# Persuasion and Matching: Optimal Productive Transport

Anton Kolotilin, Roberto Corrao, and Alexander Wolitzky

(Date: 28th August 2024)

###### Abstract.

We consider general Bayesian persuasion problems where the receiver’s utility
is single-peaked in a one-dimensional action. We show that a signal that pools
at most two states in each realization is always optimal, and that such
_pairwise_ signals are the only solutions under a non-singularity condition
(the _twist condition_). Our core results provide conditions under which
riskier prospects induce higher or lower actions, so that the induced action
is _single-dipped_ or _single-peaked_ on each set of nested prospects. We also
provide conditions for the optimality of either full disclosure or _negative
assortative disclosure_ , where all prospects are nested. Methodologically,
our results rely on novel duality and complementary slackness theorems. Our
analysis extends to a general problem of assigning one-dimensional inputs to
productive units, which we call _optimal productive transport_. This problem
covers additional applications including club economies (assigning workers to
firms, or students to schools), robust option pricing (assigning future asset
prices to price distributions), and partisan gerrymandering (assigning voters
to districts).

_JEL Classification:_ C78, D82, D83

_Keywords:_ Bayesian persuasion, information design, first-order approach,
optimal transport, duality, complementary slackness, pairwise signal, single-
dipped signal, negative assortative disclosure, club economies, option
pricing, gerrymandering

Kolotilin: School of Economics, UNSW Business School.
Corrao and Wolitzky: Department of Economics, MIT.
This paper supercedes earlier papers titled “Persuasion with Non-Linear
Preferences,” “Persuasion as Matching,” and “Assortative Information
Disclosure.” For helpful comments and suggestions, we thank Jakša Cvitanić,
Laura Doval, Piotr Dworczak, Jeffrey Ely, Drew Fudenberg, Emir Kamenica,
Elliot Lipnowski, Stephen Morris, Paula Onuchic, Eran Shmaya, and Andriy
Zapechelnyuk, as well as many seminar participants. We thank Daniel Clark and
Yucheng Shang for excellent research assistance. Anton Kolotilin gratefully
acknowledges support from the Australian Research Council Discovery Early
Career Research Award DE160100964 and from MIT Sloan’s Program on Innovation
in Markets and Organizations. Alexander Wolitzky gratefully acknowledges
support from NSF CAREER Award 1555071 and Sloan Foundation Fellowship
2017-9633.

##  1\. Introduction

Following the seminal papers of Rayo and Segal (2010) and Kamenica and
Gentzkow (2011), the past decade has witnessed an explosion of interest in the
design of optimal information disclosure policies, or Bayesian persuasion.
While significant progress has been made in the special case where the
sender’s and receiver’s utilities are linear in the unknown state (e.g.,
Gentzkow and Kamenica 2016, Kolotilin, Mylovanov, Zapechelnyuk, and Li 2017,
Kolotilin 2018, Dworczak and Martini 2019, Kleiner, Moldovanu, and Strack
2021)—so that a distribution over states can be summarized by its mean—general
results beyond this simple case remain scarce. The literature to date thus has
little to say about the qualitative implications of economically natural
curvature properties of utilities, or about the robustness of optimal
disclosure patterns uncovered in the linear case when utilities are non-
linear.

This paper studies persuasion with non-linear preferences as an instance of a
general class of economic models that we call _optimal productive transport_.
In the persuasion context, we consider a standard setting with one sender and
one receiver, where the receiver’s action and the state of the world are both
one-dimensional. We assume that the sender always prefers higher actions, the
receiver prefers higher actions at higher states, and the receiver’s expected
utility is single-peaked in his action for any belief about the state. In this
model, the receiver’s action is optimal iff his expected marginal utility from
increasing his action equals zero: that is, iff the receiver’s first-order
condition holds. This _first-order approach_ is key for tractability. We
provide three types of results, all of which have general analogues beyond
persuasion.

First, we show that it is always without loss to focus on _pairwise signals_ ,
where each induced posterior belief has at most binary support. Moreover,
under a non-singularity condition on the sender’s and receiver’s
utilities—which we call the _twist condition_ —every optimal signal is
pairwise.

Second, we ask when it is optimal for the sender to induce higher actions with
riskier or safer prospects. In particular, when the sender pools two extreme
states $x_{1}<x_{4}$ and separately pools two moderate states $x_{2}\leq
x_{3}$ such that $x_{1}<x_{2}\leq x_{3}<x_{4}$, do the extreme states induce a
higher action—in which case we say that disclosure is _single-dipped_ , as the
receiver’s action is single-dipped on the set
$\\{x_{1},x_{2},x_{3},x_{4}\\}$—or a lower action—in which case we say that
disclosure is _single-peaked_? This question turns out to be key for
understanding optimal disclosure patterns with non-linear preferences. Our
core results provide general conditions for the optimality of single-dipped
disclosure (and, similarly, single-peaked disclosure). The conditions are
based on the following simple idea. If disclosure is not single-dipped, then
there must exist a _single-peaked triple_ : a pair of pooled states
$x_{1}<x_{4}$ and an intervening state $x_{2}\in(x_{1},x_{4})$ such that the
induced action at $x_{2}$ (say, action $y_{2}$) is greater than the induced
action at $\\{x_{1},x_{4}\\}$ (say, action $y_{1}$). Our conditions ensure
that any single-peaked triple can be profitably perturbed in the direction of
single-dippedness by shifting weight on $x_{1}$ and $x_{4}$ from $y_{1}$ to
$y_{2}$, while shifting weight on $x_{2}$ in the opposite direction.

Third, we provide conditions for the optimality of either _full disclosure_ ,
where the state is always disclosed, or (more interestingly) _negative
assortative disclosure_ , where all states are paired in a negatively
assortative manner, so that all prospects can be ordered from safests to
riskiest, and only a single state “in the middle” is disclosed. Intuitively,
full disclosure and negative assortative disclosure represent the extremes of
maximum disclosure (disclosing all states) and minimal pairwise disclosure
(disclosing only one state). There is a unique full disclosure outcome, but
there are many negative assortative disclosure outcomes, depending on the
weights on the states in each pair. We further characterize the optimal
negative assortative disclosure pattern as the solution of a pair of ordinary
differential equations, and provide examples where these equations admit an
explicit solution.

While this paper is mainly motivated by Bayesian persuasion, the theory we
develop applies equally to several other applications. We consider three: club
economies (e.g., assigning workers to firms to maximize output, or assigning
students to schools to maximize welfare), robust option pricing (assigning
future asset prices to price distributions to bound the price of a
derivative), and partisan gerrymandering (assigning voters to districts to
maximize expected seat share). To facilitate the analysis of these
applications, in Section 7 we recast our model in terms of assigning general
“inputs” to “productive units.” Table 1 explains how our general model maps to
each of our applications.

Application |  Input ($x$) |  Productive Unit ($\mu$) |  Output ($y$) |  Meaning of Single-Dippedness
---|---|---|---|---
Persuasion: |  state |  posterior |  receiver action |  riskier prospects induce higher actions
Worker-Firm Matching: |  worker with ability $x$ |  firm |  intra-firm spillover |  diverse firms are more productive
Student-School Matching: |  student with ability $x$ |  school |  peer effect |  diverse schools are more desirable
Option Pricing: |  period-2 asset price |  period-2 asset price distribution |  period-1 asset price |  riskier assets are more expensive
Gerrymandering: |  voter with partisanship $x$ |  district |  probability that designer’s party wins the district |  polarized districts are stronger
Table 1. Atlas of Our Applications

Mathematically, our model combines aspects of a production problem (combining
inputs to produce output) and a transportation problem (matching inputs and
outputs to generate utility)—hence the name “optimal productive transport.”
The model is a new kind of optimal transport problem. Our key technical
results are duality and complementary slackness theorems for this problem. The
closest strand of the optimal transport literature is that on _martingale
optimal transport_ (e.g., Beiglböck, Henry-Labordere, and Penkner 2013,
Galichon, Henry-Labordere, and Touzi 2014, Beiglböck and Juillet 2016), which
we discuss in Section 3.1.111A few recent papers apply optimal transport to
persuasion, but these works are not very related to ours either
methodologically or substantively. Perez-Richet and Skreta (2023) and Lin and
Liu (2023) consider limited sender commitment; Arieli, Babichenko, and
Sandomirskiy (2023) consider persuasion with multiple receivers; Malamud and
Schrimpf (2022) focus on the question of when optimal signals partition a
multidimensional state space.

The optimal productive transport framework nests a great deal of prior work,
both in persuasion and in the contexts of the other applications we cover.
Some key prior works are Rayo and Segal (2010), Goldstein and Leitner (2018),
and Guo and Shmaya (2019) (on persuasion); Arnott and Rowse (1987) and Saint-
Paul (2001) (on matching); Beiglböck and Juillet (2016) (on option pricing);
and Friedman and Holden (2008) (on gerrymandering). In light of our analysis,
some of the main results in these papers can be viewed as showing that single-
dipped or single-peaked disclosure is optimal—that is, that riskier or safer
prospects induce higher actions—in some special settings. For instance,
Friedman and Holden’s 2008“matching slices” gerrymandering solution, where a
gerrymanderer creates electoral districts that pool extreme supporters with
similarly extreme opponents, and wins those districts with the most extreme
supporters and opponents with the highest probability, is an example of
single-dipped disclosure. Goldstein and Leitner’s 2018 non-monotone stress
tests, where a regulator designs a test that pools the weakest banks that it
wants to receive funding with the strongest banks (and subsequently pools less
weak banks with less strong ones), such that the weakest and strongest banks
receive the highest funding, is another such example. On the other hand, Guo
and Shmaya’s 2019 “nested intervals” disclosure rule, where a designer pools
favorable states with similarly unfavorable states, and persuades the receiver
to take her preferred action with higher probability at more moderate states,
is an example of single-peaked disclosure.

Organization. The paper is organized as follows. Section 2 presents our model
in the context of persuasion. Section 3 formulates primal and dual versions of
our problem and establishes strong duality and complementary slackness.
Section 4 shows that pairwise signals are without loss. Sections 5 and 6
present our main substantive results: Section 5 provides conditions for
single-dipped or single-peaked disclosure to be optimal, and Section 6
provides conditions for full disclosure or negative assortative disclosure to
be optimal. Section 7 reframes our model as “optimal productive transport,”
and applies it to matching, option pricing, and gerrymandering, as well as
some specific persuasion problems. Section 8 concludes. Additional results, as
well as all proofs, are deferred to the Appendix or Online Appendix

##  2\. Persuasion with Non-Linear Preferences

For concreteness, we exposit our model and main results in the context of
Bayesian persuasion. In Section 7.1, we rephrase the model as a general
problem of assigning inputs to productive units, which we call _optimal
productive transport_. This more general framing covers our matching, option
pricing, and gerrymandering applications.

###  2.1. Model

We consider a standard persuasion problem, where a sender chooses a signal to
reveal information to a receiver, who then takes an action. The sender’s
utility $V(y,x)$ and the receiver’s utility $U(y,x)$ depend on the receiver’s
action $y\in Y:=[0,1]$ and the state of the world $x\in[0,1]$. The sender and
receiver share a common prior $\phi\in\Delta([0,1])$, with support
$X:=\operatorname{supp}(\phi)$.222Throughout, for any compact metric space
$X$, $\Delta(X)$ denotes the set of Borel probability measures on $X$, endowed
with the weak* topology. For any $\mu\in\Delta(X)$, its support
$\operatorname{supp}(\mu)$ is the smallest compact set of measure one. A
signal $\tau\in\Delta(\Delta(X))$ is a distribution over posterior beliefs
$\mu\in\Delta(X)$ such that the average posterior equals the prior:
$\int\mu\mathrm{d}\tau=\phi$ (Aumann and Maschler 1995, Kamenica and Gentzkow
2011). An _outcome_ $\pi\in\Delta(Y\times X)$ is a joint distribution over
actions and states. As we will see, it is equivalent to view the sender as
choosing a signal $\tau$ (the _signal-based problem_) or as directly choosing
an outcome $\pi$ subject to an obedience constraint (the _outcome-based
problem_).

We impose four standard assumptions on preferences, which are similar to those
in canonical unidimensional models of communication such as signaling (Spence
1973), cheap talk (Crawford and Sobel 1982), and hard information disclosure
(Seidmann and Winter 1997). First, utilities are smooth.

######  Assumption 1.

$V(y,x)$ and $u(y,x):=\partial U(y,x)/\partial y$ are three times
differentiable.

Apart from the receiver’s marginal utility $u$, we denote partial derivatives
with subscripts: e.g., $V_{y}(y,x)=\partial V(y,x)/\partial y$.

Second, the receiver’s expected utility is single-peaked in his action for any
posterior belief.

######  Assumption 2.

$u(y,x)$ satisfies _strict aggregate single-crossing_ in $y$: for all
posteriors $\mu\in\Delta(X)$,

$\int_{X}u(y,x)\mathrm{d}\mu(x)=0\implies\int_{X}u_{y}(y,x)\mathrm{d}\mu(x)<0.$

Quah and Strulovici (*Quah2012) and Choi and Smith (*CS) characterized a weak
version of aggregate single-crossing. We provide an analogous characterization
of strict aggregate single-crossing in Appendix A. A sufficient condition is
strict monotonicity of $u$ (or equivalently strict concavity of $U$): i.e.,
$u_{y}(y,x)<0$ for all $(y,x)$. In fact, Appendix A shows that strict
aggregate single-crossing is equivalent to strict monotonicity up to a
normalization.

Third, the receiver’s optimal action satisfies an interiority condition.333The
substance of Assumption 3 is that for each $x$, there exists $y$ such that
$u(y,x)=0$. Note that it can never be optimal for the receiver to take any $y$
such that $u(y,x)$ has a constant sign for all $x$. We can then remove all
such $y$ from $Y$ and renormalize $Y$ to $[0,1]$, so that Assumption 3 holds.

######  Assumption 3.

$\min_{x\in[0,1]}u(0,x)=\max_{x\in[0,1]}u(1,x)=0$.

The key implication of Assumptions 1–3 is that for any posterior $\mu$, the
receiver’s optimal action
$\gamma(\mu):=\operatorname*{arg\,max}_{y\in[0,1]}\int U(y,x)\mathrm{d}\mu$ is
unique and is characterized by the first-order condition

$\int_{X}u(\gamma(\mu),x)\mathrm{d}\mu(x)=0.$ (1)

Our assumptions thus allow a “first-order approach” to the persuasion problem,
similar to the approach of Mirrlees (1999) and Holmström (1979) to the
classical moral hazard problem.444The first-order approach to persuasion was
introduced in Kolotilin (2018).

Uniqueness of the receiver’s optimal action implies that any signal $\tau$
induces a unique outcome $\pi_{\tau}$, and that we can define the sender’s
indirect utility from inducing posterior $\mu$ as

$W(\mu)=\int_{X}V(\gamma(\mu),x)\mathrm{d}\mu(x).$

Fourth, the sender prefers higher actions, and the receiver’s utility is
supermodular.

######  Assumption 4.

$V_{y}(y,x)>0$ and $u_{x}(y,x)>0$.

Together with Assumptions 1–3, Assumption 4 ensures that for each action $y$
there is a unique state $\chi(y)$ such that $u(y,\chi(y))=0$ (i.e., the
receiver’s optimal action at $\chi(y)$ is $y$), and that $\chi(y)$ is a
strictly increasing, continuous function with range $[0,1]$.

A common interpretation of the receiver’s action $y\in[0,1]$ is that the
receiver has a private type and makes a binary choice—say, whether to accept
or reject a proposal—and $y$ is the receiver’s choice of a cutoff type below
which he accepts. This interpretation is especially useful for some special
cases of the model, as we see next.555To spell out this interpretation, let
$g(t|x)$ be the conditional density of the receiver’s type $t\in[0,1]$ given
the state $x\in[0,1]$. The sender’s and receiver’s utilities from rejection
are normalized to zero. The sender’s and receiver’s utilities from acceptance
are functions $\tilde{v}(t,x)$ and $\tilde{u}(t,x)$, with
$\tilde{u}(t,x)g(t|x)$ satisfying Assumption 2. For $y\in[0,1]$ (interpreted
as the cutoff such that the receiver accepts iff $t\leq y$), we recover our
model with $V(y,x)=\int_{0}^{y}\tilde{v}(t,x)g(t|x)\mathrm{d}t$ and
$U(y,x)=\int_{0}^{y}\tilde{u}(t,x)g(t|x)\mathrm{d}t$.

###  2.2. Special Cases

We list some leading special cases of the model, which we return to
periodically to illustrate our results.

(1) The _linear case_ (Gentzkow and Kamenica 2016): $u(y,x)=x-y$ and
$V(y,x)=V(y)$. That is, $\gamma(\mu)=\mathbb{E}_{\mu}[x]$ and $V$ is state-
independent. This is the well-studied case where the sender’s indirect utility
$W(\mu)$ depends only on $\mathbb{E}_{\mu}[x]$.

(2) The _linear receiver case_ : $u(y,x)=x-y$ but $V$ is arbitrary (e.g.,
possibly state-dependent). Here the receiver’s preferences are as in the
linear case, while the sender’s preferences are general.

(2a) The _separable subcase_ (Rayo and Segal 2010): $V(y,x)=w(x)G(y)$ with
$w>0$ and $G>0$. An interpretation is that the receiver has a private type $t$
with distribution $G$ and accepts a proposal iff $\mathbb{E}_{\mu}[x]\geq t$,
and the sender’s utility when the proposal is accepted is $w(x)$.666Rayo and
Segal focused on the sub-subcase with the uniform distribution $G(y)=y$. They
assume that the state $(x,z)$ is two-dimensional, that the sender’s and
receiver’s marginal utilities are $V_{y}(y,x,z)=z$ and $u(y,x)=x-y$, and that
there are finitely many states $(x,z)$, so generically the sender’s utility
can be written as $V_{y}(y,x)=w(x)$. Rayo (2013), Nikandrova and Pancs (2017),
and Onuchic and Ray (2023) consider the separable subcase where $x$ is
continuous and $(x,z)$ is supported on the graph of $x\rightarrow w(x)$.
Rochet and Vila (1994), Tamura (2018), Kramkov and Xu (2022), and Dworczak and
Kolotilin (2022) allow more general distributions of $(x,z)\in\mathbb{R}^{2}$.

(2b) The _translation-invariant subcase_ : $V(y,x)=P(y-x)$. An interpretation
is that the receiver “values” the proposal at $\mathbb{E}_{\mu}[x]$, and the
sender’s utility depends on the amount by which the proposal is “over-valued,”
$\mathbb{E}_{\mu}[x]-x$. For example, a school may care about the extent to
which its students are over- or under-placed. These preferences are similar to
those in Goldstein and Leitner’s 2018 model of stress tests (see Appendix
D.3).

(3) The _state-independent sender case_ : $V(y,x)=V(y)$ but $u$ is arbitrary.
Here the sender’s preferences are as in the linear case, while the receiver’s
preferences are general.

(3a) The _separable subcase_ : $u(y,x)=I(x)(x-y)$, with $I>0$. This subcase
extends the linear case by letting the receiver put more weight on some states
than others.

(3b) The _translation-invariant subcase_ : $u(y,x)=T(x-y)$, with $T(0)=0$. An
example that fits this subcase is that the sender’s utility when the proposal
is accepted is $1$, and accepting the proposal corresponds to the receiver
undertaking a project that can either succeed or fail, where the receiver’s
payoff is $1-\kappa$ when the project succeeds and $-\kappa$ when it fails
(and $0$ when it is not undertaken), with $\kappa\in(0,1)$. The difficulty of
the project is $1-x$, the receiver’s ability is $1-t$, the receiver’s “bad
luck” $\varepsilon$ has distribution $J$, and the project succeeds iff
$1-x\leq 1-t-\varepsilon$, or equivalently $\varepsilon\leq x-t$. This example
fits the current subcase with $V$ equal to the distribution of $t$ and
$T(x-y)=J(x-y)-\kappa$.

(3c) The _quantile sub-subcase_ : $u(y,x)=\mbox{\bf 1}\\{x\geq y\\}-\kappa$,
with $\kappa\in(0,1)$. This subcase corresponds to the previous example with
$J(x-y)=\mbox{\bf 1}\\{x\geq y\\}$, so the project succeeds iff the receiver’s
ability exceeds the project’s difficulty. While $u$ is now discontinuous, we
admit this subcase as a limit of the translation-invariant case. Yang and
Zentefis (2023) also study the quantile sub-subcase.

##  3\. Optimality Conditions

This section establishes optimality conditions that form the basis for our
analysis. Section 3.1 formulates signal-based and outcome-based primal and
dual problems, and shows that they are equivalent. We will make use of both
formulations. Section 3.2 establishes our key complementary slackness theorem.

###  3.1. Primal and Dual Programs

The sender’s signal-based primal problem is to find a signal
$\tau\in\Delta(\Delta(X))$ to

$\displaystyle\text{maximize}\quad\int_{\Delta(X)}W(\mu)\mathrm{d}\tau(\mu)$
(P) $\displaystyle\text{subject
to}\quad\int_{\Delta(X)}\mu\mathrm{d}\tau(\mu)=\phi.$ (BP)

Here, the primal constraint (BP) is the usual _Bayes plausibility_ constraint
(Kamenica and Gentzkow, 2011).

Next, let $L(X)$ denote the set of Lipschitz continuous functions on $X$. The
signal-based dual problem is to find a _price function_ $p\in L(X)$ to

$\displaystyle\text{minimize}\quad\int_{X}p(x)\mathrm{d}\phi(x)$ (D)
$\displaystyle\text{subject to}\quad\int_{X}p(x)\mathrm{d}\mu(x)\geq
W(\mu),\quad\text{for all $\mu\in\Delta(X)$.}$ (ZP)

The interpretation is that $p(x)$ is the shadow price of state $x$, and the
dual constraint (ZP) is the _zero profit_ condition that the sender’s indirect
utility from inducing any posterior $\mu$ cannot exceed the expectation of
$p(x)$ under $\mu$. This interpretation will become clearer in the general
framework of Section 7.1.

A preliminary result is that strong duality holds: (optimal) solutions to (P)
and (D) exist and give the same value.

######  Lemma 1.

There exists $\tau\in\Delta(\Delta(X))$ that solves (P); there exists $p\in
L(X)$ that solves (D); and the values of (P) and (D) are equal: for any
solutions $\tau$ of (P) and $p$ of (D), we have

$\int_{\Delta(X)}W(\mu)\mathrm{d}\tau(\mu)=\int_{X}p(x)\mathrm{d}\phi(x).$

Lemma 1 follows by showing that $W(\mu)$ is Lipschitz continuous and applying
Corollary 2 of Dworczak and Kolotilin (2022), which in turn generalizes
Theorem 2 of Dworczak and Martini (2019) from linear persuasion problems to
non-linear ones.777Corollary 2 of Dworczak and Kolotilin (2022) is proved
using strong duality in an optimal transport problem, as in Villani (2009).
Further duality results for persuasion problems include those of Dizdar and
Kováč (2020), Kramkov and Xu (2022), Galperti, Levkun, and Perego (2023), and
Smolin and Yamashita (2023).

Next, the outcome-based primal problem is to find an outcome
$\pi\in\Delta(Y\times X)$ to

$\displaystyle\text{maximize}\quad\int_{Y\times X}V(y,x)\mathrm{d}\pi(y,x)$
(P’) $\displaystyle\text{subject
to}\quad\int_{Y\times\widetilde{X}}\mathrm{d}\pi(y,x)=\int_{\widetilde{X}}\mathrm{d}\phi(x),\quad\text{for
all measurable }\widetilde{X}\subset X,$ (BP’)
$\displaystyle\int_{\widetilde{Y}\times
X}u(y,x)\mathrm{d}\pi(y,x)=0,\quad\text{for all measurable
}\widetilde{Y}\subset Y.$ (OB)

Here, (BP’) is an outcome-based version of Bayes plausibility, which says that
the marginal of $\pi$ on $X$ equals the prior, $\phi$; and (OB) is the
obedience constraint that the receiver’s action at each posterior $\mu$ is
$\gamma(\mu)$. A joint distribution $\pi$ that violates (OB) is inconsistent
with optimal play by the receiver, as there exists $\widetilde{Y}\subset Y$
such that the receiver’s play is suboptimal conditional on the event
$\\{y\in\widetilde{Y}\\}$. Conversely, for any joint distribution $\pi$ that
satisfies (BP’) and (OB), if the sender designs a mechanism that draws $(y,x)$
according to $\pi$ and recommends action $y$ to the receiver, it is optimal
for the receiver to obey the recommendation. We therefore say that an outcome
$\pi$ is _implementable_ iff it satisfies (BP’) and (OB), and _optimal_ iff it
solves (P’).

Finally, letting $B(Y)$ denote the set of bounded, measurable functions on
$Y$, the outcome-based dual problem is to find $p\in L(X)$ and $q\in B(Y)$ to

$\displaystyle\text{minimize}\quad\int_{X}p(x)\mathrm{d}\phi(x)$ (D’)
$\displaystyle\text{subject to}\quad p(x)\geq V(y,x)+q(y)u(y,x),\quad\text{for
all }(y,x)\in Y\times X.$ (ZP’)

The interpretation is that $p(x)$ is the shadow price of state $x$; $q(y)$ is
the value of relaxing the obedience constraint at action $y$; and (ZP’) says
that $p(x)$ is no less than the sender’s value from assigning state $x$ to any
action $y$, where this value is the sum of the sender’s utility, $V(y,x)$, and
the product of $q(y)$ and the amount by which obedience at $y$ is relaxed when
state $x$ is assigned to action $y$, $u(y,x)$.

We now establish that the feasible (and optimal) price functions in the
signal-based and outcome-based formulations are the same. In particular, by
Lemma 1, strong duality holds in the outcome-based formulation, as well as the
signal-based one.888Strong duality in the outcome-based formulation is
established under slightly different assumptions in Lemmas 1 and 2 of
Kolotilin (2018). However, a key step in the proof—that $q$ can be taken to be
bounded—is incomplete in Kolotilin (2018).

######  Lemma 2.

A price function $p\in L(X)$ is feasible (optimal) for (D) iff there exists
$q\in B(Y)$ such that $(p,q)$ is feasible (optimal) for (D’).

###  3.2. Complementary Slackness

Letting $p$ be the optimal price function (which we will see in Remark 1 is
unique), define the set

$\displaystyle\Lambda=\left\\{\mu\in\Delta(X):\int_{X}p(x)\mathrm{d}\mu(x)=W(\mu)\right\\}.$
(2)

Note that $\Lambda$ is compact, because $\int p(x)\mathrm{d}\mu$ and $W(\mu)$
are continuous in $\mu$.

By Lemma 1, together with (BP), a signal $\tau$ is optimal iff

$\int_{\Delta(X)}\left(\int_{X}p(x)\mathrm{d}\mu(x)-W(\mu)\right)\mathrm{d}\tau(\mu)=0.$

Hence, since the integrand is non-negative by (ZP) and $\Lambda$ is compact,
$\tau$ is optimal iff $\operatorname{supp}(\tau)\subset\Lambda$. Any posterior
$\mu\notin\Lambda$ is thus excluded from the support of any optimal signal. In
analogy with the optimal transport literature (e.g., Section 3 in Ambrosio,
Brué, and Semola 2021), we refer to the set $\Lambda$ as the _contact set_.

The following is our main technical result.

######  Theorem 1.

There exists $q\in B(Y)$ such that

(1) $(p,q)$ is optimal for (D’);

(2) for all $\mu$ in $\Lambda$ (and, thus, in the support of any optimal
signal $\tau$), we have

$\displaystyle
q(\gamma(\mu))=-\frac{\int_{X}V_{y}(\gamma(\mu),x)\mathrm{d}\mu(x)}{\int_{X}u_{y}(\gamma(\mu),x)\mathrm{d}\mu(x)};$
(3)

(3) for all non-degenerate $\mu$ in $\Lambda$ (and, thus, in the support of
any optimal signal $\tau$), the function $q$ has derivative
$q^{\prime}(\gamma(\mu))$ at $\gamma(\mu)$ satisfying, for all
$x\in\operatorname{supp}(\mu)$,

$\displaystyle
V_{y}(\gamma(\mu),x)+q(\gamma(\mu))u_{y}(\gamma(\mu),x)+q^{\prime}(\gamma(\mu))u(\gamma(\mu),x)=0.$
(4)

Theorem 1 is our key tool for characterizing optimal signals. Intuitively, by
complementary slackness, the support of any optimal outcome $\pi$ is contained
in the set of points $(y,x)$ that satisfy (ZP’). Thus, if it is ever optimal
to induce action $y$ at state $x$—i.e., if $y$ maximizes
$V(y,x)+q(y)u(y,x)$—then $y$ must satisfy the first-order condition

$V_{y}(y,x)+q(y)u_{y}(y,x)+q^{\prime}(y)u(y,x)=0,$

which is just (4) with $\gamma(\mu)=y$. Moreover, taking the expectation of
this equation with respect to posterior $\mu$ yields (3). This equation simply
says that $q(y)$ equals the product of the sender’s expected marginal utility
at $y$ and the rate at which $y$ increases as obedience is relaxed, where the
latter term equals $-1/\mathbb{E}_{\mu}[u_{y}(y,x)]$ by the implicit function
theorem applied to obedience.

As shown in Appendix E.4, another implication of Theorem 1 is:

######  Remark 1.

There is a unique solution $p$ to (D).

Lemmas 1–2 and Theorem 1 can be compared to results in optimal transport. In
standard optimal transport, two marginal distributions are given (e.g., of men
and women, or workers and firms), and the problem is to find an optimal joint
distribution with the given marginals. In our problem, the marginal
distribution over states is given (by the prior $\phi$), and the problem is to
find an optimal joint distribution with this marginal, where for each action
the conditional distribution over states satisfies obedience. Strong duality
and complementary slackness theorems are likewise key tools in optimal
transport (e.g., Villani 2009, Theorem 5.10), but the relevent versions of
these results differ from ours.999For example, in standard optimal transport,
both dual variables appear in the dual objective function, and they are both
uniquely determined.

The most relevant strand of the optimal transport literature is that on
_martingale optimal transport (MOT)_. The MOT problem is to find an optimal
joint distribution of two variables (say, $y$ and $x$) with given marginals,
subject to the martingale constraint that the expectation of $x$ given $y$ is
$y$. This problem coincides with our linear receiver case, but with an
exogenously fixed distribution of the receiver’s action. Motivated by problems
in mathematical finance, Beiglböck, Henry-Labordere, and Penkner (2013) (see
also Beiglböck, Nutz, and Touzi 2017) introduce MOT and prove that the primal
and dual problems have the same value; however, they also show that their dual
problem may not have a solution, unlike in our model with endogenous actions
(or in standard optimal transport). Results in MOT also do not establish
compactness of the contact set, which holds in our model as well as in
standard optimal transport. Thus, MOT is related to our linear receiver case,
but the endogenous action distribution apparently makes our model more
tractable.101010The MOT literature uses the contact set of an outcome-based
dual problem. See Kolotilin, Corrao, and Wolitzky (2022) for an alternative
development of the results in the current paper that relies on the contact set
of our outcome-based dual, (D’). The approach in the current version, which is
based on the contact set $\Lambda$ of the signal-based dual, (D), turns out to
be simpler.

##  4\. Pairwise Disclosure and the Twist Condition

Our first substantive result is that there is always an optimal signal that
pools at most two states in every realized posterior, and that under an
additional condition every optimal signal has this property. This result
simplifies the persuasion problem to a generalized matching problem, where the
sender chooses what pairs of states to match together, and with what weights.

Formally, a set of posteriors $M\subset\Delta(X)$ is _pairwise_ if
$|\operatorname{supp}(\mu)|\leq 2$ for all $\mu\in M$. A signal $\tau$ is
_pairwise_ if $\operatorname{supp}(\tau)$ is pairwise: that is, a pairwise
signal induces posterior beliefs with at most binary support. For example,
with a uniform prior, for any cutoff $\hat{x}\in[0,1]$ the signal that reveals
states below the cutoff and pools each pair of states $x$ and $1+\hat{x}-x$
for $x\in[\hat{x},(1+\hat{x})/2]$ to induce posterior
$\mu=\delta_{x}/2+\delta_{1+\hat{x}-x}/2$ is pairwise. The special case where
$\hat{x}=1$ is full disclosure, which is also pairwise. In contrast, no
disclosure, where $\tau(\phi)=1$, is not pairwise.111111See Figure 2. The
“disclose-pair” pattern in Panel d. is reminiscent of this example, but with
different weights on the states in each pair.

If the receiver’s utility is not quasi-concave, pairwise signals may be
suboptimal. For example, suppose the sender rules three castles, one of which
is undefended. The state $x$—the identity of the undefended castle—is
uniformly distributed. Suppose the receiver can attack any two castles, and
payoffs are $(-1,+1)$ for the sender and receiver, respectively, if the
receiver attacks the undefended castle, and are $(+1,-1)$ otherwise. Then any
pairwise signal narrows the set of possibly undefended castles to at most two,
so the receiver always wins. But if the sender discloses nothing, the receiver
wins only with probability $2/3$.121212Pairwise signals are also suboptimal in
the price-discrimination problem of Bergemann, Brooks, and Morris (2015), as
well as in Brzutowski (2023), where $U(y,x)=\mathbf{1}\\{y\geq x\\}-y$. In
these models, the receiver’s utility is not quasi-concave.

In contrast, pairwise signals are without loss under Assumptions 1–3.
Moreover, equation (4) implies that if it is optimal to induce the same action
$y$ at three states $x_{1}$, $x_{2}$, and $x_{3}$, then the vector
$(V_{y}(y,x_{1}),V_{y}(y,x_{2}),V_{y}(y,x_{3}))$ must be a linear combination
of the vectors $(u(y,x_{1}),u(y,x_{2}),u(y,x_{3}))$ and
$(u_{y}(y,x_{1}),u_{y}(y,x_{2}),u_{y}(y,x_{3}))$. This observation gives a
condition—which we call the _twist condition_ —under which pooling more than
two states is suboptimal, so that every optimal signal is pairwise.131313The
term “twist condition” is in analogy to optimal transport, where the twist
condition is an analogous non-singularity condition (e.g., Definition 1.16 in
Santambrogio 2015).

Twist Condition _For any action $y$ and any triple of states
$x_{1}<x_{2}<x_{3}$ such that $x_{1}<\chi(y)<x_{3}$, we have $|S|\neq
0$,141414Here $|\cdot|$ denotes the determinant of a matrix; we also use the
same notation for the cardinality of a set. where_

$S:=\begin{pmatrix}V_{y}(y,x_{1})&V_{y}(y,x_{2})&V_{y}(y,x_{3})\\\
u(y,x_{1})&u(y,x_{2})&u(y,x_{3})\\\
u_{y}(y,x_{1})&u_{y}(y,x_{2})&u_{y}(y,x_{3})\end{pmatrix}.$ (5)

We will apply this condition extensively in Section 5.

######  Theorem 2.

For any signal $\tau$ (whether optimal or not), there exists a pairwise signal
$\hat{\tau}$ that induces the same outcome. Moreover, if the twist condition
holds, then the contact set is pairwise, and hence so is any optimal signal.

The intuition for the first part of the theorem is that for any posterior
$\mu$, there exists a hyperplane passing through it such that all posteriors
on the hyperplane induce the same action, and the extreme points of the
hyperplane in the simplex have at most binary support. Thus, any posterior
that puts weight on more than two states can be split into posteriors with at
most binary support without affecting the induced outcome. Figure 1
illustrates this argument for a posterior with weight on three states.

The intuition for the second part is that this splitting leaves an extra
degree of freedom, which can be profitably exploited under the twist
condition. Consider a posterior $\mu$ with
$\operatorname{supp}(\mu)=\\{x_{1},x_{2},x_{3}\\}$. We can split $\mu$ into
posteriors $\mu^{\prime}$ and $\mu^{\prime\prime}$ with at most binary support
that both induce action $\gamma(\mu)$. For example, suppose that
$\operatorname{supp}(\mu^{\prime})=\\{x_{1},x_{2}\\}$ and
$\operatorname{supp}(\mu^{\prime\prime})=\\{x_{1},x_{3}\\}$. Now consider a
perturbation that moves probability mass $\varepsilon$ on $x_{1}$ from
$\mu^{\prime}$ to $\mu^{\prime\prime}$. This perturbation induces non-zero
marginal changes in the action at $\mu^{\prime}$ and $\mu^{\prime\prime}$.
Under the twist condition, these changes have a non-zero marginal effect on
the planner’s expected utility, by the implicit function theorem. Therefore,
either this perturbation or the reverse perturbation, where $\varepsilon$ is
replaced with $-\varepsilon$, is strictly profitable.151515Formally, the
second part of Theorem 2 directly follows from Theorem 1.

$x_{1}$$x_{2}$$x_{3}$$\mu^{\prime}$$\mu^{\prime\prime}$$\leftarrow\gamma(\mu)$$\mu$
Figure 1. Pairwise Signals are Without Loss _Notes:_ The optimal action at any
posterior on the line between $\mu^{\prime}$ and $\mu^{\prime\prime}$ equals
$\gamma(\mu)$, so splitting $\mu$ into $\mu^{\prime}$ and $\mu^{\prime\prime}$
eliminates a non-binary-support posterior without changing the outcome.

Prior results by Rayo and Segal (2010), Alonso and Câmara (2016), and Zhang
and Zhou (2016) also give conditions under which all optimal signals are
pairwise. Theorem 2 easily implies these earlier results.161616Proposition 4
in Alonso and Câmara (2016) states that if $u(y,x)=x-y$ and there do not exist
$\zeta\leq 0$ and $\iota\in\mathbb{R}$ such that $V_{y}(y,x_{i})=\zeta
x_{i}+\iota$ for $i=1,2,3$, then it is not optimal to induce action $y$ at
states $x_{1}$, $x_{2}$, and $x_{3}$. This result is too strong as stated, and
it is not correct unless $\zeta$ is also allowed to be positive. Theorem 2
implies this corrected version of Alonso and Câmara’s result. Note that the
twist condition always fails in the linear case, where $|S|=0$. Hence, in the
linear case, Theorem 2 never rules out pooling multiple states, and indeed
pooling multiple states is often optimal (e.g., Kolotilin, Mylovanov,
Zapechelnyuk, and Li 2017).171717Of course, Theorem 2 shows that even when
pooling multiple states is optimal, there also exists an optimal pairwise
signal, where the “multi-state pool” is split into pairs. Conversely, if
multiple posteriors all induce the same action, they can be pooled without
affecting the outcome.

An immediate corollary of Theorem 2 is that no disclosure is generically
suboptimal when there are at least three states, because for a fixed action
$y$, a generic vector $(V_{y}(y,x))_{x\in X}$ with $|X|\geq 3$ coordinates
cannot be expressed as a linear combination of two fixed vectors
$(u(y,x))_{x\in X}$ and $(u_{y}(y,x))_{x\in X}$, as is required by (4).

######  Corollary 1.

For any prior $\phi$ with $|\operatorname{supp}(\phi)|\geq 3$ and any $u$, no
disclosure is suboptimal for generic $V_{y}$.

Given Kamenica and Gentzkow’s concavification result, Corollary 1 implies
that, for generic utilities, the sender’s indirect utility is not concave in
the posterior when there are more than two states. Note that Corollary 1
allows the case where $u$ and $V_{y}$ always have the opposite sign, so the
sender’s and receiver’s ordinal preferences over actions are diametrically
opposed. Hence, even in this case no disclosure is generically suboptimal.

##  5\. Single-Dipped and Single-Peaked Disclosure

The next two sections present our main results, which characterize optimal
disclosure patterns. The current section asks when it is optimal for riskier
or safer prospects to induce higher actions: that is, when optimal signals are
“single-dipped” or “single-peaked.” As we will see, this is a key question,
which unifies and generalizes much of what is known about special cases of the
persuasion problem with non-linear preferences, as well as other models that
fit our optimal productive transport framework.181818In the MOT context,
Beiglböck and Juillet (2016) argue that single-dippedness/-peakedness are
canonical properties analogous to positive/negative assortativity in standard
matching models. Mathematically, positive/negative assortativity corresponds
to monotonicity in the FOSD order, while single-dippedness/-peakedness
corresponds to monotonicity in a variability order that depends on $u$; when
$u(y,x)=x-y$, this variability order is the usual convex order.

###  5.1. Single-Dippedness/-Peakedness

A signal $\tau$ is _single-dipped_ (_-peaked_) if for any
$\mu_{1},\mu_{2}\in\operatorname{supp}(\tau)$ such that
$\operatorname{supp}(\mu_{1})$ contains $x_{1}<x_{3}$ and
$\operatorname{supp}(\mu_{2})$ contains $x_{2}\in(x_{1},x_{3})$, we have
$\gamma(\mu_{1})\geq(\leq)\gamma(\mu_{2})$. Similarly, $\tau$ is _strictly
single-dipped_ (_-peaked_) if for any
$\mu_{1},\mu_{2}\in\operatorname{supp}(\tau)$ such that
$\operatorname{supp}(\mu_{1})$ contains $x_{1}<x_{3}$ and
$\operatorname{supp}(\mu_{2})$ contains $x_{2}\in(x_{1},x_{3})$ we have
$\gamma(\mu_{1})>(<)\gamma(\mu_{2})$. We also apply these definitions to an
arbitrary set of posteriors $M\subset\Delta(X)$ by replacing
$\operatorname{supp}(\tau)$ with $M$ in the definitions. In particular, a
pairwise signal is single-dipped if the induced receiver action is single-
dipped on each set of nested pairs of states.

An equivalent definition is that a signal $\tau$ is single-dipped if it never
induces a _strictly single-peaked triple_ $(y_{1},x_{1})$, $(y_{2},x_{2})$,
$(y_{1},x_{3})$, with $x_{1}<x_{2}<x_{3}$ and $y_{1}<y_{2}$, in that there
exist $\mu_{1},\mu_{2}\in\operatorname{supp}(\tau)$ such that
$x_{1},x_{3}\in\operatorname{supp}(\mu_{1})$ and $y_{1}=\gamma(\mu_{1})$, and
$x_{2}\in\operatorname{supp}(\mu_{2})$ and $y_{2}=\gamma(\mu_{2})$.
(Otherwise, such a triple would witness a violation of single-dippedness.)

$0$$\frac{1}{2}$$1$$0$$\frac{1}{2}$$1$$x$$y$ |  $0$$\frac{1}{2}$$1$$0$$\frac{1}{2}$$1$$x$$y$
---|---
a. Full Disclosure |  b. No Disclosure

$0$$\frac{1}{3}$$1$$0$$\frac{1}{3}$$\frac{2}{3}$$1$$x$$y$ |  $0$$\frac{1}{3}$$\frac{1}{2}$$1$$0$$\frac{1}{3}$$\frac{1}{2}$$\frac{5}{6}$$1$$x$$y$
c. Negative Assortative Disclosure |  d. Disclose-Pair

$0$$\frac{2}{5}$$1$$0$$1$$x$$y$ |  $0$$\frac{1}{4}$$\frac{1}{2}$$\frac{3}{4}$$1$$0$$\frac{1}{4}$$\frac{1}{2}$$\frac{3}{4}$$1$$x$$y$
e. A Complicated Single-Dipped Set |  f. Median Matching is not Single-Dipped
Figure 2. Some Single-Dipped Disclosure Patterns

Each panel in Figure 2 illustrates a signal in the linear receiver case
($u(y,x)=x-y$). Panel a. is full disclosure, which is trivially strictly
single-dipped, as no states are paired. Panel b. is no disclosure, which is
single-dipped but not strictly single-dipped. Panels c., d., and e. are all
strictly single-dipped. Panel c. is an example of negative assortative
disclosure, where state $x=1/3$ is disclosed and the other states are paired
with weight $2/3$ on the higher state in each pair. Panel d. shows a signal
where all states below $1/3$ (as well as state $1/2$) are disclosed, and the
other states are paired with weight $3/4$ on the higher state in each pair.
This “disclose-pair” pattern is a strictly single-dipped analogue of _upper-
censorship_ , where all states below a cutoff are disclosed, and all states
above the cutoff are pooled (e.g., Kolotilin, Mylovanov, and Zapechelnyuk
2022). Upper-censorship is only weakly single-dipped, whereas disclose-pair
splits up the pooling region in upper-censorship to obtain strict single-
dippedness. Panel e. shows a more complicated strictly single-dipped signal.
While strict single-dippedness implies that each action is induced at at most
two states, Panel e. shows that more than two actions can be induced at a
single state (here, state $2/5$).191919Also, while the function $\chi_{2}$ is
always monotone under strict single-dippedness, Panel e. shows that the
function $\chi_{1}$ can be non-monotone. Finally, Panel f. shows “matching
across the median” (e.g., Kremer and Maskin 1996), which is not single-dipped,
for example because it contains the strictly single-peaked triple
$\left\\{(1/4,1/2),(1/2,3/4),(3/4,1/2)\right\\}$.

All of our results (and all proofs, except for the proof of Theorem 4) are
symmetric between the single-dipped and single-peaked cases. We thus present
our results and proofs only for the single-dipped case (except for the proof
of Theorem 4), omitting the analogous results for the single-peaked case.

######  Remark 2.

A strictly single-dipped set can be described by two functions $\chi_{1}$ and
$\chi_{2}$ that specify the states $\chi_{1}(y)$ and $\chi_{2}(y)$ which are
pooled together to induce each action $y$. Specifically, for any strictly
single-dipped set $\Lambda$, there exist unique functions $\chi_{1}$ and
$\chi_{2}$ from $Y_{\Lambda}=\\{\gamma(\mu):\mu\in\Lambda\\}$ to $X$ such that
$\operatorname{supp}(\mu)=\\{\chi_{1}(\gamma(\mu)),\chi_{2}(\gamma(\mu))\\}$
for all $\mu\in\Lambda$, $\chi_{1}(y)=\chi(y)=\chi_{2}(y)$ or
$\chi_{1}(y)<\chi(y)<\chi_{2}(y)$ for all $y$, and
$\chi_{2}(y)\leq\chi_{2}(y^{\prime})$ and
$\chi_{1}(y^{\prime})\notin(\chi_{1}(y),\chi_{2}(y))$ for all
$y<y^{\prime}$.202020This remark follows from Corollary 1.6 and Lemma A.9 of
Beiglböck and Juillet (2016). For completeness, we provide a simple self-
contained proof in the appendix.

###  5.2. Variational Theorem

The next result captures the core economic logic behind single-dippedness. It
is also our key tool for determining when optimal plans are single-dipped: we
use it to establish our main sufficient condition for single-dipped disclosure
to be optimal (Theorem 4, in the next subsection), and also use it directly to
study some applications in Appendix D.212121Theorem 3 provides conditions
under which _every_ optimal signal is single-dipped. In addition, Lemma 11 in
Appendix E.10 provides weaker conditions under which _some_ optimal signal has
this property.

######  Theorem 3.

Suppose that for any pair of actions $y_{1}<y_{2}$ and any triple of states
$x_{1}<x_{2}<x_{3}$ such that $x_{1}<\chi(y_{1})<x_{3}$, there exists a vector
$\beta\geq 0$ such that $R\beta\geq 0$ and $R\beta\neq 0$, where

$R:=\begin{pmatrix}V(y_{2},x_{1})-V(y_{1},x_{1})&-(V(y_{2},x_{2})-V(y_{1},x_{2}))&V(y_{2},x_{3})-V(y_{1},x_{3})\\\
-u(y_{1},x_{1})&u(y_{1},x_{2})&-u(y_{1},x_{3})\\\
u(y_{2},x_{1})&-u(y_{2},x_{2})&u(y_{2},x_{3})\end{pmatrix}.$

Then the contact set is single-dipped, and hence so is every optimal signal.

$y_{1}$$x_{1}$$x_{2}$$x_{3}$$y_{2}$$\beta_{1}$$\beta_{2}$$\beta_{3}$ Figure 3.
A Profitable Perturbation of a Non-Single-Dipped Signal _Notes:_ The figure
shows a perturbation that shifts weights $\beta_{1}$ and $\beta_{3}$ on
$x_{1}$ and $x_{3}$ from a posterior inducing action $y_{1}$ to a posterior
inducing action $y_{2}$, and shifts weight $\beta_{2}$ on $x_{2}$ in the
opposite direction. This perturbation is profitable if it increases the
receiver’s expected marginal utility at $y_{1}$ and $y_{2}$ and also increases
the sender’s expected utility for fixed $y_{1}$ and $y_{2}$.

The intuition behind Theorem 3 is very simple, and is illustrated in Figure 3.
The condition in Theorem 3 says that a signal that induces a strictly single-
peaked triple $(y_{1},x_{1})$, $(y_{2},x_{2})$, $(y_{1},x_{3})$ with positive
probability can be improved by re-allocating mass $\beta_{1}$ on $x_{1}$ and
mass $\beta_{3}$ on $x_{3}$ from $y_{1}$ to $y_{2}$, while re-allocating mass
$\beta_{2}$ on $x_{2}$ from $y_{2}$ to $y_{1}$. This re-allocation is
profitable for the sender, because the sender’s expected utility increases
when $y_{1}$ and $y_{2}$ are held fixed (i.e., the first coordinate of
$R\beta$ is non-negative); the receiver’s marginal utility conditional on
being recommended $y_{1}$ increases (i.e., the second coordinate of $R\beta$
is non-negative), which increases the receiver’s action, and hence increases
the sender’s expected utility; and the receiver’s marginal utility conditional
on being recommended $y_{2}$ also increases (i.e., the third coordinate of
$R\beta$ is non-negative), which again increases the sender’s expected
utility. Moreover, at least one of these improvements is strict (i.e.,
$R\beta\neq 0$). The same logic also applies for any signal that induces a
strictly single-peaked triple, even if this triple occurs with $0$
probability, except now mass must be re-allocated from small intervals around
$x_{1}$, $x_{2}$, and $x_{3}$.222222Formally, this step relies on our
complementary slackness theorem, Theorem 1.

###  5.3. Sufficient Conditions for Single-Dipped Disclosure

We can now give our main sufficient condition on utilities for single-dipped
disclosure to be optimal. This is a central result of our paper. As we will
see, our condition covers several prior models, as well as some new
applications.

######  Theorem 4.

If $u_{yx}(y,x)/u_{x}(y,x)$ and $V_{yx}(y_{2},x)/u_{x}(y_{1},x)$ are
increasing in $x$ for any $y$ and $y_{1}\leq y_{2}$, then there exists an
optimal single-dipped signal.

Moreover, if in addition either $u_{yx}(y,x)/u_{x}(y,x)$ or
$V_{yx}(y_{2},x)/u_{x}(y_{1},x)$ is strictly increasing in $x$ for any $y$ and
$y_{1}\leq y_{2}$, then the contact set is strictly single-dipped, and hence
so is every optimal signal.

The proof establishes single-dippedness by constructing perturbations that
satisfy the conditions in Theorem 3, and further establishes strict single-
dippedness by verifying the twist condition from Theorem 2.

$x_{1}$$x_{2}$$x_{3}$$V\left(y_{2},x\right)$$V\left(y_{1},x\right)$$V$$x$ |  $x_{1}$$x_{2}$$x_{3}$$u\left(y_{1},x\right)$$u\left(y_{2},x\right)$$u$$x$
---|---
a. Linear Receiver Case. | b. State-Independent Sender Case.
Figure 4. The Intuition for Theorem 4 in Two Special Cases _Notes:_ Panel a.
In the linear receiver case, when the sender’s utility increment
$V(y_{2},x)-V(y_{1},x)$ is convex in the state, more extreme states should
induce higher actions.
Panel b. In the state-independent sender case, when the receiver’s marginal
utility $u(y,x)$ is more convex in the state at higher actions, more extreme
states should induce higher actions.

The intuition for Theorem 4 is relatively straightforward in the linear
receiver and state-independent sender cases. (See Figure 4.) In the linear
receiver case, we have $u_{yx}(y,x)/u_{x}(y,x)=0$ and
$V_{yx}(y_{2},x)/u_{x}(y_{1},x)=V_{yx}(y_{2},x)$, so our sufficient conditions
for single-dipped disclosure to be optimal are satisfied iff $V_{y}$ is convex
in $x$.232323In the separable and translation-invariant subcases, convexity of
$V_{y}$ simplifies to convexity of $w$ and $P^{\prime}$, respectively. To see
why, note that for any strictly single-peaked triple $(y_{1},x_{1})$,
$(y_{2},x_{2})$, $(y_{1},x_{3})$, the perturbation that moves mass on $x_{1}$
and $x_{3}$ from $y_{1}$ to $y_{2}$ and moves mass on $x_{2}$ in the opposite
direction, so as to hold fixed the receiver’s marginal utility conditional on
being recommended either action, has the effect of also holding fixed the
probability of each recommendation, while spreading out the state conditional
on action $y_{2}$ and concentrating the state conditional on action $y_{1}$.
This perturbation is profitable when the difference $V(y_{2},x)-V(y_{1},x)$ is
convex in $x$, which holds whenever $V_{y}$ is convex in $x$.

In the state-independent sender case, we have
$V_{yx}(y_{2},x)/u_{x}(y_{1},x)=0$, so our sufficient conditions for single-
dipped disclosure to be optimal are satisfied iff $u_{x}$ is log-supermodular
in $(y,x)$, or equivalently $u$ is more convex in $x$ at higher actions
$y$.242424In the separable and translation-invariant subcases, log-
supermodularity of $u_{x}$ simplifies to $2I^{\prime}(x)^{2}\geq
I(x)I^{\prime\prime}(x)$ and log-concavity of $T^{\prime}$, respectively. To
see why, note that for any strictly single-peaked triple $(y_{1},x_{1})$,
$(y_{2},x_{2})$, $(y_{1},x_{3})$, the perturbation that moves mass on $x_{1}$
and $x_{3}$ from $y_{1}$ to $y_{2}$ and moves mass on $x_{2}$ in the opposite
direction, so as to hold fixed the receiver’s marginal utility conditional on
being recommended $y_{1}$ as well as the total probability of each
recommendation, has the effect of increasing the receiver’s marginal utility
conditional on being recommended $y_{2}$. This follows because, by log-
supermodularity of $u_{x}$, for the receiver’s expected marginal utility the
marginal rate of substitution between “shifting weight from $x_{1}$ to
$x_{2}$” and “shifting weight from $x_{2}$ to $x_{3}$” is higher at $y_{1}$
than $y_{2}$. Finally, when $V$ is state-independent, this perturbation
increases the sender’s expected utility.252525In the linear receiver and
state-independent sender cases, the sufficient conditions for the optimality
of strict single-dipped disclosure in Theorem 4 are “almost necessary,”
because the condition $|S|\neq 0$ on $Y\times[0,1]$ implies that $|S|$ has a
constant sign on $Y\times[0,1]$, which can be shown to be equivalent to strict
convexity of $V_{y}$ in the linear receiver case, and to strict log-
supermodularity of $u_{x}$ in the state-independent sender case. By Theorem 2,
a necessary condition for the optimality of strictly single-dipped disclosure
is that $|S|\neq 0$ on the restricted domain where $x_{1}<\chi(y)<x_{3}$.

As we explain in Section 7, there are close antecedents to the conditions in
Theorem 4 for the linear receiver and state-independent sender cases, in non-
persuasion settings that nonetheless fall in our general optimal productive
transport framework. In particular, results in the MOT literature (e.g.,
Theorem 6.1 of Beiglböck and Juillet 2016) can be translated to our framework
to imply the linear receiver case of Theorem 4, while results in the
gerrymandering literature (Lemma 1 of Friedman and Holden 2008) can be
translated to our framework to imply the state-independent sender case of
Theorem 4. Theorem 4 thus unifies and generalizes these disparate
contributions.

We also establish an additional result in Appendix B: under our conditions for
strictly single-dipped disclosure to be optimal (and a regularity condition),
the optimal signal is unique.262626This result is somewhat akin to _Brenier’s
theorem_ in optimal transport, which shows that the optimal transport plan is
unique under a suitable complementarity-type condition, called the twist or
generalized Spence-Mirrlees condition (Brenier 1991, Gangbo and McCann 1996;
or see Section 1.3 in Santambrogio 2015).

##  6\. Full Disclosure and Negative Assortative Disclosure

While single-dippedness is an important property, it remains important to
fully characterize optimal signals, when this is tractable.272727Recall that
many different disclosure patterns can be single-dipped, as illustrated in
Figure 2. The current section does this for the polar cases of “maximum” and
“minimal” pairwise disclosure. The former case corresponds to _full
disclosure_ , where each state is disclosed; while the latter case corresponds
to _negative assortative disclosure_ , where all states are paired in a
negatively assortative manner, so all posteriors can be ordered from least to
most extreme. Here our results on full disclosure extend existing results,
whereas our results on negative assortative disclosure are entirely novel.

###  6.1. Full Disclosure

_Full disclosure_ is the (unique) signal $\tau$ where every
$\mu\in\operatorname{supp}(\tau)$ is degenerate.

If for all states $x_{1}$ and $x_{2}$, and all probabilities $\rho$, the
sender prefers to split the posterior
$\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$ into degenerate posteriors
$\delta_{x_{1}}$ and $\delta_{x_{2}}$, then the sender prefers full disclosure
to any pairwise signal. Since pairwise plans are without loss by Theorem 2,
full disclosure is then optimal. Conversely, if the sender strictly prefers
not to split $\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$ into
$\delta_{x_{1}}$ and $\delta_{x_{2}}$ for some states $x_{1}$ and $x_{2}$ and
some probability $\rho$, then the sender strictly prefers the pairwise signal
that differs from full disclosure only in that it pools states $x_{1}$ and
$x_{2}$ into $\mu$; so full disclosure is not optimal.282828This argument is
valid when $\phi$ has finite support. The general case (Theorem 5) uses
duality and is adaptated from part (2) of Proposition 1 in Kolotilin (2018);
we give a slightly simpler proof and also establish uniqueness. Recalling that
belief $\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$ induces action
$\gamma(\mu)$ satisfying $\rho
u(\gamma(\mu),x_{1})+(1-\rho)u(\gamma(\mu),x_{2})=0$, we obtain the following
result.

######  Theorem 5.

Full disclosure is optimal iff, for all
$\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$ with $x_{1}<x_{2}$ in $X$ and
$\rho\in(0,1)$, we have

$\rho V(\gamma(\mu),x_{1})+(1-\rho)V(\gamma(\mu),x_{2})\leq\rho
V(\gamma(\delta_{x_{1}}),x_{1})+(1-\rho)V(\gamma(\delta_{x_{2}}),x_{2}).$ (6)

Moreover, full disclosure is uniquely optimal if (6) holds with strict
inequality for all such $\mu$.

In the linear case, condition (6) holds iff $V$ is convex in $y$. In the
state-independent sender case, condition (6) simplifies as follows:

######  Corollary 2.

In the state-independent sender case, full disclosure is optimal iff, for all
$\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$ with $x_{1},x_{2}\in X$ and
$\rho\in(0,1)$, we have

$V\left(\gamma\left(\mu\right)\right)\leq\rho
V\left(\gamma\left(\delta_{x_{1}}\right)\right)+\left(1-\rho\right)V\left(\gamma\left(\delta_{x_{2}}\right)\right).$
(7)

In a classical one-to-one matching model, Becker (1973) showed that if the
utility from matching two types $h\left(x_{1},x_{2}\right)$ is supermodular,
then it is optimal to match like types. Legros and Newman (2002) refer to this
extreme form of positive assortative matching as _segregation_. Their
Propositions 4 and 9 show that segregation is optimal iff
$h\left(x_{1},x_{1}\right)+h\left(x_{2},x_{2}\right)\geq
2h\left(x_{1},x_{2}\right)$ for all $x_{1},x_{2}$ (which is a strictly weaker
property than supermodularity). In the persuasion setting, segregation
corresponds to full disclosure. Note that if we fix $\rho=1/2$ and let
$h\left(x_{1},x_{2}\right)=V\left(\gamma\left(\delta_{x_{1}}/2+\delta_{x_{2}}/2\right)\right)$,
then (7) reduces to Legros and Newman’s condition. Intuitively, full
disclosure is “less likely” to be optimal in persuasion than segregation is in
classical matching, because in persuasion the designer has an extra degree of
freedom $\rho$ in designing matches.

In the linear receiver case, there is a simple sufficient condition for (6):

######  Corollary 2’.

In the linear receiver case, full disclosure is optimal if $V(y,x)$ is convex
in $y$ and satisfies $V(x_{1},x_{2})+V(x_{2},x_{1})\leq
V(x_{1},x_{1})+V(x_{2},x_{2})$ for all $x_{1},x_{2}\in X$.

A sufficient condition for $V(x_{1},x_{2})+V(x_{2},x_{1})\leq
V(x_{1},x_{1})+V(x_{2},x_{2})$ is supermodularity of $V$: for all
$x_{1}<x_{2}$ and $y_{1}<y_{2}$, $V(y_{1},x_{1})+V(y_{2},x_{2})\geq
V(y_{1},x_{2})+V(y_{2},x_{1})$. Thus, in the linear receiver case, full
disclosure is optimal whenever the sender’s utility is convex in $y$ and
supermodular in $\left(y,x\right)$. This sufficient condition for full
disclosure generalizes that given by Rayo and Segal (2010) for the separable
subcase.292929Their condition is that $w$ is increasing in $x$ and $G$ is
convex in $y$, where $V(y,x)=w(x)G(y)$. In the sub-subcase with $G(y)=y$, (6)
holds iff $w$ is increasing in $x$, because (6) simplifies to
$\rho(1-\rho)(w(x_{2})-w(x_{1}))(x_{2}-x_{1})\geq 0$.

In addition, in Appendix B we show that when the prior has full support and
the twist condition holds, full disclosure is uniquely optimal whenever it is
optimal.

###  6.2. Negative Assortative Disclosure

A pairwise signal $\tau$ is _negative assortative_ if the supports of any
$\mu,\mu^{\prime}\in\operatorname{supp}(\tau)$ are nested: that is, denoting
$\operatorname{supp}(\mu)=\\{x_{1},x_{2}\\}$ and
$\operatorname{supp}(\mu^{\prime})=\\{x^{\prime}_{1},x^{\prime}_{2}\\}$, where
without loss $x_{1}\leq x_{2}$ and $x^{\prime}_{1}\leq x^{\prime}_{2}$, we
have either $x_{1}\leq x^{\prime}_{1}\leq x^{\prime}_{2}\leq x_{2}$ or
$x^{\prime}_{1}\leq x_{1}\leq x_{2}\leq x^{\prime}_{2}$. We also apply the
same definition to an arbitrary set of posteriors $M\in\Delta(X)$ by replacing
$\operatorname{supp}(\tau)$ with $M$. In particular, a strictly single-dipped
contact set $\Lambda$ is negative assortative if $\chi_{1}:Y_{\Lambda}\to X$
is decreasing and $\chi_{2}:Y_{\Lambda}\to X$ is increasing.303030Recall that,
as defined in Remark 2, $\chi_{1}(y)$ and $\chi_{2}(y)$ are the smaller and
larger states that are pooled together to induce action $y\in Y_{\Lambda}$.

The main result of this section is that if strictly single-dipped disclosure
is optimal and the sender strictly prefers to pool any two states, then
negative assortative disclosure is optimal. Moreover, if the prior has a
density, then the optimal signal is unique (by Theorem 7 in Appendix B) and is
characterized as the solution to a system of two ordinary differential
equations.

To see the intuition, note that if strictly single-dipped disclosure is
optimal, then any two pairs of pooled states $\\{x_{1},x_{3}\\}$ and
$\\{x^{\prime}_{1},x^{\prime}_{3}\\}$ with $x_{1}<x_{3}$,
$x^{\prime}_{1}<x^{\prime}_{3}$, and $x_{1}\leq x^{\prime}_{1}$, must be
either ordered (i.e., $x_{1}<x_{3}\leq x^{\prime}_{1}<x^{\prime}_{3}$) or
nested (i.e., $x_{1}\leq x^{\prime}_{1}<x^{\prime}_{3}\leq x_{3}$). This
follows because if the pairs overlap (i.e.,
$x_{1}<x^{\prime}_{1}<x_{3}<x^{\prime}_{3}$), then either
$(x_{1},x^{\prime}_{1},x_{3})$ or $(x^{\prime}_{1},x_{3},x^{\prime}_{3})$,
together with the corresponding actions, would form a single-peaked triple.
Hence, for any pair of pooled states $\\{x_{1},x_{3}\\}$, there must exist a
disclosed state $x_{2}\in(x_{1},x_{3})$: intuitively, there must exist pairs
of pooled states in the interval $(x_{1},x_{3})$ that are closer and closer
together, until the pair degenerates into a single disclosed state. Therefore,
if any two pairs of pooled states $\\{x_{1},x_{3}\\}$ and
$\\{x^{\prime}_{1},x^{\prime}_{3}\\}$ are ordered, there would exist two
distinct disclosed states $x_{2}\in(x_{1},x_{3})$ and
$x^{\prime}_{2}\in(x^{\prime}_{1},x^{\prime}_{3})$. But if the sender strictly
prefers to pool any two states, this is impossible. Finally, if pairs of
pooled states cannot overlap or be ordered, the only remaining possibility is
that all pairs of pooled states are nested: that is, disclosure is negative
assortative.313131In this argument, the existence of the two disclosed states
relies on the assumption that $\operatorname{supp}(\phi)=[0,1]$. The formal
proof again relies on complementary slackness.

######  Theorem 6.

Assume that $X=[0,1]$. If $\Lambda$ is strictly single-dipped and for all
$x_{1}<x_{2}$ there exists $\rho\in(0,1)$ such that

$\rho V(\gamma(\mu),x_{1})+(1-\rho)V(\gamma(\mu),x_{2})>\rho
V(\gamma(\delta_{x_{1}})),x_{1})+(1-\rho)V(\gamma(\delta_{x_{2}}),x_{2}),$ (8)

with $\mu=\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}}$, then $\Lambda$ is
negative assortative. Moreover, if the prior $\phi$ has a density $f$, then
the optimal signal is unique, and the functions $\chi_{1}$ and $\chi_{2}$ are
continuous and solve the system of differential equations,

$\displaystyle
u(y,\chi_{1}(y))(-\mathrm{d}\phi([0,\chi_{1}(y)]))+u(y,\chi_{2}(y))\mathrm{d}\phi([0,\chi_{2}(y)])=0,$
(9)
$\displaystyle\begin{gathered}\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac{V_{y}(y,\chi_{1}(y))u(y,\chi_{2}(y))-V_{y}(y,\chi_{2}(y))u(y,\chi_{1}(y))}{u(y,\chi_{1}(y))u_{y}(y,\chi_{2}(y))-u(y,\chi_{2}(y))u_{y}(y,\chi_{1}(y))}\right)\\\
=\frac{V_{y}(y,\chi_{1}(y))u_{y}(y,\chi_{2}(y))-V_{y}(y,\chi_{2}(y))u_{y}(y,\chi_{1}(y))}{u_{y}(y,\chi_{1}(y))u(y,\chi_{2}(y))-u_{y}(y,\chi_{2}(y))u(y,\chi_{1}(y))},\end{gathered}$
(12)

for all $y>\underline{y}$ in $Y_{\Lambda}$, with the boundary conditions

$\begin{gathered}(\chi_{1}(\overline{y}),\chi_{1}(\underline{y}),\chi_{2}(\underline{y}),\chi_{2}(\overline{y}))=(0,\chi(\underline{y}),\chi(\underline{y}),1),\end{gathered}$
(13)

where $\underline{y}=\min Y_{\Lambda}$ and $\overline{y}=\max Y_{\Lambda}$.

Like equation (6) in the previous subsection, equation (8) simplifies in
special cases. In the linear case, (8) holds iff $V$ is strictly concave in
$y$.323232In the linear case, $V$ is strictly concave iff no disclosure is
uniquely optimal for all priors, by Corollary 1 in Kolotilin, Mylovanov, and
Zapechelnyuk (2022). In the state-independent sender case, it holds iff
$V(\gamma(\mu))>\rho
V(\gamma(\delta_{x_{1}}))+(1-\rho)V(\gamma(\delta_{x_{2}}))$. In the linear
receiver case, it holds if $V(y,x)$ is concave in $y$ and satisfies
$V(x_{1},x_{2})+V(x_{2},x_{1})>V(x_{1},x_{1})+V(x_{2},x_{2})$ for all
$x_{1}<x_{2}$; a sufficient condition for the latter property is strict
submodularity of $V$. These conditions generalize the sufficient condition for
pooling given by Rayo and Segal (2010) for the separable subcase.333333Their
condition is that $w$ is strictly decreasing in $x$ and $G$ is concave in $y$,
where $V(y,x)=w(x)G(y)$. In the sub-subcase with $G(y)=y$, (8) holds iff $w$
is strictly decreasing in $x$.

To understand the differential equations, note that if $\chi_{1}$ and
$\chi_{2}$ are differentiable then (9) can be written as

$u(y,\chi_{1}(y))f(\chi_{1}(y))\chi_{1}^{\prime}(y)=u(y,\chi_{2}(y))f(\chi_{2}(y))\chi_{2}^{\prime}(y).$

This follows from (1) and (BP), as each posterior $\mu$ inducing
$y=\gamma(\mu)$ is

$\mu=\frac{-f(\chi_{1}(y))\chi_{1}^{\prime}(y)}{-f(\chi_{1}(y))\chi_{1}^{\prime}(y)+f(\chi_{2}(y))\chi_{2}^{\prime}(y)}\delta_{\chi_{1}(y)}+\frac{f(\chi_{2}(y))\chi_{2}^{\prime}(y)}{-f(\chi_{1}(y))\chi_{1}^{\prime}(y)+f(\chi_{2}(y))\chi_{2}^{\prime}(y)}\delta_{\chi_{2}(y)}.$

In addition, (12) results from solving the system of equations (from the
sender’s FOC, (4)),

$\displaystyle
v(y,\chi_{1}(y))+q(y)u_{y}(y,\chi_{1}(y))+q^{\prime}(y)u(y,\chi_{1}(y))=0,$
$\displaystyle
v(y,\chi_{2}(y))+q(y)u_{y}(y,\chi_{2}(y))+q^{\prime}(y)u(y,\chi_{2}(y))=0,$

for $q(y)$ and $q^{\prime}(y)$, and recalling that $q^{\prime}$ is the
derivative of $q$. Finally, the boundary condition (13) follows because the
lowest induced action $\underline{y}$ is induced at the disclosed state
$\chi(\underline{y})=\chi_{1}(\underline{y})=\chi_{2}(\underline{y})$, and the
highest induced action $\overline{y}$ is induced at states
$0=\chi_{1}(\overline{y})$ and $1=\chi_{2}(\overline{y})$.353535In the linear
receiver case, (12) simplifies to
$\frac{\mathrm{d}}{\mathrm{d}y}\left(V_{y}(y,\chi_{1}(y))\frac{\chi_{2}(y)-y}{\chi_{2}(y)-\chi_{1}(y)}+V_{y}(y,\chi_{2}(y))\frac{y-\chi_{1}(y)}{\chi_{2}(y)-\chi_{1}(y)}\right)=-\frac{V_{y}(y,\chi_{2}(y))-V_{y}(y,\chi_{1}(y))}{\chi_{2}(y)-\chi_{1}(y)}.$
Geometrically, this says that the slope of the curve
$\gamma(\mu)\mapsto\mathbb{E}_{\mu}[V_{y}(\gamma(\mu),x)]$ is equal to the
negative of the slope of the secant passing through the points
$(\chi_{1}(\gamma(\mu)),V_{y}(\gamma(\mu),\chi_{1}(\gamma(\mu))))$ and
$(\chi_{2}(\gamma(\mu)),V_{y}(\gamma(\mu),\chi_{2}(\gamma(\mu))))$. Nikandrova
and Pancs (2017) derive this condition for the separable sub-subcase with
$V_{y}(y,x)=w(x)$.

We can also give primitive conditions on $V$ and $u$ for (8) to hold, and
hence for the unique optimal signal to be negative assortative.

######  Corollary 3.

Assume that the condition for strict single-dippedness given in Theorem 4
holds. Then for all $x_{1}<x_{2}$ there exists $\rho\in(0,1)$ such that (8)
holds iff, for all $y\in Y$,

$\begin{gathered}V_{yy}(y,\chi(y))\leq\tfrac{V_{y}(y,\chi(y))u_{yy}(y,\chi(y))}{u_{y}(y,\chi(y))}+2\tfrac{V_{yx}(y,\chi(y))u_{y}(y,\chi(y))-V_{y}(y,\chi(y))u_{yx}(y,\chi(y))}{u_{x}(y,\chi(y))}.\end{gathered}$
(14)

Equation (14) is a local necessary condition for (8): if (14) fails, then (8)
also fails for $x_{1}<x_{2}$ sufficiently close to $\chi(y)$. When the
condition for strict single-dippednes holds, this local necessary condition
turns out to be globally sufficient for (8). Equation (14) simplifies
dramatically in some special cases. In the linear receiver case, (14)
simplifies to $V_{yy}(y,y)+2V_{yx}(y,y)\leq 0$; in the translation-invariant
subcase of the linear receiver case, this simplifies further to
$P^{\prime\prime}(0)\geq 0$. In the separable (resp., translation-invariant)
subcase of the state-independent sender case, (14) simplifies to
${V_{yy}(y)}/{V_{y}(y)}\leq 2I^{\prime}(y)/I(y)$ (resp.,
${V_{yy}(y)}/{V_{y}(y)}\leq{T^{\prime\prime}(0)}/{T^{\prime}(0)}$).

In Appendix C, we give three examples of optimal single-dipped negative
assortative disclosure. Example 1 illustrates how the differential equations
(9)–(12) can sometimes be explicitly solved to find the optimal signal.
Example 2 characterizes the optimal signal in quantile persuasion (i.e., Case
(3c) in Section 2.2). In quantile persuasion, our sufficient conditions for
strictly single-dipped disclosure to be optimal are not satisfied, and there
are multiple optimal signals; however, one optimal signal is strictly single-
dipped negative assortative. Finally, Example 3 illustrates that in some cases
the unique optimal signal randomizes conditional on the state, even when the
prior is atomless.363636In contrast, Zeng (2023) shows that there is always a
deterministic optimal signal in the separable subcase of the linear receiver
or state-independent sender case.

##  7\. Optimal Productive Transport: Theory and Applications

We now give a general interpretation of our framework in terms of assigning
inputs to productive units, and describe the implications of our results for
matching, option pricing, and partisan gerrymandering, as well as some
specific persuasion models. We pay particular attention to the implications of
single-dippedness summarized in Table 1 in the Introduction.

###  7.1. Optimal Productive Transport

Our signal-based primal problem (P) may be generalized as follows: given a
distribution $\phi\in\Delta([0,1])$ with support $\operatorname{supp}(\phi)=X$
of _inputs_ $x$, find a distribution $\tau\in\Delta(\Delta(X))$ of _productive
units_ $\mu\in\Delta(X)$ to

$\displaystyle\text{maximize}\quad\int_{\Delta(X)}\int_{X}V(\tilde{\gamma}(\mu),x)\mathrm{d}\mu(x)\mathrm{d}\tau(\mu)$
$\displaystyle\text{subject
to}\quad\int_{\Delta(X)}\mu\mathrm{d}\tau(\mu)=\phi,$

where $\tilde{\gamma}:\Delta(X)\to Y$ is an arbitrary _production function_
that specifies the _output_ $y$ produced by unit $\mu$.373737The
interpretation of the assumption that the domain of $\tilde{\gamma}$ is
probability measures on $X$ rather than arbitrary measures is that production
exhibits constant returns to scale: i.e., nothing is gained by varying the
scale of a productive unit. This _optimal productive transport_ problem is the
same as (P), except that the value of an arbitrary production function
$\tilde{\gamma}(\mu)$ is not necessary given by the first-order condition (1)
for some function $u$.

But now suppose that $\tilde{\gamma}$ satisfies the following two properties:

Betweenness:

For any $\mu,\eta\in\Delta(X)$ satisfying
$\tilde{\gamma}(\mu)>\tilde{\gamma}(\eta)$, and any $\rho\in(0,1)$, we have
$\tilde{\gamma}(\mu)>\tilde{\gamma}(\rho\eta+(1-\rho)\mu)>\tilde{\gamma}(\eta).$

Continuity:

$\tilde{\gamma}$ is continuous on $\Delta(X)$.

The key property here is the first one, which says that mixing two units
produces an output in between those produced by each of them in isolation. The
following result—which adapts Proposition A.1 of Dekel (1986)—shows that these
properties ensure the existence of a function $u$ such that $\tilde{\gamma}$
is given by (1). Thus, under Betweenness and Continuity, the optimal
productive transport problem is exactly the same as our persuasion problem.

######  Proposition 1.

A function $\tilde{\gamma}$ satisfies Betweenness and Continuity iff there
exists a continuous function $u:Y\times X\to\mathbb{R}$ such that, for any
$\mu\in\Delta(X)$,

$\int_{X}u(y,x)\mathrm{d}\mu(x)=(<)0\iff y=(>)\tilde{\gamma}(\mu).$

In the persuasion context, the function $u$ in Proposition 1 is the receiver’s
marginal utility. More generally, $u(y,x)$ can be viewed as a measure of the
“efficacy” of input $x$ in the production of output $y$. For the remainder of
this section, we assume Betweenness and Continuity, as well as that $u$
satisfies Assumptions (1)–(4).

The general interpretation of the signal-based primal problem (P) is that a
planner obtains utility $V(y,x)$ from using input $x$ in the production of
output $y$, and assigns inputs to productive units according to a _production
plan_ $\tau\in\Delta(\Delta(X))$ to maximize the expectation of
$V(\gamma(\mu),x)$. The corresponding interpretation of the signal-based dual
problem (D) is that there is a decentralized economy with price $p(x)$
attached to input $x$, where the zero-profit condition (ZP) says that an
entrepreneur who obtains utility $V(y,x)$ from using input $x$ in the
production of output $y$ cannot create a unit $\mu\in\Delta(X)$ that leaves
her with a positive utility after paying for its inputs.

More precisely, a pair $(\tau,p)\in\Delta(\Delta(X))\times L(X)$ is a
_competitive equilibrium_ if
(i) all inputs are assigned to productive units:
$\int\mu\mathrm{d}\tau(\mu)=\phi$;
(ii) operating units make zero profits: $\int p(x)\mathrm{d}\mu=W(\mu)$ for
all $\mu\in\operatorname{supp}(\tau)$;
(iii) no entrant can make strictly positive profits: $\int
p(x)\mathrm{d}\mu\geq W(\mu)$ for all $\mu\in\Delta(X)$.
Then, by strong duality (Lemma 1), we have

######  Remark 3.

A competitive equilibrium exists, and a pair $(\tau,p)$ is a competitive
equilibrium iff $\tau$ solves (P) and $p$ solves (D).

We will make use of the interpretation of optimal plans and prices as
competitive equilibria in the matching context in Section 7.2.

Similarly, the general interpretation of the outcome-based primal problem (P’)
is that a planner chooses an outcome $\pi\in\Delta(Y\times X)$ to maximize her
expected utility, subject to the constraints that all inputs are utilized and
that each output $y$ is produced by a unit $\mu$ satisfying $\gamma(\mu)=y$.
The corresponding dual can again be interpreted as a decentralized economy,
where now the zero-profit condition (ZP’) says that an entrepreneur who
produces any output $y$ cannot profitably employ any input in the production
of this output, after accounting for the input’s price, $p(x)$, and the
marginal effect of its use on the output, $q(y)u(y,x)$. Finally, complementary
slackness (Theorem 1) says that any entrepreneur who breaks even must employ
only inputs that satisfy (ZP’) with equality.

###  7.2. Matching and Clubs

Assigning workers with heterogeneous abilities to firms with workplace peer
effects (i.e., intra-firm spillovers) is an important topic in labor economics
(Kremer and Maskin 1996, Saint-Paul 2001, Eeckhout 2018, Boerma, Tsyvinski,
and Zimin 2021). In Saint-Paul (2001), there is a continuum of workers indexed
by ability $x\in[0,1]$. The population distribution of $x$ is $\phi$, with
support $X\subset[0,1]$. Workers sort into firms, which are ex ante
homogeneous and face constant returns to scale, so that a firm can be
identified with its distribution of workers $\mu\in\Delta(X)$. Workplace peer
effects depend on the mean worker ability in a firm,
$\gamma(\mu)=\mathbb{E}_{\mu}[x]$, so the output of a worker with ability $x$
in firm $\mu$ can be written as $V(\gamma(\mu),x)$. The planner’s problem of
assigning workers to firms to maximize total output is thus precisely (P), in
the linear receiver case where $\gamma(\mu)=\mathbb{E}_{\mu}[x]$. Moreover,
the problem of finding competitive equilibrium wages $p(x)$ for workers with
ability $x$ (where, as in Section 7.1, a competitive equilibrium is an
assignment of workers to firms and wages such that operating firms make zero
profits and no entrant can make strictly positive profits), is precisely (D).
In this context, Lemma 1 says that a competitive equilibrium exists and
maximizes total output.383838This generalizes Proposition 3 of Saint-Paul
(2001).

Saint-Paul (2001) considered the special case of this model where
$V(y,x)=xG(y)$ for an increasing function $G$. In this case, the total output
of a firm $\mu$ equals $\mathbb{E}_{\mu}[x]G(\mathbb{E}_{\mu}[x])$. Since this
is a function only of $\mathbb{E}_{\mu}[x]$, Saint-Paul (2001) coincides with
the linear case of our model. We now summarize the implications of our results
for the general linear receiver case of our model in the worker assignment
context. We pay special attention to the separable case where
$V(y,x)=w(x)G(y)$ for increasing functions $w$ and $G$, which may be
particularly natural in this context.

In the worker assignment context, an assignment is pairwise if each firm
contains at most two worker types, and a pairwise assignment is strictly
single-dipped if firms with more heterogeneous workers have higher average
worker ability. Since we are in the linear receiver case, Theorem 4 implies
that a strictly single-dipped assignment is optimal whenever $V_{y}$ is
strictly convex in $x$—or, in the separable case, $w$ is strictly convex.
Intuitively, $V_{y}$ is the marginal benefit of having higher-ability
coworkers, so when this is convex in a worker’s own ability, it is optimal for
firms where the distribution of worker abilities is more spread out to have
higher average worker ability. Moreover, when in addition $V$ is strictly
increasing in both arguments and convex in $x$, firms with more heterogeneous
workers also produce higher output, as if
$\mathbb{E}_{\mu}[x]=y<y^{\prime}=\mathbb{E}_{\mu^{\prime}}[x]$ and
$\operatorname{supp}(\mu^{\prime})$ is more spread out than
$\operatorname{supp}(\mu)$, then

$\mathbb{E}_{\mu}[V(y,x)]<\mathbb{E}_{\mu}[V(y^{\prime},x)]<\mathbb{E}_{\mu^{\prime}}[V(y^{\prime},x)].$

Thus, in this case firms with more heterogeneous workforces are more
productive, and hence also pay higher average wages (by zero profit).

Our conditions for full disclosure (i.e., _segregation_ , where each firm has
a homogeneous workforce) and negative assortative disclosure/matching (where
all firms can be ordered from least to most heterogeneous) are also
interesting in the worker assignment context. By Corollary 2’, segregation is
optimal if $V$ is convex in $y$ and supermodular—or, in the separable case,
$G$ is convex. On the other hand, by Corollary 3, negative assortative
matching is optimal if $V_{y}$ is strictly convex in $x$ and
$V_{yy}+2V_{yx}\leq 0$—or, in the separable case, $w$ is strictly convex and
$w(x)G^{\prime\prime}(y)+2w^{\prime}(x)G^{\prime}(y)\leq 0$. Intuitively,
these results say that segregation is optimal if peer effects are convex, and
that negative assortative matching is optimal if single-dipped assignment is
optimal and peer effects are sufficiently concave.

We also note that $p(x)$ is convex whenever $V$ is convex in $x$—or, in the
separable case, $w$ is convex. This follows because $p(x)=\sup_{y\in
Y}V(y,x)+q(y)(x-y)$, so if $V_{xx}>0$ then $p$ is the supremum of a set of
convex functions. Recalling that $p(x)$ is the equilibrium wage of a worker
with ability $x$, this says that wages rise more than one-for-one with
ability. This result reflects the fact that higher-ability workers are not
only better workers, but also better coworkers.

A model that is equivalent to worker-firm matching can capture the assignment
of students to schools with peer effects, or more generally the assignment of
heterogeneous agents to clubs. In Arnott and Rowse (1987), there is a
continuum of students indexed by ability $x$, who must be assigned to ex ante
identical schools, which can be identified with their student bodies
$\mu\in\Delta(X)$. A student with ability $x$ who attends a school $\mu$
attains an education that she values at $V(\gamma(\mu),x)$, where again
$\gamma(\mu)=\mathbb{E}_{\mu}[x]$. Arnott and Rowse (1987) study the planner’s
problem of assigning students to schools to maximize total educational
attainment: this problem is equivalent to the linear receiver case of (P). The
“decentralized” version of this problem is considered by Epple and Romano
(1998), who study competitive equilibrium in a market for private schooling,
where a school $\mu$ with mean student ability $y=\gamma(\mu)$ charges tuition
$t(y,x)$ to students of ability $x$. Here, a competitive equilibrium may be
defined precisely as in Section 7.1, with the interpretation that $p(x)$ is
the equilibrium utility of a student with ability $x$, and that the tuition
charged to a student with ability $x$ to attend a school with mean student
ability $y$ is $t(y,x)=V(y,x)-p(x)$. (The assumption here that schools take
student utility levels as given when setting tuition is called _utility
taking_ in the literature on club economies: e.g., Ellickson, Grodal,
Scotchmer, and Zame 1999.) In this context, Lemma 1 says that a competitive
equilibrium exists and maximizes total educational attainment.404040Arnott and
Rowse (1987) additionally endogenize public spending on schools, while Epple
and Romano (1998) let students differ in income as well as ability. Arnott and
Rowse (1987) focus on the Cobb-Douglas production function
$V(y,x)=x^{\alpha}y^{\beta}$, and provide conditions for the optimality of
segregation (“perfect streaming”) or no disclosure (“complete mixing”).

The conditions on $V$ under which an optimal assignment of students to schools
is single-dipped, segregation, or negative assortative are the same as in the
worker assignment context. Indeed, bearing in mind that $p(x)$ is the
equilibrium utility of an agent with ability $x$ in either model—so that the
wage of a worker with ability $x$ is $p(x)$, and the tuition paid by a student
with ability $x$ who attends a school with mean ability $y$ is
$V(y,x)-p(x)$—the models are identical. In particular, if $V_{y}$ is strictly
convex in $x$, then a strictly single-dipped assignment is optimal, so that
schools with more heterogeneous student bodies are more desirable for all
students.

Finally, the outcome-based dual (D’) has a particularly natural interpretation
in the student assignment/club economy setting. In a competitive equilibrium,
a school with mean student ability $y$ charges tuition $t(y,x)=q(y)(y-x)$ to
students with ability $x$. (Thus, a school subsides its students with above-
average ability, and charges students with below-average ability.) A student
with ability $x$ attends the school $y$ that gives her the highest utility,
$p(x)=\sup_{y\in Y}V(y,x)-q(y)(y-x)$. All operating schools break even, and no
entrepreneur can turn a positive profit by starting a new school.

###  7.3. Option Pricing

In mathematical finance, the literature on martingale optimal transport (e.g.,
Beiglböck, Henry-Labordere, and Penkner 2013, Galichon, Henry-Labordere, and
Touzi 2014, Beiglböck and Juillet 2016) studies the following problem. An
underlying asset will be marketed in two future periods, 1 and 2. In period 0,
an exotic option is for sale, which will pay $V(y,x)$ if the realized asset
price is $y$ in period 1 and $x$ in period 2. An analyst knows the marginal
distributions of $y$ and $x$, but her only information regarding their joint
distribution is that it satisfies $\mathbb{E}[x|y]=y$ for every $y$. The
interpretation of this assumption is that there are liquid markets for
European call options on the asset price in each period, from which the
analyst can infer the marginal distributions of the asset price (by Breeden
and Litzenberger 1978); and the analyst believes that the market satisfies no-
arbitrarge, which implies that the asset price is a martingale under the risk-
neutral measure. The analyst’s problem is to find the joint distribution
$\pi\in\Delta(Y\times X)$ that maximizes the expected value of the option
(and, thus, the maximum option price consistent with no-arbitrage), subject to
the two marginal constraints and the martingale constraint.

Now consider the variant of this problem where the marginal distribution of
$y$ is also unknown. The interpretation is that there is a liquid market for
call options only on the period 2 asset price: for example, perhaps the asset
is a share in a firm that is expected to go public after period 1, and there
are only liquid options markets for the prices of publicly traded firms. Then
the analyst’s problem of determining the maximum option price, subject to
constraint that the marginal distribution of the period 2 price $x$ is $\phi$,
and the martingale constraint $\mathbb{E}[x|y]=y$, is precisely (P’), in the
linear receiver case where $\gamma(\mu)=\mathbb{E}_{\mu}[x]$. Lemma 1
establishes strong duality for this problem.414141The possibility that the
period 1 marginal may be unknown, and the resulting problem (P’), are briefly
considered in Corollary 1.5 of Acciaio, Beiglbock, Penkner, and Schachermayer
(2016). That result establishes weak duality and primal attainment, but not
dual attainment, which as we discuss is an important issue.

In this context, the optimal dual variables $(p,q)$ have an important
interpretation. Recall that the option to be priced pays $V(y,x)$ when the
asset price is $y$ in period 1 and $x$ in period 2. An alternative to buying
this exotic option is to buy a simple option that pays $p(x)$ when the period
2 asset price is $x$, and in addition to plan to sell $q(y)$ units of the
asset itself in period 1 when the period 1 asset price is $y$. Since selling
$q(y)$ units at price $y$ yields a profit of $q(y)(y-x)$ when the period 2
asset price turns out to be $x$, this alternative strategy is sure to
outperform—or _super-replicate_ —the exotic option iff

$p(x)+q(y)(y-x)\geq V(y,x)\quad\text{ for all }(y,x)\in Y\times X.$

Note that this condition is precisely (ZP’). Thus, Lemma 1 implies that the
maximum option price can be calculated as either $\int_{Y\times
X}V(y,x)\mathrm{d}\pi(y,x)$ under the joint distribution of asset prices $\pi$
that solves (P’) (i.e., the maximum expected value of the exotic option), or
as $\int_{X}p(x)\mathrm{d}\phi(x)$, for the simple option payouts $p(x)$ that
solve (D’) (i.e., the price of the cheapest simple option that super-
replicates the exotic option).

In the option pricing context, a joint distribution of asset prices is
pairwise if it is a _binomial tree_ : each period 1 price $y$ can be followed
by at most two distinct period 2 prices $x$. Moreover, a pairwise joint
distribution is strictly single-dipped if more dispersed period 2 prices
follow higher period 1 prices: that is, if riskier assets are more expensive.
Since we are in the linear receiver case, Theorem 4 implies that the option
price is maximized by a strictly single-dipped distribution whenever $V_{y}$
is strictly convex in $x$. This condition is known as the “martingale Spence-
Mirrlees condition” in the MOT literature, which Beiglböck and Juillet (2016),
Henry-Labordère and Touzi (2016), and Beiglböck, Henry-Labordère, and Touzi
(2017) show implies that a strictly single-dipped distribution (which they
call a “left-curtain coupling”) is optimal in the standard MOT problem (where
the period 1 asset price distribution is fixed
exogenously).424242Specifically, Beiglböck and Juillet (2016) show that the
unique optimal outcome is single-dipped in the translation-invariant subcase
if $P^{\prime}$ is strictly convex (Theorem 6.1), and in the separable subcase
if $w$ is strictly convex (Theorem 6.3); while Theorem 5.1 in Henry-Labordère
and Touzi (2016) and Theorem 3.3 in Beiglböck, Henry-Labordère, and Touzi
(2017) extend this conclusion to the general linear receiver case where
$V_{y}$ is strictly convex in $x$. All these papers concern the MOT context,
where the distribution of $y$ is fixed exogenously. Moreover, by Corollary 2’,
full disclosure (where $x=y$ with probability 1) is optimal if $V$ is strictly
convex in $y$ and supermodular; while Corollary 3 implies that negative
assortative matching (where higher period 1 prices are always followed by more
dispersed period 2 prices, so more expensive assets are riskier) is optimal if
$V_{y}$ is strictly convex in $x$ and $V_{yy}+2V_{yx}\leq 0$.

The formula for $q(y)$ also has an interesting interpretation in the option
pricing context. By equation (3), for every period 1 price $y$ in the support
of the marginal of an optimal joint distribution $\pi$, we have

$q(y)=\mathbb{E}_{\pi}[V_{y}(y,x)|y].$

This is a version of Shephard’s lemma: the amount of the asset sold at period
1 price $y$ under the cheapest super-replicating strategy equals the
derivative of the option price with respect to $y$. Moreover, in finance, the
derivative of the option price with respect to the underlying asset price is
known as the option’s “Delta.” Thus, in the option pricing context, $q(y)$ is
simply Delta.

###  7.4. Partisan Gerrymandering

Partisan gerrymandering—where a partisan designer assigns voters to districts
to maximize her party’s seat share—is an important feature of American
politics. Kolotilin and Wolitzky (2023) develop and calibrate a model of
partisan gerrymandering, which generalizes the leading earlier models of Owen
and Grofman (1988), Friedman and Holden (2008), and Gul and Pesendorfer
(2010). In this model, there is a continuum of voters indexed by their
partisanship $x\in[0,1]$. The population distribution of $x$ is $\phi$, with
support $X\subset[0,1]$. The designer chooses a _districting plan_
$\tau\in\Delta(\Delta(X))$ that assigns voters to equipopulous districts
$\mu\in[0,1]$, prior to the realization of an aggregate shock $y\in\mathbb{R}$
with cdf $V$. The share of type-$x$ voters who vote for the designer’s party
when the aggregate shock takes value $y$ is deterministic and is denoted by
$v(y,x)\in[0,1]$.434343Among other notational differences, the order of the
arguments of $v$ is reversed in Kolotilin and Wolitzky (2023). The function
$v(y,x)$ is assumed to be strictly decreasing in $y$ and strictly increasing
in $x$: that is, higher aggregate shocks are less favorable for the designer,
while voters with higher partisanship are more favorable. The designer wins a
district $\mu$ iff she receives a majority of the district vote. Thus,
defining $u(y,x):=v(y,x)-1/2$, note that the designer wins a district $\mu$
iff $y\leq\gamma(\mu)$, where $\gamma(\mu)$ is given by (1). The designer thus
wins a district $\mu$ with probability $V(\gamma(\mu))$. Finally, the designer
chooses $\tau$ to maximize her expected seat share, subject to the constraint
that all voters are assigned to equipopulous districts: i.e.,
$\int_{\Delta(X)}\mu\mathrm{d}\tau(\mu)=\phi$.444444As discussed in Kolotilin
and Wolitzky (2023), the equipopulation constraint is strictly enforced in
practice, while other constraints on districting (such as geographic
continuity of districts) are often relatively slack, and are thus neglected in
much of the gerrymandering literature. The designer’s problem is thus
precisely (P), in the state-independent sender case where $V(y,x)=V(y)$. Note
that the designer’s preferences are state-independent because she cares only
about the probability of winning each district, and not directly about a
district’s composition.

In the gerrymandering context, a districting plan is pairwise if each district
contains at most two voter types, and a pairwise districting plan is strictly
single-dipped if more polarized districts are more favorable for the designer
(i.e., if $\gamma(\mu)>\gamma(\mu^{\prime})$ for all
$\mu,\mu^{\prime}\in\operatorname{supp}(\tau)$ such that
$\operatorname{supp}(\mu)$ contains $x_{1}<x_{3}$ and
$\operatorname{supp}(\mu^{\prime})$ contains $x_{2}\in(x_{1},x_{3})$.) Since
$V$ is state-independent, Theorem 4 implies that strictly single-dipped
districting is optimal whenever $u_{x}$ is strictly log-supermodular. This
result generalizes a main result of Friedman and Holden (2008) (their Lemma
1), which shows that strictly single-dipped districting is optimal under an
“informative signal property” that is equivalent to log-supermodularity of
$u_{x}$.454545Friedman and Holden (2008) additionally assume that there is a
finite number of districts and that $u$ satisfies another condition called
“central unimodality.” As explained in Kolotilin and Wolitzky (2023), the
intuition for this result is that log-supermodularity of $u_{x}$ means that
moderate voters “swing more” with the aggregate shock $y$ than more extreme
voters, so that a marginal voter is less likely to be pivotal in a district
consisting of moderates than in a district that is evenly divided between
left-wing and right-wing extremists. The designer then optimally exploits this
difference in pivot probabilities by assigning more favorable marginal voters
to more polarized districts: i.e., by creating a single-dipped districting
plan.

Kolotilin and Wolitzky (2023) go on to apply the duality and complementary
slackness developed in the current paper to derive further properties of
optimal districting plans, and in particular ask whether it is optimal for the
designer to segregate the strongest opposing voters (as in traditional “pack-
and-crack” districting plans) or more moderate voters (as in an alternative
plan proposed by Friedman and Holden 2008, which resembles negative
assortative disclosure).464646As argued by Cox and Holden (2011), this is a
key question for assessing the likely consequences of restrictions on
districting such as those instituted by the Voting Rights Act of 1965. The
reader is referred to that paper for these additional results.

###  7.5. Specific Persuasion Models

Our analysis covers most persuasion models with non-linear preferences
considered to date, including Zhang and Zhou’s 2016 model of information
disclosure in contests; Guo and Shmaya’s 2019 model of persuading a receiver
with affiliated private information; and Goldstein and Leitner’s 2018 model of
optimal stress tests. The main results in the latter two papers show that
single-peaked negative assortative disclosure is optimal in their models. In
Appendix D, we describe how our analysis covers these prior models and in some
cases provides additional results.

##  8\. Conclusion

This paper has developed a general model of assigning inputs to productive
outputs, which we call _optimal productive transport_. Our leading application
is Bayesian persuasion, but the model also covers other applications including
matching, option pricing, and partisan gerrymandering. In the persuasion
context, our substantive results provide conditions for all optimal signals to
be pairwise, for riskier or safer prospects to induce higher actions, and for
full or negative assortative disclosure to be optimal. In some cases, we can
characterize optimal signals as the solution to a pair of ordinary
differential equations, or even solve them in closed form. Methodologically,
we develop novel duality and complementary slackness theorems, which form the
basis of all of our proofs.

We mention a few open issues. First, while the persuasion literature has made
progress by allowing unrestricted disclosure policies, the pairwise signals
that we highlight are not always realistic. (For example, in reality it is
probably not feasible to design a stress test that pools only the weakest and
strongest banks.) An alternative, complementary approach is to restrict the
sender to partitioning the state space into intervals, as in Rayo (2013) and
Onuchic and Ray (2023). An interesting observation is that, at least in the
separable subcase of our model considered by Rayo and Onuchic and Ray, our
condition (8) is equivalent to the condition that complete pooling is uniquely
optimal among monotone partitions for all prior distributions. This suggests
that, under our conditions for the optimality of single-dipped/-peaked
disclosure, negative assortative disclosure might be the optimal unrestricted
disclosure policy for all priors iff no disclosure is the optimal monotone
policy for all priors. More generally, analyzing the relationship between the
optimal pairwise signals we have characterized and simpler signals such as
monotone partitions is an important direction for future research.

Second, in the informed receiver interpretation of our model mentioned in
Section 2, our analysis pertains to disclosure mechanisms that do not first
elicit the receiver’s type, or _public persuasion_ in the language of
Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017). Public persuasion turns out
to be without loss in Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017), as
well as in Guo and Shmaya (2019). It would be interesting to investigate
conditions for the optimality of public persuasion in our more general model,
and in particular to see how they relate to our conditions for the optimality
of full or negative assortative disclosure.

Third, while we have taken some steps toward fully characterizing optimal
signals by deriving the differential equations (9)–(12) and solving them in a
couple examples, much more remains to be done. Equations (9)–(12) are closely
related to the optimality and Monge-Ampere equations in optimal transport
(e.g., Section 1.7.6 in Santambrogio 2015). The rich mathematical literature
on these equations may hold some insights for fully characterizing optimal
signals in certain settings.

Finally, our model could be generalized to allow multidimensional states or
actions. We suspect that our results on duality (Lemma 1), complementary
slackness (Theorem 1), and pairwise signals (Theorem 2) generalize up to some
technicalities.474747For example, our proof of Theorem 1 is facilitated by the
existence of a bijection between actions $y$ and states $\chi(y)$ such that
$u(y,\chi(y))=0$, cf. Assumption 4. Generalizing our other results would
require a more general notion of single-dippedness. With a unidimensional
action and a multidimensional state, one can still define a notion of single-
dippedness as inducing higher actions at more extreme states; with
multidimensional actions, the appropriate generalization is
unclear.484848Possibly relevant recent work on multidimensional martingale
optimal transport includes Ghoussoub, Kim, and Lim (2019) and De March and
Touzi (2019). For results on multidimensional persuasion focusing on the
linear case, see Dworczak and Kolotilin (2022).

## References

  * (1)
  * Acciaio, Beiglbock, Penkner, and Schachermayer (2016) Acciaio, B., M. Beiglbock, F. Penkner, and W. Schachermayer (2016): “A Model-Free Version of the Fundamental Theorem of Asset Pricing and the Super-Replication Theorem,” _Mathematical Finance_ , 26(2), 233–251.
  * Aliprantis and Border (2006) Aliprantis, C. D., and K. Border (2006): _Infinite Dimensional Analysis: A Hitchhiker’s Guide_. Springer.
  * Alonso and Câmara (2016) Alonso, R., and O. Câmara (2016): “Bayesian Persuasion with Heterogeneous Priors,” _Journal of Economic Theory_ , 165, 672–706.
  * Ambrosio, Brué, and Semola (2021) Ambrosio, L., E. Brué, and D. Semola (2021): _Lectures on Optimal Transport_. Springer.
  * Arieli, Babichenko, and Sandomirskiy (2023) Arieli, I., Y. Babichenko, and F. Sandomirskiy (2023): “Persuasion as Transportation,” _Working paper_.
  * Arnott and Rowse (1987) Arnott, R., and J. Rowse (1987): “Peer Group Effects and Educational Attainment,” _Journal of Public Economics_ , 32, 287–305.
  * Aumann and Maschler (1995) Aumann, R. J., and M. Maschler (1995): _Repeated Games with Incomplete Information_. MIT press.
  * Becker (1973) Becker, G. S. (1973): “A Theory of Marriage: Part I,” _Journal of Political Economy_ , 81(4), 813–846.
  * Beiglböck, Henry-Labordere, and Penkner (2013) Beiglböck, M., P. Henry-Labordere, and F. Penkner (2013): “Model-Independent Bounds for Option Prices–a Mass Transport Approach,” _Finance and Stochastics_ , 17(3), 477–501.
  * Beiglböck, Henry-Labordère, and Touzi (2017) Beiglböck, M., P. Henry-Labordère, and N. Touzi (2017): “Monotone Martingale Transport Plans and Skorokhod Embedding,” _Stochastic Processes and their Applications_ , 127(9), 3005–3013.
  * Beiglböck and Juillet (2016) Beiglböck, M., and N. Juillet (2016): “On a Problem of Optimal Transport under Marginal Martingale Constraints,” _Annals of Probability_ , 44(1), 42–106.
  * Beiglböck, Nutz, and Touzi (2017) Beiglböck, M., M. Nutz, and N. Touzi (2017): “Complete Duality for Martingale Optimal Transport on the Line,” _Annals of Probability_ , 45(5), 3038–3074.
  * Bergemann, Brooks, and Morris (2015) Bergemann, D., B. Brooks, and S. Morris (2015): “The Limits of Price Discrimination,” _American Economic Review_ , 105, 921–957.
  * Boerma, Tsyvinski, and Zimin (2021) Boerma, J., A. Tsyvinski, and A. Zimin (2021): “Sorting with Team Formation,” _Working paper_.
  * Breeden and Litzenberger (1978) Breeden, D., and R. Litzenberger (1978): “Prices of State-Contingent Claims Implicit in Option Prices,” _Journal of Business_ , 51(5), 621–651.
  * Brenier (1991) Brenier, Y. (1991): “Polar Factorization and Monotone Rearrangement of Vector-Valued Functions,” _Communications on Pure and Applied Mathematics_ , 44(4), 375–417.
  * Brzutowski (2023) Brzutowski, T. (2023): “Encouraging a Go-Getter,” _Working paper_.
  * Choi and Smith (2017) Choi, M., and L. Smith (2017): “Ordinal Aggregation Results via Karlin’s Variation Diminishing Property,” _Journal of Economic Theory_ , 168, 1–11.
  * Cox and Holden (2011) Cox, A. B., and R. R. Holden (2011): “Reconsidering Racial and Partisan Gerrymandering,” _University of Chicago Law Review_ , 78, 553–604.
  * Crawford and Sobel (1982) Crawford, V., and J. Sobel (1982): “Strategic Information Transmission,” _Econometrica_ , 50, 1431–1451.
  * De March and Touzi (2019) De March, H., and N. Touzi (2019): “Irreducible Convex Paving for Decomposition of Multidimensional Martingale Transport Plans,” _Annals of Probability_ , 47(3), 1726–1774.
  * Dekel (1986) Dekel, E. (1986): “An Axiomatic Characterization of Preferences under Uncertainty: Weakening the Independence Axiom,” _Journal of Economic Theory_ , 40(2), 304–318.
  * Dizdar and Kováč (2020) Dizdar, D., and E. Kováč (2020): “A Simple Proof of Strong Duality in the Linear Persuasion Problem,” _Games and Economic Behavior_ , 122, 407–412.
  * Dworczak and Kolotilin (2022) Dworczak, P., and A. Kolotilin (2022): “The Persuasion Duality,” _Working paper_.
  * Dworczak and Martini (2019) Dworczak, P., and G. Martini (2019): “The Simple Economics of Optimal Persuasion,” _Journal of Political Economy_ , 127(5), 1993–2048.
  * Eeckhout (2018) Eeckhout, J. (2018): “Sorting in the Labor Market,” _Annual Review of Economics_ , 10, 1–29.
  * Ellickson, Grodal, Scotchmer, and Zame (1999) Ellickson, B., B. Grodal, S. Scotchmer, and W. Zame (1999): “Clubs and the Market,” _Econometrica_ , 67(5), 1185–1217.
  * Epple and Romano (1998) Epple, D., and R. Romano (1998): “Competition between Private and Public Schools, Vouchers, and Peer-Group Effects,” _American Economic Review_ , 88(1), 33–62.
  * Friedman and Holden (2008) Friedman, J. N., and R. T. Holden (2008): “Optimal Gerrymandering: Sometimes Pack, but Never Crack,” _American Economic Review_ , 98(1), 113–44.
  * Gale (1989) Gale, D. (1989): _The Theory of Linear Economic Models_. University of Chicago press.
  * Galichon, Henry-Labordere, and Touzi (2014) Galichon, A., P. Henry-Labordere, and N. Touzi (2014): “A Stochastic Control Approach to No-Arbitrage Bounds Given marginals, with an Application to Lookback Options,” _Annals of Applied Probability_ , 24(1), 312–336.
  * Galperti, Levkun, and Perego (2023) Galperti, S., A. Levkun, and J. Perego (2023): “The Value of Data Records,” _Review of Economic Studies_ , Forthcoming.
  * Gangbo and McCann (1996) Gangbo, W., and R. McCann (1996): “The Geometry of Optimal Transportation,” _Acta Mathematica_ , 177(2), 113–161.
  * Garcia and Tsur (2021) Garcia, D., and M. Tsur (2021): “Information Design in Competitive Insurance Markets,” _Journal of Economic Theory_ , 191, 105–160.
  * Gentzkow and Kamenica (2016) Gentzkow, M., and E. Kamenica (2016): “A Rothschild-Stiglitz Approach to Bayesian Persuasion,” _American Economic Review, Papers & Proceedings_, 106, 597–601.
  * Ghoussoub, Kim, and Lim (2019) Ghoussoub, N., Y.-H. Kim, and T. Lim (2019): “Structure of Optimal Martingale Transport Plans in General Dimensions,” _Annals of Probability_ , 47(1), 109–164.
  * Goldstein and Leitner (2018) Goldstein, I., and Y. Leitner (2018): “Stress Tests and Information Disclosure,” _Journal of Economic Theory_ , 177, 34–69.
  * Gul and Pesendorfer (2010) Gul, F., and W. Pesendorfer (2010): “Strategic Redistricting,” _American Economic Review_ , 100(4), 1616–1141.
  * Guo and Shmaya (2019) Guo, Y., and E. Shmaya (2019): “The Interval Structure of Optimal Disclosure,” _Econometrica_ , 87(2), 653–675.
  * Henry-Labordère and Touzi (2016) Henry-Labordère, P., and N. Touzi (2016): “An Explicit Martingale Version of the One-Dimensional Brenier Theorem,” _Finance and Stochastics_ , 20(3), 635–668.
  * Holmström (1979) Holmström, B. (1979): “Moral Hazard and Observability,” _Bell Journal of Economics_ , 10(1), 74–91.
  * Inostroza and Pavan (2023) Inostroza, N., and A. Pavan (2023): “Adversarial Coordination and Public Information Design,” _Working paper_.
  * Kamenica and Gentzkow (2011) Kamenica, E., and M. Gentzkow (2011): “Bayesian Persuasion,” _American Economic Review_ , 101, 2590–2615.
  * Kleiner, Moldovanu, and Strack (2021) Kleiner, A., B. Moldovanu, and P. Strack (2021): “Extreme Points and Majorization: Economic Applications,” _Econometrica_ , 89(4), 1557–1593.
  * Kolotilin (2018) Kolotilin, A. (2018): “Optimal Information Disclosure: A Linear Programming Approach,” _Theoretical Economics_ , 13, 607–636.
  * Kolotilin, Corrao, and Wolitzky (2022) Kolotilin, A., R. Corrao, and A. Wolitzky (2022): “Persuasion with Non-Linear Preferences,” _Arxiv working paper_ , https://arxiv.org/abs/2206.09164.
  * Kolotilin, Mylovanov, and Zapechelnyuk (2022) Kolotilin, A., T. Mylovanov, and A. Zapechelnyuk (2022): “Censorship as Optimal Persuasion,” _Theoretical Economics_ , 17(2), 561–585.
  * Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017) Kolotilin, A., T. Mylovanov, A. Zapechelnyuk, and M. Li (2017): “Persuasion of a Privately Informed Receiver,” _Econometrica_ , 85, 1949–1964.
  * Kolotilin and Wolitzky (2023) Kolotilin, A., and A. Wolitzky (2023): “The Economics of Partisan Gerrymandering,” _Working paper_.
  * Kramkov and Xu (2022) Kramkov, D., and Y. Xu (2022): “An Optimal Transport Problem with Backward Martingale Constraints Motivated by Insider Trading,” _Annals of Applied Probability_ , 32(1), 294–326.
  * Kremer and Maskin (1996) Kremer, M., and E. Maskin (1996): “Wage Inequality and Segregation by Skill,” _Working paper_.
  * Legros and Newman (2002) Legros, P., and A. F. Newman (2002): “Monotone Matching in Perfect and Imperfect Worlds,” _Review of Economic Studies_ , 69(4), 925–942.
  * Leitner and Williams (2023) Leitner, Y., and B. Williams (2023): “Model Secrecy and Stress Tests,” _Journal of Finance_ , 78(2), 1055–1095.
  * Lin and Liu (2023) Lin, X., and C. Liu (2023): “Credible Persuasion,” _Journal of Political Economy_ , Forthcoming.
  * Malamud and Schrimpf (2022) Malamud, S., and A. Schrimpf (2022): “Persuasion by Dimension Reduction,” _Working paper_.
  * Mirrlees (1999) Mirrlees, J. A. (1999): “The Theory of Moral Hazard and Unobservable Behaviour: Part I,” _Review of Economic Studies_ , 66(1), 3–21.
  * Nikandrova and Pancs (2017) Nikandrova, A., and R. Pancs (2017): “Conjugate Information Disclosure in an Auction with Learning,” _Journal of Economic Theory_ , 171, 174–212.
  * Onuchic and Ray (2023) Onuchic, P., and D. Ray (2023): “Conveying Value via Categories,” _Theoretical Economics_ , Forthcoming.
  * Owen and Grofman (1988) Owen, G., and B. Grofman (1988): “Optimal Partisan Gerrymandering,” _Political Geography Quarterly_ , 7(1), 5–22.
  * Perez-Richet and Skreta (2023) Perez-Richet, E., and V. Skreta (2023): “Fraud-Proof Non-Market Allocation Mechanisms,” _Working paper_.
  * Quah and Strulovici (2012) Quah, J., and B. Strulovici (2012): “Aggregating the Single Crossing Property,” _Econometrica_ , 80, 2333–2348.
  * Rayo (2013) Rayo, L. (2013): “Monopolistic Signal Provision,” _BE Journal of Theoretical Economics_ , 13(1), 27–58.
  * Rayo and Segal (2010) Rayo, L., and I. Segal (2010): “Optimal Information Disclosure,” _Journal of Political Economy_ , 118, 949–987.
  * Rochet and Vila (1994) Rochet, J.-C., and J.-L. Vila (1994): “Insider Trading without Normality,” _Review of Economic Studies_ , 61, 131–152.
  * Saint-Paul (2001) Saint-Paul, G. (2001): “On the Distribution of Income and Worker Assignment under Intrafirm Spillovers, with an Application to Ideas and Networks,” _Journal of Political Economy_ , 109(1), 1–37.
  * Santambrogio (2015) Santambrogio, F. (2015): _Optimal Transport for Applied Mathematicians_ , vol. 55. Springer.
  * Seidmann and Winter (1997) Seidmann, D., and E. Winter (1997): “Strategic Information Transmission with Verifiable Messages,” _Econometrica_ , 65, 163–169.
  * Smolin and Yamashita (2023) Smolin, A., and T. Yamashita (2023): “Information Design in Games: Certification Approach,” _Working paper_.
  * Spence (1973) Spence, M. (1973): “Job Market Signaling,” _Quarterly Journal of Economics_ , 87, 355–374.
  * Tamura (2018) Tamura, W. (2018): “Bayesian Persuasion with Quadratic Preferences,” _Working paper_.
  * Villani (2009) Villani, C. (2009): _Optimal Transport: Old and New_ , vol. 338. Springer.
  * Winkler (1988) Winkler, G. (1988): “Extreme Points of Moment Sets,” _Mathematics of Operations Research_ , 13(4), 581–587.
  * Yang and Zentefis (2023) Yang, K. H., and A. Zentefis (2023): “Extreme Points and First-Order Stochastic Dominance: Theory and Applications,” _Working paper_.
  * Zeng (2023) Zeng, Y. (2023): “Derandomization of Persuasion Mechanisms,” _Journal of Economic Theory_ , 212.
  * Zhang and Zhou (2016) Zhang, J., and J. Zhou (2016): “Information Disclosure in Contests: A Bayesian Persuasion Approach,” _Economic Journal_ , 126(597), 2197–2217.

##  A. Characterization of Strict Aggregate Single-Crossing

We present two alternative conditions that are equivalent to strict aggregate
single-crossing of $u$. Condition (2) is analogous to the “signed-ratio
monotonicity” conditions for weak aggregate single-crossing in Theorem 1 of
Quah and Strulovici (2012) and Corollary 2 of Choi and Smith (2017). We give a
shorter proof based on the optimality of pairwise signals (see Appendix F.1).
Condition (3) is novel. It corresponds to strict monotonicity of $u$ (i.e.,
$u_{y}(y,x)<0$), up to a normalizing factor $g(y)>0$.

######  Lemma 3.

Let Assumption 1 hold. The following statements are equivalent:

  1. (1)

Assumption 2 holds.

  2. (2)

For all $x$, $x^{\prime}$, and $y$, we have

$\displaystyle u(y,x)=0$ $\displaystyle\implies u_{y}(y,x)<0,$ (15)
$\displaystyle u(y,x)<0<u(y,x^{\prime})$ $\displaystyle\implies
u(y,x^{\prime})u_{y}(y,x)-u(y,x)u_{y}(y,x^{\prime})<0.$ (16)

  3. (3)

There exists a differentiable function $g(y)>0$ such that
$\tilde{u}(y,x)=u(y,x)/g(y)$ satisfies $\tilde{u}_{y}(y,x)<0$ for all $(y,x)$.

##  B. Uniqueness

This appendix presents a notable technical result: under a regularity
condition, strict single-dippedness implies that there is a unique optimal
signal. Moreover, it also shows that conditions for uniqueness are much weaker
when full disclosure is optimal.

We say that a strictly single-dipped set $\Lambda$ is _regular_ if for each
$y\in Y_{\Lambda}$, there exists $\varepsilon>0$ such that either (i)
$\chi_{1}(\tilde{y})=\chi_{2}(\tilde{y})$ for all
$\tilde{y}\in(y-\varepsilon,y)\cap Y_{\Lambda}$ or (ii)
$\chi_{1}(\tilde{y})<\chi_{2}(\tilde{y})$ for all
$\tilde{y}\in(y-\varepsilon,y)\cap Y_{\Lambda}$. This regularity condition
rules out pathological cases where states switch infinitely many times from
being disclosed to being paired. This condition is satisfied in every example
in the literature that we know of.

######  Theorem 7.

If $X=[0,1]$, $\phi$ has a density, and $\Lambda$ is strictly single-dipped
and regular, then there is a unique optimal signal.

In martingale optimal transport, the optimal solution is unique under the
martingale Spence-Mirrlees condition (e.g., Proposition 3.5 in Beiglböck,
Henry-Labordère, and Touzi 2017), which coincides with our condition for the
optimality of strict single-dippedness in the linear receiver case. The key
implication of Theorem 7 is that the optimal marginal distribution of actions
is unique. There is no analogue of this result in martingale optimal
transport, where this marginal distribution is fixed.

To see the intuition for Theorem 7, consider the case where $\phi$ is discrete
and $\chi_{2}$ is strictly increasing. Let
$\overline{y}=\max_{\mu\in\Lambda}\gamma(\mu)$ be the highest action that can
be optimally induced. Since $\Lambda$ is strictly single-dipped, there is a
unique posterior $\overline{\mu}$ in $\Lambda$ inducing action $\overline{y}$,
namely $\overline{\mu}=\delta_{\chi_{2}(\overline{y})}$ if
$\chi_{1}(\overline{y})=\chi_{2}(\overline{y})$ and
$\overline{\mu}=\overline{\rho}\delta_{\chi_{1}(\overline{y})}+(1-\overline{\rho})\delta_{\chi_{2}(\overline{y})}$
where $\overline{\rho}\in(0,1)$ is uniquely determined by (1) if
$\chi_{1}(\overline{y})<\chi_{2}(\overline{y})$. Since $\chi_{2}$ is strictly
increasing, for any optimal signal $\tau$, the state $\chi_{2}(\overline{y})$
can only induce action $\overline{y}$ and thus
$\tau(\overline{\mu})=\phi(\chi_{2}(\overline{y}))$ if
$\chi_{1}(\overline{y})=\chi_{2}(\overline{y})$ and
$\tau(\overline{\mu})=\phi(\chi_{2}(\overline{y}))/(1-\overline{\rho})$ if
$\chi_{1}(\overline{y})<\chi_{2}(\overline{y})$. Working our way through the
support of $\phi$ from the highest state to the lowest in this fashion, we
obtain the unique value of $\tau(\mu)$ for each $\mu\in\Lambda$. When $\phi$
has a density, the possibility that $\chi_{2}$ may be only weakly increasing
does not threaten uniqueness of the optimal signal, because the set of states
corresponding to flat regions of $\chi_{2}$ is at most countable and thus has
$\phi$-measure $0$. Finally, our regularity condition ensures that the above
argument extends easily from the discrete case to the continuous one.

In addition, when the prior has full support and the contact set is pairwise
(e.g., the twist condition holds), full disclosure is uniquely optimal
whenever it is optimal. To see the intuition, suppose full disclosure is
optimal, and suppose there is another optimal signal that pools some states
$x_{1}$ and $x_{2}$ to induce an action $y$. Then the signal that discloses
all other states while pooling $x_{1}$ and $x_{2}$ to induce $y$ is also
optimal. But then the signal that discloses all other states while pooling
$x_{1}$, $x_{2}$, and the third state $\chi(y)\neq x_{1},x_{2}$ to induce $y$
would also be optimal—but this signal is not pairwise, which is a
contradiction.

######  Remark 4.

Assume that $X=[0,1]$. If the contact set is pairwise and full disclosure is
optimal, then it is uniquely optimal.

##  C. Examples of Negative Assortative Disclosure

######  Example 1 (Solving the Differential Equations).

Consider the linear receiver case with $Y=X=[1/e,e]$, $f(x)=1/(2x)$, and
$V(y,x)=y/x$. We claim that the unique optimal outcome matches each state
$x_{1}\in[1/e,1]$ with state $x_{2}=1/x_{1}$ with equal weights, so that the
induced action is $y={x_{1}}/{2}+{1}/({2x_{1}})$. Thus,
$\chi_{1}(y)=y-\sqrt{y^{2}-1}$, and $\chi_{2}(y)=y+\sqrt{y^{2}-1}$ for all
$y\in[1,{e}/{2}+{1}/(2e)]$.

Indeed, by Theorem 4, the optimal outcome is strictly single-dipped, since
$w(x)=1/x$ is strictly convex. By Corollary 3, (8) holds, since
$w^{\prime}<0$. Hence, by Theorem 6, the optimal outcome is single-dipped
negative assortative and satisfies (9)–(13). Now, for $x_{2}=1/x_{1}$ and
$y={x_{1}}/{2}+{1}/({2x_{1}})$, (9) holds because

$\displaystyle
u(y,x_{2})=\left(\frac{1}{2x_{1}}-\frac{x_{1}}{2}\right)=-\left(\frac{x_{1}}{2}-\frac{1}{2x_{1}}\right)=-u(y,x_{1}),$
$\displaystyle
f(x_{2})\frac{\mathrm{d}x_{2}}{\mathrm{d}y}=\frac{1}{{2}/{x_{1}}}\left(-\frac{1}{x_{1}^{2}}\frac{\mathrm{d}x_{1}}{\mathrm{d}y}\right)=-\frac{1}{2x_{1}}\frac{\mathrm{d}x_{1}}{\mathrm{d}y}=-f(x_{1})\frac{\mathrm{d}x_{1}}{\mathrm{d}y},$

(12) holds because

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}y}\left(w(x_{1})\frac{1}{2}+w(x_{2})\frac{1}{2}\right)=\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac{1}{2x_{1}}+\frac{x_{1}}{2}\right)=\frac{\mathrm{d}}{\mathrm{d}y}y=1,$
$\displaystyle\frac{w(x_{2})-w(x_{1})}{x_{2}-x_{1}}=\frac{x_{1}-{1}/{x_{1}}}{{1}/{x_{1}}-x_{1}}=-1,$

and (13) holds because $1/(1/e)=e$ and $1/1=1$.494949We can also solve this
example by directly applying Theorem 1), because, for $q(y)=y$, the function
$V(y,x)+q(y)u(y,x)={y}/{x}+y(x-y)$ is maximized at $y=x/2+1/(2x)$ for all
$x\in[1/e,e]$.

######  Example 2 (Quantile Persuasion).

Consider the quantile sub-subcase of the state-independent sender case, where
$u(y,x)=\mbox{\bf 1}\\{x\geq y\\}-\kappa$ with $\kappa\in(0,1)$. Let $\phi$
have a density on $[0,1]$. Assuming that the receiver breaks ties in favor of
the sender, we have, for $x_{1}<x_{2}$,

$\gamma(\rho\delta_{x_{1}}+(1-\rho)\delta_{x_{2}})=\begin{cases}x_{2},&\rho\leq
1-\kappa,\\\ x_{1},&\rho>1-\kappa.\end{cases}$

Note that (8) always holds for $\rho\in(0,1-\kappa)$. We claim that there
exists an optimal single-dipped negative assortative signal where the induced
distribution over actions $\alpha$ satisfies
$\alpha([y,1])=\phi([y,1])/\kappa$, and the posterior inducing any action
$y\in[\underline{y},1]$ is
$(1-\kappa)\delta_{\chi_{1}(y)}+\kappa\delta_{\chi_{2}(y)}$, where
$\chi_{2}(y)=y$, $\chi_{1}(y)$ solves
$\kappa\phi([0,\chi_{1}(y)])=(1-\kappa)\phi([y,1])$, and $\underline{y}$
solves
$\kappa\phi([0,\underline{y}])=(1-\kappa)\phi([\underline{y},1])$.505050See
Appendix F.7 for the proof. A notable feature of this signal is that, with the
informed receiver interpretation, it would remain optimal even if the sender
knew the receiver’s type and could condition disclosure on it.

######  Example 3 (A Stochastic Optimal Signal).

In the following example, for some priors negative assortative disclosure is
optimal; and for other priors, the unique optimal signal randomizes
conditional on the state, even though the prior is atomless.

$-1$$3$$-1$$1$$x$$y$ Figure 5. The Optimal Signal in Example 3 _Notes:_ The
three black line segments depict the graphs of $\chi_{1}$ and $\chi_{2}$. The
red line segments indicate pairs of states $\chi_{1}(y)$ and $\chi_{2}(y)$
that are optimal to pool to induce action $y\in(0,1]$. If the prior density
satisfies $f(-y)>3f(3y)$ for all $y\in(0,1]$, then, for each state $x<0$, the
optimal signal randomizes between disclosing $x$ (and inducing action $x$) and
pooling $x$ with state $-3x$ (and inducing action $-x$).

Consider the translation-invariant subcase of the state-independent sender
case. Let $Y=X=[-1,3]$, let $\phi$ have a density $f$ with $f(-y)\geq 3f(3y)$
for all $y\in(0,1]$, let $u(y,x)=T(x-y$) with $T(0)=0$ and strictly log-
concave $T^{\prime}$, and let $V(y,x)=T(2y)$. With the informed receiver
interpretation, this captures a case where, for example, $\kappa=1/2$, the
distribution of $\varepsilon$ is $N(0,\sigma^{2})$, and the distribution of
$t$ is $N(0,(\sigma/2)^{2})$.515151By symmetry and strict log-concavity of
$T^{\prime}$,
$V_{yy}(y)/V_{y}(y)=2T^{\prime\prime}(2y)/T^{\prime}(2y)>(<)T^{\prime\prime}(0)/T^{\prime}(0)=0$
for $0>(<)y,$ showing that (14) fails for $y<0$, and thus Theorem 6 does not
apply.

We claim that

$\chi_{1}(y)=\begin{cases}y,&y\in[-1,0],\\\
-y,&y\in(0,1],\end{cases}\quad\text{and}\quad\chi_{2}(y)=\begin{cases}y,&y\in[-1,0],\\\
3y,&y\in(0,1],\end{cases}$

so that the posterior inducing any action $y\in[-1,1]$ is
$\delta_{\chi_{1}(y)}/2+\delta_{\chi_{2}(y)}/2$, and the distribution over
actions $\alpha$ has density $a$ given by

$\displaystyle a(y)$ $\displaystyle=\begin{cases}6f(3y),&y\in(0,1],\\\
f(-y)-3f(3y),&y\in[-1,0).\end{cases}$

The unique optimal signal is single-dipped negative assortative iff
$f(-y)=3f(3y)$ for all $y\in(0,1]$. In contrast, if $f(-y)>3f(3y)$ for all
$y\in(0,1]$, then each state $x\in[-1,0)$ is mixed between inducing actions
$y=x$ and $y=-x$.525252See Appendix F.8 for the proof. See Figure 5.

##  D. Specific Persuasion Models

This appendix shows how our analysis covers some well-known prior persuasion
models, where single-dipped or single-peaked disclosure is optimal.535353The
applications in this section also illustrate some technical points. Section
D.1 illustrates how directly applying Theorem 3 can yield weaker sufficient
conditions for the optimality of single-dipped disclosure than those in
Theorem 4. Sections D.2 and D.3 illustrate how our analysis extends when some
of our assumptions are violated: in Section D.2, Assumption 3 fails, so the
receiver’s optimal action may be at the boundary and thus violate the first-
order condition; in Section D.3, Assumption 4 fails, as the sender only weakly
prefers higher actions.

###  D.1. Contests

Zhang and Zhou (2016) study information disclosure in contests. In their
model, two contestants, $A$ and $B$, compete for a prize by exerting efforts
$z_{A}$ and $z_{B}$. The probability that contestant $i=A,B$ wins is
$z_{i}/(z_{A}+z_{B})$. Everyone knows contestant A’s value $v_{A}=1$.
Contestant B’s value $v_{B}$ is known to contestant B and the designer. The
sender designs a signal about $v_{B}$ to maximize expected total effort.

It is convenient to parameterize $x=1/\sqrt{v_{B}}$ and $y=\sqrt{z_{A}}$. With
this parameterization, Zhang and Zhou’s Proposition 1 shows that, given a
posterior $\mu$, contestant A exerts effort $z_{A}^{\star}=\gamma(\mu)^{2}$
determined by
$\mathbb{E}_{\mu}\left[x-\left(1+x^{2}\right)\gamma(\mu)\right]=0$, and
contestant B (who knows $x$) exerts effort
$z_{B}^{\star}(x)=\gamma(\mu)/x-\gamma(\mu)^{2}$, so the sender’s expected
utility is
$z_{A}^{\star}+\mathbb{E}_{\mu}\left[z_{B}^{\star}(x)\right]=\mathbb{E}_{\mu}\left[\gamma(\mu)/x\right]$.
We thus recover our model with $V(y,x)=y/x$ and $u(y,x)=x-(1+x^{2})y$.

Zhang and Zhou give results on optimality of pairwise disclosure, full
disclosure, and no disclosure. Our approach easily yields the following
result, which additionally gives conditions for optimality of single-
dipped/-peaked disclosure and negative assortative disclosure (which were not
considered by Zhang and Zhou).

######  Proposition 2.

In Zhang and Zhou’s contest model where the prior $\phi$ has a positive
density on $X=[\underline{x},\overline{x}]$, where
$0<\underline{x}<\overline{x}$, if $\underline{x}\geq 1$ then the unique
optimal signal is full disclosure; and if $\overline{x}\leq 1/\sqrt{3}$
($1/\sqrt{3}\leq\underline{x}<\underline{x}\leq 1$), then the unique optimal
signal is single-dipped (-peaked) negative assortative disclosure.

The proof of single-dippedness/-peakedness uses Theorem 3 with a perturbation
that fixes both actions. In contrast, directly applying Theorem 4 would yield
only the weaker result that single-peaked negative assortative disclosure is
optimal if $1/\sqrt{2}\leq\underline{x}<\overline{x}<1$.545454To see this,
suppose $\overline{x}<1$. Then $u_{x}(y,x)=1-2xy>0$ for
$y\leq\overline{x}/(1+\overline{x}\,\\!^{2})=\max Y$. Moreover,
$u_{yx}(y,x)/u_{x}(y,x)=-2x/(1-2xy)$ is always decreasing in $x$, while
$V_{yx}(y_{2},x)/u_{x}(y_{1},x)=-1/(x^{2}-2x^{3}y_{1})$ is decreasing in $x$
iff $3\underline{x}\min Y=3\underline{x}^{2}/(1+\underline{x}\,\\!^{2})\geq
1$, or equivalently $\underline{x}\geq 1/\sqrt{2}$.

###  D.2. Affiliated Information

Guo and Shmaya (2019) consider a persuasion model with a privately informed
receiver, where it is commonly known that the receiver wishes to accept a
proposal iff $x$ exceeds a threshold $x_{0}$, and the receiver’s type $t$ is
his private signal of $x$. Letting $G(t|x)$ denote the distribution of $t$
conditional on $x$, with corresponding density $g(t|x)$, this setup maps to
our model with $V(y,x)=G(y|x)$, $u(y,x)=(x-x_{0})g(y|x)$, and $g(t|x)$
strictly log-submodular in $(t,x)$.555555The ordering convention here is that
high $t$ is bad news about $x$. This ordering is opposite to Guo and Shmaya’s,
but follows our convention that the receiver accepts for types below a
cutoff.,565656Inostroza and Pavan (2023) study robust stress test design in a
setting with multiple receivers with coordination motives. As they note, the
single-receiver version of their model is a special case of Guo and Shmaya
(2019). These preferences satisfy Assumptions 1 and 2 (see Lemma 3), but not
Assumption 3, as $u(y,x)>0$ for all $y$ when $x>x_{0}$. Nonetheless, assuming
that the receiver breaks ties in the sender’s favor, we have
$\gamma(\mu)=\max\\{y:\int u(y,x)\mathrm{d}\mu\geq 0\\}$.

Let us take for granted that Theorem 3 holds even though Assumption 3 is
violated (e.g., this is clearly true with a discrete prior). Applying Theorem
3 with a perturbation that fixes one action while increasing the other action
and the sender’s expected utility (for fixed actions), we obtain the following
result, which reproduces Guo and Shmaya’s main qualitative insight.575757When
the prior has positive density on $[0,1]$, Guo and Shmaya’s Theorem 3.1
additionally implies that the optimal signal is single-peaked negative
assortative.

######  Proposition 3.

In Guo and Shmaya’s model of persuading a privately informed receiver, every
optimal signal is single-peaked.

###  D.3. Stress Tests

Goldstein and Leitner (2018) consider a model of optimal stress tests. The
sender is a bank regulator and the receiver is a perfectly competitive market.
The bank has an asset that yields a random cash flow. The asset’s quality is
$x$, which is observed by the bank and the regulator but not the market, and
is normalized to equal the asset’s expected cash flow.585858This is the model
in Section 5 of their paper, where the bank observes $x$. The regulator
designs a test to reveal information about $x$. After observing the test
result, the market offers a competitive price $y$ for the asset. Finally, the
bank decides whether to keep the asset and receive the random cash flow, or
sell it at price $y$. Letting $z$ denote the bank’s final cash holding (equal
to either the random cash flow or $y$), the bank’s payoff equals $z+\mbox{\bf
1}\\{z\geq x_{0}\\}$, where $x_{0}$ is a constant. An interpretation is that
the bank faces a run if its cash holding falls below $x_{0}$. The regulator
designs the test to maximize expected social welfare, or equivalently to
minimize the probability of a run.

Goldstein and Leitner show that a bank with a type-$x$ asset is willing to
sell at a price $y$ iff $y$ exceeds a reservation price $\tilde{\sigma}(x)$
that satisfies $\tilde{\sigma}(x)>x$ if $x<x_{0}$, $\tilde{\sigma}(x)<x$ if
$x>x_{0}$, and $\tilde{\sigma}^{\prime}(x)\geq 0$. Intuitively, if $x<x_{0}$
then the bank demands a premium to forego the chance that a lucky cash flow
shock pushes its holdings above $x_{0}$, while if $x>x_{0}$ then the bank
desires insurance against bad cash flow shocks that push its holdings below
$x_{0}$. However, the value of the regulator’s problem is unaffected if the
reservation price is re-defined as $\sigma(x)=x$ if $x\leq x_{0}$ and
$\sigma(x)=\tilde{\sigma}(x)$ if $x>x_{0}$, because it is suboptimal for the
regulator to induce a bank to sell at a price below $x_{0}$. It is more
convenient to work with the normalized reservation price $\sigma(x)$.

It is also convenient to restrict attention to tests that, for each $x$,
either induce the bank to sell or fully disclose the bank’s value: this is
without loss because pooling two asset types that do not sell is weakly worse
than disclosing these types. For such a test, the price induced by any
posterior $\mu$ is $\gamma(\mu)=\mathbb{E}_{\mu}[x]$, so we are in the linear