}(v;\pi)=v\sum_{j=1}^{n}\operatorname*{\mathbb{1}}_{v\leq v_{j}}\pi_{j}$.

Given a segmentation $\sigma$ and a pricing rule $\phi$, we define the
consumer surplus:

$W_{C}(\sigma,\phi)=\sum_{\pi\in
supp(\sigma)}\sigma(\pi)\sum_{j=1}^{n}\pi_{j}W_{j}(\phi(\pi))=\sum_{\pi\in
supp(\sigma)}\sigma(\pi)\sum_{j=1}^{n}\pi_{j}\operatorname*{\mathbb{1}}_{\phi(\pi)\leq
v_{j}}(v_{j}-\phi(\pi))$

and the producer surplus:

$W_{S}(\sigma,\phi)=\sum_{\pi\in
supp(\sigma)}\sigma(\pi)W_{S}(\phi(\pi);\pi)=\sum_{\pi\in
supp(\sigma)}\sigma(\pi)\phi(\pi)\sum_{j=1}^{n}\operatorname*{\mathbb{1}}_{\phi(\pi)\leq
v_{j}}\pi_{j}$

The market segmentation model can naturally be extended to the case where the
determined price $v$ is not necessarily in the set of user types $V$ (that is,
$\phi:supp(\sigma)\to\operatorname*{\mathbb{R}}_{+}$), although it is clear
that any optimal pricing rule must satisfy $\forall\pi\in
supp(\sigma):\phi^{*}(\pi)\in V$. [BBM15] also characterizes the set of
feasible producer and consumer surpluses, see Figure 2.

Consumer surplusProducer surplus$A$$B$$C$ Figure 2: The feasible surplus
triangle of the market segmentation problem. No segmentation achieves point A.
[BBM15] segmentation algorithm achieves point B.

###  3.3 Optimal Platform Policy: From Probabilities To Valuations

In this section, we show a reduction from the persuasion platform problem to
the market segmentation problem. We then rely on the fact that [BBM15]
provides an algorithm for finding a consumer-surplus-maximizing segmentation
in order to solve the platform persuasion problem. The reduction is done by
defining an equivalent segmentation problem, in which the users’ valuations
for the products are the sender utilities from selling the product to users in
the persuasion problem, and the initial (aggregate) market is the prior user
distribution. We show that the games induced by the two problems are
strategically equivalent, and therefore solving the equivalent segmentation
problem imminently yields a solution for the platform persuasion problem.

######  Definition 1.

Given a persuasion platform problem instance $(\Theta,x^{*},\mu)$, the
equivalent market segmentation problem is the market segmentation problem
instance $(V,\pi^{*})$ defined as follows:

$V=\\{\mu+(1-\mu)\tau_{\theta}\\}_{\theta\in\Theta}=\\{\mu(1+\theta)\\}_{\theta\in\Theta}$

$\pi^{*}=x^{*}$

Notice that in the equivalent market segmentation problem, the valuations of
the consumers are derived from the users’ persuasion thresholds, i.e. the
probabilities of the sender recommending low-quality products which make the
users indifferent between buying and not buying the product when the sender
recommends. In other words, _probabilities are translated into valuations._ In
the equivalent market segmentation problem, each user distribution $x$ is
identified with a market $\pi$, and each sender policy $p$ is identified with
a pricing rule of the producer $\phi(\pi)=\mu+(1-\mu)p(x)$.

######  Theorem 1.

Given a persuasion platform problem instance $(\Theta,x^{*},\mu)$, consider
the equivalent market segmentation problem $(V,\pi^{*})$. For any user
distribution $x$ (and its corresponding market $\pi$) and for any sender
policy $p$ (and its corresponding pricing rule $\phi$) the following
properties hold:

  1. 1.

$U_{S}(p;x)=W_{S}(\phi(\pi);\pi)$

  2. 2.

$\forall j\in[n]$: $U_{R}^{\theta_{j}}(p;x)=W_{j}(\phi(\pi))$

###### Proof.

Let $x\in\Delta(\Theta)$ and a sender policy $p$. First, notice that
$p(x)\leq\frac{\mu}{1-\mu}\theta_{j}$ if and only if $\phi(\pi)\leq v_{j}$
Since $f(q)=\mu+(1-\mu)q$ is monotonically increasing, and
$\phi(\pi)=f(p(x))$. Therefore, from the definition of the sender and producer
utilities, it follows that:

$U_{S}(p;x)=\sum_{j=1}^{n}x_{j}\operatorname*{\mathbb{1}}_{p(x)\leq\frac{\mu}{1-\mu}\theta_{j}}(\mu+(1-\mu)p(x))=\sum_{j=1}^{n}\pi_{j}\operatorname*{\mathbb{1}}_{\phi(\pi)\leq
v_{j}}\phi(\pi)=W_{S}(\phi(\pi);\pi)$

Then, notice that if $\phi(\pi)\leq v_{j}$ the utility of consumer of type
$v_{j}$ from buying the product at price $\phi(\pi)$ is exactly:

$W_{j}(\phi(\pi))=v_{j}-\phi(\pi)=\mu+(1-\mu)(\frac{\mu}{1-\mu}\theta_{j})-\mu-(1-\mu)p(x)=(1-\mu)(\frac{\mu}{1-\mu}\theta_{j}-p(x))=\mu\theta_{j}-(1-\mu)p(x)=U_{R}^{\theta_{j}}(p;x)$

and otherwise they both equal $0$. ∎

Note that the equivalent market segmentation problem is indeed strategically
equivalent to the persuasion problem only since we have restricted the
sender’s strategy space to strategies in which the sender always recommends
buying a good product. The following corollaries, which follow immediately
from Theorem 1, will be used to solve the problem of finding an optimal
platform policy in a given persuasion platform problem:

######  Corollary 1.

For every distribution $x\in\Delta(\Theta)$ (and the corresponding
distribution $\pi\in\Delta(V)$) there exists some $k\in[n]$, such that the
optimal sender strategy and the optimal pricing rule satisfy
$\phi(\pi)=v_{k}$, $p^{*}(x)=\frac{\mu}{1-\mu}\theta_{k}$.

######  Corollary 2.

For every Bayes-plausible distribution of user-distributions
$\sigma\in\Sigma$, $U_{S}(\sigma,p)=W_{S}(\sigma,\phi)$ and
$U_{P}(\sigma,p)=W_{C}(\sigma,\phi)$, where $p$ and $\phi$ are the best-
responses to $\sigma$ at the persuasion and segmentation problems
respectively.

######  Corollary 3.

Let $\sigma^{*}$ be the segmentation that maximizes the consumer surplus at
the equivalent market segmentation problem. Then $\sigma^{*}$ also maximizes
the average user utility at the persuasion platform problem.

Corollary 1 follows directly from the first property of Theorem 1. As for the
Corollary 2, first note that the set of platform strategies is exactly
$\Sigma$, since for every Bayes-plausible distribution $\sigma\in\Sigma$ there
exists a signaling policy that induces it. The Corollary 2 then follows from
this fact combined with Corollary 1, and the second property of Theorem 1.
Corollary 3 follows directly from the Corollary 2.

We conclude this section by recalling that there exists an efficient algorithm
for finding a consumer surplus maximizing segmentation:

######  Theorem 2.

(Bergemann, Brooks, and Morris [BBM15]) There exists an algorithm that given a
market segmentation problem instance $(V,\pi^{*})$ finds a consumer surplus
maximizing segmentation $\sigma^{*}$ in $O(|V|)$. Moreover, this segmentation
satisfies $W_{S}(\sigma^{*},\phi^{*})=W_{S}(\phi^{*}(\pi^{*});\pi^{*})$ (where
$\phi^{*}$ is an optimal pricing rule).

The fact that the reduction is done in linear time (combined with Theorem 2
and Corollary 3) implies that there also exists an algorithm for finding an
average user utility maximizing platform policy $\sigma^{*}$ in every
persuasion platform problem instance $(\Theta,x^{*},\mu)$. Moreover, this
policy satisfies $U_{S}(\sigma^{*},p^{*})=U_{S}(p^{*};x^{*})$. We denote the
policy obtained by applying the algorithm (with respect to the prior user
distribution $x^{*}$) by $BBM(x^{*})$.

Achieving the Pareto-frontier. In practice, a two-sided platform may care
about the welfare of both sellers and buyers, aiming to find a balance between
the two. For example, the platform may charge fees from the sellers (which
increase when the seller’s revenue increases), or the platform wants to
preserve the satisfaction of both the users and the sellers to avoid
retention. Note that since the persuasion problem can be reduced to a problem
of market segmentation, the set of feasible platform and sender utilities in
the persuasion platform problem has a similar structure as in Figure 2.
Therefore, solving the problem of maximizing the average user utility can be
used to achieve any pair of sender and platform utilities on the Pareto-
frontier: since optimal sender utility (point C in Figure 2) can be achieved
by full revelation of the user type, and since every pair of utilities on the
Pareto-frontier is a convex combination of C and B, one can achieve any such
point by simply randomizing between the user optimal policy (derived from
[BBM15]) and the full revelation policy.

##  4 Reputation-based Persuasion Platforms

###  4.1 The Model

We now turn to extend the persuasion platform model to a repeated case, in
which myopic users, drawn i.i.d from the prior user distribution
$x^{*}\in\Delta(\Theta)$, arrive sequentially for an infinite number of time
periods. In this setting the platform has the ability to impose an
irreversible punishment on the sender. The punishment results in a loss for
both the sender and the platform. The setting may be motivated by reputation
considerations where the platform and the sender are engaged in a contract in
which the platform requires the sender to fully reveal the state on a subset
of signals. If the sender violates the contract it is thrown out of the
platform to some less desirable outside option.

Formally, on the outset, the platform commits to an information revelation
policy $\sigma:\Theta\to\Delta(S)$ which will be used for all subsequent time
periods, and additionally it specifies a subset of signal $S_{T}\subset S$
(or, equivalently, on a subset of the posteriors $X_{T}\subset supp(\sigma)$),
on which it requires from the sender to act truthfully. i.e., recommend a
product of low quality with probability zero.

The sender then commits to its own persuasion policy $p:S\to\Delta(A)$ (where
again we identify $p(s)$ with the probability of recommending a low-quality
product after observing $s\in S$), and then users begin to arrive one by one.

The platform maintains a reputation for the sender, which can be either high
or low at the beginning of each time period. We assume that at the beginning
of the first time period, the sender has a high reputation.

If at time period $t$ the reputation of the sender is high, then the
interaction between the platform, the sender, and the user at time $t$ is the
same as in the one-shot model, with the distinction that now the platform also
observes the satisfaction level of the user at the end of the interaction. We
assume that the satisfaction level of a user from the interaction is bad if it
purchased a product of low quality, and good otherwise.

If the platform provided to the sender a signal on which it requires truthful
behavior, and the satisfaction level the platform observes is bad (namely, the
sender had a high reputation, and manipulated the user into buying a product
of low quality), then the sender is being permanently moved to the low
reputation state for all future time periods. Otherwise, the sender begins the
next round with a high reputation. Formally, the sender is being moved from
high to low reputation after time period $t$, if and only if $s_{t}\in S_{T}$,
$\omega_{t}=\omega_{0}$ and $a_{t}=a_{1}$.

At the low reputation state, the sender suffers from a fixed punishment
utility $\bar{u}$, satisfying $\bar{u}<U_{S}(p^{*};x^{*})$. That is, in the
low reputation state the sender’s utility is strictly lower than in the case
where it has no user information at all. One can interpret the low reputation
state as an alternative market or an external option, that is less favorable
to the sender relative to the scenario where the platform does not provide any
information about the user’s preferences. When at low reputation, the utility
for the user is defined to be zero.

The sender then plays to optimize its discounted utility stream for some
discount factor $0<\delta<1$, while the platform optimizes the non-discounted
average user utility.

Let $\Sigma$ be the set of all Bayes-plausible distributions with respect to
the prior distribution $x^{*}$. A policy of the platform is now a tuple
$(\sigma,X_{T})$, where $\sigma\in\Sigma$ and $X_{T}\subset supp(\sigma)$ is
the subset of posterior on which the platform requires the sender to play
truthfully. Note that once fixing a platform policy $(\sigma,X_{T})$, one can
think of the sender as an agent operating in an induced Markov decision
process (MDP), as demonstrated in Figure 3.

HLOtherwiseSender lied on $X_{T}$w.p. 1 Figure 3: Reputation-based Persuasion
Platform as an MDP for the sender.

Unlike the one-shot problem, the reputation-based persuasion problem cannot be
reduced to an equivalent market segmentation problem with reputation. The
reason is that in the one-shot identification, we identified prices with the
probability of recommending a low-quality product. In the repeated market
segmentation problem, prices are verifiable and this can be used by a platform
to deter the sender. In contrast, in our case, the only verifiable information
is whether a low-quality product has been purchased. This implies that the
strategic problems in the repeated setting are no longer equivalent.

Without loss of generality, we assume that $|X_{T}|=1$, and denote by $x^{T}$
the single posterior in $X_{T}$.333This is due to the fact that for any
incentive-compatible policy satisfying $|X_{T}|>1$, one can construct a new
incentive-compatible policy providing the same average user utility, for which
$|X_{T}|=1$ by simply merging all of the posteriors in $X_{T}$. To shorten, we
refer to the tuple $(\sigma,x_{T})$ by $\sigma$ only. We further denote:

$supp(\sigma)=\\{x^{1},...x^{m},x^{T}\\}$

$\forall i\in[m]:\alpha^{i}=\sigma(x^{i})$

$\alpha^{T}=\sigma(x^{T})$

Denoting $p_{j}=\tau_{\theta_{j}}$,
$I_{j}(x,p)=x_{j}\operatorname*{\mathbb{1}}_{p(x)\leq p_{j}}$, the sender and
platform one-shot utilities at the high reputation state are given by,

$U_{S}(\sigma,p)=\alpha^{T}\sum_{j=1}^{n}I_{j}(x^{T},p)(\mu+(1-\mu)p(x^{T}))+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}I_{j}(x^{i},p)(\mu+(1-\mu)p(x^{i})))$
$U_{P}(\sigma,p)=\alpha^{T}\sum_{j=1}^{n}I_{j}(x^{T},p)(\mu\theta_{j}-(1-\mu)p(x^{T}))+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}I_{j}(x^{i},p)(\mu\theta_{j}-(1-\mu)p(x^{i})))$

We now define two types of possible sender strategies:

######  Definition 2.

A strategy of the sender $p^{*}$ is greedy if for all $x\in\Delta(\Theta)$:

$p^{*}(x)=\operatorname*{argmax}_{p}U_{S}(p;x)$

######  Definition 3.

A strategy of the sender $p_{T}$ is truthful (with respect to a given platform
policy $\sigma$) if there exists a greedy strategy $p^{*}$ such that for all
$x\in\Delta(\Theta)$:

$p_{T}(x)=\begin{cases}0&\text{$x=x^{T}$}\\\ p^{*}(x)&\text{$x\neq x^{T}$}\\\
\end{cases}$

We assume that when the sender is indifferent between targeting multiple types
of users (i.e., when $\operatorname*{argmax}_{p}U_{S}(p;x)$ is not uniquely
defined), it targets the lowest type among them. That is, we uniquely define
the optimal sender strategy to be
$p^{*}(x)=\min\operatorname*{argmax}_{p}U_{S}(p;x)$. Therefore, the truthful
strategy $p_{T}$ is also uniquely defined.

Note that without loss of generality, we can only consider platform policies
that are incentive-compatible, i.e., policies in which the sender does not
benefit from deviating from being truthful (lying with zero probability) when
it operates at the posterior $x^{T}$ (clearly, the sender will always be
greedy at any other posterior, as lying at such posterior has no consequences
at all):

######  Definition 4.

A policy of the platform $\sigma\in\Sigma$ is incentive-compatible (IC) if
$p_{T}$ is the sender’s best response with respect to $\sigma$.

For any $x\in\Delta(\Theta)$ and $k\in[n]$, we denote by
$F_{k}(x)=\sum_{j=k}^{n}x_{j}$ the mass of users in $x$ whose persuasion
threshold is weakly above $p_{k}$. Given a policy $\sigma$, we denote by
$\sigma_{F}=\sigma|_{x\neq x^{T}}$ the policy obtained by conditioning on the
posterior to be any non-truthful posterior, and denote by
$x^{F}=\operatorname*{\mathbb{E}}_{x\sim\sigma_{F}}[x]=\operatorname*{\mathbb{E}}_{x\sim\sigma}[x|x\neq
x^{T}]$ its mean. Note that one could decompose the sender and platform
utilities as follows:

$U_{S}(\sigma,p)=\alpha^{T}U_{S}(p;x^{T})+(1-\alpha^{T})U_{S}(\sigma_{F},p)$

$U_{P}(\sigma,p)=\alpha^{T}U_{P}(p;x^{T})+(1-\alpha^{T})U_{P}(\sigma_{F},p)$

We denote by $V(\sigma)=U_{S}(\sigma,p_{T})$ the sender’s one-shot utility
when playing truthfully with respect to a platform policy $\sigma$. Since
playing $p_{T}$ ensures that the sender remains in the high reputation state,
$V(\sigma)$ is also the overall, long-term utility of the sender when playing
truthfully. In particular, $V(\sigma)$ is the sender’s long-term utility when
responding optimally to an incentive-compatible platform policy $\sigma$.

Notice that in the reputation-based setting, the problem has a straightforward
solution in the following cases: first, for a given fixed punishment level
$\bar{u}$, and for large enough discount factor $\delta$, the policy where the
platform requests from the sender a completely truthful behavior on all
signals is incentive-compatible and therefore optimal. This follows from the
fact that the sender’s utility from a deviation approaches $\bar{u}$ as
$\delta$ goes to one. Similarly, for a fixed discount factor $\delta$ and for
a low enough punishment level $\bar{u}$, the platform can also require a
truthful behavior of the sender, which again will be incentive-compatible. We
aim to study the platform’s optimal signaling policy in the general case,
where $\bar{u}$ and $\delta$ are such that the sender might benefit from
deviation from the truthful strategy with respect to the platform’s request.

###  4.2 Optimal Platform Policy Characterization

In this section, we provide a useful characterization of an optimal platform
policy in the reputation-based setting, which will then be used to simplify
the platform’s optimization problem of finding an optimal policy. We start by
defining two standard properties of a platform policy:

######  Definition 5.

A platform policy $\sigma\in\Sigma$ is Pareto-efficient if
$\forall\tilde{\sigma}\in\Sigma$:

  1. 1.

$U_{S}(\sigma,p_{T})<U_{S}(\tilde{\sigma},p_{T})\Rightarrow
U_{P}(\sigma,p_{T})>U_{P}(\tilde{\sigma},p_{T})$

  2. 2.

$U_{P}(\sigma,p_{T})<U_{P}(\tilde{\sigma},p_{T})\Rightarrow
U_{S}(\sigma,p_{T})>U_{S}(\tilde{\sigma},p_{T})$

######  Definition 6.

A platform policy $\sigma\in\Sigma$ is lowest-type-targeting if $\forall x\in
supp(\sigma_{F})$:

$p^{*}(x)=\frac{\mu}{1-\mu}\theta^{x}_{min}$

where $\theta^{x}_{min}=\min supp(x)$.

That is, a platform policy is lowest-type-targeting if, for every user
distribution in its support, the sender’s optimal policy is to play according
to the lowest type’s persuasion threshold (i.e., it does not benefit from
increasing the probability of lying in the price of losing the lowest type
users). Notice that when a platform policy is lowest-type-targeting, all users
always follow the sender’s recommendation, hence all products are sold. In
Lemma 9 we show that this property is equivalent to Pareto-efficiency.

We start with a standard result showing that the set of all optimal platform
policies in the reputation-based persuasion platform setting must contain at
least one Pareto-efficient policy $\sigma$, for which
$U_{S}(\sigma_{F},p^{*})=U_{S}(p^{*};x^{F})$ (that is, the property of [BBM15]
is satisfied with respect to the policy conditioned on the fact that the
platform does not require truthful behavior). Note that we define an optimal
platform policy as an incentive-compatible policy maximizing the platform
utility (which is the average user utility) over all incentive-compatible
platform policies. That is, denoting by $\Sigma^{*}$ the set of all optimal
platform policies, we have that $\Sigma^{*}\subseteq\Sigma_{IC}$ where
$\Sigma_{IC}$ is the set of all incentive-compatible platform policies. The
optimization problem of the platform can be then written as follows:

$\displaystyle\max_{\sigma\in\Sigma_{IC}}$ $\displaystyle U_{P}(\sigma,p_{T})$
(1)

Note that the problem defined in (1) is an infinite-dimensional optimization
problem. To simplify the optimization problem, we introduce an optimal
platform policy characterization in Theorem 3:

######  Theorem 3.

Denote by $\Sigma^{*}$ the set of all optimal platform policies. Then, there
must exists a Pareto-efficient policy $\sigma\in\Sigma^{*}$ such that
$U_{S}(\sigma_{F},p^{*})=U_{S}(p^{*};x^{F})$.

Theorem 3 is then used to simplify the optimization problem of finding the
optimal platform policy as follows:

######  Corollary 4.

The platform’s optimization problem can be solved using the following two-step
procedure:

  * •

Solve the following finite-dimensional optimization problem:

$\displaystyle\min_{\sigma=(\alpha,x^{T},x^{F})}$ $\displaystyle
U_{S}(\sigma,p_{T})$ (2) s.t. $\displaystyle 0\leq\alpha\leq 1$ $\displaystyle
x^{T},x^{F}\in\Delta(\Theta)$ $\displaystyle\alpha
x^{T}+(1-\alpha)x^{F}=x^{*}$ $\displaystyle\sigma\in\Sigma_{IC}$

  * •

Apply the [BBM15] algorithm with respect to $x^{F}$.

Corollary 4 follows immediately from Theorem 3: the first stage yields the
lowest sender utility that can be achieved by an incentive-compatible platform
policy (follows from Theorem 3). Hence, the maximal platform utility is
bounded from above by the point on the Pareto frontier in which the Sender
achieves the above minimal utility. In order to complete the construction we
need to show that this platform utility is indeed achievable. To see this, we
use the same approach as in the one-shot case: we apply the reduction to
[BBM15] with respect to $x^{F}$, and achieve the Pareto-efficient utility
which is maximal for the platform and does not effect the sender’s utility.
Note that incentive compatibility does not break since the utility for the
sender and and $x^{T}$ are not being modified (follows from Corollary 5
below).

Unlike the problem defined in (1), the simplified problem in (2) is a finite-
dimensional optimization problem, and is more likely to be solved analytically
or numerically. However, the optimization problem is non-convex, and therefore
it is still generally difficult (this is due to the fact that the incentive-
compatibility constraints are non-convex, see Lemma 2).

###  4.3 Proof of Theorem 3

To show this characterization we first introduce some auxiliary lemmas which
will be used in the proof of Theorem 3 (and the proof of Theorem 3 will then
follow). To begin with, notice that for a given platform policy $\sigma$,
$p_{T}(x)=p^{*}(x)$ for any posterior $x\neq x^{T}$. Moreover, it is clear
that playing greedily at any non-truthful posterior is optimal for the sender,
as it maximizes its one-shot utility and cannot lead to punishment (i.e., the
sender being moved by the platform from a high to a low reputation state).
Therefore, any potentially profitable deviation from $p_{T}$ must be
conditioned on the sender being at the truthful signal $x^{T}$. This
observation leads to the following alternative definition of incentive
compatibility:

######  Lemma 2.

A policy of the platform $\sigma\in\Sigma$ is incentive-compatible if and only
if for all $k\in[n]:$

$V(\sigma)\geq\frac{1-\delta}{\delta}\cdot(\frac{\mu}{1-\mu}\cdot\frac{F_{k}(x^{T})-1}{F_{k}(x^{T})p_{k}}+1)+\bar{u}$

###### Proof.

Given a platform policy $\sigma$, note that a deviation from $p_{T}$ can occur
only conditional on $x=x^{T}$. It is clear that such deviation should be of
the form $p(x^{T})=p_{k}$ for some $k\in[n]$. Playing $p_{T}$ always leaves
the sender at the high reputation state, hence its expected utility from the
current timestep is $\mu$ and the expected utility from all future timesteps
is $V(\sigma)$.

On the other hand, while deviating to $p_{k}$ the expected utility from the
current timestep is $F_{k}(x^{T})(\mu+(1-\mu)p_{k})$ and the expected utility
from all future timesteps is $\bar{u}$ with probability
$(1-\mu)F_{k}(x^{T})p_{k}$ (which corresponds to the product being of low
quality, the sender decides to recommend purchasing it and the drawn user has
persuasion threshold above $p_{k}$), and $V(\sigma)$ otherwise.

Therefore, the following inequality reflects the fact that deviation from
$p_{T}$ to $p_{k}$ conditional on $x\neq x^{T}$ is not beneficial for the
sender:

$(1-\delta)\mu+\delta
V(\sigma)\geq(1-\delta)F_{k}(x^{T})(\mu+(1-\mu)p_{k})+\delta(1-\mu)F_{k}(x^{T})p_{k}\bar{u}+\delta(1-(1-\mu)F_{k}(x^{T})p_{k})V(\sigma)\Leftrightarrow$

$\delta
V(\sigma)\geq(1-\delta)(F_{k}(x^{T})\mu-\mu+F_{k}(x^{T})(1-\mu)p_{k})+\delta(1-\mu)F_{k}(x^{T})p_{k}\bar{u}+\delta(1-(1-\mu)F_{k}(x^{T})p_{k})V(\sigma)\Leftrightarrow$

$\delta(1-\mu)F_{k}(x^{T})p_{k}V(\sigma)\geq(1-\delta)(F_{k}(x^{T})\mu-\mu+F_{k}(x^{T})(1-\mu)p_{k})+\delta(1-\mu)F_{k}(x^{T})p_{k}\bar{u}\Leftrightarrow$

$V(\sigma)\geq\frac{(1-\delta)(F_{k}(x^{T})\mu-\mu+F_{k}(x^{T})(1-\mu)p_{k})+\delta(1-\mu)F_{k}(x^{T})p_{k}\bar{u}}{\delta(1-\mu)F_{k}(x^{T})p_{k}}=\frac{(1-\delta)(F_{k}(x^{T})\mu-\mu+F_{k}(x^{T})(1-\mu)p_{k})}{\delta(1-\mu)F_{k}(x^{T})p_{k}}+\bar{u}=\frac{1-\delta}{\delta}\cdot(\frac{\mu}{1-\mu}\cdot\frac{F_{k}(x^{T})-1}{F_{k}(x^{T})p_{k}}+1)+\bar{u}$

When the above inequality holds for every $k\in[n]$, $p_{T}$ is indeed a best-
response of the sender to the platform policy $\sigma$. ∎

A key corollary regarding incentive-compatibility is that if two policies
$\sigma,\tilde{\sigma}\in\Sigma$ have the same truthful distribution and
guarantee the same sender utility from being truthful, then either both are
incentive-compatible or both are not:

######  Corollary 5.

Let $\sigma,\tilde{\sigma}\in\Sigma$ such that $x^{T}=\tilde{x}^{T}$ and
$V(\sigma)=V(\tilde{\sigma})$. Then $\sigma$ is incentive-compatible if and
only if $\tilde{\sigma}$ is incentive-compatible.

We continue with the following three technical lemmas, implying that given a
platform policy $\sigma\in\Sigma$ such that
$U_{S}(\sigma_{F},p^{*})>U_{S}(p^{*};x^{F})$, there exists policies
$\sigma_{l},\sigma_{h}\in\Sigma$ such that
$V(\sigma_{l})<V(\sigma)<V(\sigma_{h})$:

######  Lemma 3.

Let $\epsilon>0$. Given a platform policy $\sigma$, define
$\Sigma_{\epsilon}=\\{\tilde{\sigma}\in\Sigma:\tilde{\alpha}^{T}=(1-\epsilon)\alpha^{T}\wedge\tilde{x}^{T}=x^{T}\\}$.
Then for any $\sigma_{\epsilon}\in\Sigma_{\epsilon}$, $x^{F}_{\epsilon}\to
x^{F}$ as $\epsilon\to 0$.

###### Proof.

Let $\sigma_{\epsilon}\in\Sigma_{\epsilon}$. From Bayes-plausibility of
$\sigma$ and $\sigma_{\epsilon}$ we get:

$x^{*}=\alpha^{T}x^{T}+(1-\alpha^{T})x^{F}$

$x^{*}=(1-\epsilon)\alpha^{T}x^{T}+(1-(1-\epsilon)\alpha^{T})x^{F}_{\epsilon}$

Combining the two equations, we get:

$(1-\epsilon)\alpha^{T}x^{T}+(1-(1-\epsilon)\alpha^{T})x^{F}_{\epsilon}=\alpha^{T}x^{T}+(1-\alpha^{T})x^{F}$

Isolating $x^{F}_{\epsilon}$ yields:

$(1-(1-\epsilon)\alpha^{T})x^{F}_{\epsilon}=(\alpha^{T}-(1-\epsilon)\alpha^{T})x^{T}+(1-\alpha^{T})x^{F}=\epsilon\alpha^{T}x^{T}+(1-\alpha^{T})x^{F}$

$x^{F}_{\epsilon}=\frac{\epsilon\alpha^{T}x^{T}+(1-\alpha^{T})x^{F}}{1-(1-\epsilon)\alpha^{T}}\to\frac{(1-\alpha^{T})x^{F}}{1-\alpha^{T}}=x^{F}$

when $\epsilon\to 0$. ∎

######  Lemma 4.

Let $\sigma\in\Sigma$ such that $U_{S}(\sigma_{F},p^{*})>U_{S}(p^{*};x^{F})$.
Then, there exists some $0<\epsilon^{*}<1$ and
$\sigma_{l}\in\Sigma_{\epsilon^{*}}$ such that $V(\sigma_{l})<V(\sigma)$.

###### Proof.

For any $\epsilon^{*}>0$, denote by $\sigma_{\epsilon}$ the unique platform
policy that satisfies $\sigma_{\epsilon}\in\Sigma_{\epsilon}$ and
$\sigma^{F}_{\epsilon}=BBM(x^{F}_{\epsilon})$. As shown in [BBM15], this
policy satisfies

$(1)\ U_{S}(p^{*};x^{F}_{\epsilon})=U_{S}(\sigma^{F}_{\epsilon},p^{*})$

We also know that,

$(2)\ U_{S}(p^{*};x^{F})<U_{S}(\sigma_{F},p^{*})$

From the assumption on $\sigma$. In addition, $U_{S}(p^{*};x)$ is continuous
in $x$, therefore from Lemma 3 we know that,

$(3)\ U_{S}(p^{*};x^{F}_{\epsilon})\to U_{S}(p^{*};x^{F})$

when taking $\epsilon\to 0$. Now, taking $\epsilon\to 0$ yields:

$V(\sigma_{\epsilon})=(1-\epsilon)\alpha^{T}\mu+(1-(1-\epsilon)\alpha^{T})U_{S}(\sigma^{F}_{\epsilon},p^{*})\underset{(1)}{=}$

$(1-\epsilon)\alpha^{T}\mu+(1-(1-\epsilon)\alpha^{T})U_{S}(p^{*};x^{F}_{\epsilon})\underset{(3)}{\to}$
$\alpha^{T}\mu+(1-\alpha^{T})U_{S}(p^{*};x^{F})\underset{(2)}{<}\alpha^{T}\mu+(1-\alpha^{T})U_{S}(\sigma_{F},p^{*})=V(\sigma)$

Therefore, for small enough $\epsilon^{*}>0$, the policy
$\sigma_{l}=\sigma_{\epsilon^{*}}$ as defined above satisfies
$V(\sigma_{l})<V(\sigma)$. ∎

######  Lemma 5.

Let $\sigma\in\Sigma$ such that $U_{S}(\sigma_{F},p^{*})>U_{S}(p^{*};x^{F})$.
Then, for any $0<\epsilon<1$, there exists $\sigma_{h}\in\Sigma_{\epsilon}$
such that $V(\sigma_{h})>V(\sigma)$.

###### Proof.

Let $0<\epsilon<1$. Define $\sigma^{\prime}$ and $\sigma^{\prime\prime}$ as
follows:

$supp(\sigma^{\prime})=\\{x^{\prime 1},...x^{\prime n},x^{\prime T}\\}$

$x^{\prime T}=x^{T},\alpha^{\prime T}=\alpha^{T}$

$\forall i,j\in[n]:x^{\prime
i}_{j}=\operatorname*{\mathbb{1}}_{i=j},\alpha^{\prime
i}=(1-\alpha^{T})x^{F}_{i}$

$supp(\sigma^{\prime\prime})=\\{x^{\prime\prime 1},...x^{\prime\prime
n},x^{\prime\prime T}\\}$

$x^{\prime\prime T}=x^{T},\alpha^{\prime\prime T}=(1-\epsilon)\alpha^{T}$

$\forall i,j\in[n]:x^{\prime\prime
i}_{j}=\operatorname*{\mathbb{1}}_{i=j},\alpha^{\prime\prime
i}=(1-\alpha^{T}+\epsilon\alpha^{T})x^{F}_{\epsilon,i}$

By construction, $\sigma^{\prime}\in\Sigma$ and
$\sigma^{\prime\prime}\in\Sigma_{\epsilon}$. First, let us show that
$V(\sigma^{\prime})<V(\sigma^{\prime\prime})$. Start by defining:

$\lambda\coloneqq\frac{\epsilon\alpha^{T}}{1-\alpha^{T}+\epsilon\alpha^{T}}$

$\forall j\in[n]:u_{j}\coloneqq\mu+(1-\mu)p_{j}$

$W_{F}\coloneqq\sum_{j=1}^{n}x^{F}_{j}u_{j},W_{T}\coloneqq\sum_{j=1}^{n}x^{T}_{j}u_{j}$

Notice that since $\sigma^{\prime\prime}\in\Sigma_{\epsilon}$,
$x^{\prime\prime F}=x^{F}_{\epsilon}=\lambda x^{T}+(1-\lambda)x^{F}$. It is
straightforward to show that $U_{S}(\sigma^{\prime}_{F},p^{*})=W_{F}$. We next
express $U_{S}(\sigma^{\prime\prime}_{F},p^{*})$ in terms of $\lambda,W_{F}$
and $W_{T}$:

$U_{S}(\sigma^{\prime\prime}_{F},p^{*})=\sum_{j=1}^{n}x^{F}_{\epsilon,j}u_{j}=\sum_{j=1}^{n}(\lambda
x^{T}+(1-\lambda)x^{F})_{j}u_{j}$

$=\lambda\sum_{j=1}^{n}x^{T}_{j}u_{j}+(1-\lambda)\sum_{j=1}^{n}x^{F}_{j}u_{j}=\lambda
W_{T}+(1-\lambda)W_{F}$

Now,

$V(\sigma^{\prime\prime})>V(\sigma^{\prime})\Leftrightarrow$
$\alpha^{\prime\prime T}\mu+(1-\alpha^{\prime\prime
T})U_{S}(\sigma^{\prime\prime}_{F},p^{*})>\alpha^{\prime
T}\mu+(1-\alpha^{\prime T})U_{S}(\sigma^{\prime}_{F},p^{*})$

$\Leftrightarrow(1-\epsilon)\alpha^{T}\mu+(1-(1-\epsilon)\alpha^{T})U_{S}(\sigma^{\prime\prime}_{F},p^{*})>\alpha^{T}\mu+(1-\alpha^{T})U_{S}(\sigma^{\prime}_{F},p^{*})$

$\Leftrightarrow(1-\alpha^{T}+\epsilon\alpha^{T})U_{S}(\sigma^{\prime\prime}_{F},p^{*})>\epsilon\alpha^{T}\mu+(1-\alpha^{T})U_{S}(\sigma^{\prime}_{F},p^{*})$

$\Leftrightarrow(1-\alpha^{T}+\epsilon\alpha^{T})(\lambda
W_{T}+(1-\lambda)W_{F})>\epsilon\alpha^{T}\mu+(1-\alpha^{T})W_{F}$

$\Leftrightarrow\epsilon\alpha^{T}W_{T}+(1-\alpha^{T})W_{F}>\epsilon\alpha^{T}\mu+(1-\alpha^{T})W_{F}$

which indeed holds because $W_{T}>\mu$. Next, notice that $V(\sigma)\leq
V(\sigma^{\prime})$ follows directly from the Blackwell Theorem [Bla53], as
$\sigma_{F}$ is a refinement of $\sigma^{\prime}_{F}$. Therefore, we overall
get $V(\sigma)\leq V(\sigma^{\prime})<V(\sigma^{\prime\prime})$, so setting
$\sigma_{h}=\sigma^{\prime\prime}$ completes the proof. ∎

We now turn to introduce a set of lemmas dealing with the connections between
Pareto-efficiency, lowest-type-targeting, and optimality of platform policies:

######  Lemma 6.

For any lowest-type-targeting $\sigma\in\Sigma$:

  1. 1.

$U_{S}(\sigma_{F},p^{*})=\mu+\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

  2. 2.

$V(\sigma)=U_{S}(\sigma,p_{T})=\mu+(1-\mu)\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

  3. 3.

$U_{P}(\sigma_{F},p^{*})=\frac{\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}x^{i}_{j}\theta_{j})-\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

  4. 4.

$U_{P}(\sigma,p_{T})=\mu\sum_{j=1}^{n}x^{*}_{j}\theta_{j}-(1-\mu)\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

###### Proof.

First, the fact that $\sigma$ is lowest-type-targeting implies that
$I_{j}(x^{i},p^{*})=x_{j}^{i}$ for any $i,j\in[n]$ (since $x_{j}^{i}>0$
implies $\operatorname*{\mathbb{1}}_{p^{*}(x^{i})\leq p_{j}}=1$). Now, we get
that:

$U_{S}(\sigma_{F},p^{*})=\sum_{i=1}^{m}\sigma_{F}(x^{i})\sum_{j=1}^{n}I_{j}(x^{i},p^{*})(\mu+(1-\mu)p^{*}(x^{i}))=\sum_{i=1}^{m}\frac{\alpha^{i}}{1-\alpha^{T}}\sum_{j=1}^{n}x^{i}_{j}(\mu+(1-\mu)p^{*}(x^{i}))=$

$\frac{1}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\mu+(1-\mu)p^{*}(x^{i}))\sum_{j=1}^{n}x^{i}_{j}=\frac{1}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\mu+(1-\mu)p^{*}(x^{i}))=$

$\mu\frac{1}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}+(1-\mu)\frac{1}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})=\mu+\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

$U_{P}(\sigma_{F},p^{*})=\sum_{i=1}^{m}\sigma_{F}(x^{i})\sum_{j=1}^{n}I_{j}(x^{i},p^{*})(\mu\theta_{j}-(1-\mu)p^{*}(x^{i}))=\sum_{i=1}^{m}\frac{\alpha^{i}}{1-\alpha^{T}}\sum_{j=1}^{n}x^{i}_{j}(\mu\theta_{j}-(1-\mu)p^{*}(x^{i}))=$

$\frac{1}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}\sum_{j=1}^{n}x^{i}_{j}(\mu\theta_{j}-(1-\mu)p^{*}(x^{i}))=\frac{\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}x^{i}_{j}\theta_{j})-\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})\sum_{j=1}^{n}x^{i}_{j}=$

$\frac{\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}x^{i}_{j}\theta_{j})-\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

Now, plugging into $U_{S}(\sigma,p_{T})$ and $U_{P}(\sigma,p_{T})$, we get:

$U_{S}(\sigma,p_{T})=\alpha^{T}\mu+(1-\alpha^{T})U_{S}(\sigma_{F},p^{*})=\alpha^{T}\mu+(1-\alpha^{T})\mu+(1-\alpha^{T})\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})=\mu+(1-\mu)\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

$U_{P}(\sigma,p_{T})=\alpha^{T}\mu(\sum_{j=1}^{n}x^{T}_{j}\theta_{j})+(1-\alpha^{T})U_{P}(\sigma_{F},p^{*})=\alpha^{T}\mu(\sum_{j=1}^{n}x^{T}_{j}\theta_{j})+(1-\alpha^{T})\frac{\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}x^{i}_{j}\theta_{j})-(1-\alpha^{T})\frac{1-\mu}{1-\alpha^{T}}\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})=$

$\mu(\alpha^{T}\sum_{j=1}^{n}x^{T}_{j}\theta_{j}+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}x^{i}_{j}\theta_{j}))-(1-\mu)\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})=$

$\mu\sum_{j=1}^{n}x^{*}_{j}\theta_{j}-(1-\mu)\sum_{i=1}^{m}\alpha^{i}p^{*}(x^{i})$

where the last equality comes from Bayes-plausibility. ∎

######  Lemma 7.

For any platform policy $\sigma\in\Sigma$ which is not lowest-type-targeting,
there exists a platform policy $\tilde{\sigma}\in\Sigma$ such that
$x^{T}=\tilde{x}^{T}$, $\alpha^{T}=\tilde{\alpha}^{T}$,
$U_{P}(\tilde{\sigma},p_{T})\geq U_{P}(\sigma,p_{T})$ and
$U_{S}(\tilde{\sigma},p_{T})>U_{S}(\sigma,p_{T})$.

###### Proof.

Let $\sigma\in\Sigma$ be some platform policy which is not lowest-type-
targeting, i.e., there exists $x\in supp(\sigma_{F})$ such that
$p^{*}(x)=\frac{\mu}{1-\mu}\theta^{x}_{opt}>\frac{\mu}{1-\mu}\theta^{x}_{min}$.
Denote by $i$ and $j$ the indices satisfying $\theta_{i}=\theta^{x}_{min}$ and
$\theta_{j}=\theta^{x}_{opt}$, and notice that $i<j$. Now, define a new policy
$\tilde{\sigma}$ which is similar to $\sigma$, except $x$ is decomposed into
$y$ and $z$ as follows:

$y=\frac{1}{F_{j}(x)}(0,...0,x_{j},x_{j+1},...x_{n})$

$z=\frac{1}{1-F_{j}(x)}(x_{1},...x_{j-1},0,...0)$

$\tilde{\sigma}(y)=F_{j}(x)\sigma(x),\tilde{\sigma}(z)=(1-F_{j}(x))\sigma(x)$

Notice that $\tilde{\sigma}$ is Bayes-plausible by construction. Clearly
$x^{T}=\tilde{x}^{T}$, $\alpha^{T}=\tilde{\alpha}^{T}$. Now, notice that
$p^{*}(z)<p^{*}(x)$ (as $\theta^{x}_{opt}>\max supp(z)$) and
$p^{*}(y)=p^{*}(x)$:

$j=\operatorname*{argmax}_{l=1,...n}\sum_{k=l}^{n}x_{k}(\mu+(1-\mu)p_{l})=$

$\operatorname*{argmax}_{l=j,...n}\sum_{k=l}^{n}x_{k}(\mu+(1-\mu)p_{l})=$

$\operatorname*{argmax}_{l=j,...n}\frac{1}{F_{j}(x)}\sum_{k=l}^{n}x_{k}(\mu+(1-\mu)p_{l})=$

$\operatorname*{argmax}_{l=j,...n}\sum_{k=l}^{n}y_{k}(\mu+(1-\mu)p_{l})=j^{\prime}$

where $j^{\prime}$ is the index for which
$p^{*}(y)=\frac{\mu}{1-\mu}\theta_{j^{\prime}}$.

Now, notice that users’ utility weakly increases in $\tilde{\sigma}$ with
respect to $\sigma$, since users moved from $x$ to $z$ can only benefit while
users moved from $x$ to $y$ remain with the same utility. Therefore,
$U_{P}(\sigma,p_{T})\leq U_{P}(\tilde{\sigma},p_{T})$.

As for the sender’s utility, notice that:

$U_{S}(\tilde{\sigma},p_{T})>U_{S}(\sigma,p_{T})\Leftrightarrow
U_{S}(\tilde{\sigma}_{F},p^{*})>U_{S}(\sigma_{F},p^{*})\Leftrightarrow$

$\tilde{\sigma}(y)\sum_{k=1}^{n}y_{k}\operatorname*{\mathbb{1}}_{p^{*}(y)\leq
p_{k}}(\mu+(1-\mu)p^{*}(y))+\tilde{\sigma}(z)\sum_{k=1}^{n}z_{k}\operatorname*{\mathbb{1}}_{p^{*}(z)\leq
p_{k}}(\mu+(1-\mu)p^{*}(z))>\sigma(x)\sum_{k=1}^{n}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))\Leftrightarrow$

$F_{j}(x)\sum_{k=1}^{n}y_{k}\operatorname*{\mathbb{1}}_{p^{*}(y)\leq
p_{k}}(\mu+(1-\mu)p^{*}(y))+(1-F_{j}(x))\sum_{k=1}^{n}z_{k}\operatorname*{\mathbb{1}}_{p^{*}(z)\leq
p_{k}}(\mu+(1-\mu)p^{*}(z))>\sum_{k=1}^{n}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))\Leftrightarrow$

$\sum_{k=j}^{n}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))+(1-F_{j}(x))\sum_{k=1}^{n}z_{k}\operatorname*{\mathbb{1}}_{p^{*}(z)\leq
p_{k}}(\mu+(1-\mu)p^{*}(z))>\sum_{k=1}^{n}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))\Leftrightarrow$

$(1-F_{j}(x))\sum_{k=1}^{n}z_{k}\operatorname*{\mathbb{1}}_{p^{*}(z)\leq
p_{k}}(\mu+(1-\mu)p^{*}(z))>\sum_{k=1}^{j-1}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))\Leftrightarrow$

$\sum_{k=1}^{j-1}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(z)\leq
p_{k}}(\mu+(1-\mu)p^{*}(z))>\sum_{k=1}^{j-1}x_{k}\operatorname*{\mathbb{1}}_{p^{*}(x)\leq
p_{k}}(\mu+(1-\mu)p^{*}(x))$

Now, notice that the left-hand side is strictly greater than zero since
$p^{*}(z)$ is the optimal sender strategy at segment $z$, and
$z=(x_{1},...x_{j-1},0...0)$ up to the constant $(1-F_{j}(x))$. On the other
hand, the right-hand side equals zero since for all
$k\in\\{1,...j-1\\}:p^{*}(x)>p_{k}$. Therefore the condition holds and we get
$U_{S}(\tilde{\sigma},p_{T})>U_{S}(\sigma,p_{T})$.

∎

######  Lemma 8.

There exists an optimal policy $\sigma\in\Sigma^{*}$ which is lowest-type-
targeting.

###### Proof.

Let $\sigma\in\Sigma^{*}$ be some optimal platform policy. If $\sigma$ is
lowest-type-targeting then we are done. Otherwise, from Lemma 7 there exists a
platform policy $\tilde{\sigma}$ such that $x^{T}=\tilde{x}^{T}$,
$\alpha^{T}=\tilde{\alpha}^{T}$, $U_{P}(\tilde{\sigma},p_{T})\geq
U_{P}(\sigma,p_{T})$ and $U_{S}(\tilde{\sigma},p_{T})>U_{S}(\sigma,p_{T})$.
From Corollary 5 combined with the fact that $\sigma$ is incentive-compatible,
it follows that $\tilde{\sigma}$ is also incentive-compatible. Therefore it
follows that $U_{P}(\tilde{\sigma},p_{T})=U_{P}(\sigma,p_{T})$, otherwise it
contradicts the fact that $\sigma\in\Sigma^{*}$, hence
$\tilde{\sigma}\in\Sigma^{*}$. Now, if $\tilde{\sigma}$ is lowest-type-
targeting then we’re done, otherwise, we repeat the same process. Note that
this process can only be repeated a finite number of times, therefore we must
end up with an optimal lowest-type-targeting platform policy. ∎

######  Lemma 9.

A platform policy $\sigma\in\Sigma$ is lowest-type-targeting if and only if it
is Pareto-efficient.

###### Proof.

Lemma 7 implies that any platform policy which is not lowest-type-targeting is
also not Pareto-efficient. To show the opposite direction, we now show that
the sum of the sender and platform utility of any platform policy is maximal
for lowest-type-targeting policies: From Lemma 6, the sum of utilities for any
lowest-type-targeting $\tilde{\sigma}$ is:

$U_{P}(\tilde{\sigma},p_{T})+U_{S}(\tilde{\sigma},p_{T})=\mu(1+\sum_{j=1}^{n}x^{*}_{j}\theta_{j})$

And for any $\sigma\in\Sigma$ the sum of utilities is:

$U_{P}(\sigma,p_{T})+U_{S}(\sigma,p_{T})=\alpha^{T}\mu(1+\sum_{j=1}^{n}x^{T}_{j}\theta_{j})+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}I_{j}(x^{i},p^{*})(1+\theta_{j}))\leq$

$\alpha^{T}\mu(1+\sum_{j=1}^{n}x^{T}_{j}\theta_{j})+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}(1+\theta_{j}))=$

$\mu+\alpha^{T}\mu(\sum_{j=1}^{n}x^{T}_{j}\theta_{j})+\sum_{i=1}^{m}\alpha^{i}(\sum_{j=1}^{n}(\theta_{j}))=\mu(1+\sum_{j=1}^{n}x^{*}_{j}\theta_{j})$

where the last equality comes from Bayes-plausibility. Now, assume by
contradiction that there exists a lowest-type-targeting policy that is not
Pareto-efficient. Then there exists some other policy such that the sum of
utilities is strictly larger, in contradiction to the claim we just proved. ∎

The following corollary then follows directly from Lemma 8 and Lemma 9:

######  Corollary 6.

Denote by $\Sigma_{P}$ the set of all Pareto-efficient platform policies.
Then, the set of all Pareto-efficient optimal platform policies
$\Sigma^{*}\cap\Sigma_{P}$ is nonempty.

We now turn to prove Theorem 3, which provides a characterization of the
optimal platform policy:

###### Proof of Theorem 3..

Assume by contradiction that this is not the case. That is, for any
$\sigma\in\Sigma^{*}$ we have $U_{S}(\sigma_{F},p^{*})>U_{S}(p^{*};x^{F})$.
Denote by $\Sigma_{P}$ the set of all Pareto-efficient platform policies. From
Corollary 6, the set $\tilde{\Sigma}\coloneqq\Sigma^{*}\cap\Sigma_{P}$ is not
empty.

Let
$\sigma\in\operatorname*{argmin}_{\sigma^{\prime}\in\tilde{\Sigma}}{\alpha^{\prime
T}}$. From the assumption we know that
$U_{S}(\sigma_{F},p^{*})>U_{S}(p^{*};x^{F})$. We will show that there exists
$\hat{\sigma}\in\tilde{\Sigma}$ such that $\hat{\alpha}^{T}<\alpha^{T}$, in
contradiction to the minimality of $\alpha^{T}$.

For any $\epsilon>0$, let us define the following set of platform policies:

$\Sigma_{\epsilon}=\\{\tilde{\sigma}\in\Sigma:\tilde{\alpha}^{T}=(1-\epsilon)\alpha^{T}\wedge\tilde{x}^{T}=x^{T}\\}$.

and note that for every $\sigma_{\epsilon}\in\Sigma_{\epsilon}$,
$x^{F}_{\epsilon}=\operatorname*{\mathbb{E}}_{x\sim\sigma^{F}_{\epsilon}}[x]=\operatorname*{\mathbb{E}}_{x\sim\sigma_{\epsilon}}[x|x\neq
x^{T}]$ is uniquely defined. From Lemma 4 there exist $\epsilon^{*}>0$ and
$\sigma_{l}\in\Sigma_{\epsilon^{*}}$ such that $V(\sigma_{l})<V(\sigma)$. From
Lemma 5, for the same $\epsilon^{*}$ there exists
$\sigma_{h}\in\Sigma_{\epsilon^{*}}$ such that $V(\sigma_{h})>V(\sigma)$. $V$
is a continuous function and $\Sigma_{\epsilon^{*}}$ is a convex set,
therefore one can take a convex combination of the two policies and construct
a new policy $\tilde{\sigma}\in\Sigma_{\epsilon^{*}}$ such that
$V(\tilde{\sigma})=V(\sigma)$. From Corollary 5 $\tilde{\sigma}$ is incentive-
compatible (as $\sigma$ is incentive compatible, $\tilde{x}^{T}=x^{T}$ and
$V(\tilde{\sigma})=V(\sigma)$). It is now left to show that $\tilde{\sigma}$
provides the same average user utility as $\sigma$ does.

First, we recall that Bergemann et. al. [BBM15] showed that for every
segmentation $\tilde{\sigma}_{F}\in\Sigma$ there exists a Pareto-efficient
segmentation $\hat{\sigma}_{F}\in\Sigma$ such that
$W_{S}(\tilde{\sigma}_{F},\phi^{*})=W_{S}(\hat{\sigma}_{F},\phi^{*})$. Using
their claim it can be shown that there exists a platform policy
$\hat{\sigma}\in\Sigma_{\epsilon^{*}}$ such that
$U_{S}(\tilde{\sigma}_{F},p^{*})=U_{S}(\hat{\sigma}_{F},p^{*})$, and
$\hat{\sigma}$ is Pareto-efficient. Note that $\hat{\sigma}$ is still
incentive-compatible from the same arguments concerning $\tilde{\sigma}$.

Using the fact that both $\sigma$ and $\hat{\sigma}$ are Pareto-efficient, we
get that:

$V(\hat{\sigma})=V(\sigma)\Leftrightarrow
U_{P}(\hat{\sigma},p_{T})=U_{P}(\sigma,p_{T})$

therefore, $\hat{\sigma}\in\tilde{\Sigma}$ and
$\hat{\alpha}^{T}=(1-\epsilon^{*})\alpha^{T}<\alpha^{T}$, in contradiction to
the fact that
$\sigma\in\operatorname*{argmin}_{\sigma^{\prime}\in\tilde{\Sigma}}{\alpha^{\prime
T}}$.

∎

##  5 Conclusions

In this paper, we introduced a bi-level Bayesian persuasion model in which a
third-party platform controls the information available to the sender about
user preferences. We characterized the optimal information disclosure policy
of the platform, which maximizes average user utility, in a subgame-perfect
Bayesian equilibrium. We then introduced the reputation-based persuasion
platform problem in which myopic users arrive sequentially and analyzed the
equilibrium behavior of the platform and sender in this setting, and
simplified the optimization problem of the platform to a finite dimension
optimization problem. However, a complete solution to the platform’s
optimization problem in the reputation-based setting is left for future work.
Overall, our results suggest that introducing a platform to control the
information available to the sender can incentivize it to take a more truthful
strategy and protect users from being recommended low-quality products. These
findings have potential applications in economic situations in which the
seller is provided with information regarding user preferences by some third-
party entity, which is interested in maximizing users’ welfare.

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