The Optimal Timing of Persuasion∗
Jacopo Bizzotto†,
Jesper Rüdiger‡ and
Adrien Vigier§
December 2015
PRELIMINARY DRAFT
Abstract
We study the effect of the arrival of exogenous news in dynamic games of Bayesian
persuasion. A receiver chooses what action to take, and when to act. If Receiver waits,
exogenous news may be observed. A sender chooses a first information structure that
generates a signal before the news and, if Receiver waits, a second signal structure after
the news. Receiver observes the signal structures, as well as the corresponding signal
realized. We show that, in the sense of Blackwell, information provided by Sender before
the news is non-monotonic in the quantity of exogenous news. For very small and very
large quantities of exogenous news, Sender provides enough information for Receiver to
act before the news with probability one. By contrast, for intermediate quantities of
exogenous news, Sender curbs the information she provides, in a way that minimizes the
chances of Receiver acting early against Sender’s interest. In this case, Sender may even
communicate nothing before the news, and concentrate her persuasion effort after the
news.
JEL classification: D72, D82, D83
Keywords: Dynamic Bayesian Persuasion, Exogenous News, Information Disclosure
∗
This paper was born from discussions spurred by a series of lectures on Bayesian Persuasion by Jeff Ely
in the spring of 2015. We would like to thank Bård Harstad, Meg Meyer, and seminar participants at the
BI Norwegian Business School in Oslo, and the University of Copenhagen for their helpful comments. All
remaining errors are our own.
†
Department of Economics, University of Oslo. Email: [email protected].
‡
Department of Economics, University of Copenhagen. Email: [email protected].
§
Department of Economics, Oxford University. Email: [email protected].
1
Introduction
An entrepreneur seeks funding from an investor to commercialize a product, a new drink,
say. Investing early gives higher returns –should the project turn out profitable– but implies
missing the opportunity to learn more about the profitability of the project, whether from the
entrepreneur or other sources. The entrepreneur appoints a market research firm to carry out
trials; he chooses the test markets’ size, and how to time these tests. The test results then
become available to both entrepreneur and investor. This paper is a first step in the study
of a class of dynamic Bayesian persuasion games, of which the example above is a particular
application.
Our model is a dynamic version of the baseline model of Kamenica and Gentzkow (2011)
who explore information disclosure in a broader – but static – setting. The players are Sender
(‘she’) and Receiver (‘he’). There are two states of the world, and Receiver chooses among
two actions. Receiver’s preferences over actions depend on the realized state, while Sender’s
preferences are state-independent. Receiver chooses when to act: before or after observing
exogenous public news.1 The game ends as soon as Receiver chooses an action. At any moment
in the game, Sender filters information by publicly selecting an information structure, i.e. a
state-contingent distribution over signal realizations. The distinguishing features of the model
are:
1. Receiver chooses the time of his action;
2. Factors beyond Sender’s control affect public information over time.
Alone, neither of the features above has much bearing. Absent external factors affecting
the flow of information, Receiver has no reason to delay his action.2 Similarly, if Receiver
were forced to wait until external factors have played, then Sender would have no reason to
provide information early on.
The interplay between these two features, on the contrary, raises novel issues. As Sender
could lose control over the flow of information, she may prefer to persuade Receiver to act
before the news. Yet, Receiver internalizes the value from waiting and, as a consequence,
persuading early is harder than persuading late. Hence, Sender may instead prefer delaying
1
Specifically, we explore both ‘uncertain’ and ‘imperfect’ news. In the model, exogenous news is characterized by two parameters: the probability that it is observed and, conditional on being observed, the precision
of the news. The ‘quantity’ of exogenous news increases as either parameter rises.
2
This contrasts with Brocas and Carrillo (2007). There, constraints placed on the quantity of information
which Sender can generate each period induce delay.
1
her persuasion effort after the news. Our paper explores this tension, and analyses the resulting
dynamics of Sender’s disclosure of information.
We first explore the relationship between equilibrium outcome and the quantity of exogenous news. The first main result of our analysis is the following. In the sense of Blackwell,
information provided by Sender before the news is non-monotonic in the quantity of exogenous
news. For very small and very large quantities of exogenous news, Sender provides enough
information for Receiver to act before the news with probability one. By contrast, for intermediate quantities of exogenous news, Sender curbs the information she provides, in a way
that minimizes the chances of Receiver acting early against Sender’s interest.
We then explore the relationship between equilibrium outcome and the qualitative nature of
exogenous news. The effect of uncertain news differs sharply from that of imperfect news. Most
surprisingly, near perfect news of intermediate certainty induces Sender to remain completely
idle before the news. By contrast, near certain news of intermediate precision always induces
Sender to filter some information early on. Hence, in the former case Sender concentrates her
persuasion effort later on whereas, in the latter case, Sender’s persuasion effort takes place
gradually over time.
We next develop the intuition behind our results, with a simple example based on the
application introduced earlier.
1.1
An Example with an Entrepreneur and an Investor
An entrepreneur seeks funding from an investor to commercialize a product. The prior probability that the project is profitable is 30%. The discount factor is 9/10 for the investor, and
1 for the entrepreneur.3 The investor doubles his stake if the project is profitable, and loses
everything otherwise. In the absence of exogenous news, therefore:
1. The investor always acts early on, and provides funding if and only if he believes the
project to be profitable with probability greater than 50%.
2. The entrepreneur, for his part, filtering information optimally, secures funding with 60%
probability.4
3
In the main analysis, we allow Sender to discount time as well. Here, we ignore Sender’s discounting
to simplify the exposition. The qualitative message is the same, with or without discounting on the part of
Sender. Receiver’s discounting however is clearly essential, as otherwise Receiver would never act before the
news.
4
Suppose that the entrepreneur sets up the following information structure. If the state is ‘profitable’ then
2
Reduction in chance of funding
In the rest of this example, we illustrate the loss to the entrepreneur from exogenous news,
for two prominent strategies of the entrepreneur: ‘early persuasion’ and ‘late persuasion’. To
keep the example as simple as possible, we assume in what follows that the news is observed
with probability q but that, if it is observed, then the news reveals the state with probability
1.
60%
30%
27%
q
0
1/9
1
7/27
‘early persuasion’ strategy
‘late persuasion’ strategy
‘late persuasion’ + investor waits
Figure 1: Loss to Entrepreneur from exogenous news
The dashed curve on Figure 1 shows the loss from trying to persuade the investor before
the news, i.e. when the entrepreneur’s filtering of information induces the investor to make his
call before the news with probability 1 (‘early persuasion’ strategy); the lined curve shows the
a signal ‘go’ is generated with probability 1. If the state is ‘unprofitable’ then a signal ‘don’t go’ is generated
with probability 4/7, and the signal ‘go’ is generated with probability 3/7. Signal ‘go’ is then realized with
3/10
3
7 3
3
3
6
probability 10
+ 10
. 7 = 10
+ 10
= 10
, and the posterior induced is 3/10+3/10
= 12 . This example is chosen to
parallel –and allow comparison with– the well-known ‘judge-prosecutor’ example developed in Kamenica and
Gentzkow (2011).
3
loss from trying to persuade the investor after the news, i.e. when the entrepreneur remains
idle before the news (‘late persuasion’ strategy).
For very low values of the parameter q, the investor never waits: he does not expect
to learn much, and waiting is costly. Hence exogenous news has no bite, for low q: the
investor behaves as he would have without the news. By trying to persuade the investor
before the news, the entrepreneur remains able to secure funding with probability 60%. ‘Late
persuasion’, by contrast, induces zero payoff: as the prior is strictly below 50%, the investor
chooses to retract immediately, if the entrepreneur remains idle early on. In fact, a yet stronger
observation can be made. Even if the investor waited with probability 1, then the entrepreneur
would be better off trying to persuade the investor early on. The reason is that –so long as q
is strictly greater than zero– exogenous news implies a strict loss of control over the flow of
information, from the perspective of the entrepreneur. This intuitive remark is captured in
Figure 1 by the dotted curve, showing the loss from trying to persuade the investor after the
news, but conditional on Receiver waiting with probability 1.5
As q reaches 1/9, the behavior of the investor starts changing: the investor now can find
it worth waiting. Specifically, to be made to invest early, the investor now requires beliefs
strictly greater than 50%. If q were 1/3, for instance, the investor would require beliefs above
75%.6 Figure 2 shows the investor’s threshold belief to invest early on, as q increases from 1/9
to 1. From the entrepreneur’s viewpoint, therefore, persuading the investor early on becomes
harder as q increases beyond 1/9. The entrepreneur now incurs a loss from exogenous news
either way, whether he attempts ‘early persuasion’ or ‘late persuasion’. In the former case,
the loss is due to the added difficulty of persuading the investor to fund the project early
on; in the latter case, as before, the loss follows from missing some control over the flow of
information. However, straightforward continuity arguments establish that the loss from ‘late
persuasion’ is greater than the loss from ‘early persuasion’, for q values close to 1/9 (see Figure
1). ‘Early persuasion’ thus dominates ‘late persuasion’ for low values of q.
Notice however, using Figure 2, that the investor’s threshold belief increases at a high rate
for q (greater, but) close to 1/9. The intuition is the following. At the threshold belief, the
5
The dotted curve thus lies (weakly) below the solid curve. The two curves join as, given the prior of 30%,
the investor finds it worth his while to wait. This occurs for q = 7/27.
6
If the investor’s belief is 75% then, by funding now, he will be taking the correct action with probability
75%. If he waits, then there is probability 1/3 that he will observe the state (and take the correct action
for sure), and probability 2/3 that he will not observe the state (and thus take the correct action with 75%
probability). Therefore, he will take the correct action with probability 83% (5/6), if he just waits. But the
9
investor discounts time, and 10
· 83%=75%.
4
90%
50%
q
0 1/9
1
Figure 2: Threshold belief for early funding
investor is just indifferent between investing before the news, or waiting to hopefully observe
the news. When q increases, waiting becomes more attractive. To remain indifferent between
waiting and investing now, the investor’s confidence in the profitability of the project must
rise. But this, in turn, raises the investor’s expected payoff from waiting: in cases where no
news is ever observed –which occurs with probability close to 8/9– then today’s belief is also
tomorrow’s belief. So, to remain indifferent between waiting and investing now, the belief
of the investor must rise further, and so on. A ‘multiplier effect’ emerges, causing a sharp
response of the investor’s threshold belief to increases in q. As this effect is driven by the
possibility of not observing exogenous news, its strength decreases as q becomes large. The
loss from trying to persuade the investor before the news is thus large for intermediate values
of q. In this case, the entrepreneur may prefer saving himself until after the news. ‘Late
persuasion’ thus dominates ‘early persuasion’ for intermediate values of q (see Figure 1).
Consider finally values of q close to 1. The entrepreneur can remain idle early on, but the
investor then almost surely learns the state. In this case, therefore, ‘late persuasion’ is akin
to foregoing the chance of ever manipulating the information flow. The entrepreneur then
secures funding with probability equal to the prior, i.e. 30%. But clearly, the entrepreneur
can do better by filtering information early on. As the investor discounts time, a project
profitable with 90% probability early on is worth to him as much as a project with guaranteed profitability later on. Suppose that the entrepreneur sets up the following information
structure: if the state is ‘profitable’ then a signal ‘go’ is generated with probability 1; if the
state is ‘unprofitable’ then a signal ‘don’t go’ is generated with probability 20/21, and the
signal ‘go’ is generated with probability 1/21. The signal ‘go’ is then realized with probability
7
1
3
+ 10
· 21
= 1/3, while the posterior belief it induces is 3/10
=90%. The entrepreneur can thus
10
1/3
5
secure funding with 33% probability, using ‘early persuasion’. In particular, ‘early persuasion’
dominates ‘late persuasion’, for q close to 1 (see Figure 1).
The insights emerging from this example and the figures accompanying it are thus (i) that
information provided by Sender before the news is non-monotonic in the quantity of exogenous
news, and (ii) that –in cases– remaining idle before the news can be part of Sender’s optimal
dynamic strategy. The rest of the paper fleshes out the former insights in broader settings,
and uncovers the role played by the assumptions of the example above. We show, for instance,
that most forms of exogenous news induce Sender to spread his persuasion effort over time.
This occurs, in particular, for news observed with high probability, but whose precision is
neither too low, nor too high.
1.2
Related Literature
This paper studies persuasion in a dynamic setting. We model persuasion resulting from the
ability of a sender to commit to arbitrary information structures, i.e. functions mapping states
of the world to signal distributions, following the pioneering work of Rayo and Segal (2010)
and Kamenica and Gentzkow (2011).
The state of the world is fixed, in the settings we consider. The dynamics springs from
the arrival of exogenous news, and the possibility for the receiver to ‘time’ his decision. Our
approach therefore is entirely distinct from existing dynamic models of persuasion, including
seminal contributions from Renault, Solan and Vieille (2014) and Ely (2015). As opposed to
us, in these papers (a) the sender controls throughout the flow of information to the receiver
and (b) there is no timing on the part of the receiver. Instead, the dynamics in these papers
springs from the changing state of the world. The focus and the results of our paper are, not
surprisingly, very different.7
Dynamic information disclosure has been studied in a variety of settings, and our paper
contributes to this literature. Recent examples include Che and Hörner (2013) and Kremer,
Mansour and Perry (2014), which both study models where a principal uses information
disclosure to incentivize socially beneficial experimentation; Bergemann and Wambach (2014)
consider a seller’s optimal sequential information revelation in an ascending-price auction
with independent values; Hörner and Skrzypacz (2014) model an agent who is informed about
7
Kolotilin (2015) explores persuasion in a model with exogenous news, but without dynamics. Essentially,
his model resembles what ours would look like without period 1. He derives the conditions under which players’
welfare is monotonic in information.
6
his type but face a hold-up problem, inducing him to sell his information to the principal
gradually.
Finally, we are also related in spirit to models of dynamic communication in settings of
asymmetric information. Several papers (Aumann and Hart, 2003; Krishna and Morgan, 2004;
Forges and Koessler, 2008; Goltsman, Hörner, Pavlov and Squintani, 2009) investigate ‘long
conversation’. In these models, conversation is used to induce a jointly controlled lottery over
outcomes, which in many cases allows both the informed and the uninformed party to be
better off. Golosov, Skreta, Tsyvinski and Wilson (2014) analyze repeated cheap talk in a
model where the receiver must make a decision at each stage, and find that full information
revelation may be possible.
The paper is organized as follows. Section 2 describes the model. Section 3 lays down the
preliminary steps of the analysis. Section 4 contains the main result, and Section 5 focuses
on two worked examples. Section 6 concludes.
2
Model
We study a dynamic model of ‘Bayesian persuasion’ with exogenous news. Receiver (‘he’)
must take an action, but can choose when to act. Knowing this, Sender (‘she’) chooses how
to filter information, and when to do it.
Payoffs. The state of the world is ω ∈ Ω = {L, R}, and Receiver chooses an action in {L, R}.
In each state of the world, the ‘correct’ action yields Receiver (instantaneous) payoff 1; the
‘incorrect’ action yields Receiver 0. Sender’s payoffs on the other hand are state independent.
Sender’s (instantaneous) payoff is 1 if action R is played, and 0 otherwise. The action R is
thus Sender’s preferred action.
Duration. Time is discrete, and the game lasts two periods at most, t ∈ {1, 2}. The actual
duration of the game is determined by the timing of Receiver’s action. Players share a common
discount factor δ < 1.
We turn next to players’ information concerning the realized state. All along the game
the players remain symmetrically informed about the state, and players thus have common
7
beliefs throughout. The initial probability that ω = R is µ01 . Players’ belief at the outset is
thus µ01 , though, as time unfolds, the players learn about the state:
• First, learning occurs autonomously through exogenous news.
• Second, Sender filters information about the state, before and/or after the news.
We next describe in detail the two mechanisms.
Exogenous news. The news becomes available between period 1 and period 2, with probability q, and, if available, has precision π > 1/2. This process is subsumed by the random
signal z, with values in {∅, l, r}:
P(z = ∅|ω) = 1 − q, ∀ ω,
P(z = l|ω = L, z 6= ∅) = π = P(z = r|ω = R, z 6= ∅).
Information filtering. The prior belief at the beginning of a period is µ0t . Each period
Sender selects, unconstrained, an information structure {σt (.|ω)}ω∈Ω .8 Receiver observes both
the structure selected and the signal realized. We let µt denote Bayes’ posterior belief after
the signal is observed. As usual in the literature we henceforth identify Sender’s chosen
information structures with the distributions over Bayes’ posteriors they induce. Hence, in
µ0
this interpretation, each period Sender selects a distribution over beliefs, τt t (.), consistent
µ0
with µ0t .9 The belief at the end of period t, µt , is then drawn from the distribution τt t (.).
Strategies. A strategy for Receiver is a sequence of functions mapping beliefs (at the end of
a period) to decisions: d1 : ∆(Ω) → {L, R, W } and d2 : ∆(Ω) → {L, R}. While Receiver may
choose to wait (d1 = W ) in the first period, that option is unavailable in the second period.
8
An information structure specifies a finite set of signal realizations and a probability distribution over that
set for each state of the world.
9
A distribution over beliefs is consistent with µ0t if it has mean µ0t . In particular, a distribution over beliefs
is consistent with µ0t if and only if an information structure {σt (.|ω)}ω∈Ω induces it by Bayesian updating,
given the prior µ0t (e.g. Aumann, Maschler and Stearns (1995)).
8
A strategy for Sender is a sequence of filtering rules, denoted τt , for t = 1, 2. A filtering
rule maps beliefs (at the beginning of a period) to (consistent) distributions over beliefs:
τt : ∆(Ω) → ∆(∆(Ω)).
µ01
µ1
Sender
chooses τ1
Receiver makes
decision d1
Exogenous news
µ02
µ2
Sender
chooses τ2
Receiver makes
decision d2
Figure 3: Timing
Timing. The complete timing of the game is summarized on Figure 3. First, nature draws ω.
µ0
Sender then chooses a distribution τ1 1 (.), consistent with µ01 . A posterior belief µ1 is realized,
drawn from this distribution. Receiver now makes a decision dµ1 1 . If he takes an action, L
or R, payoffs are realized and the game terminates immediately. If Receiver chooses to wait,
the game proceeds past period 1, and the players observe z. The posterior induced is µ02 , and
µ0
the game enters period 2. Sender then chooses a distribution τ2 2 (.), consistent with µ02 . A
posterior belief µ2 is realized, drawn from this distribution. Receiver makes a decision dµ2 2 ,
payoffs are realized, and the game finally terminates.
Equilibrium. The equilibrium concept we use is that of Perfect Bayesian Equilibrium.10 To
circumvent multiple equilibria inducing identical outcomes, we impose:11
Assumption 1. In any period and given any history, if two consistent distributions over
beliefs are ordered according to Blackwell’s criterion then, if Sender is indifferent between
them, he chooses the least informative one.
10
A Perfect Bayesian Equilibrium requires Sender to choose consistent distributions over posterior beliefs
which maximize his expected payoff conditional on Receiver’s equilibrium strategy, and Receiver to choose a
strategy that consists of a set of history-dependent actions that maximize his expected payoff conditional on
Sender’s equilibrium strategy. Furthermore, since neither player has private information, the common belief
of the two players is a sufficient statistic for all payoff relevant information at any instant of the game.
11
Note that this assumption is natural if, for instance, small costs are associated with filtering information.
9
3
Preliminaries
In this section, we examine players’ equilibrium behavior in period 2, players’ expected payoffs
coming out of period 1, as well as Receiver’s equilibrium decision in period 1. The study of
Sender’s filtering rule in period 1 is delayed until Section 4.
We start with some definitions and notation. The ‘splitting’ of a prior µ0t onto posteriors
0
µt1 , ..., µtN is denoted τ̂ µt [µt1 , ..., µtN ]. We can now define a central class of filtering rules.
Definition 1. Let 0 = a1 ≤ b1 ≤ ... ≤ aM ≤ bM = 1 and, for each m, Im = [am , bm ]. We say
S
that the filtering rule τt is regular, with supports M
m=1 Im , if and only if:
µ0
0
µ0
0
1. τt t = τ̂ µt [µ0t ] for µ0t ∈
Sn
m=1 Im ,
2. τt t = τ̂ µt [bm−1 , am ] for µ0t ∈ (bm−1 , am ).
The regular filtering rule with union of supports
SM
m=1 Im is denoted τ̂
hS
M
m=1 Im
i
.
S
µ0
Regular filtering rules are simple objects. They are fully characterized by µ0t ∈[0,1] supp τt t ,
the union of all supports, as the prior spans the unit interval. If µ0t belongs to this union then
Sender provides no information; otherwise Sender splits µ0t onto the elements of the union
closest to the prior µ0t on each side.
3.1
Last Period
Once in period 2, our game coincides with the static games analyzed in Kamenica and
Gentzkow (2011). We first briefly summarize the properties of their unique equilibrium, as
they translate in the context of our own game.
Lemma 1. [Kamenica and Gentzkow (2011)] In period 2, in equilibrium:
1. Receiver chooses R if µ2 ≥ 1/2 and L if µ2 < 1/2;
h
i
2. Sender’s filtering rule is regular, with union of supports 0 ∪ [ 21 , 1]: τ2 = τ̂ 0 ∪ [ 21 , 1] .
Players’ equilibrium expected payoffs coming into period 2 with belief µ02 are, respectively,
(
v R (µ02 ) =
1 − µ02 if µ02 < 1/2
µ02
if µ02 ≥ 1/2,
10
(1)
and
(
v S (µ02 ) =
2µ02 if µ02 < 1/2
1 if µ02 ≥ 1/2.
(2)
If the prior belief induces Receiver to take Sender’s preferred action, then Sender provides
no information. If, on the other hand, the prior belief induces Receiver to take Sender’s least
preferred action, then Sender splits the prior onto 0 and 1/2. Thus, Receiver either learns
that the state is L and takes action L, or is just indifferent between the two actions and takes
action R.12
Two central features are worth stressing. First, Receiver chooses the ‘correct’ action with
probability 1 conditional on state R, but may choose the ‘incorrect’ action conditional on
state L. Second, Receiver never benefits from information provided by Sender in period 2:
v R (.) are exactly the payoffs Receiver would achieve alone.13
3.2
First Period: Preliminary Steps
We now move backwards in time, and consider players’ equilibrium expected payoffs after
µ1 is realized, but before the decision of Receiver in that period. These payoffs are denoted
V R (µ1 ) for Receiver and V S (µ1 ) for Sender, respectively. We derive them in turn.
Observe first that as Receiver never benefits from the information provided by Sender in
period 2, his expected payoffs coming out of period 1 depend only on exogenous news. To
illustrate, let µ1 ≤ 1/2. With this prior, Receiver’s optimal action is L. Receiver can act
immediately, or wait for exogenous news.14 His expected payoff is 1 − µ1 in the first case, and
δ qπ + (1 − q)(1 − µ1 ) in the second case. Acting early thus dominates waiting if and only if
δqπ
µ1 < 1 − 1−δ(1−q)
; this threshold and its symmetric counterpart play a key role in the analysis
of the model. The following corollary summarizes the remarks above.
12
In the unique equilibrium, whenever indifferent between two decisions, Receiver chooses the one which
Sender prefers.
13
Note that as Receiver can always ignore the information provided by Sender’s, he can never be worse off
as a result of that information. Here, the key feature is that Receiver is never made better off as a result of
that information.
14
Receiver could also wait and ignore the exogenous news, but clearly that is never optimal.
11
Lemma 2. In equilibrium, Receiver waits if µ1 ∈ [µ, µ) and takes an early action if µ1 < µ
or µ1 ≥ µ, where
δqπ
µ := 1 −
,
(3)
1 − δ(1 − q)
µ := 1 − µ.
(4)
Hence,


1 − µ1



 δ qπ + (1 − q)(1 − µ )
1
V R (µ1 ) =

δ qπ + (1 − q)µ1




µ1
if
if
if
if
µ1 < µ
µ ≤ µ1 ≤ 1/2
1/2 < µ1 < µ.
µ1 ≥ µ.
(5)
Observe that our dynamic game reduces to a static one unless µ < µ. This condition may
be viewed as either imposing sufficient patience on part of the Receiver, or a lower bound on
the quantity of exogenous news. Since the focus of our analysis is on (q, π)-space we formulate
this condition as a constraint on δ:
Assumption 2. δ > δ :=
1
.
1−q+2qπ
We next compute V S , Sender’s equilibrium expected payoff after µ1 is realized. If µ1 < µ
or µ1 ≥ µ then payoffs are trivial, as Receiver acts immediately. The case µ1 ∈ [µ, µ) is solved
using Lemma 1. In that case, either exogenous news is favorable to Sender or it is not. If it
is, then µ02 > 1/2 and Sender achieves payoff 1. If it is not, then µ02 < 1/2 and Sender, by
(2), can achieve expected payoff 2µ02 . Appropriately weighing each case gives V S ; Lemma 3
summarizes.
Lemma 3.


0



 δ[q(1 − π + µ ) + (1 − q)2µ ]
1
1
V S (µ1 ) =

δ[q(1 − π + µ1 ) + (1 − q)]




1
if
if
if
if
µ1
µ1
µ1
µ1
<µ
∈ [µ, 1/2)
∈ [1/2, µ)
≥ µ.
(6)
The next section completes our equilibrium analysis, by examining Sender’s optimal filtering rule in period 1.
12
4
Main Result
We characterize in this section the complete equilibrium. Section 4.1 establishes that the
equilibrium filtering rule in period 1 takes one of (only) three forms. Section 4.2 explores the
link between the qualitative nature of exogenous news and Sender’s choice of filtering rule in
period 1.
4.1
Optimal Filtering Rules
In Section 3, we derived Sender’s equilibrium expected payoff at the end of period 1. Let
V̂ S (µ01 ) denote Sender’s equilibrium expected payoff at the beginning of period 1. As Sender
chooses the distribution of µ1 amongst all distributions consistent with µ01 , we have
V̂ S = cav V S ,
(7)
where cav V S refers to the concavification of the function V S (c.f. Aumann et al. (1995), and
Kamenica and Gentzkow (2011)). Equation (7) is the basis of this section’s analysis.
Lemma 4. In equilibrium:
h
i
τ1 = τ̂ x : V (x) = V̂ (x) .
S
S
(8)
Moreover, V S (x) = V̂ S (x) if x ∈ [µ, 1]. Hence, Sender provides no information in period 1
for priors above µ. Finally, V S (x) < V̂ S (x) if x ∈ (0, µ) ∪ (1/2, µ). Thus, the set of priors for
which Sender provides some information in period 1 is non-empty and, in particular, contains
all priors in (0, µ) ∪ (1/2, µ).
Equation (8) follows from (7) and the definition of regular filtering rules. It amounts to the
observation that the equilibrium strategy can be ‘read off’ the graphs of V S and V̂ S . Where
V S = V̂ S , Sender provides no information; where instead V̂ S > V S , Sender splits the prior
onto the closest points satisfying V S = V̂ S .
The intuition for the rest of the lemma is the following. If the prior is µ or above, Receiver
prefers to take action R in the first period based on prior information. Clearly, in that case
Sender has no incentive to provide any information.
Next, by ruling out posteriors in (0, µ) ∪ (1/2, µ), the lemma sets a ‘lower bound’ on
Sender’s provision of information in period 1. Posteriors in (0, µ) induce Receiver to choose
action L in state R; but Sender can do better in that case, by just revealing the state.15 The
15
See Kamenica and Gentzkow (2011) for a counterpart of this result.
13
fact that posteriors in (1/2, µ) should be ruled out is more subtle. On the one hand, these
posteriors are ‘too informative’, since in the absence of exogenous news they provide Receiver
with strictly more evidence than needed for him to take action R in period 2. On the other
hand, these posteriors are ‘not informative enough’, as they do not induce Receiver to take
action R in period 1. Thus, Sender can profitably split these posteriors onto the edges of the
interval, 1/2 and µ.
We proceed with three lemmas, showing that the equilibrium filtering rule in period 1
takes one of three forms.
Lemma 5. The following are equivalent:
i. V̂ S (µ) > V S (µ);
µ
µ
ii. V̂ S (µ) = (1 − µ )V S (0) + µ V S (µ) > V S (µ);
h
i
iii. In equilibrium: τ1 = τ̂ 0 ∪ [µ, 1] .
The equivalence between (i) and (ii) says that if Sender optimally provides some information in period 1 given prior µ, then she splits the prior onto 0 and µ. The equivalence between
(ii) and (iii) establishes the link between the ‘local’ condition (ii) and the equilibrium filtering
rule in period 1. Sender splits the prior onto 0 and µ if the prior lies in (0, µ), and does
not provide any information otherwise. Hence, Sender maximizes chances of Receiver acting
early in her favor. In fact, Receiver acts before exogenous news is observed with probability
1. These remarks motivate our next definition.
h
Definition 2. We say that Sender engages in early persuasion if and only if τ1 = τ̂ 0 ∪
i
[µ, 1] , i.e. if and only if Sender maximizes chances of Receiver acting early in her favor.
The next result examines the situation arising when early persuasion is not an equilibrium.
Lemma 6. Suppose that the conditions of Lemma 5 are violated. The following are then
equivalent:
i. V̂ S (1/2) > V S (1/2);
ii. V̂ S (1/2) =
V S (µ)+V S (µ)
2
> V S (1/2);
h
i
iii. In equilibrium: τ1 = τ̂ 0 ∪ µ ∪ [µ, 1] .
14
The equivalence between (i) and (ii) says that if Sender optimally provides some information in period 1 given prior 1/2 (but providing information is sub-optimal given prior µ)
then she splits the prior onto µ and µ. The equivalence between (ii) and (iii) pins down the
equilibrium filtering rule in period 1. Sender splits all priors in (µ, µ) onto the edges of the
interval and all priors below µ onto 0 and µ. Sender thus maximizes chances of Receiver
acting early in her favor, subject to minimizing chances of Receiver acting early against her
interest. It should be noted that Receiver never chooses action L in period 1 when the prior
in that period is above µ. Furthermore, for all priors below µ, Sender induces Receiver to act
with positive probability in each period. These remarks motivate our next definition.
Definition
3. iWe say that Sender engages in multiperiod persuasion if and only if τ1 =
h
τ̂ 0 ∪ µ ∪ [µ, 1] , i.e. if and only if Sender maximizes chances of Receiver acting early in her
favor, subject to minimizing the probability that Receiver acts early against her interest.
The last lemma examines the situation arising when neither early nor multiperiod persuasion is an equilibrium.
Lemma
i conditions of Lemmas 5 and 6 are violated. Then, in equilibrium,
h 7. Suppose that the
τ1 = τ̂ 0 ∪ [µ, 1/2] ∪ [µ, 1] .
Combined with Lemma 4, Lemma 7 thus shows that if providing information is sub-optimal
given prior µ as well as given prior 1/2 then, in equilibrium, Sender minimizes the information
she provides, subject to minimizing chances of Receiver acting early against her interest. It
should be noted, in particular, that for priors in [µ, 1/2]: (a) Sender provides no information
in period 1, and (b) Receiver waits with probability 1. These remarks motivate our last
definition.
h
Definition 4. We say that Sender engages in late persuasion if and only if τ1 = τ̂ 0 ∪
i
[µ, 1/2] ∪ [µ, 1] , i.e. if and only if provision of information in period 1 is kept at its lowest
bound (set in Lemma 4), subject to minimizing the probability that Receiver acts early against
Sender’s interest.
This section characterized all possible equilibrium outcomes. The next section explores
the link between parameters and equilibrium outcomes.
15
4.2
Main Theorem
Section 4.1 greatly simplified our problem. By Lemmas 5 - 7, all that is now left to consider
is whether Sender wishes to split µ onto 0 and µ and, if not, whether Sender wishes to split
1/2 onto µ and µ. The reader less concerned with details can skip directly to Theorem 1 and
Figure 4.
Using (6), Sender is indifferent between splitting µ onto 0 and µ and not providing any
information if and only if
µ
δ[q(1 − π + µ) + (1 − q)2µ] = .
(9)
µ
The right-hand side is the probability of R when Sender splits the prior µ onto 0 and µ.
Similarly, Sender is indifferent between splitting 1/2 onto µ and µ and not providing any
information if and only if
δq(1 − π) + δ(2 − q)µ = 1.
(10)
Equation (9) reduces to (1 + δ)µ(1 − µ) + δµ − 1 = 0. Let m(δ) and M (δ) denote the real
roots of this equation, where m(δ) ≤ M (δ). Whenever δ ≥ √23 , these roots are well-defined.
Moreover, m(δ) > 12 and M (δ) < 1.
Next, define
π
h(q) := .
µ
Functions πk (q) := kh(q) thus correspond to iso-µ curves of level k, in (q, π)-space. The
function h(q) is positive, decreasing and convex in q.
The level curve π1/2 establishes the frontier above which the dynamics of our model bites.
Moreover, by earlier remarks, Sender engages in early persuasion if and only if π < πm(δ) (q)
or π > πM (δ) (q). In the alternative scenario, the equilibrium outcome is determined by (10).
That equation simplifies to π = π̌(q) := (1 − δq)h(q)/[(1 + δ) − 2δqh(q)].
Define next qm(δ) and qM (δ) such that πm(δ) (qm(δ) ) = 1 = πM (δ) (qM (δ) ). Hence qm(δ) < qM (δ) ,
and qM (δ) < 1. Moreover, the following properties hold:
1. πm(δ) (q) ≤ π̌(q) ≤ πM (δ) (q), for all q ∈ [qm(δ) , qM (δ) ]
2. π̌(qm(δ) ) = 1 = π̌(qM (δ) )
3. π̌ 0 (qm(δ) ) < 0 < π̌ 0 (qM (δ) )
4. π̌ 0 (.) changes sign exactly once in [qm(δ) , qM (δ) ].
16
Figure 4: Main Theorem
The functions π1/2 , πm(δ) , πM (δ) , and π̌ thus define, for δ ≥ √23 , a partition of (q, π)-space
containing 5 non-empty regions, illustrated in Figure 4. Region ‘0’ is the part of (q, π)-space
lying below π1/2 (i.e. the region ruled out by Assumption 2). Region I lies between π1/2 and
πm(δ) . The union of Regions II and IV constitutes the part of (q, π)-space lying between πm(δ)
and πM (δ) , while π̌ separates the two regions. Finally, Region III lies above πM (δ) . We can
now state our main theorem.
Theorem 1. There exists a unique equilibrium. If δ <
persuasion. If δ ≥ √23 , then:
√2 ,
3
then Sender engages in early
i. Sender engages in early persuasion in regions I and III;
ii. Sender engages in multiperiod persuasion in region II;
iii. Sender engages in late persuasion in region IV.
Figure 4 highlights our two principal results. First, information provided by Sender before
the news is non-monotonic in the quantity of exogenous news (this statement is formalized
in the next corollary). For very small and very large quantities of news, Sender provides
enough information for Receiver to act before the news with probability one. By contrast, for
17
intermediate quantities of exogenous news, Sender curbs the information she provides, in a
way that minimizes the chances of Receiver acting early against Sender’s interest. Second, in
the intermediate region, Sender may even –depending on the qualitative nature of the news–
remain idle in period 1, and concentrate her persuasion effort in period 2.
To qualify our first remark, consider the exogenous news structures of our model in light
of Blackwell’s ordering. Let z and z 0 denote two random exogenous signals of our model,
respectively parameterized by (q, π) and (q 0 , π 0 ). Note then that (q 0 , π 0 ) ≥ (q, π) implies that
signal z 0 is more informative in the sense of Blackwell than signal z.16 Similar remarks apply
0
to Sender’s provision of information. The splitting τ µ1 [c, d] is more informative in the sense
0
of Blackwell than the splitting τ µ1 [a, b] whenever c ≤ a ≤ b ≤ d. In view of Theorem 1, this
yields:
Corollary 1. In the sense of Blackwell, Sender’s period 1 communication is non-monotonic
as a function of the quantity of exogenous news.
The next section explores the contrasting effects of the certainty of the news, q, and of the
precision of the news, π.
5
Two Examples
We now explore the ramifications of Theorem 1, by focusing on two examples. Throughout
this section, the prior belief µ01 is fixed at 1/2. Recall that, in the absence of exogenous news,
Receiver never waits and 1/2 is the threshold belief inducing him to take action R. We thus
refer to the latter scenario as the static benchmark, and investigate Sender’s utility loss from
introducing exogenous news into the model.
5.1
A Decomposition of Sender’s Loss from Exogenous News
We start with this section’s core definitions. By Lemma 4, Sender splits in period 1 the
prior belief onto at most two posteriors, in equilibrium. Furthermore, if Sender provides any
information, then the upper posterior must be µ. Let µ̂ denote the lower posterior of the
equilibrium split. The interpretation is the following: µ̂ is the belief from Sender’s failed
attempt to persuade Receiver to act in her favor early on. We therefore refer to µ̂ as the
failed persuasion belief, and to µ as the early threshold belief. Note that splitting the prior
16
Here ‘≥’ is used to denote the standard (partial) order on R2 .
18
2µ−1
1/2 onto µ and µ̂ implies placing weight φ(µ̂) := 2(µ−µ̂)
onto the posterior µ̂. Moreover,
1
φ(0) = 1− 2µ , φ(µ) = 1/2 and φ(1/2) = 1. We next define three ‘costs’ to Sender, representing
the different components of his loss from exogenous news –relative to the static benchmark
payoff v S (1/2) = 1.
First, if Sender tries to persuade Receiver to act in her favor early on then –unless µ = 1/2–
she must accept that her effort might fail. If so, the belief at the end of period 1 will fall to
µ̂. The indirect ‘persuasion cost’ P (µ̂) is defined as the cost of sometimes having to incur
the failed persuasion belief:
P (µ̂) := φ(µ̂) × (1 − 2µ̂).
The first factor is the probability that Sender fails to persuade Receiver to act in her favor
early on. The second factor is the cost of entering period 2 with belief µ̂ instead of 1/2, i.e.
v S (1/2) − v S (µ̂).
Notice that the indirect persuasion cost indeed captures an indirect cost of exogenous news:
it occurs through the impact of the news on the early threshold belief of Receiver. Naturally,
Sender also incurs a direct cost of exogenous news, occurring through the impact of exogenous
news on realized posterior beliefs. The direct ‘leakage cost’ L(µ̂) is defined accordingly:
L(µ̂) := φ(µ̂) × q(µ̂ − (1 − π)).
The first factor is again the probability that Sender fails to persuade Receiver to act in her
favor early on, ending up at µ̂ by the end of period 1. The second factor is the difference in
expected payoffs with and without exogenous news, coming out of period 1 with belief µ̂, i.e.
v S (µ̂) − V S (µ̂), times the probability of actually observing the news.
Finally, the discounting cost D(µ̂) captures the cost of the payoffs being realized later
on:17
D(µ̂) := φ(µ̂) × (1 − δ)[q(1 − π + µ̂) + (1 − q)2µ̂].
The first factor is, as in the first two costs above, the probability that Sender fails to persuade
Receiver to act in her favor early on, ending up at µ̂ by the end of period 1. The second factor
is simply (1 − δ)V S (µ̂) where, recall, V S (µ̂) is Sender’s expected payoff from ending period 1
at belief µ̂.
17
Notice that if we defined separate discount factors δR and δS for Receiver and Sender, then δR < 1 and
δS = 1 would imply that D(µ̂) = 0 whereas P (µ̂), L(µ̂) > 0.
19
5.2
Example I: Certain News
Let q = 1 and π < 1, such that news is always observed, but the state is revealed with noise.
Note then that µ = δπ. As shown in Section 4, Sender has three relevant options: (a) late
persuasion, (b) multiperiod persuasion, or (c) early persuasion. We consider them one by one.
Late persuasion. Under late persuasion, Sender reveals no information in period 1. This,
in turn, induces Receiver to defer his action until period 2. Let UaI = δ[1 − π + 12 ] denote the
corresponding expected payoff for Sender. Then:
UaI = v S (1/2) − L(1/2) − D(1/2).
Note that as Sender makes no attempt to persuade Receiver early on, she incurs no indirect
persuasion cost. However, this comes at the price of a high direct leakage cost, since in this
case exogenous news becomes observed with probability 1.
Multiperiod persuasion. With multiperiod persuasion, Sender sets µ̂ = µ. Receiver, in
turn, either takes Sender’s preferred action early on, or waits until the second period. Sender’s
expected payoff is then UbI = 21 + 12 × δ(1 − π + 1 − δπ). In particular:
UbI = v S (1/2) − P µ − L µ − D µ .
Here Sender incurs each of the three costs. However, relative to late persuasion, both leakage
and discounting costs are smaller. Discounting costs are smaller because Receiver chooses
R in period 1 with some probability. Similarly, direct leakage costs are smaller, as Sender
sometimes avoids the news being observed. Moreover, when it is observed, the belief is µ
rather than 1/2, and leakage costs are decreasing in the belief.
Early persuasion. Under early persuasion, finally, Sender sets µ̂ = 0, and induces Receiver
1
to act early with probability 1. Sender’s expected payoff is then UcI = 2µ
. In particular:
UcI = v R (1/2) − P (0) .
Here Sender eliminates discounting and leakage costs, since Receiver always acts before the
news. The price she pays, of course, is a greater indirect persuasion cost.
20
We can now state:
Proposition 1. Suppose µ01 = 12 and q = 1. There exists a unique equilibrium. Late persua√
√
sion is never used. If δ ≤ 23 , then Sender engages in early persuasion. If δ > 23 , then there
exist π ∗ and π ∗∗ with π ∗∗ > π ∗ such that
1. if π < π ∗ or π > π ∗∗ then Sender engages in early persuasion,
2. if π ∗ ≤ π ≤ π ∗∗ then Sender engages in multiperiod persuasion.
Furthermore, as δ → 1, then π ∗ →
1
2
and π ∗∗ → 1.
The proposition first establishes that Sender is never idle early on. The intuition is the
following. When q = 1, then using (5), Receiver’s equilibrium payoff coming out of period
1 with any belief in the interval [µ, µ) is just δπ. Receiver thus chooses the correct action
with (constant) probability π. Now suppose that instead, Sender splits µ01 = 1/2 onto the end
points µ and µ. If the realized posterior is µ1 = µ, then the probability of the correct action
being taken stays at π. If, on the other hand, the realized posterior is µ1 = µ, then Receiver
takes action R immediately. In that case, as µ = δπ, the probability that the correct action is
taken is δπ. Splitting the prior belief in this way thus has two effects: (i) actions are moved
forward in time with some probability, and (ii) the probability that the correct action is taken
is lowered. The first effect reduces Sender’s discounting cost. The second effect implies that
Sender is more efficient in his manipulation of Receiver’s belief. Indeed in equilibrium, during
period 2 Receiver is always induced to choose action R conditional on state R. Hence, lower
probability that the correct action is taken amounts to greater probability that the chosen
action is R. Both effects thus raise Sender’s payoff, and splitting the prior onto the end points
µ and µ always dominates providing no information early on.
Second, the proposition says that whenever the quantity of news is either very small or
very large, then Sender engages in early persuasion, whereas otherwise, he uses multiperiod
persuasion. The intuition for the first part is the following. For π close to 1, Sender must give
up the hope of manipulating the flow of information, if play moves into period 2. So early
persuasion is necessarily best, in that case. For π close to 1/2δ, by contrast, the early threshold
belief µ approaches 1/2. But the indirect persuasion cost then vanishes, by definition. Yet
1/2δ is strictly above 1/2. Hence, leakage costs remain (strictly) positive, for π close to 1/2δ.
Again, early persuasion is thus necessarily best.
21
The case of intermediate π values is more subtle. First, we showed in the paragraph
below the proposition that splitting the prior 1/2 onto µ and µ is the minimum amount
of information which –in equilibrium– Sender can convey during period 1. The question
remaining is thus: does Sender prefer providing information in period 1 when the prior is µ,
and π intermediate? The answer to this question is no, and the intuition is the following.
As π increases, µ decreases, while µ increases. The gap between them, therefore, rapidly
grows. Eventually, µ overtakes the posterior induced by the ‘favorable’ outcome of the news
(z = r). At that point, the information leakage from exogenous news becomes less than the
information which Sender must release if she wants Receiver to act in her favor early on. For
yet greater π, although it implies incurring a discounting cost, providing no information beyond
µ yields higher expected payoff than engaging in early persuasion. Multiperiod persuasion thus
dominates early persuasion, for intermediate quantities of exogenous news.
5.3
Example II: Perfect News
Consider now q < 1 but π = 1. Hence, the news perfectly reveals the state, but is unobserved
with some probability. Sender, as in example I, has three relevant options: (a) late persuasion,
(b) multiperiod persuasion, or (c) early persuasion. The corresponding payoffs are UaII =
δ(2−q)µ
1
, and UcII = 2µ
. The payoffs’ decomposition from example I
δ[1 − 2q ], UbII = 12 +
2
δq
hold here too, though now µ = 1−δ(1−q) . Proposition 2 characterizes the equilibrium in this
example.
√
Proposition 2. Suppose µ01 = 21 and π = 1. There exists a unique equilibrium. If δ ≤ 23
√
then Sender engages in early persuasion. If δ > 23 then there exist q ∗ and q ∗∗ with q ∗∗ > q ∗
such that
1. if q < q ∗ or q > q ∗∗ then Sender engages in early persuasion,
2. if q ∗ ≤ q ≤ q ∗∗ then Sender engages in late persuasion.
Furthermore, as δ → 1, then q ∗ → q ∗∗ and q ∗∗ → 1.
Unlike example I, late persuasion is now optimal for some parameter values. The reason
is the following. In example I, Receiver’s equilibrium payoffs coming out of period 1, V R ,
were constant over the interval [µ, µ). This was due to the fact that Receiver always observed
exogenous news. But this is no longer true, in example II. Now, with positive probability,
Receiver gets no news. Information provided by Sender early on is then paramount, as that
22
information then determines Receiver’s payoff in period 2. This, in turn, triggers feedback
effects as Sender tries to persuade Receiver in period 1. A Receiver with belief 1/2 would
happily take action R early on, as long as this gives him payoff V R (1/2) = δ(1 + q)/2 or
more. But if Sender releases information to induce belief δ(1 + q)/2 > 1/2 then, at the new
belief, Receiver’s expected payoff from waiting is now higher: V R (δ(1 + q)/2) = δ[q + (1 −
q)δ(1 + q)/2] > δ(1 + q)/2. Hence, further information must be released to induce Receiver
to take action R early on. For strong enough feedback effects, remaining idle early on is part
of Sender’s optimal strategy. Late persuasion then ensues.
The result that for either very small or very large quantities of news then Sender engages
in early persuasion is akin to the corresponding result obtained in the context of example I.
We do not repeat the arguments.
6
Conclusion
We have presented a dynamic game of Bayesian persuasion between a sender and a receiver
with partially misaligned preferences. In this game, exogenous public news is observed in the
course of the game. Moreover, Sender can filter information at any time during the game and
Receiver can decide what action to take and when to take his action.
We have shown that when exogenous news is likely and precise, then Sender provides
sufficient information early on to induce Receiver to take an action before the news. Sender
follows the same strategy in the case of either unlikely or imprecise exogenous news. In
intermediate cases, on the contrary, the optimal strategy involves limited persuasion early on.
Thus, the information provided early on is often insufficient to induce Receiver to act early.
In the latter case, persuasion is sensitive to the nature of news. Limited early persuasion can
take different forms. Perhaps the most striking observation in this respect holds whenever
exogenous news is unlikely but (if observed) very precise. In this case, even though Sender
has access to the largest possible set of information structures, she nevertheless prefers to
provide the bare minimum amount of information needed to dissuade Receiver from choosing
Sender’s least favorite action early on.
Dynamic Bayesian persuasion has so far received limited attention and, more specifically,
Bayesian persuasion with evolving exogenous information has not yet been considered. We
show that even a stylized dynamic setting generates a rich set of novel predictions that are
missing in static frameworks. We are nevertheless aware of the limitations of our framework
23
and hope our results with spur further contributions along this line of research.
Our framework could be generalized in several directions. It may be fruitful to consider a
more general characterization of the nature of the exogenous news, or allow Sender to commit
ex ante to a multi-period signal structure. Other research on dynamic Bayesian persuasion
has focused on settings in which the state of the world evolves over time Renault et al. (2014);
Ely (2015). A dynamic framework in which both the state of the world and the exogenous
information about the state of the world evolve over time might generate new insights.
7
Appendix - Proofs
Proof of Lemma 4.
Equation (8) follows from (7) and the definition of regular filtering rules.
By Lemma 3, V S (x) = maxµ∈[0,1] V S (µ) for any x ∈ [µ, 1]. As a result, in equilibrium, for any
x ∈ [µ, 1], Sender chooses τ1x = τ̂ x [x], which implies V S (x) = V̂ S (x).
By Lemma 3, (a) τ1x = τ̂ x [0, 1] ensures a higher expected payoff to Sender than τ1x = τ̂ x [x]
for any x ∈ (0, µ), and (b) τ1x = τ̂ x [1/2, µ] ensures a higher payoff to Sender than τ1x = τ̂ x [x]
for any x ∈ (1/2, µ). As a consequence of (a) and (b), for any x ∈ (0, µ) ∪ (1/2, µ), Sender
chooses τ1x 6= τ̂ x [x], which implies V S (x) < V̂ S (x).
Proof of Lemma 5. We begin with a few ancillary observations.
(a) by Lemma 4, in period 1 Sender chooses a regular filtering rule.
n
1
Let Um=1
Im
denote the union of the supports of τ1 in equilibrium. Then:
1
1
n
n
, and [µ, 1] ∈ Um=1
,
(b) by Lemma 4, (0, µ) ∪ (1/2, µ) ∈ [0, 1] \ Um=1
Im
Im
µ
µ
µ
µ
(c) by Lemma 3, Sender (weakly) prefers τ = τ̂ [µ] to τ = τ̂ [0, x] for any x ∈ (µ, 1/2).
(d) τ1x = τ̂ x [x] and τ1x = τ̂ x [µ, 1/2] ensure the same expected payoff to Sender for any x ∈
(µ, 1/2).
µ
µ
Observations (a)-(c) imply that V̂ S (µ) > V S (µ) if and only if V̂ S (µ) = (1− µ )V S (0)+ µ V S (µ).
As a consequence, i) implies ii).
µ
µ
Moreover, observations (a)-(d) imply that whenever V̂ S (µ) = (1 − µ )V S (0) + µ V S (µ) then,
h
i
in equilibrium, Sender chooses τ x = τ̂ x [0, µ] ∀x ∈ [0, µ] and τ1 = τ̂ 0 ∪ [µ, 1] . As a result, ii)
implies iii).
By definition, ii) implies i) and iii) implies ii). By transitivity, i), ii) and iii) are equivalent.
24
Proof of Lemma 6.
Two ancillary observations are useful.
(a’) If the conditions of Lemma 5 are violated, then V̂ S (µ) = V S (µ).
(b’) Observations (a)-(d) (see proof of Lemma 5) ensure that whenever V̂ S (1/2) > V S (1/2)
then V̂ S (x) > V S (x) ∀x ∈ (µ, 1/2).
Observations (a)-(d), together with (a’) and (b’) ensure that whenever the conditions of
Lemma 5 are violated, it is then case that i) implies both ii) and iii). As by definition
iii) implies ii) and ii) implies i), then by transitivity i), ii) and iii) are equivalent.
Proof of Lemma 7.
n
1
Let Um=1
Im
denote the union of the supports of τ1 in equilibrium. Whenever the conditions
1
n
. Whenever instead the conditions of Lemma 6 are
Im
of Lemma 5 are violated µ ∈ Um=1
1
n
. Therefore, whenever the conditions of Lemma 5 and Lemma
Im
violated, then 1/2 ∈ Um=1
n
1
6 are violated at the same time, observation (d) ensures that [µ, 1/2] ∈ Um=1
Im
. Then by
n
1
observations (a)-(d) it must be the case that Um=1
Im
= 0 ∪ [µ, 1/2] ∪ [µ, 1].
Proof of Theorem 1.
Sender’s optimal strategy in period 2 is defined in Lemma 1. We thus focus on period 1.
Either the conditions in Lemma 5 hold (call this case A), or those conditions are violated (call
this case B), and one of two cases arises. Either conditions in Lemma 6 hold (call this case
B.1) or they do not (call this case B.2).
Whether case A or case B hold is determined by comparing Sender’s payoff from splitting µ
onto {0, µ} vs Sender’s payoff from waiting when the belief is µ. The corresponding indifference
condition is
µ
V S (µ) = ,
(11)
µ
which, using (6) and multiplying by µ on both sides may be rewritten as
δµ[q(1 − π + µ) + (1 − q)2µ] = µ.
25
(12)
Multiplying through by [1 − δ(1 − q)]2 yields
h
i
δ 2 qπ (1 − π)q 1 − δ(1 − q) + (2 − q) 1 − δ(1 − q) − δqπ = [1 − δ(1 − q) − δqπ][1 − δ(1 − q)],
which, eliminating terms on the LHS and simplifying leads to
h
i
δ 2 qπ 1 − δ(1 − q) − qπ = [1 − δ(1 − q) − δqπ][1 − δ(1 − q) − δ 2 qπ].
Rewriting in polynomial form yields
h
i
h
i h
i2
π 2 q 2 δ 2 (1 + δ) − π δq 1 − δ(1 − q) (2δ + 1) + 1 − δ(1 − q) = 0,
the roots of which are πm(δ) (q) and πM (δ) (q).
Finally, cases B1 and B2 are determined by (10). That equation simplifies to π = π̌(q) :=
(1 − δq)h(q)/[(1 + δ) − 2δqh(q)].
Proof of Proposition 1
It is immediate from comparing UaI and UbI that deferring persuasion is never optimal. Fur√
thermore, since maxπ {UbI − UcI } = 21 + d − 1 + d, multi-stage persuasion is never optimal for
√
δ ≤ 23 .
As π → 1, we have UbI → 2δ (1 − δ) + 21 and UcI → 2δ1 = 1−δ
+ 12 > 2δ (1 − δ) + 12 . Similarly,
2δ
for π = π := 2δ1 , UcI = 1 and UbI = 2δ [ 32 − π] + 21 < 1. So clearly early persuasion is optimal
for very high or very low π. Since UcI is decreasing and convex in π and UbI is decreasing and
√
linear in π, then for δ > 23 , the difference between the two will shift sign at two values: π ∗
√
√
4δ 2 −3
4δ 2 −3
and π ∗∗ . We can solve to get π ∗ = 1+2δ−
and π ∗∗ = 1+2δ+
. Checking the limit as
2δ(1−δ)
2δ(1−δ)
δ → 1 gives the final part of the proposition.
Proof of Proposition 2
Multi-stage persuasion is never optimal. This is the case as UbII ≥ UaII if and only if µ ≥
1
1 − δ(2−q)
, but whenever this inequality holds, UcII ≥ UbII .
1
Furthermore, since maxq {UaII −UcII } = 2δ + 41 − 12 ( 2δ−1
), multi-stage persuasion is never optimal
√
for δ ≤ 23 .
26
δ > δ implies that q ∈ (q, 1], where q ≡ 1−δ
. As q → 1, we have (UcII − UaII ) → ( 2δ1 − 2δ ) > 0.
δ
Similarly, as q → q, UcII = 1 and UbII = 3δ2 − 21 ≤ 1. So clearly early persuasion is optimal
for very high or very low q. Since UcII is decreasing and convex in q and UaII is decreasing
√
and linear in q, then for δ > 23 , the difference between the two will shift sign at at most two
√
√
4δ 2 −3
1+2δ+ 4δ 2 −3
∗∗
values: q ∗ and q ∗∗ . We can solve to get q ∗ = 1+2δ−
and
q
=
. Checking the
2δ(1−δ)
2δ(1−δ)
limit as δ → 1 gives the final part of the proposition.
27
References
Aumann, R. J. and Hart, S. (2003) Long cheap talk, Econometrica, pp. 1619–1660.
Aumann, R. J., Maschler, M. and Stearns, R. E. (1995) Repeated games with incomplete
information, MIT press.
Bergemann, D. and Wambach, A. (2014) Sequential information disclosure in auctions, Journal of Economic Theory, 1, 1–22.
Brocas, I. and Carrillo, J. D. (2007) Influence through ignorance, The RAND Journal of
Economics, 38, 931–947.
Che, Y.-K. and Hörner, J. (2013) Optimal design for social learning, Tech. rep.
Ely, J. (2015) Beeps, Mimeo.
Forges, F. and Koessler, F. (2008) Long persuasion games, Journal of Economic Theory, 143,
1–35.
Golosov, M., Skreta, V., Tsyvinski, A. and Wilson, A. (2014) Dynamic strategic information
transmission, Journal of Economic Theory, 151, 304–341.
Goltsman, M., Hörner, J., Pavlov, G. and Squintani, F. (2009) Mediation, arbitration and
negotiation, Journal of Economic Theory, 144, 1397–1420.
Hörner, J. and Skrzypacz, A. (2014) Selling information, Mimeo.
Kamenica, E. and Gentzkow, M. (2011) Bayesian Persuasion, American Economic Review,
101, 2590–2615.
Kolotilin, A. (2015) Experimental design to persuade, Games and Economic Behavior, 90,
215–226.
Kremer, I., Mansour, Y. and Perry, M. (2014) Implementing the wisdom of the crowd, Journal
of Political Economy, 122, 988–1012.
Krishna, V. and Morgan, J. (2004) The art of conversation: eliciting information from experts
through multi-stage communication, Journal of Economic theory, 117, 147–179.
28
Rayo, L. and Segal, I. (2010) Optimal information disclosure, Journal of political Economy,
118, 949–987.
Renault, J., Solan, E. and Vieille, N. (2014) Optimal dynamic information provision, Mimeo.
29