Discussion Paper No. 1006

MULTISTAGE
INFORMATION TRANSMISSIONWITH VOLUNTARYMONETARY TRANSFER

Hitoshi Sadakane

Secondary Revised January 2019
Revised June 2018
June 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Multistage Information Transmission with
Voluntary Monetary Transfer∗
Hitoshi Sadakane†
Institute of Economic Research, Kyoto University
November 1, 2018
Abstract
We analyze a cheap talk model in which an informed sender andan uninformed receiver engage in finite-period communication before
the receiver chooses a project. During the communication phase, ineach period, the sender sends a cheap talk message and the receivervoluntarily pays money for the message she receives. Our results
show that combining multistage information transmission with the receiver’s voluntary payments can improve welfare. Moreover, we findan upper bound of the receiver’s equilibrium payoﬀ and provide a
suﬃcient condition for it to be approximated by the receiver’s equilibrium payoﬀ. This result shows that multistage information transmission with voluntary monetary transfer can be more beneficial forthe receiver than a wide class of other communication protocols (e.g.,
mediation and arbitration).
JEL Classification: C72; C73; D82; D83
Keywords: Incomplete information; Cheap talk; Multistage strategic communication;
Voluntary monetary transfer∗

This paper is based on the second chapter of my Ph.D. dissertation at Kobe University.
An earlier version of this paper (ISER Discussion Paper No. 1006) was awarded the 19th
ISER-Moriguchi Prize. I am grateful to Yasuyuki Miyahara for his invaluable support. I
also thank Hideo Suehiro, Hideshi Itoh, Hisao Hisamoto, Kohei Kawamura, Shinsuke Kambe,
Shintaro Miura, Takashi Shimizu, Takeshi Murooka, and seminar participants at the Contract
Theory Workshop, Contract Theory Workshop summer camp, KIER, and ISER for their helpfulcomments and suggestions. I gratefully acknowledge financial support from the Grant-in-Aid
for Young Scientists (Start-up) (17H06778). Any remaining errors are my own.
†
Institute of Economic Research, Kyoto University, Sakyo-ku, Kyoto 6068501, Japan
E-mail address: sadakane@kier.kyoto-u.ac.jp

1. Introduction
A lack of information typically leads to ineﬃcient decisions. Therefore, in many
economic situations, decision makers need to gather the relevant information
before making their decisions. One canonical way of gathering informationis consulting informed experts. For example, CEOs consult management consultants; politicians seek advice from strategic planners; and law enforcementoﬃcers hire informants. In the abovementioned examples, the individuals who
supply information are often paid for doing so.
Contract theory indicates that a properly designed contract containing information contingent payments helps the decision maker to screen the information possessed by the informed expert. However, if information is transmitted
through ordinary and informal talk, or equivalently, through “cheap talk,” contractibility does not always exist. In such situations, the decision maker cannot
commit to information-contingent payments. Hence, it seems that allowing
the decision maker to make “voluntary” payments does not aﬀect informationtransmission. Nevertheless, the information transmitted via cheap talk is oftenbought and sold without signing a contract.
Can voluntary payments by the decision maker facilitate cheap talk communication? If they can, how should the decision maker pay for cheap talkmessages? To address this question, we enrich the canonical cheap talk model
originally provided by Crawford and Sobel (1982) (hereafter, CS). Specifically,
we analyze a sender–receiver game in which an informed expert (sender orhe) and an uninformed decision maker (receiver or she) engage in finite-period
communication. During the communication phase, in each period, the sendersends a cheap talk message to the receiver, and then the receiver pays money
to the sender voluntarily. Once the communication phase is over, the receiverchooses a project.
In the CS model, the project choice and underlying asymmetric information
are one-dimensional. Moreover, the sender’s most desirable project is always
higher than that of the receiver to a certain degree. Hence, the sender has an1incentive to cheat the receiver into choosing a higher project than the receiver’s
most profitable one. This fact prevents detailed information transmission. By
contrast, if the receiver can make message-contingent payments, by paying
more money for the messages inducing the lower projects, the receiver canweaken the sender’s exaggeration incentive. However, when the informationtransmission is one shot, the receiver never pays since making payments after
receiving a message is a waste of money. In the present study, we consider ascenario in which information is conveyed in a gradual fashion and show that bycombining multistage information transmission with the receiver’s voluntary
payments, a message-contingent payment scheme can be self-enforcing.1 As aresult, information transmission can be improved even in situations in which
there is no contractibility.
We find that under some conditions (i) the receiver can obtain more detailed
information from the sender than in the CS model2 and (ii) an equilibrium whose
outcome Pareto-dominates all the equilibrium outcomes in the CS model can
exist.3 We also show that no fully separating equilibrium exists in our model.
This result implies that information transmission is still limited even in thecommunication procedure that we describe. By considering the well-known
uniform-quadratic model, i.e., with quadratic preferences regarding the project
and a uniform type distribution, we find an upper bound of the receiver’s
equilibrium payoﬀ and provide a suﬃcient condition for it to be approximatedby the receiver’s payoﬀ under an equilibrium.
To demonstrate the benefit of multistage information transmission with
1

Without monetary transfer, allowing multiple rounds of unilateral (one-sided) communication in the CS model does not aﬀect the set of equilibria identified by the original model.
Krishna and Morgan (2004) show that allowing multiple rounds of “bilateral” (face-to-face or
two-sided) communication in the CS model leads to Pareto improvements.
2
This result means that there exists an equilibrium whose partition has a greater number of
elements than that achieved in any equilibrium in the CS model.
3
In our model, there always exists an equilibrium in which the receiver never pays money
to the sender. For instance, irrespective of the number of periods in the communication phase,
there exists an equilibrium in which the sender sends an informative message to the receiver
only in the first period and the receiver never pays. The equilibrium partition achieved insuch an equilibrium is achievable in the CS model. Obviously, players waste time on pointlesscommunication; in other words, the receiver does not use long-term communication eﬀectively.
Therefore, by constructing equilibria inducing Pareto improvements, we show the benefit ofmultistage information transmission.

2

voluntary transfer payments, we construct an interval partition equilibrium inwhich information about the state of the world is conveyed in order from theright-most interval on the state space. Specifically, in the first period, if thesender sends a message that means that the true state belongs to the right-mostinterval, the receiver will neither pay money nor obtain additional informationin the future. Otherwise, the receiver pays a certain amount of money to thesender. After this payment, in the second period, the sender conveys whether
the true state belongs to the second right-most interval that is the neighbor to
the left of the first one. If the receiver learns that the true state belongs to thesecond right-most interval, she will neither pay money nor obtain additional
information in the future. Otherwise, the receiver pays money to the senderand then the sender conveys additional information in the next period. This
information elicitation is repeated in the communication phase. If the receiverdeviates in terms of payment in a period, the sender conveys no information
thereafter. Once communication is over, the receiver chooses her best projectbased on the information she has.
The logic underlying this equilibrium is as follows. First, under the information elicitation explained above, the receiver pays money to the senderwhenever the information opposite to the sender’s bias is conveyed. As a result,
the receiver makes message-contingent payments on the equilibrium path: ahigher payment for information inducing a lower project. As noted earlier, thispayment scheme weakens the sender’s exaggeration incentive. Second, since
the sender can gradually convey his information, he can punish the receiverfor not paying by babbling. Thus, the receiver makes a payment in the current period to prevent the sender’s babbling in the future. Roughly speaking,similar to Benoit and Krishna (1985), the dependence of the selection of the
future equilibrium on players’ past behavior constructs punishments for theirdeviation. This fact enables the receiver to make message-contingent payments
to some extent during the communication phase.
Our model is potentially applicable for studying the eﬀective use of informants. The Federal Bureau of Investigation (FBI) mentions that the “use of3

informants to assist in the investigation of criminal activity may involve anelement of deception, ... or cooperation with persons whose reliability andmotivation may be open to question.“4 This statement suggests that informants
are often biased and that their information might neither be credible nor certifiable. Alemany (2002) indicates that cooperation agreements between the Drug
Enforcement Agency (DEA) and informants are often silent with respect to thecompensation of the latter. This fact implies that the parties may not always beable to sign a contract containing information-contingent payments. Indeed,
there are numerous cases of oral promises made by DEA agents to informants
subsequently being broken.5 The present study shows that by using multistage
information elicitation and voluntary transfer payments, information transmission can be improved even in situations in which there is no contractibility.
Our results have important implications for the theory of organizational
economics regarding designing communication protocols and organizational
structures. We show that multistage information transmission with voluntarytransfer payments can be more beneficial for the receiver than a wide range ofother communication protocols. It is well known that information transmission can be improved when more general communication protocols (i.e., noisy
communication) are available.6 By considering a mediation model under theuniform-quadratic assumption,7 Goltsman et al. (2009) characterize the optimallevel of noise in the communication. We compare our communication procedure with the optimal mediation that maximizes the receiver’s ex ante expected
payoﬀ and show that under some conditions, the receiver prefers the former tothe latter.
4

FBI, Frequently Asked Questions, “What is the FBI’s policy on the use of informants?”(https://www.fbi.gov/about/faqs/what-is-the-fbis-policy-on-the-use-of-informants).
5
For details, see Alemany (2002).
6
Many studies highlight that noisy communication leads to improved information transmission (e.g., Krishna and Morgan, 2004; Blume et al., 2007; Goltsman et al., 2009; Ivanov, 2010;
and Ambrus et al., 2013). Goltsman et al. (2009) characterize the optimal mediation mechanismthat controls the noise in communication. Blume et al. (2007) and Krishna and Morgan (2004)
show that the optimal mediation mechanism can be implemented under some communication
protocols without monetary transfer.
7
Under mediation analyzed by Goltsman et al. (2009), a neutral third party (mediator) asks
the sender for information and advises the receiver who chooses a project.

4

Dessein (2002) studies a simple delegation problem8 and establishes the remarkable result that the receiver prefers full delegation to communication aslong as the incentive conflict is not too large. Since the work of Dessein (2002),
designing “who decides what” has been extensively studied. Many works
investigate general settings in which the parties can commit to an informationcontingent decision rule.9 Under the uniform-quadratic assumption, Goltsman
et al. (2009) characterize an optimal information-contingent decision rule, theoptimal arbitration.10 Although, under arbitration, players benefit from a “formal
contract” that forces them to commit to the predetermined decision rule, surprisingly, our results show that the receiver can obtain a higher ex ante expected
payoﬀ in our communication procedure than under the optimal arbitration.
Related Literature

A seminal analysis of the strategic information transmis-

sion between an informed sender and an uninformed receiver was provided
by CS. In the CS model, the sender sends a costless and unverifiable11 message
about his private information to the receiver, who then decides on the projectthat aﬀects the payoﬀs of both players. CS obtain a complete characterization of
the set of equilibria in their model and show that the existence of the incentive
conflict prevents the full revelation of information. In the present study, weinvestigate how information transmission can be improved under multistage
information transmission with voluntary monetary transfers.
Krishna and Morgan (2008) study an amendment to the CS model by allow8

The receiver chooses whether to communicate with the sender. She decides herself after
cheap talk communication or fully delegates the decision-making authority to the sender.
9
One simple decision rule for the receiver is to delegate authority to the sender, but possibly toconstrain the set of available decisions. This class of mechanisms (analyzed by Holmström, 1977;
Melumad and Shibano, 1991; and Alonso and Matouschek, 2008) is called delegation mechanism.
Goltsman et al. (2009) show that the optimal arbitration mechanism is deterministic as a consequence and that the optimal arbitration includes the optimal delegation mechanism.
10
Under arbitration, a neutral third party (arbitrator) asks the sender for information andchooses a project according to a predetermined potentially stochastic decision rule.
11
Seidmann and Winter (1997) and Mathis (2008) study the sender–receiver game in which themessage sent by the sender is (partially) verifiable, that is, the set of available messages depends
on the sender’s type. These authors provide the suﬃcient conditions (Mathis (2008) provides thenecessary and suﬃcient conditions) for the existence of a fully revealing equilibrium. Forges
and Koessler (2008) study a multistage sender–receiver game with certifiable messages andgeometrically characterize the set of equilibrium payoﬀs.

5

ing the parties to write a contract containing message-contingent payments.
They show that full information revelation is feasible but not optimal and they
characterize the optimal contract. In their model, there is a crucial assumptionthat the receiver can commit herself to compensate the sender for his message. We show that when the communication phase has multiple periods, thereceiver can control the sender’s incentive through voluntary payments eventhough there is no contractibility.
Our results are closely related to those of Krishna and Morgan (2004). Boththeir study and our analysis investigate how information transmission canbe improved through the receiver’s active participation in the communication
process. Krishna and Morgan (2004) add a long communication protocol to
the CS model.12 They show that if bilateral (face-to-face) communication between the receiver and sender is possible before the sender sends a message
about his private information to the receiver, there exists an equilibrium whose
outcome Pareto-dominates all the equilibrium outcomes in the CS model. The
key factor to their results is that after the sender conveys some informationin the face-to-face communication, multiple equilibria exist in the remaining
game. The outcome of this face-to-face communication, which could be random, determines which of these equilibria is played in the future. This aﬀects
what the sender conveys during the face-to-face communication. Therefore, in
Krishna and Morgan (2004), the receiver tries to control the sender’s incentive
by controlling the degree of uncertainty associated with the outcome of theface-to-face communication. By contrast, in our model, the receiver tries tocontrol the sender’s incentive directly through voluntary transfer payments.
Spence (1973) shows that costly signaling helps people convey their private
information credibly. In the framework of the CS model, Austen-Smith andBanks (2000), Kartik (2007), and Karamychev and Visser (2016) show that in12

Aumann and Hart (2003) study a finite simultaneous-move (long conversation) game inwhich there are two players, one being better informed than the other. They provide a complete
geometrical characterization of the set of equilibrium payoﬀs when the state of the world isfinite and long communication is possible. In this study, the state space and players’ action
space must be finite. Therefore, we cannot directly apply the results of Aumann and Hart (2003)
to the model in the present paper.

6

formation transmission can be improved when the sender can send a costly
message (money burning, or equivalently, paying money to the receiver) tosignal information.13 In their settings, a fully separating equilibrium that is optimal from the receiver’s perspective can exist. However, in the equilibrium thatmaximizes the sender’s ex ante expected payoﬀ, the sender does not pay money
to separate an interval of states. Karamychev and Visser (2016) show that in
the sender’s optimal equilibrium, he pays to adjust the pooling intervals. In thepresent study, we focus on the situation in which the sender cannot pay money(or equivalently, cannot send a costly signal) to the receiver and show that thesignaling structure can be endogenously generated by the receiver’s voluntary
payment. Moreover, Section 4.4 shows that under the uniform-quadratic assumption, the receiver can obtain the higher ex ante expected payoﬀ than that
under the sender’s optimal equilibrium in the model analyzed in Karamychev
and Visser (2016). This result suggests that in some cases, it might be betterfor the receiver to generate the signaling structure by herself through voluntary
payments rather than to rely on the sender’s costly signaling.
Hörner and Skrzypacz (2016) study a model of gradual persuasion in which
the sender is paid and gradually reveals “certifiable” information. They showthat the sequential revelation of partially informative signals can increase payments to the sender who is trying to sell his information to the receiver. In
their model, cheap talk communication is of no help. In the present paper, weshow that gradual information transmission can perform well with voluntarytransfer payments even in the situation in which the sender can send only cheap
talk messages.
In all the abovementioned studies, once the communication phase is over,
the receiver chooses a project; that is, the project choice is once and for all.
By contrast, in the studies mentioned hereafter, there are multiple rounds ofcommunication and actions. More precisely, in each period, the sender sends
a message and the receiver chooses a project. Hence, these models diﬀer from
13

Relatedly, Kartik et al. (2007) and Kartik (2009) study amendments to the CS model withother means of costly signals such as lying costs.

7

mine.
Golosov et al. (2014) study strategic information transmission in a finitely
repeated cheap talk game. Only the sender knows the state of the world,which remains constant through out the game. They show that the sender cancondition his message on the receiver’s past actions; in addition, the receivercan choose actions that reward the sender for following a path of messages thateventually leads to the full revelation of information. In contrast to this result,
there is no fully revealing equilibrium in our model.
Kolotilin and Li (2017) investigate the optimal relational contracts in an
infinitely repeated cheap talk game. In their model, both the sender and receivercan pay each other. Therefore, there are equilibria in which the sender always
reveals his private information completely. They show that full separation
can be attained in the equilibrium, whereas partial or complete pooling is
optimal if preferences are divergent. In contrast to our study, the sender’sprivate information is not persistent in their model. Hence, gradual informationtransmission does not appear.
Paper Outline

The rest of the paper is organized as follows. Section 2 in-troduces the model. Section 3 derives the general properties of the perfect
Bayesian equilibria in the model. Section 4 analyzes the uniform-quadraticmodel and shows the benefits of multistage information transmission withvoluntary monetary transfers. Section 4.1 shows the two main results by constructing an equilibrium in which information is transmitted within two periods. Section 4.2 shows the benefit of long-term communication. Section 4.3
discusses the implications for organization design. In Section 4.4, we compare
our communication procedure with the sender’s optimal signaling. Section 5
generalizes the players’ payoﬀ functions and prior probability distribution, anddescribes two results that correspond to the results in Section 4.1. Section 6 gives
some concluding remarks.

82. Model
There are two players, a sender (S) and a receiver (R). R has the authority tochoose a project y ∈ Y ≡ R+ , but the outcome produced by project y depends
on S’s private information, θ ∈ Θ ≡ [0, 1], which is distributed according to adiﬀerentiable distribution function G with density g.
Before R chooses a project, R and S engage in T-period communication. Eachperiod consists of two stages, stage 1 and stage 2. At stage 1, S sends a costlessand unverifiable message to R. Let M ≡ [0, 1] be S’s message space. A message
sent by S at stage 1 in period t is denoted by mt . At stage 2, R voluntarily pays
money to S. Let wt ∈ W ≡ R+ be a payment amount R pays at stage 2 in periodt. After T-period communication, the game proceeds to period T + 1, in which
R chooses a project.
Let w be a sequence of transfers, w ≡ (w1 , . . . , wT ) ∈ W T . The players’ payoﬀ
functions UR : Y × Θ × W T → R and US : Y × Θ × W T → R are defined as follows:
U (y, θ, w) ≡ r · u (y, θ) −
R

R

T
∑

wt

t=1

U (y, θ, w) ≡ s · u (y, θ, b) +
S

S

T
∑

wt

t=1

where r, s, and b are positive constants. The term∑T

t=1 wt represents the total

amount of payments.
Here, r · uR (y, θ) and s · uS (y, θ, b) denote utilities from project y for R and S,
respectively. The functions uR and uS satisfy CS’s assumptions:• uR (y, θ) ≡ uS (y, θ, 0);• uS is twice-continuously diﬀerentiable in y, θ, and b for all y ∈ R+ , θ ∈ Θ,and b ∈ R+ ;• for all θ ∈ Θ and b ∈ R+ , there exists y ∈ R+ such that uS1 (y, θ, b) ≡∂uS(y, θ, b) = 0; and∂y∂ u• uS11 (y, θ, b) ≡ ∂∂yu2 (y, θ, b) < 0, uS12 (y, θ, b) ≡ ∂y∂θ(y, θ, b) > 0, and uS13 (y, θ, b) ≡
2 S

2 S

9

∂2 uS(y, θ, b) > 0 for all y ∈ R+ , θ ∈ Θ, and b ∈ R+ .
∂y∂b

Under these assumptions, for each given (θ, b), there exists a unique maximizing
project: yR (θ) = arg max y uR (y, θ) and yS (θ, b) = arg max y uS (y, θ, b). Parameterb > 0 represents “bias,” which measures how much S’s interest diﬀers from R’s.
Since uS13 (y, θ, b) > 0 and b > 0, we obtain yR (θ) < yS (θ, b). Constants r > 0 ands > 0 are scalar parameters that measure the relative importance of the projectchoice versus transfer payments.
The timing of game is summarized as follows:
1. Before the game starts, nature randomly draws a state θ ∈ Θ with common
prior G, and S observes θ privately.
2. R and S engage in T-period communication.
• At stage 1 in period t, S sends a message mt to R,• At stage 2 in period t, R voluntarily pays wt to S.
3. After T-period communication, R chooses a project y and the game ends.
Hereafter, Γ(b, s, r, T) denotes this T-period communication game.

2.1. History and Strategies
A (public) history h(t, j) is defined as a sequence of players’ past actions realized
until the beginning of stage j in period t.





(m1 , w1 , . . . , mt−1 , wt−1 ) if j = 1,
(t, j)h ≡(m1 , w1 , . . . , mt−1 , wt−1 , mt ) if j = 2.
A (public) history hT+1 is defined as a sequence of players’ past actions realized
until the beginning of period T + 1, in which R chooses a project.
hT+1 ≡ (m1 , w1 , . . . , mT , wT ).

10

Let H(t, j) and HT+1 be the set of h(t, j) and hT+1 , respectively. We assume that
H(1,1) is a singleton set {ϕ}. The set of all histories at stage j is denoted by∪(t,1)
H j ≡ Tt=1 H(t, j) . Let h(t,1)∈ Θ × H(t,1) ≡ HΘbe S’s private history at stage 1 inθperiod t. Let HΘ1 be the set of all private histories of S: HΘ1 ≡ Θ × H 1 .
S’s behavior strategy σ specifies a probability distribution of messages that
S of type θ sends at stage 1 in period t, σ : HΘ1 → ∆M.14 R’s pure strategy is afunction ρ : H 2 ∪HT+1 → R+ ,15 which specifies the payment amount and project.
Note that ρ(h(t,2) ) ∈ W, and ρ(hT+1 ) ∈ Y.16 A belief system, f : H 2 ∪ HT+1 → ∆Θ,
specifies R’s belief about S’s types at history h ∈ H 2 ∪ HT+1 .

3. Equilibrium
We analyze (weak) perfect Bayesian equilibria17 : both players’ strategies must
maximize their expected payoﬀs after all histories, and the system of beliefs
f must be consistent with the regular conditional probability derived from((σ, ρ), f ) and G. The formal definition of perfect Bayesian equilibria can befound in Appendix 3.A. Hereafter, we call a perfect Bayesian equilibrium simply
equilibrium. In this section, we derives the general properties of the equilibria.

3.1. Relationship to the CS Model
We discuss the relationship between the equilibria in the CS model and those inΓ(b, s, r, T). Since R cannot obtain additional information about θ after stage 2
in period T, she has no incentive to choose wT > 0. Therefore, wT must be
equal to 0 in any equilibrium. Consequently, Γ(b, s, r, 1) is essentially equivalent
Let B(X) be the Borel algebra on a set X. S’s behavior strategy σ is defined as {σt }Tt=1 , where(t,1)
σt : B(M) × HΘ → [0, 1] holds the following two properties: (1) for every M̃ ∈ B(M), a function
(t,1)
(t,1)
(t,1)
(t,1)
σt (M̃, ·) : HΘ → [0, 1] is (B(HΘ ), B([0, 1]))-measurable, (2) for every hθ ∈ HΘ , function(t,1)σt (·, hθ ) : B(M) → [0, 1] is a probability measure. The definition of σ originates from Milgrom
and Weber (1985).
15
More precisely, ρ is defined as {ρt }T+1
such that ρt : Ht → R+ is (B(Ht ), B(R+ ))-measurable.
t=1
Note that Ht = H(t,2) for t ≤ T, and that Ht = HT+1 for t = T + 1.
16
Due to the strict concavity of R’s preference over projects, she never mixes projects inperiod T + 1.
17
There always exists an equilibrium that is essentially equivalent to a perfect Bayesianequilibrium in the CS model. Hence, in this study, we do not prove the existence theorem.
14

11

to the CS model, and we call it the one-shot cheap talk game. CS have shownthat in the one-shot cheap talk game, for every b > 0, there exists a positive
integer ñ(b) such that, for every n ∈ {1, . . . , ñ(b)}, there exists at least one equilibrium with an n-element partition: {[an , an−1 ), [an−1 , an−2 ), . . . , [a1 , a0 ]}. In thisequilibrium, S’s type θ ∈ [ai+1 , ai ) conveys that his type belongs to this interval, and after receiving the message that “θ belongs to [ai+1 , ai ),” R chooses the∫ aiproject y(ai+1 , ai ) = arg max y a uR (y, θ)g(θ)dθ. We define y(ai+1 , ai ) = yR (a) for
i+1

ai+1 = ai = a. Since u is strictly concave, y(ai+1 , ai ) is uniquely determined.
R

Moreover, since uR12 (y, θ) > 0, y(ai+1 , ai ) is strictly increasing in both of its arguments. S whose type falls on a boundary between adjacent intervals must be
indiﬀerent between the associated values of y. Therefore, we have the following
conditions.18 For i = 1, . . . , n − 1,s · uS (y(ai+1 , ai ), ai , b) − s · uS (y(ai , ai−1 ), ai , b) = 0;(1)an = 0;(2)

a0 = 1.

(3)

We call a sequence a ≡ {a0 , . . . , an } a (backward) solution of (1) if a satisfies(1)–(3). We impose the following monotonicity condition on a solution of (1).
Condition M . If a′ and a′′ are two solutions of (1) with a′0 = a′′0 and a′1 > a′′1 ,then a′i > a′′i for all i ≥ 2.
This condition is met by standard versions of the CS model, such as theuniform-quadratic case: s · uS (y, θ, b) ≡ −s(y − (θ + b))2 , r · uR (y, θ) ≡ −r(y − θ)2 ,and G(θ) is uniform distribution over [0, 1]. CS show that Condition M also
holds for more general specifications.
Consider a strategy profile such that S sends an informative message onlyin period 1 and R pays nothing to S at any payment stage. If both S’s strategy
regarding m1 and R’s strategy regarding project choice are the same as anequilibrium in the CS model, then this strategy profile constitutes an equilibriumin Γ(b, s, r, T). This outcome immediately yields the following Fact 1.
18

See the condition (A) on page 1437 in CS (1982).

12Fact 1. Any equilibrium partition achieved in the CS model can be achieved under an
equilibrium in Γ(b, s, r, T).

3.2. Relationship to Direct Contract
In this subsection, we first discuss the relationship between equilibria in Γ(b, s, r, T)
and those in a case in which R can sign a contract that specifies the transfer andproject as functions of messages sent by S.
Fix an equilibrium ξ = ((σ, ρ), f ). Let µξ : Θ → ∆(MT ) be a probability
distribution over MT derived from (σ, ρ). Given ρ and a sequence of messagesm ∈ MT , a sequence of payments w ∈ W T and a project y are determined. Let ωξ :MT → W T and yξ : MT → Y be the functions derived from ρ, respectively. Notethat fξ denotes the belief system that is consistent with the regular conditional
probability derived from µξ and G.19
Consider the case in which R can write an indirect contract (MT , ωξ , yξ ) thatspecifies w and y dependently on m. Obviously, sending m ∈ supp{µξ (θ)}
is optimal for S of type θ, and yξ (m) maximizes R’s expected payoﬀ underfξ (·|m). Furthermore, the assessment (µξ , (MT , ωξ , yξ )) and the given equilibriumξ induce a same probability distribution on W T × Y for any θ.
Next, we discuss the relationship between (Bayesian Nash) equilibria under an indirect contract (MT , ωξ , yξ ) and those under a direct contract in which
R can sign a contract that specifies transfers and projects as functions of thedirect message µ : Θ → Θ. Let (Θ, ωΘ , yΘ ) be a direct contract under which
R pays ωΘ (µ) and chooses yθ (µ) as S reports µ ∈ Θ. According to the standard revelation principle for Bayesian games, when R can commit to both thepayment scheme ωξ (m) = (w1 (m), . . . , wT (m)) and a project yξ (m), there is a
Precisely, µξ : B(MT ) × Θ → [0, 1] is a function with the properties such that µξ (M̃T , ·) :
Θ → [0, 1] is (B(Θ), B([0, 1]))-measurable for any M̃T ∈ B(MT ); and µξ (·, θ) : B(MT ) → [0, 1]T
is a probability measure. Hence, the∫ joint probability P(Θ̃, M̃ ) that defines the simultaneous behavior of (θ, m) is given by θ∈Θ̃ µξ (M̃T , θ)G(dθ). Fix Θ̃ ∈ B(Θ), then the probability
measure P(Θ̃, ·) : B(MT ) → [0, 1] on (MT , B(MT )) is absolutely continuous with respect toP(Θ, ·) : B(MT ) → [0, 1]. Therefore, R’s posterior
belief Prob(Θ̃|m) is defined as a Radon∫
Nikodym derivative fξ such that P(Θ̃, M̃T ) = m∈M̃T fξ (Θ̃|m)P(Θ, dm) for any Θ̃ ∈ B(Θ) andM̃T ∈ B(MT ).
19

13direct contract (Θ, ωΘ , yΘ ) under which there exists a truth telling equilibrium∑such that µ(θ) = θ, ωΘ (θ) = Tt=1 wt (m), and yΘ (θ) = yξ (m) for any θ ∈ Θand m ∈ suppµξ (·|θ). Because of the definition of (µξ , ωξ , yξ ), we immediately
obtain the following Proposition 1. This result shows that the outcome of ξ canbe replicated by a direct contract (Θ, ωΘ , yΘ ).
Proposition 1. Fix an equilibrium ξ in Γ(b, s, r, T). Then, there is a direct contract(Θ, ωΘ , yΘ ) under which a truth telling equilibrium that is outcome equivalent to ξ
exists.

3.3. Partition Equilibrium
As is the case in the CS model, all the equilibria in Γ(b, s, r, T) are interval
partitional, that is, all the equilibria are partition equilibria.
Definition 1 (Partition Equilibrium). Fix an equilibrium ξ in Γ(b, s, r, T). Consider a truth telling equilibrium under a direct contract (Θ, ωΘ , yΘ ) which isoutcome equivalent to ξ. If there exists a family of sets {Iλ }λ∈Λ over Θ such that
1. {Iλ }λ∈Λ constitutes an interval partition20 over Θ;
2. yΘ (θ) = yΘ (θ′ ) and ωΘ (θ) = ωΘ (θ′ ) for all θ, θ′ ∈ Iλ ; and3. yΘ (θ) , yΘ (θ′ ) for all θ ∈ Iλ and θ′ ∈ Iλ′ ,λ ; thenwe call ξ partition equilibrium, and {Iλ }λ∈Λ equilibrium partition.
First, we show the following Proposition 2.
Proposition 2. All equilibria under a direct contract are partition equilibria.
The proof is in Appendix 3.B. As shown in Subsection 3.2, any equilibriumoutcome in Γ(b, s, r, T) is also achieved in equilibrium under a corresponding
direct contract. Therefore, Proposition 2 means that all equilibria in Γ(b, s, r, T)
are partition equilibria.
Corollary 1. All equilibria in Γ(b, s, r, T) are partition equilibria.
20

For all λ , λ′ , Iλ ∩ Iλ′ = ∅. For all λ ∈ Λ, Iλ is convex, and14∪

λ∈Λ Iλ = Θ.

Fix an equilibrium ξ, then we have an equilibrium partition {Iλ }λ∈Λ . For anyθ ∈ Iλ ≡ [aλ , aλ−1 ] and any m ∈ supp{µξ (θ)}, we obtain fξ (θ|m) = g(θ)/[G(aλ−1 )−G(aλ )]. Hence, it must be satisfied that for any θ ∈ Iλ ≡ [aλ , aλ−1 ],yΘ (θ) = y(aλ , aλ−1 ) ≡ arg maxy∈R∫ aλ−1 [aλ]g(θ)
uR (y, θ) dθ.
G(aλ−1 ) − G(aλ )The following Proposition 3 shows that there is no fully separating equilibrium in Γ(b, s, r, T).
Proposition 3. There exists no fully separating equilibrium in Γ(b, s, r, T).
The proof is in Appendix 3.C. If R can commit herself to compensating for
S’s message, fully separating equilibria (full revelation contracts) are always
feasible. However, in our model, since there is neither commitment nor contractibility, R pays money to S only when paying money is optimal for her. For
S’s truth telling (µ(θ) = θ) to be optimal, the total amounts of monetary transfers
must be diﬀerent for each θ ∈ Θ. Precisely, ωΘ (θ) must be strictly decreasingin θ ∈ Θ. This means that if the given R’s payment strategy leads to S’s truth
telling, R almost certainly reaches a history where she pays a certain amountof money to S even though she has already detected the true state. At such ahistory, R has no incentive to pay. For this reason, there is no fully separating
equilibrium.
Whether the cardinality of the equilibrium partition is finite remains an open
question. Next, we provide a suﬃcient condition (Assumptions 1 and 2) for the
cardinality of the equilibrium partition to be finite.
Assumption 1. S’s utility function uS satisfiesuS (y, θ, b) = ψ(|y − θ − b|),where ψ′′ (·) < 0 and ψ′ (0) = 0.
Assumption 2. The distribution G and R’s utility function uR jointly satisfy: for any

15

colsed interval [a, a) with 0 ≤ a ≤ a ≤ 1,∫ a[y(a, a) = arg max
y∈R

a

]
a+a
g(θ)
R
u (y, θ) dθ <+ b.
2
G(a) − G(a)(4)

Assumption 2 is mild. For example, suppose that uR (y, θ, b) = l(|y−θ|), wherel′′ (·) < 0 and l′ (0) = 0, and that G is non increasing. Then, the inequality (4)holds.
Proposition 4. Under Assumption 1 and 2, in any equilibrium, the equilibrium
partition has a finite number of elements.
The proof is in Appendix 3.D. Proposition 4 shows that under Assumptions 1
and 2, the equilibrium partition is a finite set. In Appendix 4.E, we discuss thefact that an equilibrium which has separating intervals in its partition might
exist if Assumption 2 is not satisfied.
Hereafter, [aλ , aλ−1 ) denotes Iλ , and ωλ denotes ωΘ (θ) for θ ∈ [aλ , aλ−1 ). In
any equilibrium, there must exist λ̃ ∈ Λ such that ωλ̃+1 ≤ ωλ̃ .21 From S’s
incentive compatibility condition,
ψ(|y(aλ̃+1 , aλ̃ ) − aλ̃ − b|) ≥ ψ(|y(aλ̃ , aλ̃−1 ) − aλ̃ − b|).

(5)

Figure 1 illustrates the inequality (5). The blue curve is ψ(|y(aλ̃+1 , aλ̃ ) − θ − b|),and the red curve is ψ(|y(aλ̃ , aλ̃−1 ) − θ − b|). Note that yλ̃+1 = y(aλ̃+1 , aλ̃ ); yλ̃ =y(aλ̃ , aλ̃−1 ); ψλ̃+1 = ψ(|yλ̃+1 − aλ̃ − b|); and ψλ̃ = ψ(|yλ̃ − aλ̃ − b|).
Since y(aλ̃+1 , aλ̃ ) < aλ̃ , the left-hand side of the inequality (5) is less than ψ(b).
Moreover, from Assumption 2, the right-hand side of the inequality (5) is higher
than ψ([aλ̃−1 − aλ̃ ]/2). Therefore, we must have aλ̃−1 − aλ̃ > 2b irrespective of thelength of the communication phase.
This result implies that in any equilibrium, at history hT+1 where R believes
that θ ∈ [aλ̃ , aλ̃−1 ), R’s conditional expected utility from project is strictly less21

Suppose that this condition does not hold. If the true state belongs to the leftmost element of
the equilibrium partition, R almost certainly reaches a history where she pays a certain amountto S even though she does not obtain additional information in the future.

16ψ(y, θ, b|y)
6

aλ̃+1

aλ̃
yλ̃+1

aλ̃−1
-

yλ̃

θ

•ψλ̃+1ψλ̃•Figure 1: (ωλ̃ − ωλ̃+1 )/s = ψλ̃+1 − ψλ̃ ≥ 0than the optimal:∫ aλ̃−1 [

]
g(θ)
R
r
u (y(aλ̃ , aλ̃−1 ), θ) dθG(aλ̃−1 ) − G(aλ̃ )aλ̃]∫ aλ̃−1 [
g(θ)
R R
<r
u (y (θ), θ) dθ.
G(aλ̃−1 ) − G(aλ̃ )
aλ̃

(6)

Moreover, R reaches such a history with probability G(aλ̃−1 ) − G(aλ̃ ). Hence, inany equilibrium, R’s expected payoﬀ is strictly less than U:U≡r∫ aλ̃−1 [
aλ̃

<r

∫ 1[

∫
]
g(θ)u (y(aλ̃ , aλ̃−1 ), θ) dθ + r[

R]g(θ)uR (yR (θ), θ) dθ.

θ<[aλ̃ ,aλ̃−1 ]]g(θ)uR (yR (θ), θ) dθ

0

To make the characterization more specific, we assume the following.
Assumption 3. R’s utility from project uR satisfiesuR (y, θ) = l(|y − θ|),where l′′ (·) < 0 and l′ (0) = 0.
Assumption 4. The distribution G is the uniform distribution.
Under Assumptions 3 and 4, Assumption 2 is satisfied.
Proposition 5. Under Assumptions 1, 3, and 4, the upper bound of R’s equilibrium

17

payoﬀ is given by∫
U(b, r) = r

θ∈[0,4b]l(|2b − θ|)dθ.

The proof is in Appendix 3.F. One of the main findings in our analysis is thatwhen T is suﬃciently high and s/r is small enough, this upper bound U(b, r)
can be approximated by R’s equilibrium payoﬀ. For the details of this result,
see Proposition 10 in Section 4.

4. The Uniform-quadratic CaseIn this section, we show the benefits of multistage information transmission
with voluntary payments, concentrating on the well-known uniform-quadratic
case: r · uR (y, θ) = −r(y − θ)2 , s · uS (y, θ, b) = −s(y − (θ + b))2 , and G(θ) is a uniform
distribution over Θ.

4.1. Two-period Information Elicitation
The key idea on which we build the analysis is that the dependence of future information on past payments ensures that R makes message-contingent
payments. To understand the intuition behind this idea, we construct an equilibrium in which information is transmitted within two periods and R pays apositive amount of money to S on the equilibrium path. By constructing such anequilibrium, we show that multistage information transmission with voluntary
payments can be more beneficial for both S and R than the one-shot cheap talkcommunication. In Section 5, we generalize the players’ payoﬀ functions andprior probability distribution and show the results that correspond to those inthis subsection.
Suppose that b ∈ (1/12, 1/4). Then, there are two equilibria in the oneshot cheap talk game. One is the uninformative equilibrium: the babbling
equilibrium. The other is a partially informative equilibrium: a0 = 1, a1 =1/2 − 2b and a2 = 0. CS have shown that both S and R prefer the partially18

informative equilibrium to the uninformative equilibrium. In the partiallyinformative equilibrium, the ex ante expected payoﬀ of R is −r(1/48+b2 ) whereasthat of S is −s(1/48 + b2 ) − sb2 .
The first result establishes that if T ≥ 2 and r is large relative to s, there exists
an equilibrium whose partition has more steps than the one-shot cheap talkgame does.
Proposition 6. Fix b ∈ (1/12, 1/4). If s/r < (1 − 4b)/(1 + 12b), there is a continuumof 3-element partition equilibria.
We characterize a class of 3-element partition equilibria in which informationis transmitted in order from the rightmost element of the equilibrium partition.
In the equilibrium, S gradually conveys his information within the first andsecond period. If S conveys information contrary to his bias in the first period,
then R pays to S in order to extract more precise information in the second
period. If R does not pay in the first period, then S never gives additional
information. As s becomes smaller, the necessary payment becomes smaller
since the eﬀect of the message-contingent payment on S’s incentive becomeslarger. Furthermore, as r becomes higher, the punishment by babbling message
becomes more severe. This is the reason why s/r needs to be small enough.
Proof. Consider a strategy profile under which the information is transmitted
in the following steps. At stage 1 in period 1, S of type θ < a1 randomly sends
a message m1 according to a uniform distribution over [0, a1 ), and S of typeθ ≥ a1 randomly sends a message m1 according to a uniform distribution over
[a1 , 1]. If R receives m1 < a1 at stage 1 in period 1, then she pays w1 = w to S.
Otherwise, she pays nothing to S at stage 2 in period 1. At stage 1 in period 2,
if m1 < a1 and w1 ≥ w, then S of type θ < a2 randomly sends a message m2
according to a uniform distribution over [0, a2 ), and S of type θ ≥ a2 randomly
sends a message m2 according to a uniform distribution over [a2 , 1]. Otherwise,
S conveys no information, i.e., any type of S randomly sends a message m2
according to a uniform distribution over [0, 1]. In period t ≥ 2, R pays nothing
to S. In period t ≥ 3, S conveys no information.
19

Once communication is over, R chooses her best project based on the information she has. At hT+1 such that m1 ≥ a1 , since R believes θ is uniformly
distributed over [a1 , 1], the optimal project for R is y1 = (a1 + 1)/2. At hT+1 suchthat m1 < a1 , w1 ≥ w, and m2 ≥ a2 , since R believes θ is uniformly distributed
over [a2 , a1 ), the optimal project for R is y2 = (a2 +a1 )/2. At hT+1 such that m1 < a1 ,w1 ≥ w, and m2 < a2 , since R believes θ is uniformly distributed over [0, a2 ), theoptimal project for R is y3 = a2 /2. At hT+1 such that m1 < a1 and w1 < w, since
R believes θ is uniformly distributed over [0, a1 ), the optimal project for R isỹ = a1 /2. Figure 2 illustrates the equilibrium strategy.

a3 0- a1  m1- a0

m1?

w1 = w(w1 < w)
m2 =babbling(w2 = 0)?
ỹ = a21?

θ1(w1 = 0)?
m2 =babbling(w2 = 0)?
1
y1 = 1+a2?

a3  m2 - a2
0?
w2 = 0?a2

y3 = 2

m2

(a1 )

?

w2 = 0

- a0

1

θ

?

2
y2 = a1 +a2

Figure 2: Equilibrium Strategy

In what follows, we ensure that by taking a1 , a2 and w suitably, we canconstruct an equilibrium in which S and R follow the abovementioned strategy
profile.
In period t ≥ 2, R always pays nothing to S. Therefore, the partition{[0, a2 ), [a2 , a1 )} must coincide with the 2-element equilibrium partition achievedin the one-shot cheap talk game in which θ is drawn from the uniform distribution over [0, a1 ). By CS, the following must be satisfied:
a2 = a1 /2 − 2b.

(7)

Since we now focus on a 3-element partition equilibrium, we must have a2 > 0.
20

Hence, a1 > 4b must be satisfied.
Under the abovementioned strategy profile, S of type θ ∈ (ai , ai−1 ) sends
messages so that yi would be chosen by R. Hence, S’s payoﬀ is derived asfollows:− s(y3 − (θ + b))2 + w for θ ∈ [0, a2 );− s(y2 − (θ + b))2 + w for θ ∈ [a2 , a1 );− s(y1 − (θ + b))2 for θ ∈ [a1 , 1].
Since we suppose that a2 = a1 /2 − 2b, we obtain− s(y3 − (θ + b))2 > −s(y2 − (θ + b))2 for θ ∈ [0, a2 );− s(y3 − (θ + b))2 < −s(y2 − (θ + b))2 for θ ∈ (a2 , 1];− s(y3 − (θ + b))2 = −s(y2 − (θ + b))2 for θ = a2 .
Clearly, at stage 1 in period 2 such that m1 < a1 and w1 ≥ w, S has no incentive todeviate from the given strategy. Moreover, if m1 < a1 and w1 < w, or if m1 ≥ a1 ,S sends a babbling message. Therefore, S has no incentive to deviate at such ahistory. The same can be said in period t ≥ 3. Hence, we conclude that S has
no incentive to deviate in period t ≥ 2 when a2 = a1 /2 − 2b.
At stage 1 in period 1, if S of type θ sends m1 ≥ a1 , then he obtains −s(y1 −(θ + b))2 . Otherwise, S of type θ ≥ a2 obtains −s(y2 − (θ + b))2 + w, and S of type
θ < a2 obtains −s(y3 − (θ + b))2 + w. If the following equation (8) holds, then theinequalities (9) and (10) hold.
− s(y1 − (a1 + b))2 = −s(y2 − (a1 + b))2 + w;

(8)

− s(y1 − (θ + b))2 ≥ max{−s(y j+1 − (θ + b))2 + w} for θ ≥ a1 ;(9)j∈{1,2}− s(y j+1 − (θ + b))2 + w > −s(y1 − (θ + b))2 for j = {1, 2} and θ ∈ [a j+1 , a j ). (10)If (9) and (10) hold, S has no incentive to deviate at stage 1 in period 1.
21

By equation (8), we obtainw = w(a1 ) ≡ s[(2 + 4b − a1 )(−2 + 12b + 3a1 )]/16.

(11)

Since w(a1 ) is strictly increasing in a1 ∈ [4b, 1], we have an inverse function of w(·)
such that w−1 (w) ≡ a1 (w) is strictly increasing in w ∈ [w(4b), w(1)]. Moreover,since we suppose that b ∈ (1/12, 1/4), R’s payment is nonnegative: w(4b) =}{√s(12b − 1)/4 > 0. Note that a1 (w) = 32 2 − (1 + 6b)2 − 12w/s and a1 (w) ∈ (4b, 1)
where w ∈ (w(4b), w(1)).
In summary, we conclude that S has no incentive to deviate from the given
strategy when the boundaries of the partition satisfy the following conditions:
1 for i = 0,





}
{
√


2
2 − 12w/s

(1
+
6b)
for i = 1,2−3
ai (w) ≡ 
}
{

√

1


(1 + 6b)2 − 12w/s − 2b for i = 2,
2
−

3






0 for i = 3.

(12)where w ∈ (w(4b), w(1)). Figure 3 illustrates S’s incentive compatibility conditions.
US (y, θ, w|y)
6
6

6

?

?

w

0

w

y3

a2

y2

a1

y1

1θ

Blue curve: s · uS (y3 , θ, b) + w
Red curve: s · uS (y2 , θ, b) + w
Black curve: s · uS (y1 , θ, b)
Figure 3: S’s payoﬀ on the equilibrium pathAt any h(t,2) , R has no incentive to increase the amount of payment because22

it would not aﬀect S’s behavior. Therefore, we have only to ensure that paying
w is optimal for R after receiving m1 < a1 .
If R pays w1 ≥ w after receiving m1 < a1 , then she obtains u∗ (w1 ):

1 ∑u (w1 ) = −w1 −a1 i=1
2∗∫ ai []2
ai+1 + air− θ dθ
2ai+1

r ∑(ai − ai+1 )3= −w1 −
12a1 i=1r= −w1 −
(a3 + (a1 − a2 )3 ).
12a1 2
2

On the other hand, by paying w1 < w, R obtains u(w1 ):
∫ a1

(
)
1 R a1u(w1 ) = −w1 +U, θ dθ
a1
2
0∫ a1 ()2ra1= −w1 −− θ dθ
a1 0
2
a21= −w1 − r .
12
The payoﬀs u∗ (w1 ) and u(w1 ) have a unique maximum at w1 = w and w1 = 0,
respectively. Thus, paying w is an optimal decision for R if and only if u∗ (w) ≥u(0). Using condition (12) yields(){a1 (w)}2
2
u (w) ≥ u(0) ⇐⇒ r
− b ≥ w.
16
∗

(13)

{
}
√
Since a1 (w) ≡ 23 2 − (1 + 6b)2 − 12w/s , for any w ∈ (w(4b), w(1))• a1 (w) is strictly increasing in w;•{a1 (w)}2{a (w(4b))}2− b2 > 0 and 1 16
− b2 = 0;
16
2

d
2
• dw
2 {a1 (w)} > 0.

Hence, if r({a1 (w(1))}2 /16−b2 ) > w(1), then the function r({a1 (w)}2 /16−b2 ), where{}√
a1 (w) = 32 2 − (1 + 6b)2 − 12w/s , has a fixed point w(s/r) ∈ (w(4b), w(1)) such23

that for all w ∈ [w(s/r), w(1)), the inequality (13) holds. Note that w(s/r) ↓ w(4b)as s/r ↓ 0.
Since a1 (w(1)) = 1 and w(1) = s(1+4b)(1+12b)/16, the inequality r({a1 (w(1))}2 /16−b2 ) > w(1) can be simplified intos
1 − 4b<.
r 1 + 12b
Therefore, if s/r < (1 − 4b)/(1 + 12b), then the given strategy profile andthe system of beliefs constitute an equilibrium when w ∈ [w(s/r), w(1)) and theboundaries of partition satisfy the condition (12).

□Remark 1. In the equilibrium, meaningful information transmission must occur
after R pays w. For this reason, in the equilibrium outlined above, it is necessarythat a2 = a1 /2 − 2b > 0. Hence, both 4b < a1 and b < 1/4 must be satisfied.
There is a possibility of the existence of a 3-element partition equilibrium
in which S conveys information in a diﬀerent order. For example, consider thefollowing strategy profile. In period 1, S reveals whether θ ≥ a2 . If θ ≥ a2 ,then R pays w̃, and then, S reveals whether θ < a1 . Note that a2 < a1 . The
following Proposition 7 shows that there is no equilibrium where informationis transmitted in such a way.
Proposition 7. Fix b ∈ (1/12, 1/4). There exists no 3-element partition equilibriumsuch that information is transmitted in order from the leftmost element of the equilibrium
partition.
The proof of is in Appendix 4.A. Under the abovementioned strategy profile,
R pays w̃ > 0 only when she receive the message that means θ ≥ a2 . Intuitively,
this payment strategy aﬀects S’s incentive for misrepresentation negatively,
since it strengthens the exaggeration incentive of S whose type is close to theright boundary of the interval. Hence, even if we consider the equilibrium
in which information is transmitted in order from the leftmost element of theequilibrium partition, we cannot increase the elements of equilibrium partition.
It can be confirmed that if r is large relative to s, R can obtain the greater
expected “revenue from the project” under a 3-element partition equilibrium
24

constructed in Proposition 6 than under the 2-element partition equilibriumin the one-shot cheap talk game. This result is due to the fact that R canobtain more detailed information about S’s type. However, since R has tomake a payment to improve information transmission, multistage informationtransmission with voluntary transfer payments is not always beneficial to R.
We now show the second result that when r is large relative to s, multistage
information transmission with voluntary transfer payments is more beneficial
to both R and S than the one-shot cheap talk communication.
In the one-shot cheap talk game, both players always strictly prefer the 2element partition equilibrium to the babbling equilibrium from the ex-ante perspective. Let EÛκ be the ex ante expected payoﬀ of κ ∈ {R, S} under the 2-element
partition equilibrium in the one-shot cheap talk game, where {[ã2 , ã1 )[ã1 , ã0 ]} isκthe equilibrium partition. As noted earlier, ã1 = ã0 /2 − 2b = 1/2 − 2b. Let EU (x)be the ex ante expected payoﬀ of κ ∈ {R, S} in the 3-element partition equilibrium with the partition {[a3 , a2 ), [a2 , a1 )[a1 , a0 ]}, where x = a1 and a2 = x/2 − 2b.
Since w ∈ [w(s/r), w(1)), we must have x ∈ [a1 (s/r), 1), where a1 (s/r) = a1 (w(s/r)).
Recall that w(s/r) ↓ w(4b) as s/r ↓ 0. Hence, a1 (s/r) ↓ 4b as s/r ↓ 0.
The following lemma shows that if r is large relative to s, there exists a 3element partition equilibrium that R prefers to all the equilibria in the one-shotcheap talk game.
Lemma 1. There exists a positive value η∗ (b) such that if s/r < η∗ (b),R

EU (x) > EÛR for some x ∈ [a1 (s/r), 1).
Proof. Suppose that s/r < (1−4b)/(1+12b). Fix a 3-element partition equilibrium
constructed in the proof of Proposition 6. By the definition of a1 (w), we havew(x) ≡ a−1
1 (x) = s[(2 + 4b − x)(−2 + 12b + 3x)]/16 for x = a1 ∈ [a1 (s/r), 1).
Hereafter, the function w(x) is denoted by s·α(b, x). Recall that s·α(b, x) is strictly
greater than zero for x ∈ [a1 (s/r), 1).
25

R

R’s expected payoﬀ EU (x) is given by]2
3 ∫ ai−1 (x) [∑ai−1 (x) + ai (x)EU (x) = −r− θ dθ − xs · α(b, x)
2
a
(x)
i
i=1
{ 3
}
x
r
+ xb2 − (1 − x)3 − xs · α(b, x).
= −r
48
12
R

CS show thatEÛ

2 ∫ ãi−1 []2∑ãi−1 + ãi= −r− θ dθ2ãii=1r= − − rb2 .
48

R

R

Let δ(b, x) ≡ 1r {EU (x) − EÛR }. We obtain
δ(b, x) = −1xs(1 − x3 ) + (1 − x) + b2 (1 − x) − xα(b, x).
16
4r

δ(b, x) > 0 holds if and only if∗η (b, x) ≡1
− 16
(1 − x3 ) + x4 (1 − x) + b2 (1 − x)xα(b, x)

s
> .
r

∂η∗

We obtain ∂x |x=1 < 0, η∗ (b, 1) = 0, and infx∈[a1 (s/r),1) xα(b, x) = a1 (s/r)α(b, a1 (s/r)) >0. Therefore, η∗ (b, x) > 0 for some x ∈ (4b, 1), and η∗ (b, x) < +∞ for any x ∈[a1 (s/r), 1).
For any x ∈ (4b, 1), if s/r < a−1(x), it is satisfied that x ∈ [a1 (s/r), 1). Hence, by
1[]∗taking η∗ (b) as supx∈X min{a−1
(x),
η
(b,
x)}
where X ≡ {x : η∗ (b, x) > 0} ∩ (4b, 1),1
we can conclude that if s/r < η∗ (b), then δ(b, x) > 0 for some x ∈ [a1 (s/r), 1). This
completes the proof of Lemma 1.

□

Remark 2. Note that x is almost equal to 1. Then, boundaries of the 3-element
partition equilibrium almost coincide with boundaries of the 2-element partition
equilibrium in the one-shot cheap talk game. Nevertheless, the payment ofmonetary transfer is strictly higher than 0. Therefore, if s/r < (1 − 4b)/(1 + 12b),
there always exists a 3-element partition equilibrium that is unfavorable to26

R. Namely, for any s/r < (1 − 4b)/(1 + 12b), there exists ε̂(s/r) > 0 such thatη∗ (b, x) < s/r for all x ∈ (1 − ε̂(s/r), 1).
Next, we show the following lemma.
RS

Lemma 2. EU (x) > EÛR implies that EU (x) > EÛS .
Proof. Recall that EUS (x) denotes the ex ante expected payoﬀ of S under the
3-element partition equilibrium with a1 = x ∈ [a1 (s/r), 1). We obtain]2
3 ∫ ai−1 (x) [∑ai−1 (x) + ai (x)s
EU (x) = −− θ dθ − sb2 + x · w(x)2
ai (x){i=1}
Rs=
EU (x) + x · w(x) − sb2 + x · w(x).
rS

CS show that
EÛ

R

S

2 ∫ ãi−1 []2∑ãi−1 + ãi= −− θ dθ − sb2
s
2
ãi
i=1
s
= EÛR − sb2 .
r
S

Clearly, if EU (x) > EÛR , then EU (x) > EÛS .

□From Lemma 1 and 2, we immediately obtain the following result.
Proposition 8. Fix b ∈ (1/12, 1/4). Then, there exists a positive value η∗ (b) such that
if s/r < η∗ (b), there exists a 3-element partition equilibrium whose outcome ex ante
Pareto-dominates all the equilibrium outcomes in the one-shot cheap talk game.
It is known that the existence of a non-strategic mediator leads to improved
information transmission. Now, we compare the information elicitation described in Figure 2 with the optimal mediation. In the mediation model, S cansend a message to an impartial mediator, who then passes on a recommendation to R according to some predetermined stochastic rule. R chooses her best
project based on the recommendation from mediator. Goltsman et al. (2009)
characterize the optimal mediation under which R’s ex ante expected payoﬀ is27

−rb(1 − b)/3. The following corollary shows that in two-period information elicitation with voluntary monetary transfer, R can obtain a higher ex ante expected
payoﬀ than that under the optimal mediation.
√
Corollary 2. Fix b ∈ (1/12, (4 + 3)/26).22 Then, there exists η′ (b) such that if
s/r < η′ (b),Rr
EU (x) > − b(1 − b) for some x ∈ [a1 (s/r), 1).
3

Since this corollary can be proved in the same way as the proof of Lemma 1,
the formal proof is omitted. When b is almost equal to 1/4, boundaries of the
3-element partition equilibrium almost coincide with those of the 2-element
partition equilibrium in the one-shot cheap talk game: a1 ≈ 1 and a2 ≈ 1/2 − 2b.
The value of −rb(1 − b)/3 is always strictly higher than R’s equilibrium payoﬀ
under the 2-element partition equilibrium in the one-shot cheap talk game.
Therefore, the parameter b needs to be strictly less than 1/4.

4.2. Eﬀective Long-term Communication
In the previous subsection, we restrict attention to the equilibrium in which
information is transmitted within only two periods, regardless of the lengthof communication. It seems that R does not use T-period communicationeﬀectively. In this subsection, we show the benefit of long-term communication.
Recall the earlier discussion of the upper bound of R’s equilibrium payoﬀ.
Proposition 5 provides it as∫U(b, r) = r=−

θ∈[0,4b]3l(|2b − θ|)dθ

16rb
.
3

One of the main findings in the present paper is that when T is long enough,this upper bound U(b, r) = −16rb3 /3 can be approximated by R’s equilibrium
22

( √ )
1
Note that 51 < 26
4 + 3 < 14 .

28

payoﬀ.
First, we demonstrate that under a certain condition, there exists an equilibrium in which information is transmitted within the whole T-period in orderfrom the rightmost element of the equilibrium partition. Specifically, we consider the following information elicitation. In period 1, S conveys whether thevalue of θ is less than a1 . If θ < a1 , then R pays a certain amount of money. After
that, in period 2, S conveys whether the value of θ is less than a2 . If θ < a2 , thenR pays again. This information elicitation is repeated until the last period in thecommunication phase. In the last period, S of type θ < aT−1 conveys whether
the value of θ is less than aT . Under this communication process, R eventuallylearns to which element of a partition {[at+1 , at )}Tt=1 ∪ [a1 , a0 ] the state θ belongs.
We call this communication process (monotone) eﬀective T-period communication.23
Figure 4 illustrates this information elicitation.
aT+1

0
aT+1

0
aT+1

0

a1

a0
θ
w
>
0
1
1
?
a2 a1 a0-θ
w2 > 01?
..
.
at at−1 · · · a2 a1 a0
-
θ
w
>
0
1
t
..?
.
wT−1 > 0?
aT+1 aT
aT−1···
a2 a1 a0θ
 -0 ?wT = 0 ?wT = 0
1
aT+1 aT
aT−1 · · · a4 a3 a2 a1 a0
yT
y3 y3 y2 y1 1 θ
0 yT+1- -Figure 4: Eﬀective T-period communication

Proposition 9. Fix b ∈ (0, 1/4). If s/r < (1−4b)/(1+12b), there exists an equilibriumwith eﬀective T-period communication.
Under the eﬀective T-period communication, the information is transmitted
in the following steps. At h(t,1) such that mt′ < at′ and wt′ ≥ w∗t′ for all t′ < t, S of
23

This information elicitation is similar to that in Ivanov (2015) and Hörner and Skrzypacz(2016).

29

type θ < at randomly sends a message mt according to a uniform distribution
over [0, at ), and S of type θ ≥ at randomly sends a message mt according to a
uniform distribution over [at , 1]. Otherwise, any type of S randomly sends a
message according to the same distribution, a uniform distribution over [0, 1].
If S conveys that θ < at at stage 1 in period t, then R pays w∗t to S at stage 2.
Otherwise, he pays nothing.
Let I(hT+1 ) be the closure of {θ ∈ Θ : f (θ|hT+1 ) > 0}. Under the abovementioned strategy profile, for any hT+1 , the closed set I(hT+1 ) belongs to {[at , at−1 ]}T+1∪
t=1{[aT+1 , at−1 ]}Tt=2 , and R’s posterior belief f (θ|hT+1 ) is a uniformly distribution on
I(hT+1 ). Therefore, R chooses y = {min I(hT+1 ) + max I(hT+1 )}/2 at hT+1 .
At h(T,2) such that mt′ < at′ and wt′ ≥ w∗t′ for all t′ < T, since R does not
obtain additional information after making a payment, w∗T must be equal to 0.
Therefore, {[aT+1 , aT ), [aT , aT−1 )} coincides with the 2-element equilibrium partition achieved in the one shot cheap talk game where Θ = [0, aT−1 ). Hence, we
obtain
aT =

aT−1
− 2b.
2

This implies that aT−1 > 4b. Define at and w∗t as follows:

at

w∗t





1 − ta for t ∈ {1, . . . , T − 1},





 1−(T−1)a≡ − 2b for t = T,

2






0 for t = T + 1.
2bsa for t ∈ {0, . . . , T − 2},





s
≡ 
{1 + 12b − a(T + 1)}{1 + 4b − a(T − 3)}

16






0 for t = T.

(14)

for t = T − 1,(15)

Suppose that a < (1 − 4b)/(T − 1). Then, 4b < aT−1 and at−1 − at = a > 0 fort ∈ {1, . . . , T − 1}. Note that w∗T−1 > 0 if a < min{(1 + 12b)/(T + 1), (1 + 4b)/(T − 3)}.
Since we suppose that T ≥ 3, we obtain (1 + 4b)/(T − 3) > (1 − 4b)/(T − 1).
Therefore, if a < min{(1 − 4b)/(T − 1), (1 + 12b)/(T + 1)}, the given boundaries
30and payments are well-defined. Moreover, for any t ∈ {1, . . . , T − 1}, w∗t becomesa solution to an equation,
(

)2
(
)2
at + at−1
at+1 + at−s− (at + b) = −s
− (at + b) + w∗t ,22
derived from S’s incentive compatibility condition: S whose type falls on theboundaries between adjacent intervals is indiﬀerent between the associatedvalues of y.
The abovementioned strategy profile and system of beliefs, hereafter ξT ,cannot always be an equilibrium. Whether it is so depends on the value of a.
We ensure that ξT can be an equilibrium when a is small enough. R’s paymentw∗t in each t ≤ T −2 goes to 0 as a goes to 0. Consider a history at stage 2 in periodT − 1 such that mt < at for all t ≤ T − 1 and wt ≥ w∗t for all t < T − 1. Then, there
are two cheap talk equilibria in the remaining game: the babbling equilibrium
and the 2-element partition equilibrium. Since we now suppose that a ≈ 0,
if the 2-element partition equilibrium is chosen in period T, R’s continuation
payoﬀ is approximated by −r(b2 − 1/48). Otherwise, R’s continuation payoﬀ
is approximated by −r/12. Moreover, w∗T−1 ≈ s(1 + 12b)(1 + 4b)/16. Since wesuppose that s/r < (1 − 4b)/(1 + 12b), we have
) (
)
(
r
s
1
− −
> (1 + 12b)(1 + 4b).
−r b2 −
48
12
16
Thus, R has an incentive to pay w∗T−1 at this history so that the babbling equilibrium would not be chosen in the last period. Furthermore, at h(t,2) where R
pays w∗t , if w∗t is small enough, R pays to ensure that the babbling equilibrium
would not be chosen in the future. Hence, by taking a small enough, we canconstruct an equilibrium with eﬀective T-period communication. The formal
proof is found in Appendix 4.B.
Proposition 9 shows only the possibility of the eﬀective T-period communication. In order for ξT to be an equilibrium, it might be necessary for aT−1 to beclose to 1. If aT−1 is close to 1, R reaches a history h(T,1) at which I(h(T,1) ) = [0, aT−1 ]with a high probability on the equilibrium path. Moreover, {[aT+1 , aT ), [aT , aT−1 )}31almost coincides with the 2-element equilibrium partition achieved in the oneshot cheap talk game. In such a case, the initial (T − 1)-period communication
does not have much meaning from ex ante perspective. However, as S becomesless concerned with the project, the eﬀects of monetary transfer on S’s incentive
becomes larger. In other words, the necessary payments for controlling S’s
incentive becomes smaller as s goes to 0. Hence, if s is small enough, it is notnecessary for aT−1 to be close to 1. This fact suggests that long-term communication becomes more beneficial for R as s becomes smaller. To see this, we show
the following Proposition 10.
Proposition 10. Fix b ∈ (0, 1/4). For any d > 0, there exists T(b, d) and η(b, d) suchthat if T ≥ T(b, d) and s/r < η(b, d), R can obtain a higher ex ante expected payoﬀ than
−16rb3 /3 − rd.
The proof is in Appendix 4.C. We earlier show that an upper bound of R’sequilibrium payoﬀ is −16rb3 /3. This Proposition 10 shows that if the communication phase has a suﬃciently large number of periods and S weighs transfer
payments more heavily than the project choice, this upper bound can be approximated by R’s equilibrium payoﬀ.

4.3. Comparison with Predetermined Decision Rules
Now, under the uniform-quadratic assumption, we compare the gradual information elicitation, which we have described in previous subsections, with bothdelegation and arbitration. When R delegates control, her payoﬀ is given by−rb2 since S always chooses the most desirable project for him: yS (θ, b) = θ + b.
As shown by CS, the ex ante expected payoﬀ of R under the one-shot cheap talkcommunication is given by
(
R
EUCS
= −r

)
b2 (n2 − 1)
1
+
,
12n2
3

32

where n ∈ {1, . . . , ñ}. The maximum number of partition equilibrium outcomesñ is given by) 12 
 1 1 (2ñ ≡ − + 1 +, 2 2b 

where ⌈x⌉ denotes the smallest integer greater than or equal to x. Dessein (2002)
R< −rb2 for n ≥ 2, and thus, R prefers delegation to the one-shotshows that EUCS

cheap talk communication whenever informative communication is possible,b < 1/4.
By contrast, in our model, if T ≥ T(b, d) and s/r < η(b, d), R can obtain thehigher equilibrium payoﬀ than −16rb3 /3 − rd. If b < 3/16 and d < b2 − 16b3 /3,
the inequality −16rb3 /3 − rd > −rb2 holds.
Next, consider the situation in which arbitration is available. Under arbitration, S sends a message to a neutral third party (arbitrator), and after receiving
the messages, the arbitrator announces a project. This announcement serves asa binding recommendation to R. In other words, R cannot choose any actionthat is diﬀerent from the recommended one. Goltsman et al. (2009) characterize the optimal arbitration rule and show that R’s ex ante expected payoﬀ
under optimal arbitration is −rb2 (1 − 4b/3).24 We immediately verify that if
d < b2 (1−4b/3)−16b3 /3 and b < 3/20, the inequality −16rb3 /3−rd > −rb2 (1−4b/3)
holds.
Therefore, Proposition 10 implies that when the communication phase hasa suﬃciently large number of periods and R places greater importance on the
project than S does, R can obtain higher ex ante expected payoﬀ than underdelegation and arbitration.25

4.4. Comparison with Sender-optimal Signaling
As noted in Section 1, costly signaling helps people convey their private information credibly. Naturally enough, even in our setting, if S can send a costly
24

Having restricted attention to deterministic mechanism, Melumad and Shibano (1991)
provide the optimal arbitration (optimal delegation) rule.
25
Since the optimal arbitration rule dominates the optimal mediation rule, our communication
protocol could strictly dominate the optimal mediation rule.

33

message (paying money to R) to signal information, a fully separating equilibrium that is optimal from R’s perspective can exist. However, it is known that
under general assumptions, the perfect separation is never optimal from S’s
perspective although it is feasible.
Karamychev and Visser (2016) study an amendment to the CS model by
allowing S to use both costless and costly messages. They show that in S’s
optimal equilibrium, S pays to adjust the pooling intervals.26 Moreover, under
the uniform-quadratic assumption, they characterize Sender-optimal equilibria
whose partition has at most ñ + 1 steps.27 In such equilibria, R’s expected payoﬀ⌈() ⌉√is less than −r/{12(ñ + 1)2 }. Since ñ ≡ −1 + 1 + 2/b /2 , the integer ñ satisfies
that 2ñ(ñ + 1) ≤ b ≤ 2ñ(ñ − 1).
Therefore, if ñ > 4 holds, we obtain
−

16rb3
r
r
≥−
>−
.
3
3
12(ñ + 1)12(ñ + 1)2

This inequality and Proposition 10 suggest that in some cases, it might be betterfor R to generate the signaling structure by herself through voluntary payment
rather than to rely on S’s costly signaling.

5. Generalization of Proposition 6 and Proposition 8
In this section, under the more general settings where the players’ payoﬀ function and the prior probability of the state are kept as is in Section 2, we show
three results that correspond to the results in Section 4.1.
Recall that ñ (≡ ñ(b) in Section 3.1) denotes the maximum number of elementsof equilibrium partition achievable in the one-shot cheap talk game. As can beobserved from the uniform-quadratic case, after S conveys some information inperiod 1, there must be multiple equilibria in the remaining game. Therefore,
we assume that ñ ≥ 2. In the one-shot cheap talk game, if Condition M holds,
26

de Haan et al. (2015) experimentally study the strategic information transmission in a setting where both cheap talk and burning money are available, and they find that the individuals
who supply information prefer to communicate through cheap talk.
27
See Proposition 4 in Karamychev and Visser (2016).

34

then the most informative equilibrium is ñ-element partition equilibrium where{[ãñ , ãñ−1 ), . . . [ã1 , ã0 ]}, and 0 = ãñ < ãñ−1 < · · · < ã1 < ã0 = 1.
The following Proposition 11 establishes that an equilibrium whose partition
has more steps that that in the one-shot cheap talk game exists.
Proposition 11. Fix b > 0 and suppose that ñ ≥ 2. Then, there exists a positive value
η(b) such that if s/r < η(b), there is a continuum of (ñ + 1)-element partition equilibria.
To prove this Proposition, we construct a strategy profile that induces a(ñ + 1)-element partition: {[âñ+1 , âñ ), . . . [â1 , â0 ]}, and 0 = âñ+1 < âñ < · · · < â1 <â0 = 1. The following strategy profile is an extension of the strategy profile that
we construct in Section 4.1.
At stage 1 in period 1, S conveys whether θ < â1 . If θ < â1 , then R pays acertain amount of money, w∗ , to S at stage 2 in period 1. Otherwise, she pays
nothing to S. If θ < â1 and w1 ≥ w∗ , at stage 1 in period 2, S conveys to whichelement of {[âñ+1 , âñ ), . . . [â2 , â1 )} the true state θ belongs. Otherwise, S conveys
no information regardless of his type. In period t ≥ 2, R always pays nothing
to S. In period t ≥ 3, S conveys no information. In period T + 1, R chooses a∫
project ρ(hT+1 ) = arg max y uR (y, θ) f (θ|hT+1 )dθ. In the rest of this section, (σ̂, ρ̂)
denotes the strategy profile defined above, and denote by fˆ the belief systemderived from (σ̂, ρ̂).

âñ+1 
0?

w1 = w∗?

âñ+10m2 â m2- âñ

wt = 0?

- â1  m1- â0

m1

ñ−1

(w1 < w∗ )mt =babbling(wt = 0)?
y(0, â1 )
?

···

â2

m2

(â1 )

wt = 0?

?

y(0, âñ ) y(âñ+1 , âñ )y(â2 , â1 )Figure 5: Equilibrium35θ1(w1 = 0)?
mt =babbling(wt = 0)?
y(â1 , 1)
- â0

1

θ

Under the strategy profile outlined above, we have to take an equilibrium
partition whose boundaries {âñ+1 , . . . , â1 } coincide with those of the ñ-element
partition equilibrium in the one-shot cheap talk game, where the state is distributed on [0, â1 ) with the density g(·)/G(â1 ). Let {[âxñ+1 , âxñ ), . . . , [âx1 , âx0 ]} be apartition that satisfies (1) for i = 2, . . . , ñ with âx0 = 1, âxñ+1 = 0, and âx1 = x ∈ (ã1 , 1).
The following inequality must hold for R’s payment w∗ to be optimal.
∫ âxñ ∫ x1r ∑ âi R
r
x
x
u
(y(
uR (y(0, âx1 ), θ)g(θ)dθ ≥ w∗ , (16)
â
,
â
),
θ)g(θ)dθ−i+1 i
G(âx1 ) i=1 âx
G(âx1 ) 0
i+1

∫ âx
where y(âxi+1 , âxi ) = arg max y âx i uR (y(âxi+1 , âxi ), θ)g(θ)dθ. The left-hand side of
i+1

this inequality represents the value of additional information, that is, the valueof the partition {[0, âxñ ), . . . , [âx2 , âx1 )} that R receives in period 2 by paying w∗ after
receiving a message that means θ < âx1 . It is obvious that R always strictlyprefers partition {[0, âxñ ), . . . , [âx2 , âx1 )} to partition {[0, âx1 )}, which implies that theleft-hand side of the inequality (16) is positive and increasing in r when ñ ≥ 2.
Since w∗ must be equal to s · uS (y(âx1 , 1), âx1 , b) − s · uS (y(âx2 , âx1 ), âx1 , b),28 the righthand side of the inequality (16) is decreasing in s and goes to 0 as s goes to 0.
Therefore, if r is large enough relative to s, then paying w∗ is optimal for R.
In Appendix 5.A, we ensure that there exists η(b) > 0 such that if rs < η(b),by taking the boundaries of partition {[âxñ+1 , âxñ ), . . . .[âx1 , âx0 ]} suitably, ((σ̂, ρ̂), fˆ)constitutes an equilibrium.
Next, we show that under some conditions, multistage information transmission with voluntary monetary transfer is more beneficial to both R and S
than one-shot cheap talk communication. To observe this, we restrict attention
to the equilibrium, ((σ̂, ρ̂), fˆ), which we construct in Proposition 11.
Let EÛR (x) be the ex ante expected payoﬀ of R under ((σ̂, ρ̂), fˆ) with thepartition {[âxñ+1 , âxñ ), . . . [âx1 , âx0 ]}. Define a1 (s/r) as the infimum value of z such that(16) holds for all x ∈ [z, 1).

28

Recall Figure 3 in Section 4.1.

36

We have
EÛR (x) = Ŵ(x) − E[w∗ ],where Ŵ(x) denotes R’s ex ante expected utility from project:Ŵ(x) ≡ rñ+1 ∫ âx
∑
i−1
i=1

âxi

uR (y(âxi , âxi−1 ), θ)g(θ)dθ.

CS show that in the one-shot cheap talk game, under Condition M, R alwaysR
strictly prefers ñ-element partition equilibrium to any other equilibria. EUCS
de-notes the ex ante expected payoﬀ of R under the ñ-element partition equilibriumin the one-shot cheap talk game. We obtain
R
EUCS

= r

ñ ∫ ãi−1∑
i=1uR (y(ãi , ãi−1 ), θ)g(θ)dθ.

ãi

For any i ∈ {1, . . . , ñ}, the boundary âxi can be made to be as close to ãi−1 as desiredRby making x suﬃciently close to 1. Therefore, we have limx↑1 Ŵ(x) = EUCS.
R
This implies that if the following Condition C holds, Ŵ(x) > EUCS
for somex ∈ (a1 (s/r), 1).
Condition C . ddxŴ x=1 < 0.
Under Condition C, for some x ∈ (a1 (s/r), 1), the partition {[âxñ+1 , âxñ ), . . . [âx1 , âx0 ]}is finer than the partition {[ãñ , ãñ−1 ), . . . [ã1 , ã0 ]}. Hereafter, we restrict attentionto ((uR , uS ), G) under which Condition C holds. Note that there exists a pairof players’ payoﬀ functions and the prior distribution of state under which
Condition C holds. It is not true that Condition M implies that Condition C. In
Remark 6 in Appendix 5.C, we provide an example in which Condition M issatisfied, while Condition C is not.
We now show the following Proposition 12.
Proposition 12. Fix b > 0 and suppose that ñ ≥ 2 and Condition C holds. Then, there
exists a positive value η̃(b) such that if s/r < η̃(b), there exists a (ñ+1)-element partition37

equilibrium whose outcome ex ante Pareto-dominates all the equilibrium outcomes in
the one-shot cheap talk game.
We prove Proposition 12 by three steps. Let ((σ̂, ρ̂), fˆ) be a partition equilibrium constructed in Proposition 11. First, we show that if s/r < η(b), the set of(ñ + 1)-element partition equilibria that S prefers to all equilibria in Γ(b, s, r, 1)
is nonempty. Second, we show that there exists a positive value η(b) such that
if s/r < η(b), the set of (ñ + 1)-element partition equilibria that R prefers to all
the equilibria in Γ(b, s, r, 1) is nonempty. Finally, we show that there exists a
positive value η̃(b) such that if s/r < η̃(b), the intersection of the above two sets
is nonempty: The formal proof is in Appendix 5.B.
Finally, we show that Condition C is not necessary for a Prato improvement.
Proposition 13. Fix b > 0 and suppose that ñ ≥ 3. Then, there exists a positive
value η̈(b) such that if s/r < η̈(b), there exists a ñ-element partition equilibrium whose
outcome ex ante Pareto-dominates all the equilibrium outcomes in the one-shot cheap
talk game.
Under the strategy profile on which we focus here, information is elicitedin the same way as the previous Proposition 11 and 12, whereas the numberof elements of the equilibrium partition is ñ. Let {[ãxñ , ãxñ−1 ), . . . , [ãx1 , ãx0 ]} be theequilibrium partition with a˜1 x = x ∈ (ã1 , 1). The boundaries {ãxñ , . . . , ãx1 } coincidewith those of the (ñ−1)-element partition equilibrium in the one-shot cheap talkgame, where the state space is [0, x). By the definition, if x = ã1 , the boundaries{ãxñ , . . . , ãx0 } coincide with those of the ñ-element partition equilibrium in the
one-shot cheap talk game. In Appendix 5.D, we show that the above strategy
profile can constitute an equilibrium that leads to a Pareto improvement.

6. Concluding Remarks
In this study, we analyzed a cheap talk game in which an informed senderand an uninformed receiver engage in finite-period communication before thereceiver makes a decision. During the communication phase, the sender sends38

a (cheap talk) message more than once and the receiver can pay money to thesender whenever she receives a message. We have shown that the dependenceof future information on past payments creates an incentive for the receiverto pay money. This result ensures that the receiver makes message-contingent
payments to some extent even in the situation in which there is no contractibility,
and consequently, information transmission can be improved relative to the
one-shot cheap talk communication.
Under the assumption of quadratic preferences and a uniform type distribution, we found an upper bound of the receiver’s equilibrium payoﬀ, andprovided a suﬃcient condition for it to be approximated by the receiver’s payoﬀ under a certain equilibrium. Consequently, when the communication phasehas a suﬃciently large number of periods and the receiver places greater importance on the project than the sender does, multistage information transmission
with voluntary payments can be more beneficial for the receiver than a wideclass of other communication protocols (e.g., mediation, arbitration, and thesender’s optimal signaling).
In this paper, we focused on the multistage unilateral communication. Intuitively, it seems that the sender’s punishment by babbling message can create
the receiver’s payment incentive even in situations in which players engage inmore general communication protocols such as multistage bilateral communication. Hence, a natural question to ask is whether the receiver’s voluntary
payment can work jointly with such general communication protocols? Considering such a model remains for further research.

Appendix
Appendix 3.A

Perfect Bayesian Equilibria

Let H ≡ Θ × M1 × W1 × · · · × MT × WT × Y be the set of sequences of the realized state and players’ actions, (θ, m1 , w1 , . . . , mT , wT , y).29 Let B(H) be the29

In order to avoid confusion, we add a time operator to the players’ action space.

39

Borel algebra on H. Let P denote a probability measure on the measurable
space (H, B(H)) derived from ((σ, ρ), G). Given h ∈ H, the values of players’
payoﬀs, both UR and US , are uniquely determined. Moreover, UR : H → R
and US : H → R are Borel functions. Therefore, the players’ ex ante expected payoﬀs E[UR (y, θ, w)|(σ, ρ)] and E[UR (y, θ, w)|(σ, ρ)] are well-defined.
Let V S (σ, ρ|h(t,1), mt ) and V R ((σ, ρ), f |h(t,2) , wt ) be the continuation payoﬀ of S afterθ
sending mt at h(t,1)
and the continuation payoﬀ of R after paying wt at historyθh(t,2) , respectively.
Definition 2. A strategy profile (σ, ρ) and a belief system f constitute a perfect
Bayesian equilibrium if the following conditions hold. For any t ∈ {1, . . . , T},(t,1)
1. for any h(t,1)∈ HΘ
and mt ∈ supp{σ(·|h(t,1))},θθ

mt ∈ arg max
V S (σ, ρ|hθ(t,1) , m′t ),′mt

2. for any h(t,2) ∈ H(t,2) ,
{
}
R
(t,2)
′
′
ρ(h(t,2) ) ∈ arg max
V
((σ,
ρ),
f
|h
,
w
)
−
w
t
t ,
′
wt

3. for any hT+1 ∈ HT+1 ,
∫
ρ(h

T+1

) ∈ arg maxr′yuR (y′ , θ) f (dθ|hT+1 ),4. the belief system f is consistent with (σ, ρ).
Consistency of the belief system
Given h(t,2) , the belief system induces a probability measure f (·|h(t,2) ) on (Θ, B(Θ)).
Moreover, since S’s behavior strategy σ(M̃, ·|h(t,2) , wt ) : Θ → [0, 1] is measurable for any M̃ ∈ B(Mt+1 ) and wt ∈ Wt , we can define a probability measure P̂(·|h(t,2) , wt ) on (Θ × Mt+1 , B(Θ) ⊗ B(Mt+1 )) as follows: for Θ̃ ∈ B(Θ) and40

M̃ ∈ B(Mt+1 ),∫P̂(Θ̃ × M̃|h(t,2)

, wt ) ≡Θ̃σ(M̃, θ|h(t,2) , wt ) f (dθ|h(t,2) ).

Therefore, we calculate the conditional probability: if P̂(Θ × M̃|h(t,2) , wt ) > 0,thenProb(Θ̃|h(t,2) , wt , M̃) =P̂(Θ̃ × M̃|h(t,2) , wt )P̂(Θ × M̃|h(t,2) , wt ).

We would like to define a conditional probability Prob(Θ̃|h(t,2) , wt , mt+1 ). FixΘ̃ ∈ B(Θ). Then, P̂(Θ̃ × M̃|h(t,2) , wt ) and P̂(Θ × M̃|h(t,2) , wt ) induce probability
measures ν̃ and ν on (Mt+1 , B(Mt+1 )), respectively. Since ν̃ is absolutely continuous with respect to ν, there exists a Radon–Nikodym derivative g(mt+1 ) suchthat for any M̃ ∈ B(Mt+1 ),∫ν̃ =g(mt+1 )ν(dmt+1 ).
M̃Hence, we require that for mt+1 ∈ supp(ν),
f (Θ̃|h(t+1,2) ) = g(mt+1 ),where h(t+1,2) = (h(t,2) , wt , mt+1 ).
If mt+1 < supp(ν), the conditional probability f (·|h(t+1,2) ) can be arbitrary.

Appendix 3.B

Proof of Proposition 2

Consider a truth telling (Bayesian Nash) equilibrium under a direct contract(Θ, ωΘ , yΘ ). Then, the existence of a partition {Iλ }λ∈Λ that satisfies the conditions
2–3 in Definition 1 is trivial. Hence, we have only to ensure that Iλ is convex
for each λ ∈ Λ. First, we show that yΘ : Θ → Y satisfies the following property.
Lemma 3. yΘ (θ) is nondecreasing in θ.

41

Proof of Lemma 3. From S’s incentive compatibility condition, for any θ, θ′ ∈ Θ,uS (yΘ (θ), θ, b) + ωΘ (θ) ≥ uS (yΘ (θ′ ), θ, b) + ωΘ (θ′ ), anduS (yΘ (θ′ ), θ′ , b) + ωΘ (θ′ ) ≥ uS (yΘ (θ), θ′ , b) + +ωΘ (θ).
These inequalities can be simplified intouS (yΘ (θ), θ, b) − uS (yΘ (θ′ ), θ, b) ≥ uS (yΘ (θ), θ′ , b) − uS (yΘ (θ′ ), θ′ , b)
The assumption uS12 (y, θ, b) > 0 yields yΘ (θ) ≥ yΘ (θ′ ) for θ > θ′ .

□From Lemma 3, we immediately obtain the following lemma.
Lemma 4. In a truth telling equilibrium under a direct contract (Θ, ωΘ , yΘ ), if yΘ (θ) =yΘ (θ) for θ < θ, then yΘ (θ) = yΘ (θ) = yΘ (θ) for all θ ∈ [θ, θ]. Moreover, ωΘ (θ) =ωΘ (θ) = ωΘ (θ) for all θ ∈ [θ, θ].
Lemma 4 implies the convexity of Iλ . ^Appendix 3.C

Proof of Proposition 3

Suppose that a fully separating equilibrium ξF exists. Let (Θ, ωF , yF ) be a directcontract under which there exists a truth telling strategy equilibrium that is
outcome equivalent to ξF . Obviously, yF (θ) = yR (θ) = arg max y uR (y, θ). For
truth telling to be incentive compatible, it is necessary to satisfy the following
condition:s · uS (yR (θ), θ, b) + ωF (θ) ≥ s · uS (yR (θ′ ), θ, b) + ωF (θ′ )for all θ′ , θ.

From the first-order condition, we obtain the diﬀerential equationdd
ωF (θ) = −s · uS1 (yR (θ), θ, b) yR (θ).
dθ
dθ
d R
y (θ) > 0, S’s incentive compatibility
Since uS1 (yR (θ), θ, b) > 0 and y′R (θ) ≡ dθ

condition requires that the compensation schedule that induces full revelation42

is strictly decreasing in θ. We obtain∫ 1
ωF (θ) = ωF (1) +θs · uS1 (yR (z), z, b)y′R (z)dz.

(17)

We now show that R’s payment strategy satisfying condition (17) neversatisfies the equilibrium condition. Let H(θ) denote supp{P̃(·|θ)}, where P̃(·|θ)
is the probability measure on (HT+1 , B(HT+1 )) derived from (σ, ρ) given θ: theset of hT+1 that has a positive probability under the given ξF when the true stateis θ.
Step 1: Fix θ ∈ (0, 1) and (m1 , w1 , . . . , mT , wT ) ∈ H(θ). Then, there exists t < T∑such that wt > 0 and wt = 0 for any t > t. Moreover, tt=1 wt = ωF (θ) holds.
If supp{ f (·|(m1 , w1 , . . . , mt )} = {θ}, R has no incentive to pay wt at this history.
Therefore, there must exist at least one θ ∈ supp{ f (·|(m1 , w1 , . . . , mt )} such thatθ , θ. Furthermore, since ωF (θ) < ωF (θ) for θ > θ, we must have θ < θ. Thisimplies that there exists (m1 , . . . , wT ) ∈ H(θ) such that (m1 , . . . , wt ) = (m1 , . . . , wt ),and wt > 0 for some t ∈ {t + 1, T − 1} ≡ T1 .
Step 2: Let t be the maximum number that satisfies wt > 0. From the∑tdefinition of t, we have t=1 wt = ωF (θ). Similar to Step 1, there must exista θ̃ ∈ supp{ f (·|(m1 , w1 , . . . , mt )} such that θ̃ < θ. Furthermore, there exists(m̃1 , . . . , w̃T ) ∈ H(θ̃) such that (m̃1 , . . . , w̃t ) = (m1 , . . . , wt ), and w̃t > 0 for somet ∈ {t + 1, T − 1} ≡ T2 .
For ξF to be an equilibrium, the above operation must be repeated infinitely
regardless of its start point θ. However, this is impossible in the set of finite
numbers. Hence, we conclude that there exists no fully separating equilibrium.
^

Appendix 3.D Proof of Proposition 4
Fix an equilibrium ξ. Let (Θ, ωΘ , yΘ ) be a direct contract under which there exists
a truth telling strategy equilibrium that is outcome equivalent to ξ. Proposition 2shows that ξ is a partition equilibrium.

43

Let [a′ , a′′ ] be an element of the equilibrium partition30 such that a′ < a′′ .
Then, we have
• limθ↓a′ s · uS (yΘ (θ), θ, b) + ωΘ (θ) = s · ψ(|y(a′ , a′′ ) − a′ − b|) + ωΘ (a′ ), and• s · ψ(|y(a′ , a′′ ) − a′′ − b|) + ωΘ (a′′ ) = limθ↑a′′ s · uS (yΘ (θ), θ, b) + ωΘ (θ),
where yΘ (θ) = y(a′ , a′′ ) for any θ ∈ [a′ , a′′ ]. Moreover, since we assume thaty(a′ , a′′ ) < (a′ + a′′ )/2 + b, we obtains · ψ(|y(a′ , a′′ ) − a′ − b|) > s · ψ(|y(a′ , a′′ ) − a′′ − b|).

(18)

Let Θ̂ be the set of all boundaries of equilibrium partition. First, we show
the following Claim 1.
Claim 1. If there exists closed intervals [ak+1 , ak ] and [a j , a j−1 ] such that ak+1 < ak <a j < a j−1 and [ak+1 , ak ], [a j , a j−1 ] ⊂ Θ̂,31 then• ω is strictly decreasing in θ over [ak+1 , ak ] and [a j , a j−1 ], and• limθ↑ak ωΘ (θ) ≡ ω > ω ≡ limθ↓a j ωΘ (θ).
Proof of Claim 1. Since [ak+1 , ak ], [a j , a j−1 ] ⊂ Θ̂, under the given S’s strategy, S oftype θ ∈ [ak+1 , ak ]∪[a j , a j−1 ] reveals the true state. Hence, we obtain yΘ (θ) = yR (θ)
for θ ∈ [ak+1 , ak ] ∪ [a j , a j−1 ].
The first-order condition for S of type θ ∈ [ak+1 , ak ] results in the diﬀerential
equationddωΘ (θ) = −s · uS1 (yR (θ), θ, b) yR (θ).
dθ
dθ
d R
Since uS1 (yR (θ), θ, b) > 0, and y′R (θ) ≡ dθy (θ) > 0, S’s incentive compatibility

condition requires thatωΘ (θ) = ωΘ (ak ) +∫ ak
θ

s · uS1 (yR (z), z, b)y′R (z)dz.

(19)

The same argument holds for the cases of [a′ , a′′ ), (a′ , a′′ ], and (a′ , a′′ )If (ak+1 , ak ) ⊂ Θ̂ is satisfied, the [ak+1 , ak ] ⊂ Θ̂ is also satisfied since Θ̂ is the set of theboundaries of equilibrium partition.
30
31

44

The same argument holds for interval [a j , a j−1 ]. Hence, we obtainωΘ (θ) = ωΘ (a j−1 ) +∫ a j−1θs · uS1 (yR (z), z, b)y′R (z)dz.

(20)

From conditions (19) and (20), the given compensation schedule is strictly decreasing in θ over [ak+1 , ak ] and [a j , a j−1 ].
To simplify the proof, we now suppose that there exists no closed interval[a, a] ⊂ (ak , a j ) such that [a, a] ⊂ Θ̂. The equilibrium payoﬀs of S of type ak and a j
are s · ψ(b) + ωΘ (ak ) and s · ψ(b) + ωΘ (a j ), respectively.
From condition (18), we conclude that s · uS (yΘ (θ), θ, b) + ωΘ (θ) is strictly
decreasing in θ over [ak , a j ] ∩ Θ̂. Therefore, the following must be satisfied
lim s · uS (yR (θ), θ, b) + ωΘ (θ) = s · ψ(b) + ωθ↑ak> s · ψ(b) + ω = lim s · uS (yR (θ), θ, b) + ωΘ (θ).
θ↓a jThis outcome completes the proof of Claim 1.

□

Now, we suppose that there exists an interval which is subset of Θ̂. Let[ak+1 , ak ] be the leftmost interval such that [ak+1 , ak ] ⊂ Θ̂ and ak+1 < ak . By
Claim 1, for almost every θ ∈ [ak+1 , ak ], there is no θ̃ ∈ Θ \ [ak+1 , ak ] such thatωΘ (θ̃) = ωΘ (θ). In the same way as the proof of Appendix 3.C, we can prove that
this result contradicts the fact that the given strategy profile is an equilibrium.32
Therefore, the equilibrium partition does not include any separating interval.
Next, we show that the cardinality of Θ̂ is finite. We prove this by contradiction. Suppose that the cardinality of Θ̂ is countably infinite: {Iλ }λ∈N . Let[an+1 , an ) and [an , an−1 ) be adjacent elements of equilibrium partition. The payment amount S of type θ ∈ [a j , a j−1 ) receives is denoted by ω j . We have thefollowing Claim 2.
Claim 2. # {[an , an−1 ) ∈ {Iλ }λ∈N : ωn ≥ ωn+1 } < +∞.

32

The proof is a straightforward application of each step in Appendix 3.C. Therefore, it isomitted.

45

Proof of Claim 2. Since S of type θ = an is indiﬀerent between y(an+1 , an ) andy(an , an−1 ), the following must be satisfied:s · ψ(|y(an+1 , an ) − an − b|) + ωn+1 = s · ψ(|y(an , an−1 ) − an − b|) + ωn .
Hence, if ωn ≥ ωn+1 holds, we have s · ψ(|y(an+1 , an ) − an − b|) ≥ s · ψ(|y(an , an−1 ) −an − b|). Since s · ψ(|y(an+1 , an ) − an − b|) is increasing in an+1 ∈ [0, an ], if ωn ≥ ωn+1
holds, we must haves · ψ(b) ≥ s · ψ(|y(an , an−1 ) − an − b|).
ψ(|y − an − b|) is strictly increasing in y ∈ [0, an + b] and strictly decreasing iny ∈ [an + b, ∞), and an < y(an , an−1 ) < (an + an−1 )/2 + b. Therefore, if an−1 − an ≤ bholds, we obtain s · ψ(b) < s · ψ(|y(an , an−1 ) − an − b|). This means that if ωn ≥
ωn+1 holds, we must have an−1 − an > b. Therefore, it must be satisfied that
# {[an , an−1 ) ∈ {Iλ }λ∈N : ωn ≥ ωn+1 } < 1/b. This completes the proof of Claim 2.
□
Claim 2 implies that if the cardinality of Θ̂ is countably infinite, there exists
an infinite sequence {[a j , a j−1 )} j∈N ⊂ {Iλ }λ∈N such that ω j < ω j+1 , and ω j , ωΘ (θ)
for θ ∈ [0, 1] \ {[a j , a j−1 )} j∈N . In the same way as the proof of Appendix 3.C, we
can prove that this result contradicts the fact that the given strategy profile is anequilibrium. Therefore, the cardinality of Θ̂ must be finite. Claim 1 and Claim 2
conclude that all equilibria are finite partition equilibria. ^Appendix 3.E

Discussion of Assumption 2

Assumption 2 guarantees Claim 1 that plays a critical role to prove the finitenessof equilibrium partition. To see this, suppose that Assumption 2 does not hold.
Then, if there is a pair of a, a1 , and a such that
1. 0 < a < a1 < a < 1, and 1 − a = a;46

2. ωΘ (θ) = ω1 for θ ∈ (a, a), and

∫1



ω̂
+
s · uS1 (yR (z), z, b)y′R (z)dz for θ ∈ [a, 1],θωΘ (θ) = ∫aω̂ + s · uS1 (yR (z), z, b)y′R (z)dz for θ ∈ [0, a],θwhere uS = ψ;33
3. s·ψ(|y(a, a1 )−θ−b|) = s·ψ(|y(a1 , a)−a1 −b|), s·ψ(b)+ω̂ = s·ψ(|y(a, a1 )−a−b|)+ω1 ,∫1and s · ψ(|y(a1 , a) − a − b|) + ω1 = s · ψ(b) + ω̂ + a s · uS1 (yR (z), z, b)y′R (z)dz;
then
the following strategy profile can be constitute an equilibrium, which has separating intervals.
In the first period, S reveals whether θ belongs to (a, a). If θ ∈ (a, a), R pays ω1 ,and then, S reveals whether θ < a1 . If θ < (a, a), R pays ŵ in period 2. After thispayment, S’s types {ϵ, ϵ + a} pool together and send message mϵ . After receiving∫amϵ , R pays ω(mϵ ) = ϵ s · uS1 (yR (z), z, b)y′R (z)dz in period 2. After receiving ω(mϵ ),S reveals whether θ = ϵ or ϵ + a. If R deviates in terms of payment in a period,
S conveys no information thereafter. S’s incentive compatibility condition is
met by the second and third condition of the abovementioned requirements.
Moreover, if s/r is small enough, R makes a payment to prevent S’s babbling.
However, even if the Assumption 2 is not satisfied, the existence of the pair(a, a1 , a) is not guaranteed. It remains an open question.

Appendix 3.F

Proof of Proposition 5

Since G is the uniform distribution and uR (y, θ, b) = l(|y − θ|), the optimal projectfor R is given by y(aλ , aλ−1 ) = (aλ + aλ−1 )/2 for any [aλ , aλ−1 ) ∈ {Iλ }λ∈Λ . Recall that
there must exists λ̃ ∈ Λ such that ωλ̃+1 ≤ ωλ̃ . Therefore, we obtainψ(|(aλ̃+1 − aλ̃ )/2 − b|) − ψ(|(aλ̃−1 − aλ̃ )/2 − b|) = (ωλ̃ − ωλ̃+1 )/s ≥ 0.

33

Clearly, the given ωΘ (θ) does not hold Claim 1.

47

(21)

Since aλ̃+1 < aλ̃ , the inequality (21) can be simplified intoψ(b) ≥ ψ(|(aλ̃−1 − aλ̃ )/2 − b|).

(22)

Moreover, since aλ̃ < aλ̃−1 , we obtainb ≤ (aλ̃−1 − aλ̃ )/2 − b⇔aλ̃−1 − aλ̃ ≥ 4b.

(23)

Therefore, we obtain∫ aλ̃−1raλ̃

g(θ)u (y(aλ̃ , aλ̃−1 ), θ)dθ = rR∫ aλ̃−1aλ̃l(|(aλ̃−1 − aλ̃ )/2 − θ|)

∫ 4b
<r

l(|2b − θ|).
0

Appendix 4.A

Proof of Proposition 7

For S’s incentive compatibility condition to be satisfied, the partition {[a2 , a1 ), [a1 , 1]}
must coincide with the 2-element equilibrium partition achieved in the one-shotcheap talk game where θ is drawn from the uniform distribution over [0, a1 ).
By CS, the boundary a1 satisfies that
(

)2
(
)2
a1
a1 + a2−s− a1 − b = −s
− a1 − b .
22This equation implies that
1 − a1 = a1 − a2 + 4b.

(24)

Moreover, similar to the condition (8), the incentive compatibility condition
for S of type θ = a2 induces the following equation:w̃ = s{(a2 + a1 )/2 − (a2 + b)}2 − s(a2 /2 − (a2 + b))2 .
The value of w̃ is positive if and only if a1 − a2 > a2 + 4b. This means that48

a1 − a2 > 4b. Hence, we obtain(a2 − 0) + (a1 − a2 ) + (1 − a1 ) = 2(a1 − a2 ) + 4b + a2
> 12b + 3a2 .
Since we now suppose that b ∈ (1/12, 1/4), we obtain 12b + 3a2 > 1. Therefore,boundaries of the partition and the payment w̃ are not well defined. Thisoutcome implies that we cannot construct a 3-element partition equilibrium
described in Proposition 7. ^Appendix 4.B

Proof of Proposition 9First, we now ensure of the optimality of S’s strategy. At history h(t,1) such thatwt′ < w∗t′ or mt′ ≥ at′ for some t′ < t, any type of S randomly sends a message
according to the same distribution, a uniform distribution over [0, 1]. Therefore,
there is no profitable deviation for S at such a history.
At history h(1,1) or h(t,1) such that mt′ < at′ and wt′ ≥ w∗t′ for all t′ < t, if S of typeθ sends mt ≥ at , then he will obtain −s ((at + at−1 )/2 − (θ + b))2 in the future. Oth∑erwise, the continuation payoﬀ of S can be −s ((at̃+1 + at̃ )/2 − (θ + b))2 + t̃l=t w∗lfor some t̃ ∈ {t, . . . , T}. Since at and w∗t satisfy (14)–(15), it is easy to verify thatfor any t̃ ∈ {t, . . . , T},
−s

(

t̃
)2
(
)2 ∑
a + at̃at + at−1− (θ + b) > −s t̃+1
− (θ + b) +w∗l for any θ > at , (25)
2
2
l=t

t̃
)2
(
)2 ∑
at̃+1 + at̃
at + at−1− (θ + b) < −s
− (θ + b) +w∗l for any θ ∈ [at̃+1 , at̃ ),
−s
2
2

(

l=t

(

)2
(
)2
at + at−1
at+1 + at−s− (θ + b) = −s
− (θ + b) + w∗t for θ = at .
22
Moreover, take θ = at , then t solves


t̃
(
)2 ∑




at̃+1 + at̃

∗
−s
.
max 
−
(θ
+
b)
+
w
l



2
t̃∈{t,...,T} 
l=t

49

(26)
(27)

Hence, (25)–(27) imply that there is no profitable deviation for S from ξT .
Next, we ensure of the optimality of R’s strategy. At any history hT+1 ∈HT+1 , the posterior belief f (θ|hT+1 ≡ (h(T,2) , wT )) = f (θ|h(T,2) ) is a uniform distribution supported on an interval whose mid-point is equal to {min I(hT+1 ) +max I(hT+1 )}/2. Therefore, y = {min I(hT+1 ) + max I(hT+1 )}/2 is an optimal projectfor R at any hT+1 ∈ HT+1 .
Consider a history h(t,2) for t ∈ {1, . . . , T − 1}. If wt′ < w∗t′ or mt′ ≥ at′ forsome t′ < t, then R has no chance to obtain additional information in the future.
Therefore, she must pay nothing to S at such a history. If mt′ < at′ and wt′ ≥ w∗t′
for all t′ < t and mt < at ,34 by paying wt ≥ w∗t , R obtains u∗t (wt ):u∗t (wt )= −wt −

T
∑
i=t+1
T
∑

ai
w∗i − rat

T ∫ ai(∑1 aai+1 ati=t(

i+1 + ai2)2
− θ dθ

)
3
3
2
(a
)
(a)
a
a
b
T−1
i
T−1
= −wt −++ (T − 1 − t).
w∗i − r
at
at
48at
48at
i=t+1

On the other hand, by paying wt < w∗t , she obtains ut (wt ):
∫ at

(
)2
1 at
ut (wt ) = −wt − r− θ dθ
0 at 2(at )2= −wt − r.
12
Clearly, u∗t (wt ) and ut (wt ) have a maximum at wt = w∗t and wt = 0, respectively.
Therefore, paying w∗t is optimal for R if and only if u∗t (w∗t ) ≥ ut (0)) ∑T(a)3 (at )2
ai
aT−1 b2 (aT−1 )3−− (T − 1 − t)
+
≥
w∗i .
⇐⇒ r −
at
48at
48at
12
at
i=t
(

(28)

By making a suﬃciently close to 0, the left-hand side of this inequality canbe made as close to r(1/16 − b2 ) as desired and the right-hand side of thisinequality can be made as close to s(1 + 12)(1 + 4b)/16 as desired. It is obviousthat if s/r < (1 − 4b)/(1 + 12b), there exists ã(b, T) > 0 such that if a < ã(b, T),34

R learns θ < at at the immediately preceding stage.

50

{
}
1−4b
then u∗t > ut for any t ∈ {1, . . . , T − 1}. Take a < min 1+12b
,
,
ã(b,
T)
. Then, ξT
T+1
T−1
constitutes an equilibrium. ^Appendix 4.C

Proof of Proposition 10

We now impose a condition, a = {1 − (4b + ε)}/(T − 1), on ξT . Since aT−1 = 4b + ε ∈(4b, 1), ε ∈ (0, 1 − 4b) must be satisfied. Moreover, a = {1 − (4b + ε)}/(T − 1) <(1 − 4b)/(T − 1) < (1 + 4b)/(T − 3) holds. Therefore, if a < (1 + 12b)/(T + 1), ai
and w∗t are well defined. We now suppose that T > T̃(b) ≡ 1/8b + 1/2, and thena < (1 + 12b)/(T + 1) for any ε ∈ (0, 1 − 4b). Let ξε be this modified strategy profileand system of beliefs. The following lemma shows that if r is large relative to s,then ξε can be an equilibrium.
Lemma 5. Fix b ∈ (0, 1/4), and T ≥ T̃(b). Then, for any ε ∈ (0, 1 − 4b), there exists
η(b, T, ε) such that if s/r < η(b, T, ε), then ξε constitutes an equilibrium.
Proof of Lemma 5. It is obvious that the restriction a = {1 − (4b + ε)}/(T − 1) aﬀects
only R’s optimal decision at h(t,2) such that mt′ < at′ and wt′ ≥ w∗t′ for all t′ < t − 1
and mt < at . Therefore, we have only to ensure whether the inequality (28)holds.
The left-hand side of the inequality (28) can be simplified into
aT−1
r
at

{

}
4(at )3 − (aT−1 )3 − (T − 1 − t)a3 aT−1 2−b .
48aT−1at

Since at = aT−1 + (T − 1 − t)a, we obtain4(at )3 − (aT−1 )3 − (T − 1 − t)a3
(at )3(at )2
aT−1 2>>> b2 >
b.
48aT−116aT−1
16
atThis implies that
aT−1
at

{

}
4(at )3 − (aT−1 )3 − (T − 1 − t)a3 aT−1 2−
b > 0.
48aT−1atMoreover, since w∗t > 0 , the right-hand side of the inequality (28) is higher than51

0. Therefore, we obtainu∗t (w∗t ) ≥ ut (0)

⇐⇒
s
<
r

aT−1
at

{ 4(a )3 −(a33T−1 ) −(T−1−t)at

48aT−11s∑T− aT−1b2
at∗ ai
i=t wi at

}
.

(29)

Note that the value of w∗i /s does not depend on s. Now, we conclude that there
exists η(b, T, ε) such that if s/r < η(b, T, ε), then the inequality (29) holds and ξεconstitutes an equilibrium.

□

EUR (ε) denotes the ex ante expected payoﬀ of R under a strategy profile ξε .
EU (ε) = rW(ε) −RT∑w∗i ai .

i=1

rW(ε) denotes the expected revenue from the project under ξε :rW(ε) = −r

T+1 ∫ ai−1 (
∑
a +a
i

i−1

)2
− θ dθ

2
ai
[ i=1
]
(4b + ε)3
1 {1 − (4b + ε)}32
= r −(4b + ε)b −
−
.
48
48
(T − 1)2

There exists ε(b, d) > 0 such that if ε ∈ (0, ε(b, d)), then][
(4b + ε)316
2
> − rb3 − rd.
r −(4b + ε)b −48
3This implies that for any ε ∈ (0, ε(b, d)), there exists T(b, ε, d) such that for any
T ≥ T(b, ε, d),rW(ε) > −16 3
rb − rd.
3(30)

Recall that w∗i is linearly increasing in s for all i ∈ {1, . . . , T}. Suppose thatT ≥ T(b, ε, d). Then, for any ε ∈ (0, ε(b, d)), there exists η̂(b, T, ε, d) such thats/r < η̂(b, T, ε, d) ≡W(ε) + 16b3 /3 + d∑T ∗1
i=1 wi ais52=⇒

EUR (ε) > −16rb3 /3 − rd.

By Lemma 5, if s/r < η(b, T, ε), then ξε constitutes an equilibrium. Therefore,
if s/r < η̃(b, T, ε, d) ≡ min{η̂(b, T, ε, d), η(b, T, ε)}, the strategy profile ξε constitutes
an equilibrium under which EUR (ε) > −16rb3 /3 − rd.
Define T(b, d) and E(b, d) as follows.
{}
T(b, d) ≡ max min T(b, ε, d), T̃(b) , andε∈(0,ε(b,d)){}E(b, d) ≡ ε ∈ (0, ε(b, d)) : T(b, ε, d) = T(b, d) .
Define η(b, T, d) as follows:η(b, T, d) ≡ sup η̃(b, T, ε, d).
ε∈E(b,d)Suppose that ξε constitutes an equilibrium of Γ(b, s, r, T) where R obtains
EUR (ε) > −16rb3 /3 − rd. Consider Γ(b, s, r, T′ ) where T′ > T. Now, construct a
strategy profile ξ′ε by modifying ξε . In particular, under ξ′ε , players follow ξεuntil period T, and then S conveys no information and R never pays money to S
in the future. It is obvious that ξ′ε constitutes an equilibrium of Γ(b, s, r, T′ ) andR’s equilibrium payoﬀ is equal to EUR (ε) > −16rb3 /3 − rd. Hence, taking η(b, d)as η(b, T(b, d), d) completes the proof. ^Appendix 5.A

Proof of Proposition 11

Formally, the strategy profile (σ̂, ρ̂) is defined as follows. At stage 1 in period 1, Sof type θ ≥ âx1 sends a message m1 randomly according to a uniform distribution
over [âx1 , 1], and S of type θ < âx1 sends a message m1 randomly according to a
uniform distribution over [0, âx1 ). If m1 < âx1 , then R pays a certain amount ofmoney, w∗ , to S at stage 2 in period 1. Otherwise, she pays nothing to S. If
m1 < âx1 and w1 ≥ w∗ , then, at stage 1 in period 2, S of type θ ≥ âx2 randomly
sends a message m2 according to a uniform distribution over [âx2 , 1], and S oftype θ ∈ [âxi+1 , âxi ), for i ∈ {2, . . . , ñ}, randomly sends a message m2 according to a
uniform distribution over [âxi+1 , âxi ). Otherwise, S randomly sends a message m2
according to uniform distribution over [0, 1] regardless of his type. In period t ≥53

2, R always pays nothing to S. In period t ≥ 3, S always sends babbling message.
∫
In period T + 1, R chooses a project ρ(hT+1 ) ≡ arg max y uR (y, θ) fˆ(θ|hT+1 )dθ.
∪Let H be the set of all histories where R makes a decision, H ≡ { Tt=1 H(t,2) } ∪HT+1 . The closure of the set {θ ∈ Θ : fˆ(θ|h ∈ H) > 0} is denoted by I(h).
Under the belief system fˆ that is consistent with (σ̂, ρ̂), we obtain I(hT+1 ) ∈{[âxñ+1 , âxñ ], . . . [âx1 , âx0 ], [âxñ+1 , âx1 ]} for any hT+1 ∈ HT+1 . Therefore, at hT+1 such that
I(hT+1 ) = [âxi+1 , âxi ] for i ∈ {0, . . . ñ}, the optimal project for R is y(âxi+1 , âxi ) =∫ âx
arg max y âx i uR (y, θ)g(θ)dθ, and at hT+1 such that I(hT+1 ) = [âxñ+1 , âx1 ], the opi+1∫ âx
timal project for R is y(âxñ+1 , âx1 ) = arg max y âx 1 uR (y, θ)g(θ)dθ. Hence, ρ̂(hT+1 )ñ+1

becomes y(âxi+1 , âxi ) at hT+1 such that I(hT+1 ) = [âxi+1 , âxi ] for i ∈ {0, . . . ñ}, and ρ̂(hT+1 )becomes y(âxñ+1 , âx1 ) at hT+1 such that I(hT+1 ) = [âxñ+1 , âx1 ].
In period t ≥ 2, R always pays nothing to S, which implies that {[âxi+1 , âxi )}ñi=1
must coincide with the ñ-element equilibrium partition achieved in a modelwith one-shot information transmission where θ is drawn from a distribution
with density {g(θ)/G(âx1 )} · 1[0,âx1 ] (θ). Therefore, the boundaries of this partition,{[âxi+1 , âxi )}ñi=1 , must be solutions to the following non-linear diﬀerence equation
whose initial and terminal conditions are âx1 = x and âxñ+1 = 0: for i = 2, . . . , ñ,s · uS (y(âxi+1 , âxi ), âxi , b) − s · uS (y(âxi , âxi−1 ), âxi , b) = 0.

(31)

When âx1 = 1, the solution to (31) induces a partition that coincides with ñelement equilibrium partition in the one-shot cheap talk game.35 Moreover,the solution to (31) varies continuously with respect to initial condition âx1 = x.
Recall that a solution to (1)–(3) in Section 3.1 induces a partition: 0 = ãñ < · · · <ã1 < ã0 = 1. Therefore, there exists x ∈ (ã1 , 1) such that (31) is well defined for
all â1 ∈ (x, 1). Let a1 be the minimum value of x such that for all â1 ∈ (x, 1), thesolution to (31) induces an ñ-element partition: 0 = âxñ+1 < âxñ < · · · < âx1 = âx1 .
Since the solution to (31) does not depends on both s and r, the value of a1 also
does not depends on both s and r.
Suppose that {âx2 , . . . , âxñ+1 } is a solution to (31) where âx1 ∈ (a1 , 1). Then, there
35

Condition M ensures that the above diﬀerence equation has at most one solution for the
given ñ. See pages 1444–1445 of CS (1982).

54

is no profitable deviation for S from σ̂ at any h(2,1)
such that m1 < âx1 and w1 ≥ w∗ .
θMoreover, S conveys no information at any h(2,1)
such that m1 ≥ âx1 , or m1 < a1θ
and w1 < w∗ . The same can be said at any h(t,1)
for t ≥ 3. This implies that ifθ{âx1 , . . . , âxñ+1 } is a solution to (31) where âx1 ∈ (a1 , 1), then σ̂ is optimal for S at anyhθ(t,1) for t ≥ 2.
At stage 1 in period 1, if S of type θ sends m1 ≥ âx1 , then he obtains s ·uS (y(âx1 , 1), θ, b). Otherwise, S of type θ ≥ âx2 obtains s · uS (y(âx2 , âx1 ), θ, b) + w∗ , andS of type θ ∈ [âxi+1 , âxi ), for i ≥ 2, obtains s · uS (y(âxi+1 , âxi ), θ, b) + w∗ . We assume
that uS11 (y, θ, b) < 0 and uS12 (y, θ, b) > 0. Moreover, y(âxi+1 , âxi ) > y(âxi , âxi−1 ) holds.
Therefore, if the following is satisfied
s · uS (y(âx1 , 1), âx1 , b) − s · uS (y(âx2 , âx1 ), âx1 , b) = w∗ , then(32)s · uS (y(âx1 , 1), θ, b) ≥ max s · uS (y(âxj+1 , âxj ), θ, b) + w∗ for θ ≥ âx1 , and(33)

j∈{1,...,ñ}s · uS (y(âxi+1 , âxi ), θ, b) + w∗ > s · uS (y(âx1 , 1), θ, b) for i ≥ 1 and θ ∈ [âxi+1 , âxi ). (34)
When (33) and (34) hold, S has no incentive to deviate from σ̂ at stage 1 inperiod 1.
If w∗ (x) = 0 for some x ≡ âx1 ∈ (a1 , 1), then (1)–(3) has a solution: 0 = âxñ+1 <âxñ < · · · < âx0 = 1. This result means that an equilibrium whose partition
has ñ + 1 steps exists in the one-shot cheap talk game. This is incompatible
with the definition of ñ. Hence, R’s payment, w∗ (x) ≡ s · uS (y(x, 1), x, b) − s ·uS (y(âx2 , x), x, b), which holds for equation (32), is positive for any x ∈ (a1 , 1].
Note that w∗ (1) = s · uS (yR (1), 1, b) − s · uS (y(ã1 , 1), 1, b) > 0 since uS11 (y, θ, b) < 0
and y(ã1 , 1) < yR (1) < yS (1, b).
At any h(t,2) , R has no incentive to increase the amount of payment becausethat does not aﬀect S’s behavior. Therefore, we have only to ensure the optimality of ρ at h(1,2) such that m1 < x. At this history, if R pays w1 < w∗ , then she
obtains u(w1 ):ru(w1 ) = −w1 +
G(x)

∫ x
0

55

)
(
uR y(0, x), θ g(θ)dθ.

On the other hand, by paying w1 ≥ w∗ at history h(1,2) such that m1 < x, sheobtains u∗ (w1 ):r ∑u (w1 ) = −w1 +G(x) i=1
ñ

∗

∫ âx
i

âxi+1uR (y(âxi+1 , âxi ), θ)g(θ)dθ.

Therefore, paying w∗ is an optimal decision for R at h(1,2) such that m1 < x if
and only if u∗ (w∗ ) ≥ u(0) ⇐⇒r ∑G(x) i=1
ñ

∫ âx
i

u

R

âxi+1(y(âxi+1 , âxi ), θ)g(θ)dθ −

r
G(x)

∫ x

(
)
uR y(0, x), θ g(θ)dθ ≥ w∗ . (35)0

Let denote r · V(x) be the left-hand side of the inequality (35). V(x) is continuousRR
in x ∈ (a1 , 1], and V(x) > 0 for x ∈ (a1 , 1]. Moreover, V(1) = EUCS,−EUCS,ui
where
ñ
∫
∫
∑
1
ã
i−1
R
R
≡ 0 uR (y(0, 1), θ)g(θ)dθ.
EUCS,≡ ñi=1 ã uR (y(ãi , ãi−1 ), θ)g(θ)dθ and EUCS,ui
ñ
i
(
)
(
)
Let α(x) be uS y(x, 1), x, b − uS y(âx2 , x), x, b . In the following part, s · α(x) de-notes R’s payment, w∗ (x), which holds for equation (32). Inequality (35) can be
simplified into s/r ≤ V(x)/α(x). Obviously, V(x)/α(x) is continuous in x ∈ (a1 , 1],
and

R
R
EUCS,
− EUCS,ui
V(1)
ñ
= S ∗
> 0.
α(1)
u (y (1), 1, b) − uS (y(ã1 , 1), 1, b)

Therefore, if s/r < η(b) ≡ V(1)/α(1), then {x ∈ (a1 , 1) : s/r ≤ V(x)/α(x)} , ∅. Thisoutcome implies that if s/r < η(b), there exists a non-empty set {x ∈ (a1 , 1) :s/r ≤ V(x)/α(x)} such that ((σ̂, ρ̂), fˆ) constitutes an (ñ + 1)-element partition
equilibrium when x ∈ {x ∈ (a1 , 1) : s/r ≤ V(x)/α(x)}. ^Remark 3. Since V(x)/α(x) > 0 for x ∈ (a1 , 1] and V(x)/α(x) is continuous inx ∈ (a1 , 1], there exists z ∈ (a1 , 1) such that s/r ≤ V(x)/α(x) holds for any x ∈ (z, 1).
Let a1 (s/r) be the infimum value of z. Then, the value of a1 (s/r) is strictly
decreasing and goes to a1 as s/r goes to 0.

Appendix 5.B

Proof of Proposition 12

First, we show the following Lemma 6.
Lemma 6. Fix b > 0 and suppose that ñ ≥ 2. If s/r < η(b), there exists a (ñ + 1)56

element partition equilibrium ((σ̂, ρ̂), fˆ) such that S always strictly prefers ((σ̂, ρ̂), fˆ)
to any equilibrium in the one-shot cheap talk game.
S
Proof of Lemma 6. Now, EUCS
denotes the ex ante expected payoﬀ of S under theñ-element partition equilibrium with {ãñ , . . . , ã0 } in the one-shot cheap talk game.
Let EÛS (x) be the ex ante expected payoﬀ of S under the (ñ+1)-element partition
equilibrium ((σ̂, ρ̂), fˆ) with (ñ + 1)-element partition: {[âxñ+1 , âxñ ), . . . [âx1 , âx0 ]} whereâx1 ≡ x ∈ (a1 (s/r), 1).
In the one-shot cheap talk game, under Condition M, S always strictly prefers
ex ante ñ-element partition equilibrium to any other equilibria. We have
S
EUCS

= s

ñ ∫ ãi−1∑
i=1uS (y(ãi , ãi−1 ), θ, b)g(θ)dθ.

ãi

By Proposition 11, it must be satisfied that s/r < η(b) in order for an equilibrium ((σ̂, ρ̂), fˆ) to exist. Therefore, in what follows, we suppose that s/r < η(b).
The ex ante expected payoﬀ of S under ((σ̂, ρ̂), fˆ) is ñ+1 ∫ x
∑ âi−1
uS (y(âxi , âxi−1 ), θ, b)g(θ)dθ + G(x) · α(x) .
EÛS (x) = s 
âx
i=1

i

(
)
Recall that s · α(x) ≡ w∗ (x) = s · uS y(x, 1), x, b − s · uS (y(âx2 , x), x, b) is positivefor x > a1 , and s · α(x) is continuous in x > a1 .
S
Let ∆(x) denote EÛS (x) − EUCS
. Since limx↑1 ∆(x) = α(1) > 0 and ∆(x) is

continuous in x ∈ (a1 , 1], there exists d < 1 such that d ≥ a1 (s/r) and∆(x) > 0 for all x ∈ (d, 1) .
□This completes the proof of Lemma 6.

Remark 4. Define d(s/r) ≡ inf{d : d ≥ a1 (s/r) and ∆(x) > 0 for all x ∈ (d, 1)}.
Remark 3 shows that a1 (s/r) is decreasing as s/r is decreasing. Furthermore,∆(x) does not depend on both s and r, the Hence, d(s/r) is weakly decreasing as
s/r is decreasing.
Next, we show the following Lemma 7.
57

Lemma 7. Fix b > 0 and suppose that ñ ≥ 2. Then, there exists a positive value η(b)such that if s/r < η(b), there exists x ∈ (a1 (s/r), 1) such that
R
EÛR (x) > EUCS.

Intuitively, R seems to prefer the (ñ + 1)-element partition with {âxñ+1 , . . . , âx0 }to the ñ-element partition with {ãñ , . . . , ã0 } since the former has more steps thanthe latter. As we earlier show, if Condition C holds, then there exists x < 1 such
R
R
that Ŵ(x) > EUCS
. Fix x, then Ŵ(x) − EUCS
is increasing in r. Moreover, sincew∗ is decreasing and goes to 0 as s goes to 0, the expected payment E[w∗ ] is also
decreasing and goes to 0 as s goes to 0. Thus, if r is large enough relative to s,R.
then there exists x such that EÛR (x) > EUCS

Proof of Lemma 7. In common with the proof of Lemma 6, we suppose thats/r < η(b).
R
Let δ(x, s, r) denote {EÛR (x) − EUCS}/r. We obtainñ ∫ ãi−1∑sδ(x, s, r) = Ŵ(x) − G(x) · α(x) −uR (y(ãi , ãi−1 ), θ)g(θ)dθ.
r
ãi
i=1

δ(x, s, r) > 0 holds if and only if

η(b, x) ≡Ŵ(x) −∑ñ ∫ ãi−1i=1 ãiuR (y(ãi , ãi−1 ), θ)g(θ)dθG(x) · α(x)

s
> .
r

Since x belongs to (a1 (s/r), 1) and infx∈(a1 (s/r),1) G(x)α(x) > 0, η(b, x) has a least
upper bound η(b|s/r) = supx∈(a (s/r),1) η(b, x). Under Condition C, η(b, x) > 0 for1

some x ∈ (a1 (s/r), 1). This implies that η(b|s/r) > 0 for s/r < η(b). Moreover,since a1 (s/r) is not increasing as s/r is decreasing, η(b|s/r) is not decreasing as
s/r is decreasing. Therefore, we can take a supremum of s/r ∈ (0, η(b)), which
satisfies η(b|s/r) > s/r. We denote this supremum by η(b).36 This completes theproof of Lemma 7.

36□

Note that η(b) < +∞, since Ŵ(x) −∑ñ ∫ ãi−1i=1 ãiuR (y(ãi , ãi−1 ), θ)g(θ)dθ < +∞ for any x ∈ (a1 , 1).

58

Remark 5. Suppose that x almost equal to 1. Then, the partition under the (ñ+1)element partition equilibrium almost coincides with the partition under the ñelement partition equilibrium in the one-shot cheap talk game. Nevertheless,the expected payment of monetary transfer is high (almost coincides with sα(1)).
RThis means that EÛR (x) ≈ EUCS
− sα(1) for x ≈ 1. Therefore, if s/r < η(b), there

always exists a (ñ + 1)-element partition equilibrium that is unfavorable to R.
Finally, we complete the proof of Proposition 12 by demonstrating that if r
is large enough relative to s, then we can take x ∈ (a1 (s/r), 1) such thatR
S
EÛR (x) > EUCS
and EÛS (x) > EUCS.

Pproof of Proposition 12

Suppose that s′′ /r′′ < s′ /r′ < η(b). In the proof

of Lemma 7, we show that {x ∈ (a1 (s′ /r′ ), 1) : δ(x, s′ , r′ ) > 0} , ∅ and {x ∈(a1 (s′′ /r′′ ), 1) : δ(x, s′′ , r′′ ) > 0} , ∅. Since a1 (s/r) is decreasing as s/r is decreasing,{x ∈ (a1 (s′ /r′ ), 1) : δ(x, s′ , r′ ) > 0} ⊂ {x ∈ (a1 (s′′ /r′′ ), 1) : δ(x, s′′ , r′′ ) > 0}.
dηMoreover, since ddxŴ x=1 < 0, dx

x=1< 0. Furthermore, η(b, 1) = 0 and η(b, x) is

continuous in x ∈ (a1 , 1). Therefore, we obtainlim sup{x ∈ (a1 (s/r), 1) : δ(x, s, r) > 0} = 1.

s/r↓0

Since ∆(x) > 0 for x ∈ (d(s/r), 1) and d(s/r) is not increasing as s/r is decreasing, there exists η̃(b) such that if s/r < η̃(b), then {x ∈ (a1 (s/r), 1) : δ(x, s, r) >0} ∩ (d(s/r), 1) , ∅. This completes the proof of Proposition 12. ^Appendix 5.C

Condition C

Suppose that s · uS (y, θ, b) ≡ −s(y − (θ + b))2 , r · uR (y, θ) ≡ −r(y − θ)2 , and G(θ)
is uniform distribution over [0, 1]. In this case, the boundaries of the partition

59

derived from ((σ̂, ρ̂), fˆ) are given by




1









x
x
âi =  ñ+1−ix − 2b(ñ + 1 − i)(i − 1)

ñ






0

for i = 0,
for i = 1,
for i = 2, . . . , ñ,
for i = ñ + 1.

Proposition 11 shows that for ñ ≥ 2, there exists η(b) such that if s/r < η(b), then((σ̂, ρ̂), fˆ) constitutes an equilibrium whose partition is induced by âxi wherex ∈ (a1 (s/r), 1). Note that y(âxi+1 , âxi ) = (âxi+1 + âxi )/2 for i = 0, . . . , ñ.
The envelope theorem yields
ñ
∑
dâx
d
Ŵ(x) =g(âxi ) i [uR (y(âxi+1 , âxi ), âxi ) − uR (y(âxi , âxi−1 ), âxi )].
dx
dx
i=1

Since limx↑1 âxi = ãi−1 , we obtainñ−1∑dâxj+1dŴ
RR=g(ã j )[u (y(ã j+1 , ã j ), ã j ) − u (y(ã j , ã j−1 ), ã j )]dx x=1dx x=1j=1+g(ã0 )[u (y(ã1 , ã0 ), ã0 ) − u (y (ã0 ), ã0 )]
R

R

R

dâx1
dx x=1.

Therefore, we obtain
 (
) (
)
ñ−1
∑
−ã j + ã j−1 2  ñ − j ( 1 − ã1 )2
 ã j+1 − ã j 2
dŴ

−
+
=
.
 ñ −
dx x=12
2
2
j=1
Since ã j =ñ− j− 2bj(ñ − j),
ñ

) (
)
ã j+1 − ã j 2−ã j + ã j−1 2+> 0 for j = 1, . . . , ñ − 1.
−
2
2
(

60

(36)

Moreover,
 (
 (
) (
)
) (
)
ñ−1
ñ−1
∑
∑
−ã j + ã j−1 2  ñ − j−ã j + ã j−1 2  ã j+1 − ã j 2
 ã j+1 − ã j 2

−

−
<
+
+

 ñ


2
2
2
2
j=1

j=1

<

(

)
1 − ã1 2.
2This establishes ddxŴ x=1 < 0.
Remark 6. Suppose that s·uS (y, θ, b) ≡ −s(y−(θ+b))2 , r·uR (y, θ) ≡ −r(y−θ)2 , andG(θ) is a distribution over [0, 1] with a density g(θ) = −2θ + 2. By Theorem 2 inCS, any solution to (1) satisfies Condition M. By Condition M and uS13 (y, θ, b) > 0,
we obtain dâxi /dx > 0 anduR (y(ã j+1 , ã j ), ã j ) − uR (y(ã j , ã j−1 ), ã j ) > uS (y(ã j+1 , ã j ), ã j , b) − uS (y(ã j , ã j−1 ), ã j , b) ≥ 0.
Since g(1) = 0 and âx1 = x, the second term of (36) is equal to 0 and the first term
is strictly positive. This result means that ddxŴ x=1 > 0.

Appendix 5.D

Proof of Proposition 13

Under the given strategy profile, If S reveals that θ < x in the first period, R
pays w̃(x) = s · uS (y(x, 1), x, b) − s · uS (y(ãx2 , x), x, b). Since each element of theboundaries {ãxñ , . . . , ãx0 } is continuous in x and converges to a corresponding
element of {ãñ , . . . , ã0 } as x goes to ã1 , we must have w̃(ã1 ) = 0.
{}ñ−1ñ−1 ñ−1
Let [ãñ−1
,
ã
),
.
.
.
,
[
ã
,
ã
]
be the equilibrium partition of (ñ − 1)-element0ñ−1 ñ−21partition equilibrium in the one-shot cheap talk game. For any i ∈ {1, . . . , ñ}, theboundary ãxi can be made to be as close to ãñ−1
as desired by making x suﬃciently
i−1, 1), 1, b). Note thatclose to 1. We obtain w̃(1) = s · uS (yR (1), 1, b) − s · uS (y(ãñ−11w̃(1) > 0 since uS11 (y, θ, b) < 0 and y(ã1 , 1) < yR (1) < yS (1, b). Condition M
requires the uniqueness of the solution to the deference equation in Section 3.1.
The necessary payment w̃(x) is strictly positive in x ∈ (ã1 , 1).

61

The following inequality must hold for R’s payment w̃(x) to be optimal:ñ−1 ∫ ãxir ∑G(x) i=1uRãxi+1(y(ãxi+1 , ãxi ), θ)g(θ)dθ −

r
G(x)

∫ x

(
)
uR y(0, x), θ g(θ)dθ ≥ w̃(x). (37)0The left-hand side of (37) is strictly positive when ñ ≥ 3 and x > ã1 . Recall thatw̃(x) ≡ s · α̃(x, b) = s · uS (y(x, 1), x, b) − s · uS (y(ãx2 , x), x, b) > 0 for any x ∈ (ã1 , 1).
Summarizing the above, we conclude that for any x ∈ (ã1 , 1), there exists η̈(b, x)
such that if s/r < η̈(b, x), the given strategy profile constitutes a ñ-element
partition equilibrium.
The players’ equilibrium payoﬀs are given as follows:
EŨR (x) = rW̃ R (x) − s · α̃(x, b),
where W̃ R (x) =∑ñ ∫ ãxi−1
i=1 ãxiuR (y(ãxi , ãxi−1 ), θ)g(θ)dθ, andEŨS (x) = sW̃ S (x) + s · α̃(x, b),

where W̃ S (x) = s∑ñ ∫ ãxi−1
i=1 ãxiuS (y(ãxi , ãxi−1 ), θ)g(θ)dθ. By the definition of W̃ R (x) andS
R
W̃ S (x), W̃ R (ã1 ) = EUCS
and W̃ S (ã1 ) = EUCS
, respectively.
κW̃Now, we ensure that if Condition M holds, ddx> 0 for κ ∈ {R, S}. The
x=ã1

envelope theorem yields
ñ−1
∑
dãx
d R
W̃ (x) =g(ãxi ) i [uR (y(ãxi+1 , ãxi ), ãxi ) − uR (y(ãxi , ãxi−1 ), ãxi )],
dx
dx
i=1

ñ−1
∑
dãxi
dW̃ R
R
R
=
g(ãi )[u (y(ãi+1 , ãi ), ãi ) − u (y(ãi , ãi−1 ), ãi )].
dx x=ãdx x=ãi=1
1

1Condition M guarantees that for i ∈ {1, . . . , ñ − 1}, dãxi /dx > 0 and the following

62inequality:37
uR (y(ãi+1 , ãi ), ãi ) − uR (y(ãi , ãi−1 ), ãi )> uS (y(ãi+1 , ãi ), ãi ) − uS (y(ãi , ãi−1 ), ãi ) = 0.
R

> 0.
Therefore, we obtain dW̃
dx x=ã1

The total derivative of W̃ S (x) isñ−1∑dãxi sd SxW̃ (x) =g(ãi )[u (y(ãxi+1 , ãxi ), ãxi ) − us (y(ãxi , ãxi−1 ), ãxi )]
dx
dx
i=1
∫ x
ñ−1
∑
]
dy(ãxi+1 , ãxi ) ãi [ S
u1 (y(ãxi+1 , ãxi ), θ, b)g(θ)dθ .
+
dx
ãx
i=0
i+1

Since ãi is a solution of the deference equation in Section 3.1, the first term isequal to 0 at x = ã1 . Hence,∫ ãi [ñ−1∑]dy(ãxi+1 , ãxi )dW̃ S=+
uS1 (y(ãi+1 , ãi ), θ, b)g(θ)dθ .
dx x=ã
dx
ãi+1
x=ã
i=0
1

1

Condition M guarantees that dy(ãxi+1 , ãxi )/dx > 0. Moreover, y(ãi+1 , ãi ) maximizes
∫ ãi
∫ ãi
W̃ S
R
> 0 byu(y,θ)g(θ)dθ≡uS (y, θ, 0)g(θ)dθ. Therefore, we obtain ddx
ã
ã
x=ã
i+1

i+1

1

our assumption that uS13 (y, θ, b) > 0.
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conclude that there exists η̈(b) such that if s/r < η̈(b), for κ ∈ {R, S} EŨκ (x) > EUCS

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37

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