PERSUASION OF A PRIVATELY INFORMED RECEIVER
Anton Kolotilin, Tymofiy Mylovanov, Andriy Zapechelnyuk
and Ming Li
Abstract. We study persuasion mechanisms in linear environments. A receiver
has a private type and chooses between two actions. A sender designs a persuasion mechanism or an experiment to disclose information about a payoff-relevant
state. A persuasion mechanism conditions information disclosure on the receiver’s
report about his type, whereas an experiment discloses information independent of
the receiver’s type. We establish the equivalence of implementation by persuasion
mechanisms and by experiments, and characterize optimal persuasion mechanisms.
JEL Classification: D82, D83, L82
Keywords: Bayesian persuasion, information disclosure, information design, mechanism design without transfers, experiments, persuasion mechanisms

Date: July 12, 2017.
Kolotilin: School of Economics, UNSW Australia, Sydney, NSW 2052, Australia. E-mail:akolotilin@gmail.com.
Mylovanov : University of Pittsburgh, Department of Economics, 4714 Posvar Hall, 230 South
Bouquet Street, Pittsburgh, PA 15260, USA. E-mail: mylovanov@gmail.com.
Zapechelnyuk : School of Economics and Finance, University of St Andrews, Castlecliffe, the Scores,
St Andrews KY16 9AR, UK. E-mail: azapech@gmail.com.
Li: Concordia University and CIREQ, 1455 Boulevard de Maisonneuve Ouest, Department of
Economics, Concordia University, Montréal, H3G 1M8, Canada. Email: ming.li@concordia.ca.
The authors would like to express their gratitude to anonymous referees for their very helpful comments and to Navik Kartik for a thoughtful discussion at the Boston ASSA meetings.
They are also grateful to Ricardo Alonso, Dirk Bergemann, Patrick Bolton, Alessandro Bonatti,
Rahul Deb, Péter Esö, Johannes Hörner, Florian Hoffman, Roman Inderst, Emir Kamenica, DanielKräehmer, Marco Ottaviani, Mallesh Pai, Larry Samuelson, and Ilya Segal for their excellent
comments. Kolotilin acknowledges financial support from the Australian Research Council.
Zapechelnyuk acknowledges financial support from the Economic and Social Research Council(grant no. ES/N01829X/1).
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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

1. Introduction
In numerous situations, an uninformed sender (she) wishes to influence an actionof a receiver (he) who privately knows his preference type but is uncertain about apayoff-relevant state. The sender has full control over information disclosure about
the state but cannot use any other incentive tools such as monetary transfers. If thesender knew the receiver’s type, she would tailor information disclosure to the type.
However, the sender can only ask the receiver to report his type, which the receiver
may misreport in order to affect the information he receives. This begs the questionof whether the sender can benefit by designing a complex persuasion mechanism that
tailors information disclosure to the receiver’s report, as compared to an experimentthat provides the same information about the state, regardless of the receiver’s report.
In our model, the receiver must choose one of two actions. The sender’s andreceiver’s utilities depend linearly on the state and receiver’s type.1 The sender andreceiver share a common prior about the state; the receiver privately knows his type.
The sender commits to a persuasion mechanism that asks the receiver to report his
type and returns a stochastic message that depends on the state and report. After
receiving the message, the receiver updates his beliefs about the state and takes anaction that maximizes his expected utility.
We characterize the set of the receiver’s interim utilities implementable by persuasion mechanisms.2 The upper and lower bounds are achieved by full disclosure and no
disclosure of information about the state. For any mapping from messages to actions
under a persuasion mechanism, the receiver’s expected utility is linear in his type.
Because each type chooses an optimal mapping, the receiver’s interim utility is an upper envelope of linear functions and, hence, is convex. To sum up, an implementableinterim utility is necessarily a convex function that lies between the upper and lower
bounds generated by full and no disclosure.
Our main theorem shows that (1) this necessary condition is also sufficient and that
(2) any implementable interim utility can be attained by an experiment. Moreover,in our model the receiver’s interim utility uniquely determines the sender’s expectedutility. Therefore, there is no loss of generality in restricting attention to experiments.
In particular, the sender need not consider more complex persuasion mechanisms thatcondition information disclosure on the receiver’s report.
We now outline the argument behind this result. Because the utilities are linear in
the state, the only relevant content of a message about the state is the posterior mean.
Therefore, an experiment can be fully described by the distribution of the posteriormean state H that it generates. By Blackwell (1953), there exists an experiment thatgenerates H if and only if the prior distribution of the state is a mean-preserving
spread of H. By linearity, the receiver’s interim utility U can be represented as an
1

For our analysis to apply, we only need to assume that the utilities are linear in some transformation of the state and are arbitrary functions of the receiver’s type. The state and receiver’s typeare one-dimensional independent random variables.
2
The receiver’s interim utility is the mapping from the receiver’s type to his expected utility given
optimal behavior under a persuasion mechanism.

PERSUASION OF A PRIVATELY INFORMED RECEIVER

3appropriate integral of H. Using the integral form of the mean-preserving spreadcondition as in Blackwell (1951), we show that the prior is a mean-preserving spreadof H if and only if U is convex and lies between the upper and lower bounds generated
by full and no disclosure. Therefore, any U that satisfies the necessary conditions that
we identified above is implementable by an experiment.
This characterization allows us to formulate the sender’s problem as the maximization of a linear functional on a bounded set of convex functions. We derive ageneral structure of optimal persuasion mechanisms and use it to simplify the sender’s
problem into a finite-variable optimization problem.
Related Literature. Our model is a variation of Kamenica and Gentzkow (2011),
who show that an optimal experiment corresponds to a point on the concave closureof the sender’s value function. This concavification approach does not rely on any
structure of the problem such as the linearity of utility functions or the two actions of
the receiver. However, as Gentzkow and Kamenica (2016) point out, this approach has
limited applicability when the state space is infinite because it requires to calculate aconcave closure on the infinite-dimensional space of distributions over the state space.
Rayo and Segal (2010) impose more structure on the receiver’s utility and the distribution of the receiver’s type than our paper. At the same time, they allow thesender’s utility to be nonlinear in the state. Rayo and Segal (2010) partially characterize optimal experiments and represent the sender’s problem as the maximization
problem of a concave objective function subject to linear constraints. In our setting,their assumptions about the receiver would imply that either full or no disclosure is
optimal.
Kolotilin (2016) allows for nonlinear sender’s and receiver’s utility functions andan arbitrary joint distribution of the state and the receiver’s type. The linearprogramming approach in Kolotilin (2016) permits to verify whether a candidate
experiment is optimal. But this approach has limited applicability because it does
not allow to directly characterize the optimal experiments.
The three papers above study experiments.3 In contrast, we consider persuasion
mechanisms, which can tailor information disclosure to the receiver’s report.4 Linear
utility functions with two possible actions of the receiver enable us to use the envelope representation of incentive compatibility as in Myerson (1981). However, thecharacterization of implementable mechanisms in our setting differs from Myerson(1981) because there are no transfers between the sender and receiver, and there are
3There is a rapidly growing Bayesian persuasion literature that studies optimal experiments.

Bayesian persuasion with a privately informed sender is considered in Gill and Sgroi (2008, 2012),
Perez-Richet (2014), Alonso and Câmara (2016b), and Hedlund (2017). Bayesian persuasion withmultiple senders is analyzed in Ostrovsky and Schwarz (2010), Board and Lu (2016), Gentzkow and
Kamenica (2017a, 2017b), and Li and Norman (2017). Dynamic Bayesian persuasion is examined inAu (2015), Ely et al. (2015), and Ely (2017). Information acquisition and the value of informationin Bayesian persuasion are explored in Gentzkow and Kamenica (2014, 2017c), Kolotilin (2015),
Alonso and Câmara (2016a), and Bergemann and Morris (2016).
4Focusing on experiments is with loss of generality in settings with multiple receivers in Bergemannand Morris (2013), Alonso and Câmara (2016c), Mathevet et al. (2017), and Taneva (2016).

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

obedience constraints instead of participation constraints.5 The equivalence between
persuasion mechanisms and experiments relies on the majorization theory of Hardyet al. (1929) and, thus, relates to the equivalence between Bayesian and dominantstrategy implementation in Manelli and Vincent (2010) and Gershkov et al. (2013).
2. Model2.1. Setup. There are two players: sender and receiver. The receiver makes a choicea ∈ A = {0, 1} between action (a = 1) and inaction (a = 0). There are two payoffrelevant random variables: the state of the world ω ∈ Ω and the receiver’s private
type r ∈ R, where Ω and R are intervals in the real line. Random variables ω and r
are independent and have distributions F and G.
Let the receiver’s and sender’s utilities beu(ω, r, a) = a · (ω − r),(1)v (ω, r, a) = a + ρ (r) u(ω, r, a),where ρ is a bounded measurable function.6
The receiver’s utility from inaction (a = 0) is normalized to zero, whereas his utility
from action (a = 1) equals the benefit ω less the private cost r. The sender’s utility
is a weighted sum of the receiver’s utility and action. The sender is biased towards
the receiver’s action but also puts a type-specific weight ρ(r) on the receiver’s utility.
In particular, if the weight ρ is large, then the sender’s and receiver’s interests arealigned, whereas if the weight is zero, then the sender cares only about whether thereceiver acts or not.
We assume that the set of states is Ω = [0, 1] and E[ω] ∈ (0, 1). Because the optimal
action of the receiver of any type above 1 or below 0 is independent of the state, we
assume that the set of types is R = [0, 1]. These assumptions allow for elegance ofpresentation; relaxing them poses no difficulty.
2.2. Persuasion Mechanisms. In order to influence the action taken by the receiver, the sender can design a mechanism that asks the receiver to report his private
information and sends a message to the receiver conditional on his report and therealized state.
A persuasion mechanism π asks the receiver to report r̂ ∈ R and then recommends
him to take one of two actions: for every ω ∈ Ω, it recommends to act (â = 1) withprobability π (ω, r̂) and not to act (â = 0) with complementary probability 1−π (ω, r̂),where π : Ω × R → [0, 1] is a measurable function.
A persuasion mechanism is incentive compatible if the receiver finds it optimal toreport his true type and to follow the mechanism’s recommendation. Denote the
5Bayesian persuasion with monetary transfers is investigated in Bergemann and Pesendorfer(2007), Esö and Szentes (2007), Bergemann et al. (2015, 2017a, 2016b), Li and Shi (2016), andHörner and Skrzypacz (2016).
6For our analysis, it is sufficient to assume that u(ω, r, a) and v (ω, r, a) are linear in (a transformation of) the state ω. The unidimensionality of the type r and the linearity of the utilities in thetype are assumed only for elegance of exposition. We further comment on this in Section 3.3, andrefer an interested reader to Appendix C of the Supplemental Material for technical details.

PERSUASION OF A PRIVATELY INFORMED RECEIVER

5interim utility of a receiver of type r ∈ R who reports r̂ ∈ R and takes actions
a0 ∈ {0, 1} and a1 ∈ {0, 1} after recommendations â = 0 and â = 1, respectively, byZ
Uπ (r, r̂, a0 , a1 ) =(a0 (1 − π(ω, r̂)) + a1 π(ω, r̂)) (ω − r)dF (ω).
Ω

The interim utility of the truthful (r̂ = r) and obedient (a0 = 0 and a1 = 1) receiveris equal toZ
Uπ (r) = Uπ (r, r, 0, 1) =Ωπ(ω, r)(ω − r)dF (ω).

We consider mechanisms that satisfy the incentive compatibility constraint
Uπ (r) ≥ Uπ (r, r̂, a0 , a1 ) for all r, r̂ ∈ R and a0 , a1 ∈ A.

(2)

It is convenient to introduce the notation for the interim utility of the obedient receiverof type r, who reports r̂ and then obeys the recommendation of the mechanism:
Uπ (r, r̂) = Uπ (r, r̂, 0, 1) = pπ (r̂) − qπ (r̂) r,

where qπ (r̂) is the interim action and pπ (r̂) is the expected state after action conditional on a report r̂:ZZqπ (r̂) =π(ω, r̂)dF (ω) and pπ (r̂) =ωπ(ω, r̂)dF (ω) .
ΩΩWith these notations, we can draw the parallel to the standard mechanism design
problem with transfers, where r is a private value, r̂ is a reported value, qπ (r̂) is theprobability of transaction, and pπ (r̂) is the expected monetary transfer. The classicalenvelope argument yields the following lemma.
Lemma 1. A mechanism π is incentive compatible if and only if
qπ is nonincreasing,
Z 1
Uπ (r) =qπ (s)ds,

(3)
(4)

r

Uπ (0) = E[ω].

(5)

The proof is in the Appendix.
Interestingly, the obedience constraints for the intermediate types are implied by
the boundary conditions, Uπ (1) = 0 and Uπ (0) = E[ω], and truth telling, Uπ (r) ≥Uπ (r, r̂). To disobey by ignoring the recommendation, that is, to act (not to act)
irrespective of what is recommended, is not better than pretending to be the lowest
type r̂ = 0 (the highest type r̂ = 1, respectively). Due to the linearity of the receiver’s
utility, to misreport and disobey by taking the opposite action to the recommended
one is always inferior to reporting either the lowest type, r̂ = 0, or the highest type,r̂ = 1, and then following the recommendation.
In our model, there are no transfers, and there are obedience constraints insteadof individual rationality constraints. These differences between our and the standard
environment with transfers translate into the following differences in characterization.
First, there are two boundary conditions, Uπ (1) = 0 and Uπ (0) = E[ω]:6

KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI(a) We have ω − r ≤ 0 for all ω ∈ Ω = [0, 1] and r = 1. Hence, type 1’s utility is
maximized by not acting for any belief about the state, implying Uπ (1) = 0.
This is (4) evaluated at r = 1.
(b) We have ω − r ≥ 0 for all ω ∈ Ω = [0, 1] and r = 0. Hence, type 0’s utility
is maximized by acting for any belief about the state, implying Uπ (0) = E[ω].
This is (5).
Second, not every pair (q, U ) that satisfies conditions (3)–(5) is feasible, that is, amechanism π that implements such a pair need not exist. For
R 1 example, if F assigns
probability 1 on ω = 1/2, then every nonincreasing q with 0 q(r)dr = 1/2 satisfies(3)–(5). Among these functions, q is feasible if and only if it satisfies q(r) = 1 forr < 1/2 and q(r) = 0 for r > 1/2.
2.3. Experiments. An experiment communicates a one-way message to the receiverand does not require the receiver to report his type. Formally, an experiment σ sends
to the receiver a random message m from a message space M = [0, 1] that depends
on a realized state ω. Denote by σ(m|ω) the distribution of message m conditionalon state ω.
A persuasion mechanism can be interpreted as a menu of experiments from which
the receiver can freely choose one experiment. For a given mechanism π, the corresponding menu of experiments is {σr̂ }r̂∈R , where, conditional on a state ω, an experiment σr̂ sends messages 1 and 0 with probabilities π(ω, r̂) and 1−π(ω, r̂). Conversely,by the revelation principle, any equilibrium outcome induced by any menu of experiments offered to the receiver can be replicated by a persuasion mechanism.
For a given experiment σ, each message m induces a posterior belief of the receiver
about the state. Because the receiver’s utility is monotonic in his type, we can identify
every message m with the cutoff type r who is indifferent between the two actions
conditional on receiving this message. An experiment is direct if its messages areequal to the cutoff types, m = E[ω|m]. Without loss of generality, we focus on direct
experiments (as in, e.g., Kamenica and Gentzkow, 2011).
A persuasion mechanism π is equivalent to an experiment if, conditional on eachstate, the behavior of the receiver as stipulated by π is a best-reply behavior to theposterior beliefs generated by some direct experiment σ,π(ω, r) ∈ 1 − σ(r|ω), 1 − σ(r− |ω) for all ω ∈ Ω and all r ∈ R,
(6)
where σ(r− |ω) denotes the left limit of σ(.|ω) at r. To understand (6), note that,upon receiving a direct message m, every type r < m strictly prefers to choose a = 1,
and type r = m is indifferent between a = 0 and a = 1 and may optimally choose
any lottery over the two actions. Consequently, the probability that type r acts
conditional on state ω can take any value in the interval [1 − σ(r|ω), 1 − σ(r− |ω)].
3. Implementation Equivalence
In this section, we characterize the pairs of the interim utility U and action q
implementable by persuasion mechanisms, and show that the same pairs of the interim
utility and action are implementable by experiments.

PERSUASION OF A PRIVATELY INFORMED RECEIVER

7This is a step towards solving the sender’s optimization problem, because thesender’s interim utility Vπ (r) when the receiver’s type is r is a weighted sum of thereceiver’s interim utility and action,ZVπ (r) = (1 + ρ(r)(ω − r))π(ω, r)dF (ω) = qπ (r) + ρ(r)Uπ (r).
(7)Ω

3.1. Bounds on Receiver’s Interim Utility. Consider two simple mechanisms.
The full disclosure mechanism informs the receiver about the state, so the receiveracts (π(ω, r) = 1) if ω > r and does not act (π(ω, r) = 0) if ω < r. Therefore, theinterim utility is
Z 1(ω − r) dF (ω) .
U (r) =r

The no disclosure mechanism does not convey any information to the receiver, so thereceiver acts if E[ω] > r and does not act if E[ω] < r. Therefore, the interim utility is
U (r) = max {E[ω] − r, 0} .
Thus, U (r) is the receiver’s interim utility based on the prior information about ω asgiven by F , whereas U (r) is the receiver’s interim utility if he observes ω.
Note that every mechanism π must satisfy
U (r) ≤ Uπ (r) ≤ U (r) for all r ∈ R.

(8)

The left-hand side inequality of (8) is implied by incentive compatibility: the receivercannot be better off by ignoring the sender’s recommendation. The right-hand side
inequality of (8) is the feasibility constraint: the receiver’s utility cannot exceed theutility attained under full disclosure of ω.
3.2. Implementable Interim Utility and Action. The receiver’s interim utility
U and action q are implementable if there exists an incentive-compatible persuasion
mechanism π such that U (r) = Uπ (r) and q(r) = qπ (r) for all r ∈ R. Moreover, U
and q are implementable by an experiment if π is equivalent to an experiment.
Let U be the set of all convex functions bounded by U and U (see Fig. 1).
Theorem 1. The following statements are equivalent:(a) U is a convex function between U and U ;(b) U is implementable;(c) U is implementable by an experiment.
Proof. Observe that (a) states the necessary conditions for the implementation of U .
The incentive-compatibility constraint requires the convexity of U by Lemma 1, andthe feasibility constraint (8) requires U (r) ≤ U (r) ≤ U (r). Hence, (b) ⇒ (a). Also,
the implication (c) ⇒ (b) is trivial by definition. It remains to show that (a) ⇒ (c).
Let U ∈ U. Define H(r) = 1 + U 0 (r+ ), where U 0 (r+ ) denotes the right-derivative of
U at r, so that H is right-continuous. Since U (0) = U (0) = E[ω] and U 0 (0+ ) = −1, we
0have H(0) = 0; since U (1) = U (1) = 0 and U 0 (1+ ) = U (1+ ) = 0, we have H(1) = 1.

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

U (r)
U (r)

U (r)
r

0

1

Figure 1. Set U contains every convex function between U and U .
Also, since U is convex, H is nondecreasing. Hence, H is a distribution function.
Next, observe that
Z 1
Z 1
(1 − H(s))ds = U (r) ≤ U (r) =(1 − F (s))ds
(9)
r

r

for all r, with equality at r = 0, because U (0) = U (0) = E[ω]. That is, F is a meanpreserving spread of H. By Blackwell (1953), there exists a joint distribution function
P (ω, m) such that the marginal distribution of ω is F , the marginal distribution ofm is H, and EP [ω|m] = m for all m.
Now, consider an experiment σ(m|ω) = P (m|ω). By construction, E[ω|m] = m forall m of σ, and the probability that σ generates message m ≤ x, for any given value x,is H(x). Thus, σ induces type r to act with probability qσ (r) ∈ [1 − H(r), 1 − H(r− )],where indeterminacy arises at each discontinuity point r of H, because type r is
indifferent between the two actions and can, therefore, optimally choose any lottery
over these actions. Finally, (4) implies that σ implements U .
Theorem 1 yields the following characterization of implementable interim actions.
Corollary 1. The following statements are equivalent:(a) q is a nonincreasing function that satisfies
Z 1
Z 1
q(ω)dω ≤(1 − F (ω))dω for all r, with equality at r = 0;
r

(10)

r

(b) q is implementable;(c) q is implementable by an experiment.
The heart of the characterization result, Theorem 1, is that every implementable
pair of U and q is implementable by an experiment. This result relies on the following
connection between the Mirrlees (1971) representation of incentive compatibility andPERSUASION OF A PRIVATELY INFORMED RECEIVER

9the Blackwell (1953) representation of garbling. Incentive compatibility (4) and feasibility (8) imply that every implementable interim utility U and action q must satisfy
the requirements in parts (a) of Theorem 1 and Corollary 1. Part (a) of Corollary 1
states that F is a mean-preserving spread of 1−q. But any such q can be implemented
by an appropriate garbling of the full disclosure experiment.7 Because each interim
utility U is pinned down by some interim action q through (4), we obtain the result.
We now highlight a connection to the literature on the equivalence of Bayesianand dominant-strategy incentive compatibility in linear environments with transfers (Manelli and Vincent, 2010, and Gershkov et al., 2013). Using Gutmann etal. (1991), Gershkov et al. (2013) show that for a given monotonic expected allocation (Bayesian incentive-compatible mechanism) there exists a monotonic ex-post allocation (dominant-strategy incentive-compatible mechanism) that delivers the same
expected allocation. Relatedly, using Blackwell (1953), we show that for a given nonincreasing q that satisfies (10) there exists an incentive compatible π(ω, r) that isnonincreasing in r for each ω ∈ Ω and satisfiesZ
π(ω, r)dF (ω) = q(r) for each r ∈ R.
Ω

Since π(r, ω) ∈ [0, 1] is nonincreasing and, w.l.o.g., right-continuous in r, we caninterpret the function σ(r|ω) = 1−π(r, ω) as a distribution of messages for every given
state ω. That is, σ is an experiment, and mechanism π is equivalent to experimentσ. Both Blackwell (1953) and Gutmann et al. (1991) are based on the majorization
theory initiated by Hardy et al. (1929).
3.3. Discussion. We now discuss extensions and limitations of Theorem 1, when thesender’s and receiver’s utility functions are nonlinear and the receiver has more than
two actions.
Linearity in Type. The result in Theorem 1 can be easily extended to the casewhere the utilities are linear in (a transformation of) the state ω, but are arbitrary
functions of the type r. Suppose thatu(ω, r, a) = a · (c(r) + b(r)d(ω))

and v(ω, r, a) = a + ρ(r)u(ω, r, a),where b, c, d, and ρ are bounded measurable functions. In addition, assume thatb(r) is strictly positive, and c(r)/b(r) is strictly monotone.8 All other details of themodel remain as in Section 2; in particular, the receiver chooses one of two actions,a ∈ {0, 1}. We now demonstrate how a change of variables gives back the problem
formulated in Section 2. Define new variablesω̃ = d(ω) andr̃ = −c(r)/b(r),7Suppose that an experiment induces the distribution of posterior means equal to H = 1 − q.

It is optimal for a type r to act if the posterior mean is at least r, which occurs with probability
1 − H(r) = q(r). Therefore, this experiment implements q.
8These assumptions are unnecessary for our results and are made for elegance of exposition. A
more general case of utility functions linear in ω is analyzed in Appendix C of the Supplemental
Material.

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

and letu(ω, r, a)ũ(ω̃, r̃, a) =
=a·
b(r)



c(r)
+ d(ω)b(r)= a · (ω̃ − r̃) .

The utility of each type r is now multiplied by the factor 1/b(r), which has no effecton the optimal choice of action a ∈ {0, 1} for this type. Next, define ρ̃(r̃) = ρ(r)b(r)and letu(ω, r, a)ṽ(ω̃, r̃, a) = v(ω, r, a) = a + ρ(r)b(r)= a + ρ̃(r̃)ũ(ω̃, r̃, a).
b(r)Thus, after this change of variables, the sender’s and receiver’s utilities have thefunctional forms identical to (1).
Linearity in State. The linearity of the utilities in (a transformation of) the state
implies that a posterior distribution of the state can be summarized into a scalar, theposterior mean. For any message of any experiment, the receiver acts if and only if his
type is below the cutoff equal to the posterior mean state conditional on that message.
An experiment induces a probability distribution over these cutoffs. A persuasion
mechanism, which is a menu of experiments, induces a compound distribution overcutoffs, which is itself an experiment.
Without the linearity in the state, a posterior distribution of the state cannot besummarized into a scalar. Even if the receiver’s utility is increasing in the state anddecreasing in the type, different types of the receiver may have different attitudes torisk. Thus, there may be a value of screening types by offering the receiver a menu
of experiments.
For example, consider an experiment that sends one message if ω ≥ ω 0 and anothermessage otherwise. For a type r0 that is exactly indifferent between a = 0 and
a = 1 upon learning that ω ≥ ω 0 , this experiment leaves type r0 with zero surplus
and uniquely maximizes the probability that type r0 acts. Under this experiment, anearby type r00 < r0 also acts if and only if ω ≥ ω 0 . Without linearity, it is possible
to design another experiment, preferred by r00 but not by r0 , which induces type r00
to act with a different probability. Thus, the persuasion mechanism corresponding
to the menu of these two experiments implements the interim actions that cannot be
implemented by any experiment.
In Appendix B of the Supplemental Material, we formulate the model in which
the utilities are nonlinear in the state, characterize persuasion mechanisms that areequivalent to experiments, and demonstrate that the equivalence of implementation
by persuasion mechanisms and by experiments generally fails.
Binary Action Space. The assumption that the receiver chooses one of two
actions is crucial for Theorem 1 to hold. Relaxing this assumption will affect theresult in two ways.
First, the number of actions affects the complexity of the obedience constraints.
With two actions, there are only two relevant ways to disobey a mechanism, byignoring the mechanism’s message and either always act or never act. However,
the outcomes of these deviations can be achieved by reporting the extreme types,r̂ = 0 and r̂ = 1, which always act and never act, respectively. Thus, the incentive

PERSUASION OF A PRIVATELY INFORMED RECEIVER

11

to report the type truthfully automatically implies obedience. With more than two
actions, there are more ways to disobey than just ignore the messages. The obedience
constraints are nontrivial and are not implied by the truthtelling. Consequently, theimplication (a) ⇒ (b) of Theorem 1 (that every convex function between U and
U is implementable) is false, because the obedience constraints impose additional
necessary conditions for implementability.
Second, with more than two actions, there exist outcomes implementable by persuasion mechanisms but not by experiments. For example, suppose that there arethree actions available to the receiver, A = {0, 1, 2}, and consider a persuasion mechanism described by a menu of experiments that send binary messages, “high” or “low”.
Under this mechanism, each type of the receiver will choose a preferred experimentfrom the menu, and then choose one of two best-response actions to the messages.
Generally, this mechanism is not implementable by an experiment. If an experiment sends a binary message (the same for all types), then, generally, there will betypes who would have chosen a different experiment from the menu and would haveacted differently under the original persuasion mechanism. If, instead, an experimentsends multiple messages, then, generally, there will be types that have more than
two best-response actions to those messages, thus behaving differently than under
the original mechanism. Consequently, the implication (b) ⇒ (c) of Theorem 1 (thatevery implementable U is implementable by an experiment) does not hold.
In Appendix A of the Supplemental Material, we formulate the model with multiple
actions of the receiver, characterize the implementable receiver’s interim utilities, andshow that the sender can generally implement a strictly larger set of the receiver’s
interim utilities and actions by persuasion mechanisms than by experiments.
4. Optimal Mechanisms
In this section, we use Theorem 1 to characterize the persuasion mechanisms thatare optimal for the sender under additional smoothness assumptions. We assume thatthe weight ρ in the sender’s utility is continuous in the receiver’s type and that the
distribution of types G admits a continuously differentiable density g.
4.1. Sender’s Problem. The sender seeks an incentive-compatible persuasion mechanism π that maximizesZ
Vπ (r)dG(r),R

where Vπ (r) is the sender’s interim utility when the receiver’s type is r, as definedby (7). The following lemma is a useful tool for finding optimal persuasion mechanisms. It expresses the sender’s expected utility as a function of the receiver’s interim
utility.
Lemma 2. For every incentive-compatible mechanism π,ZZVπ (r)dG(r) = g(0)E[ω] +Uπ (r)I(r)dr,
R

R

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

where
I(r) = g 0 (r) + ρ(r)g(r) for all r ∈ R.
This formulation of the sender’s objective is similar to the Myerson (1981) formulation of the seller’s objective in terms of virtual valuations. Both formulations
account for the designer’s biased preferences and information rents left to the privately informed parties. Lemma 2 relies on the assumption that the sender’s utility
is a weighted sum of the receiver’s utility and action. By incentive compatibility(Lemma 1), the receiver’s interim action is the derivative of his interim utility. The
lemma then follows by integration by parts.
Proof. Observe that, by (4) and (7), we haveZ
Z
Z
Vπ (r)dG(r) =(qπ (r) + ρ(r)Uπ (r)) g(r)dr =(−Uπ0 (r+ ) + ρ(r)Uπ (r)) g(r)dr.
R

R

R

Using integration by parts, we obtainZ
Z
10− Uπ (r+ )g(r)dr = −Uπ (r)g(r) 0 +Uπ (r)g 0 (r)dr.
R
R
1
Since −Uπ (r)g(r) 0 = E[ω]g(0) by (5), the lemma follows.



By Theorem 1, the receiver’s interim utility is implementable by some persuasion
mechanism if and only if it is in U. Hence, the sender’s problem can be expressed asZ(11)max U (r)I(r)dr.
U ∈URWe say that U is optimal if it solves the above problem. For a given optimal U , thecorresponding optimal experiment is defined by the distribution H of the posteriormean state,
H(r) = 1 + U 0 (r+ ) for all r ∈ [0, 1].
4.2. Structure of Optimal Mechanisms. We characterize the structure of optimalmechanisms under the assumption that function I is nonzero almost everywhere andchanges sign n ≥ 0 times.9 Let {r1 , r2 , . . . , rn } ⊂ (0, 1) be the set of types at which Ichanges its sign, and let r0 = 0 and rn+1 = 1.
Clearly, as follows from (11), on any interval (ri , ri+1 ) where I(r) is positive, forany given values of U (ri ) and U (ri+1 ), the optimality requires that U (r) is pointwise
maximized subject to feasibility (U ≤ U ) and the convexity of U on [ri , ri+1 ]. That is,
the interim utility U on [ri , ri+1 ] is a straight line that passes through the endpoints
U (ri ) and U (ri+1 ), unless U (r) = U (r) for some r ∈ [ri , ri+1 ], as shown in Fig. 2.
Formally,(P1 ) On every interval (ri , ri+1 ) where I(r) is positive, U is the greatest convex
function that passes through the endpoints U (ri ) and U (ri+1 ) and does notexceed U .
9If there are intervals where I(r) = 0, then on those intervals the sender is indifferent about the

choice of U . Hence, multiple solutions emerge in this case. Characterization of these solutions is astraightforward but tedious extension of the result in this section.

PERSUASION OF A PRIVATELY INFORMED RECEIVER

13

Tangency pointsof U and U

U (r)

U (r)
U (r)

0

−

U (r)

I(r)

I(r)

r
−

+

r
0

1

+

−

−

Figure 2. The optimal interim utility on the interval where I(r) is
positive.

U (r)
U (r)
I(r)
r
0

+

−

+

1

Figure 3. The optimal interim utility on the interval where I(r) is
negative.
Similarly, on any interval (ri , ri+1 ) where I(r) is negative, for any given pairs of(U (ri ), U 0 (ri )) and (U (ri+1 ), U 0 (ri+1 )), the optimality requires that U (r) is pointwise
minimized subject to U ≥ U and the convexity of U on [ri , ri+1 ]. That is, the interim
utility U on [ri , ri+1 ] is an upper envelope of two straight lines that pass through theendpoints U (ri ) and U (ri+1 ) and have slopes U 0 (ri ) and U 0 (ri+1 ), as shown in Fig. 3.
Formally,(P2 ) On every interval (ri , ri+1 ) where I(r) is negative, U is piecewise linear withat most one kink and satisfies
U (r) = max U (ri ) + U 0 (ri )(r − ri ), U (ri+1 ) + U 0 (ri+1 )(r − ri+1 ) .

1

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

Therefore, we can reduce the sender’s problem (11) of optimization on the function
space U to an n-variable optimization problem. The optimal U is pinned down byproperties (P1 ) and (P2 ) within each interval (ri , ri+1 ) where the sign of I is constant.
Thus, the optimal U is fully defined by the utility values at the points {r1 , ..., rn }where I changes its sign.
For every vector ȳ = (y1 , ..., yn ) ∈ Rn , consider the set of all U ∈ U that satisfy
U (ri ) = yi for all i = 1, ..., n. If this set is nonempty, the properties (P1 ) and (P2 )uniquely determine the optimal U on this set, denoted by Uȳ∗ . For completeness,
define Uȳ∗ = U if this set is empty. Thus, we obtain:
Theorem 2. The sender’s problem (11) is an n-variable optimization problem,Z
maxn
Uȳ∗ (r)I(r)dr.
ȳ∈R

R

Properties (P1 ) and (P2 ) imply that the optimal U is piecewise linear except for theintervals where U (r) = U (r), as shown in Figs. 2–3. Recall that, for a given optimal
U , the corresponding optimal experiment induces the distribution of the mean state
given by H(r) = 1 + U 0 (r+ ) for all r ∈ [0, 1]. Therefore, the intervals on which U islinear correspond to constant H (i.e., zero density). The kinks of U correspond tothe mass points of the distribution equal to the difference between the right and leftderivatives of U at those points. The intervals on which U (r) = U (r) correspond to
the full disclosure intervals, H(r) = F (r).
Example 1. Suppose that I(r) = g 0 (r) + ρ(r)g(r) crosses the horizontal axis at mostonce and from above. This is the case, for instance, if the density g(r) is single-peakedand ρ(r) = 0. Then, Theorem 2 implies that there exists a threshold ω ∗ such thatthe optimal experiment reveals the state if it is below ω ∗ and sends the same poolingmessage for all states above ω ∗ .
5. Conclusion
In this paper, we study the model of persuasion of a privately informed receiver inlinear environments, and show the equivalence of persuasion by experiments and by
persuasion mechanisms. A privately informed receiver can be interpreted as a randomdraw from a heterogeneous population of receivers. Within this interpretation, a persuasion mechanism establishes a private communication channel with each individualin the population; it asks each individual to report his type, and returns a private
message about the state. In contrast, an experiment broadcasts the same message
to the entire population. Our equivalence result can be viewed as the equivalence
between persuasion via private communication to receivers and by broadcasting information to the whole population. In particular, so long as the receivers’ types areunobserved by the sender, there is no loss of optimality in publicity of messages andfree information sharing among the receivers.
Within the above interpretation, it would be natural to assume that the receivers’
payoffs depend not only on their own actions, but also on collective actions in the

PERSUASION OF A PRIVATELY INFORMED RECEIVER

15

population, thus being affected by network externalities. This extension is beyond
the scope of this paper and it is left open for future research.
Appendix. Proof of Lemma 1
Necessity. Incentive compatibility (2) requires that for each r̂ > r, both r and r̂
prefer truth telling

Therefore,Uπ (r) ≥ Uπ (r, r̂) = Uπ (r̂) + qπ (r̂)(r̂ − r),
Uπ (r̂) ≥ Uπ (r̂, r) = Uπ (r) + qπ (r)(r − r̂).

−qπ (r)(r̂ − r) ≤ Uπ (r̂) − Uπ (r) ≤ −qπ (r̂)(r̂ − r),which implies (3). Letting r̂ → r and then integrating from r to 1 gives
Z 1
Uπ (1) − Uπ (r) = −qπ (s)ds.
r

Also, observe that type r = 1 can secure his maximal attainable utility of 0 by always
acting (irrespective of a recommendation); so Uπ (1) = 0 and (4) follows. Finally,
the maximal attainable utility of type r = 0 is E[ω], which can be secured by neveracting; so (5) follows.
Sufficiency. It remains to show that (3)–(5) imply (2). If either r̃ ≥ r̂ ≥ r orr̃ ≤ r̂ ≤ r, then (3) and (4) imply thatZ 1
Uπ (r, r̂) = Uπ (r) + qπ (r̂)(r̂ − r) =qπ (s)ds + qπ (r̂)(r̂ − r)
r̂
Z r̃
Z 1
≥
qπ (r̃)ds +qπ (s)ds + qπ (r̃)(r̂ − r)
r̂
r̃
Z 1
=
qπ (s)ds + qπ (r̃)(r̃ − r) = Uπ (r, r̃),r̃

meaning that Uπ (r, r̂) is single-peaked in r̂, with the peak located at r̂ = r. Therefore,Uπ (r) ≥ Uπ (r, r̂) = Uπ (r, r̂, 0, 1) for all r, r̂ ∈ R.

Moreover, letting r̂ = 1 and r̂ = 0 givesUπ (r) ≥ Uπ (r, 1) = Uπ (1) + qπ (1)(1 − r) ≥ 0 = Uπ (r, r̂, 0, 0) for all r, r̂ ∈ R,
Uπ (r) ≥ Uπ (r, 0) = Uπ (0) − qπ (0)r ≥ E[ω] − r = Uπ (r, r̂, 1, 1) for all r, r̂ ∈ R.

Thus, we are left to show that Uπ (r) ≥ Uπ (r, r̂, 1, 0) for all r, r̂ ∈ R. Notice thatZ
Uπ (r, r̂, 1, 0) = (1 − π(ω, r))(ω − r)dF (ω) = E[ω] − r − Uπ (r, r̂).
Ω

Since Uπ (r, r̂) is single-peaked, we haveUπ (r, r̂, 1, 0) ≤ E[ω] − r − min{Uπ (r, 0), Uπ (r, 1)} = max{0, E[ω] − r}
= max{Uπ (r, 1), Uπ (r, 0)} ≤ Uπ (r) for all r, r̂ ∈ R.

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KOLOTILIN, MYLOVANOV, ZAPECHELNYUK, LI

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