Communication with Language Barriers�

Francesco Giovannoni† Siyang Xiong‡

February 23, 2017

Abstract

We consider a general communication model with language barriers, and study

whether language barriers harm welfare in communication. Contrary to the negative

result in Blume and Board (2013), we provide two positive results. First, the negative

effect of any language barriers can be completely eliminated, if we introduce a new

communication protocol called N-dimensional communication. Second, even if we

stick to the classical 1-dimensional communication (as in Crawford and Sobel (1982)),

for any payoff primitive, there exists some language barriers whose maximimal equi-

librium welfare dominates any cheap-talk equilibrium under no language barriers.

�We thank....†Department of Economics, University of Bristol, Francesco.Giovannoni@bristol.ac.uk‡Department of Economics, University of Bristol, siyang.xiong@bristol.ac.uk

1

Government officer: ”Why don’t they just speak English?”

Dr. Eleanor Arroway: ”Maybe because 70% of the planet speaks other lan-

guages? Mathematics is the only truly universal language. It’s no coincidence

that they are using primes.”

— Contact (the movie) 1997

1 Introduction

Communication is about transmission of information, so that a natural question to ask is

“what” information is actually transmitted and this has been the focus of the literature on

strategic communication, or “cheap talk.” This literature, however, has typically ignored

the issue of “how” information is transmitted. Yet, everyday experience suggests the

intuition that how information is transmitted may hinder or help communication. For in-

stance, it is notoriously hard to convey humor or any other emotion in modern electronic

communication but once electronic communication became sufficiently important, emo-

tions were developed to deal with this very issue (Curran and Casey (2006)). Similarly,

there is a debate on the appropriateness of releasing medical records to patients where

one of the concerns is that patients may not be able to understand medical jargon and so

the common suggestion is to avoid such jargon when likely to cause misunderstandings

(see Ross and Lin (2003) for a survey). In general, the importance of being able to com-

municate effectively is amply recognized in many fields. For example, good rhetorical

skills are considered crucial for modern politicians to the point that there is now concern

that comprehensive access to the media has made the rhetorical component of political

communication much more important than its substantial content (see Spence (1973) and

McNair (2011)). But the ability to communicate effectively is also obviously very impor-

tant in many other fields such as marketing and sales, law and, obviously, academia.1.

So, in order to fully understand the process of communication and to determine to what

extent it can be successful, it is very important to not just to determine what the parties

involved want to say, but also how they say it. In this paper, we take the “how” issue

1A large part of Thomson (2001) is dedicated to discussing issues that ultimately boil down to how to

communicate research findings in Economics.

2

seriously and study “language barriers”, as introduced in Blume and Board (2013), in

one-shot communication games. These language barriers allow us to model the possibil-

ity that in situations of strategic communication individuals may not be able to send or

understand certain messages.

To get a more precise intuition for our results, consider a standard sender-receiver

model, where a sender (S) privately observes the payoff-relevant state t 2 T, and then

sends a message m 2 M to the receiver (R), where M denotes the set of all possible mes-

sages. The receiver cannot observe t, but she has to take a payoff-relevant action a 2 A

upon receiving m. However, each player i may understand only a subset of the messages,

denoted by λi � M. Following Crawford and Sobel (1982), almost all of previous papers

implicitly assume common knowledge of λi = M for every i: we say “language barriers”

do not exist, if this holds, and exist otherwise. When language barriers exist, a subset

λi � M denotes a language type of player i, while ΛS and ΛR represent the sets of all

language types of the sender and the receiver, respectively. Then, a common prior on

T �ΛS �ΛR defines a standard Bayesian game, which is the “language barriers” model

proposed in Blume and Board (2013). This provides a parsimonious way to study a fun-

damental question: do language barriers improve or harm welfare of communication, or

equivalently, is equilibrium welfare under language barriers bigger or smaller than that

under no language barriers? A first answer is provided in Blume and Board (2013), who

show that in the presence of language barriers with language types being private infor-

mation, there will necessarily be indeterminacies of meaning in common-interest games.2 A

direct consequence of their result is that though an efficient equilibrium always exists in

common-interest games without language barriers, efficiency is impossible in the pres-

ence of private information over language types. Facing this negative result, we pursue

this fundamental question in two directions, a normative one and a positive one. We first

ask: is there a natural communication protocol that can eliminate the negative welfare

effect of any language barriers? Our second question is: for any payoff primitive, can

we find some language barriers that (weakly) improve equilibrium welfare? We provide

positive answers to both questions.

2Indeterminacies of meaning arise when, in the presence of language barriers, players’s equilibrium

strategies are such that they would want to deviate if they knew their opponent’s language type.

3

Our first main result is inspired by a phenomenon we observe in real-life commu-

nication, which is that messages are formed by combining basic units to make complex

structures that convey meaning; such structures can always achieve ever higher levels of

complexity, depending on how complex the meaning is. So, the English language can

form a relatively simple sentence structure to convey a simple message such as “close the

door”, but can build much more complex structures if communication requires it.3 Thus,

it seems restrictive in modeling communication to assume a fixed number of messages,

each with a predetermined level of complexity rather than assuming that such messages

can always be used as building blocks capable of forming more complex structures. The

communication protocol in Crawford and Sobel (1982) or Blume and Board (2013) which

is 1-dimensional, implicitly forbids forming such more complex structures, so we relax

this assumption in the simplest way possible by assuming that the set of available mes-

sages extends to a multi-dimensional set MN (for some finite integer N). This simple

change in assumptions allows us to describe to some extent this self-generating property

of real-life communication (aside from languages, think of binary codes in computer sci-

ence and Morse code) and yet, to the best of our knowledge, we are the first paper to

formalize this in the literature on strategic communication.

To be more precise, in 1-dimensional communication, as in Crawford and Sobel

(1982) or Blume and Board (2013), for a given M, the sender is allowed to send a mes-

sage m 2 M . Under our N-dimensional communication protocol, the sender is allowed

to send a message m 2 MN.4 In addition, N-dimensional communication must respect

language barriers, if they exist. In particular, type λS (� M) of the sender can send only

messages in (λS)N, and type λR (� M) of the receiver can only understand messages

in (λR)N. Our first main result is that any (finite) equilibrium which would obtain in a

game with one-dimensional communication and no language barriers can be mimicked

by an equilibrium of the same game if we added any language barriers but allowed for

N-dimensional communication (for sufficiently large N). In this sense, the negative ef-

3Chrystal (2006) discusses how the one of the fundamental characteristics of any language is a hierar-

chical structure of its syntax.4We consider only one-shot messages, and as a result, N-dimensional communication is not a conver-

sation (i.e., N-round communication). Furthemore, it is different from multi-dimensional cheap talk as in

Battaglini (2002) and Levy and Razin (2007), where the multiple dimensions there refers to the dimensions

of the payoff types.

4

fect of language barriers can be completely eliminated, if we allow for multi-dimensional

communication.

Technically, there are three obstacles for effective communication in the presence of

language barriers: (1) the sender may not know the receiver’s language type, and hence

may not know what messages to send; (2) the receiver may not know the sender’s lan-

guage type, and hence may not know how to interpret a received message; (3) there may

not be enough commonly known messages to transmit all the information. In Section

4.2.1, we show all of the three obstacles can be overcome with multi-dimensional com-

munication but there is one important point that is worth emphasizing. It is obvious that

multi-dimensional communication enlarges the set of possible messages available to the

sender, which may lead one to wonder whether this is all that matters. In fact, this point

resolves only the third technical obstacle mentioned above, but does not eliminate asym-

metric information regarding the sender’s and the receiver’s language types. In section

4.2.1, we show through a couple of examples that it is the (multiple) dimensionality of the

communication that overcomes asymmetric information about language types.

In the second part of the paper, we tackle the second question in the context of

1-dimensional communication. In particular, we follow Goltsman, Horner, Pavlov, and

Squintani (2009) and Blume and Board (2010) in comparing welfare across several proto-

cols for cheap-talk communication: arbitration, mediation, language barriers and noisy

talk. Our main result is a linear ranking of the maximal welfare achieved in these different

protocols:

ΦLB � ΦM � ΦILB � ΦN,

where ΦLB, ΦM, ΦILB, ΦN are the maximal equilibrium welfare achieved in a generic

sender-receiver game under language barriers, mediation, language barriers with the re-

striction that language types are distributed independently of payoff states (we refer to

these as independent language barriers from now on), and noisy talk, respectively. One

immediate implication is that, for any payoff primitive, there exist some language barri-

ers whose maximal equilibrium welfare (weakly) dominates any equilibrium in the cor-

responding game without language barriers.

While both Goltsman, Horner, Pavlov, and Squintani (2009) and Blume and Board

5

(2010) ask a very similar question, methodologically our approach is quite different. Golts-

man, Horner, Pavlov, and Squintani (2009) and Blume and Board (2010) consider the case

of quadratic preferences and uniform payoff distribution; they first argue mediation pro-

vides an upper bound to the welfare achievable under language barriers with indepen-

dence and noisy talk, and then construct a specific equilibrium under such language bar-

riers and under noisy talk which achieve the welfare upper bound, i.e., they establish an

equivalence result on (maximal) welfare for the three protocols. Instead, we go to the

roots of the incentives underneath each protocol, and show that equilibria with language

barriers, mediation, independent language barriers and noisy talk correspond to a series

of increasingly restrictive incentive compatibility conditions in that order, which gener-

ates the welfare order described above. Because of this approach, our results go beyond

the environment with quadratic preferences and uniform payoff distribution and indeed

hold for any general preference and distributional assumptions. We provide two further

results. Firstly, we consider two possible notions of arbitration, which are simply forms

of mediation where one of the two incentive constraints - the sender’s incentive to reveal

the truth to the mediator and the receiver’s incentive to follow the mediator’s suggested

actions - are relaxed. The first notion of arbitration corresponds to the one defined in

Goltsman, Horner, Pavlov, and Squintani (2009), where it is assumed that the receiver

must play the strategies recommended by the arbitrator whereas the sender must still be

incentivised to reveal the payoff state. Compatibly with Myerson (1991)’s terminology

we call this arbitration with adverse selection. The second notion of arbitration is absent in

the previous literature but is a modification of mediation where it is assumed the sender

must truthfully report the payoff state to the arbitrator whereas the receiver must still

be incentivised to follow the arbitrator’s recommended action. We call this arbitration

with moral hazard. Given these definitions, we show that the maximal equilibrium wel-

fare achieved with language barriers dominates that under arbitration with moral hazard

whereas no such ranking can be established with regard to arbitration with adverse se-

lection. This immediately establishes that both arbitration and language barriers welfare-

dominate mediation, but a general ranking between arbitration and language barriers is

not possible.

Our final result shows, through an example, that the welfare equivalence between

mediation, independent language barriers and noisy talk established by Goltsman, Horner,

6

Pavlov, and Squintani (2009) and Blume and Board (2010) is not robust if we relax the

uniform-distribution assumption on payoff states.

The remainder of the paper proceeds as follows: we discuss the literature in Section

2; we describe the model in Section 3; Section 4 shows how N-dimensional communica-

tion can always replicate equilibria obtained without language barriers, no matter what

these are; Section 5 shows how some language barriers can improve welfare even under

1-dimensional communication and compares such language barriers to other noisy com-

munication protocols; Section 6 concludes and Appendix contains all the proofs not in

the main part of the paper.

2 Literature Review

The literature on communication in games of asymmetric information is very large. Craw-

ford and Sobel (1982) introduced the canonical “cheap talk” setting, with an informed

“sender” who sends (costless) messages to an uninformed “receiver” who, in turn, takes

an action which affects them both. Since then, a vast literature has developed, which ex-

tended the analysis in many different directions. For example, beginning with Milgrom

(1981), there is significant amount of work that considers communication when messages

are (costless) evidence so that lying is not allowed, including Kartik (2009) where ly-

ing is arbitrarily costly. Another important areas of research are those where the anal-

ysis has been extended to multiple senders or multi-dimensional payoff state spaces (e.g.

Battaglini (2002), Chakraborthy and Harbaugh (2007) and Levy and Razin (2007)) or to is-

sues of commitment amongst the parties: Dessein (2002) and Krishna and Morgan (2008)

focus on various types of commitment on the part of the receiver while Kamenica and

Gentzkow (2011) assume the sender commits ex-ante to an informational mechanism.

Finally, there are important extensions which consider the dynamics of interactions be-

tween senders and receivers when their preferences differ (e.g. Sobel (1985) and Morris

(2001)) or when there is uncertainty about the quality of the sender’s information (e.g.

Scharfstein and Stein (1990) and Ottaviani and Sorensen (2006)).

In all of this literature, one assumption is that language is never an issue. A signifi-

7

cant exception is Farrell (1993), where the issue of how exactly information is transmitted

is taken seriously. There is a “rich language assumption”, which excludes language barri-

ers, and the crucial restriction is that messages come with some intrinsic meaning. Thus,

for Farrell (1993), the restriction is not that players cannot use or understand some mes-

sages but rather that, whenever credible, messages should be taken literally.

Still, a few authors have argued that language is necessarily too coarse for com-

munication in certain environments. For example, Arrow (1975) discusses the reasons of

organizational codes and both Cremer, Garicano, and Prat (2007) and Sobel (2015) model

such codes by using a setting where messages are too few to avoid ambiguity. While

our results suggest that N-dimensional communication can overcome all such issues, in

those environments there may be reasons, such as the complexity or the time needed to

develop and understand such messages, that pose substantial limits on how much can be

done with them. In other contexts, on the other hand, it is likely that successful commu-

nication is so important that such complex messaging strategies are worth pursuing. For

example, in the Arecibo Message Project a message was broadcast from Earth to potential

intelligent alien civilizations. This message contains information about our DNA and our

solar system and is encoded using a binary system not dissimilarly from our equilibrium

construction. The science fiction novel Contact (by Sagan (1985)) also addresses the is-

sue of one-shot communication in the presence of language barriers and it too provides

a solution where a common language is established before the content of the message is

delivered.5

As already discussed, the closest work to ours is Blume and Board (2013) who in-

troduce the notion of language types and use it to describe language barriers. The focus

in Blume and Board (2013) is on describing how even in common interest games, several

inefficiencies do arise as a result of language barriers. We adopt the same framework but

consider any communication game (not just common-interest) and introduce the notion

of N-dimensional communication. We show that such communication protocol can repli-

cate any equilibrium of the corresponding game without language barriers. Blume (2015)

looks again at the issues raised by language barriers in a sender-receiver context where

5Sagan also participated in the design of the various messages attached to the two Pioneer and two

Voyager probes. For a scientific discussion of communication with extra-terrestials, see D.A. Vakoch (2011).

8

the sender still has private information about her language type but there is no common

prior on it. We do not focus on higher-order uncertainty but due to the ex-post nature of

our results, these would be robust in such settings.

In our paper, we also look at whether particular language barriers can improve upon

communication in non-common interest setting. A few papers have particular relevance

to our work here. Krishna and Morgan (2004) show that more (Pareto) efficient equilibria

may be obtained by allowing for the informed sender and uninformed receiver to ex-

change messages at a first stage and then allowing the sender sends a second message.

The N-dimensional messages in our setting should not be interpreted as a conversation

as all communication takes place in a single stage. Blume, Board, and Kawamura (2007)

show that the exogenously given possibility of an error in communication actually im-

proves communication in equilibrium, while in our setting it is exogenous language bar-

riers that provide such results. In fact, Goltsman, Horner, Pavlov, and Squintani (2009)

provide an upper bound on ex-ante efficiency if mediation is introduced in the model and

show that both Krishna and Morgan (2004) and Blume, Board, and Kawamura (2007) at

best can reach, but not surpass that bound.6 Blume and Board (2010) study language bar-

riers under the assumption of independence between the priors on language types and

payoff types and argue that the efficiency bound can be reached by language barriers. We

extend those results to a class of much more general communication games and provide

a linear ranking amongst all these communication protocols. In particular, we show that

under the independence assumption but in this general setting, optimal language barri-

ers will always do no worse than optimal noisy talk and provide an example where they

do strictly better. This implies that, in general and in contrast with the conclusions drawn

in Goltsman, Horner, Pavlov, and Squintani (2009) and Blume and Board (2010), noisy

communication cannot always achieve the efficiency bound obtained through mediated

communication. Finally, we go beyond the independence assumption between payoff

and language types and show that the optimal such language barriers can do better than

mediation, whereas we show with an example that a comparison with arbitration cannot

6Ganguly and Ray (2011) argue that any noisy communication protocol requires a larger set of messages

than those used in the standard Crawford and Sobel (1982) setting. They show that simple mediation,

where no more messages can be used than in the corresponding Crawford and Sobel (1982) setting, does

not improve on such setting.

9

be made without specifying the welfare function.

3 Model

Let I denote a finite set of agents, and for every agent i 2 I, we use Ai and Ti to denote

the sets of actions and payoff states of agent i, respectively. Throughout the paper, we

utilize the notational convention that a subscript i refers to agent i whereas no subscript

refers to all agents. Thus, A � ∏i2I Ai and T � ∏i2I Ti. Agent i has the utility function

ui : T � A �! R.

Let M denote the set of all possible messages. For every agent i 2 I, we use a non-

empty Λi � 2M� f?g to denote the set of language types of agent i. Each language type

λi 2 Λi is defined as the set of messages that agent i understands. There is a common

prior π on T � Λ, and let πT and πΛ denote the marginal distributions on T and Λ,

respectively. We will sometimes impose the following assumption, and we will state it

explicitly, if we do.

Assumption 1 T and Λ are independently distributed.

We use jXj to denote the cardinality of of a set X. Throughout the paper, we assume

jMj > 1 and jΛj < ∞. As usual, �i represents I�fig, and x�i represents�xj�

j2I�fig.

For any positive integer N, we define a N-dimensional communication game. Before

the game starts, nature chooses a state-type profile (t, λ) according to π. Then, upon

privately observing (ti, λi), every agent i 2 I sends an N-dimensional message mi ��m1

i , ..., mNi�2 (λi)

N. Finally, upon observinghti, , λi,

�mj�

j2I

i, every agent i 2 I takes an

action ai 2 Ai.7

7This setting allows for everyone to be both a “sender” and a “receiver” but can easily be accommodated

to allow for the cases where only some players are senders and/or only some players are receivers. For the

former, it suffices to impose that some players (the non-senders) have a singleton payoff state space and for

the latter, it suffices to impose that some players (the non-receivers) have a singleton action space.

10

Thus, a game is defined by a tuple

I, M, T, Λ, π, A, (ui : T � A �! R)i2I , N�, and

players’ strategies in the game are8��σi : Ti �Λi ! MN

�,�

ρi : Ti �Λi ��

MN�I! Ai

��, 8i 2 I,

such that σi and ρi are measurable with respect to Λi. More precisely, the measurability

of σi means σi (ti, λi) 2 (λi)N for every (i, t, λ) 2 I � T �Λ. The interpretation is that a

language type λi understands only the messages with which he is endowed, and hence

this type can send a string of messages only in λi, i.e., the restriction σi (ti, λi) 2 (λi)N

defines “language barriers” for agents when sending messages.

We define the measurability of ρi as follows. Define

x �λi y if and only if x = y 2 λi or fx, yg \ λi = ?,

i.e., type λi can distinguish two messages with which he is endowed, but treats all the

other messages as a single and distinct “nonsense” message. Then, for any positive inte-

ger K, define�x1, ..., xK

��λi

�y1, ..., yK

�if and only if xk �λi yk, 8k 2 f1, ..., Kg .

The measurability of ρi means

�mj�

j2I �λi

�m0

j

�j2I=) ρi

hti, λi,

�mj�

j2I

i= ρi

�ti, λi,

�m0

j

�j2I

�,

8�i, t, λ, m, m0� 2 I � T �Λ� MN � MN.

We use “m �λi m0” to denote that “m �λi m0 is false”. This measurability requirement

captures “language barriers” for agents when receiving messages.

Given a strategy profile (σ, ρ) � (σi, ρi)i2I , the final utility of agent i given a state-

type profile (t, λ), denoted by Ui (σ, ρjt, λ), is defined as follows.

Ui (σ, ρjt, λ) = ui

� htj, ρj

�tj, λj, [σl (tl, λl)]l2I

�ij2J

�. (1)

8For notational ease, we focus on pure strategies. The analysis can be extended to mixed strategies in a

straightforward way, but requires much more messy notation.

11

Define

Ui (σ, ρ) =Z

T�ΛUi (σ, ρjt, λ)π (dt, dλ) , 8i 2 I,

i.e., Ui (σ, ρ) is player i’s expected payoff given the strategy profile (σ, ρ).

Instead of considering (language-)interim equilibria as in Blume and Board (2013), we

adopt the stronger solution concept of (language-)ex-post equilibrium.9 Given any (ti, λ) 2Ti �Λ, let π (�jti, λ) denote the distribution of t�i conditional on (ti, λ).

Definition 1 (σ, ρ) is an equilibrium ifZT�i

�Ui (σ, ρ j ti, t�i, λ)�Ui

��σ0i, σ�i

�,�ρ0i, ρ�i

�j ti, t�i, λ

��π (dt�ijti, λ) � 0, (2)

8i 2 I, 8 (ti, λ) 2 Ti �Λ, 8�σ0i, ρ0i

�.

Note that equation (2) describes the (language-)ex-post incentive compatibility of

agent i in the equilibrium: knowing (ti, λ), agent i chooses the best strategy. In Blume and

Board (2013), (language-)interim equilibria are instead defined as: knowing (ti, λi), agent

i chooses the best strategy. For them, “indeterminacies of meaning” arise when there is

a (language-)interim equilibrium that is not a (language-)ex-post equilibrium. Therefore,

“determinacy of meaning” is embedded in our equilibrium notion.

Throughout the paper, we impose the following necessary assumption for informa-

tive communication.

8i 2 I, jλij � 2, 8λi 2 Λi, (3)

π��(t, λ) 2 T �Λ : λi \ λj 6= ?, 8i, j 2 I with i 6= j

�= 1. (4)

(3) says that every player is able to transmit non-trivial information (i.e., jλij �2). (4) says that, any two language types of two distinct agents must have non-empty

intersection, because communication is not informative otherwise.10

9The “ex-post” is defined with respect to realization of language types (rather than payoff states). That

is, our equilibrium provides best replies for all agents, even if all the agents’ language types were truthfully

revealed. Clearly, this is much stronger than the corresponding “interim” and “ex-ante” equilibria.10Blume and Board (2013) assume existence of a common message for all language types of all players,

which, clearly, is stronger than (4).

12

4 Main Results: N-dimensional Communication

In this section, we show that any equilibrium in a communication game with no language

barriers can be replicated by an equilibrium of the corresponding game if we introduce

language barriers. In Section 4.1, we first define what this means formally; in Section 4.2,

we prove our main result for this section and illustrate some of its implications.

4.1 Similar games and outcome-equivalent equilibria

We will compare equilibria between communication games which only differ in whether

language barriers exist or not. To make the comparison between two such games mean-

ingful, they must be “similar” i.e., they must share the same primitives in terms of agents,

actions, payoffs, etc. Rigorously, we apply the following definition

Definition 2 Two games bG and eGbG =

DbI, bM, bT, bΛ, bπ, bA,�bui : bT � bA �! R

�i2I

, bNE ,

eG =DeI, eM, eT, eΛ, eπ, eA,

�eui : eT � eA �! R�

i2I, eNE ,

are similar if DbI, bM, bT, bA, (bui)i2I

E=

DeI, eM, eT, eA, (eui)i2I

E,

and bπbT = eπeT.

That is, two similar games may differ only in language types and the dimension of

messages they send. We now define outcome-equivalent equilibria in two similar games.

Definition 3 Given two similar games, bG and eG, an equilibrium (bσ,bρ) in bG is outcome-equivalent

to and an equilibrium (eσ,eρ) in eG if

bρi

�ti, bλi,

�bσj

�tj, bλj

��j2I

�= eρi

�ti, eλi,

�eσj

�tj, eλj

��j2I

�, 8�

i, t, bλ, eλ� 2 I� T� bΛ� eΛ,

13

Outcome-equivalent equilibria in similar games induce the same action profile for

any given profile of payoff types, regardless of language types. As a result,

Ui (bσ,bρ) = Ui (eσ,eρ) , 8i 2 I,

i.e., they induce the same expected utility for every player.

4.2 Outcome-equivalence for similar games

For any game G =

I, M, T, Λ, π, A, (ui)i2I , N�, define

N� � 1, λ�i � M, Λ

�= ∏

i2I

hΛ�i � fλ�i g

i;

π� h

E�Λ�i= πT (E) , 8E � T,

i.e., G� =D

I, M, T, Λ�, π

�, A, (ui)i2I , N

�= 1

Eis the standard communication game with

1-dimensional messages and no language barriers, which is similar to G. Our first main

result says that for any equilibrium of G�, there exists an outcome-equivalent equilibrium

of G, if N is sufficiently large. In this sense, language barriers bring no harm to welfare.

Let (σ�, ρ�) denote an equilibrium of G�. Let E (σ�,ρ�)

i denote the set of messages

agent i sends in the equilibrium, i.e.,

E (σ�,ρ�)

i � fσ�i (ti, λ�i ) : ti 2 Tig .

(σ�, ρ�) is called a finite-message equilibrium if���∏i2I E

(σ�,ρ�)i

��� < ∞ and an infinite-message

equilibrium otherwise. For notational ease, we focus on finite-message equilibria but the

analysis can be easily extended to infinite-message equilibria.

Theorem 1 Suppose Assumption 1 holds. Then, for any finite-message equilibrium (σ�, ρ�) in

any game without language barriers G� =D

I, M, T, Λ�, π

�, A, (ui)i2I , N

�= 1

E, a positive in-

teger bN exists, such that in any similar game G =

I, M, T, Λ, π, A, (ui)i2I , N�

with N � bN,

there exists an equilibrium (σ, ρ) of G that is outcome-equivalent to (σ�, ρ�).

Recall that (σ�, ρ�) and (σ, ρ) being outcome-equivalent means that the two games

induce the same equilibrium action profile for every t 2 T, regardless of λ 2 Λ. In this

14

sense, we say (σ, ρ) replicates (σ�, ρ�). In 4.2.1 and 4.2.2 we prove this result, focusing

first on the role of N-dimensionality in overcoming language barriers, absent the issue of

incentive compatibility and then showing how incentive compatibility is assured.

4.2.1 The role of N-dimensionality in Theorem 1

In this section, we leave incentive compatibility aside, and show that multiple dimensions

of messages suffice for effective communication. We return to incentive compatibility in

the next section. To prove Theorem 1, we need to overcome three technical obstacles: the

first is that senders may not know receivers’ language types; the second is that receivers

may not know senders’ language types; finally, we need to show how players transmit

information using their endowed messages given that they know each others’ language

types. We show that all of these can be achieved by utilizing messages with multiple

dimensions.

We first tackle the problem that senders do not know the receivers’ language types.

With multiple dimensions, this type of asymmetric information is easily eliminated, be-

cause, we can break a N-dimensional message into

�����[i2I

Λi

����� strings, with each string in-

tended for a receiver’s language type. Specifically, suppose N = N0 ������[i2I

Λi

����� for some

integer N0, and a sender’s N-dimensional message is m =�mλi

�i2I,λi2Λi

2h

MN0ij[i2IΛij

,

where mλi 2 MN0is the intended message from the sender to λi. Upon receiving m, the

language type λi just goes to his designated string to retrieve his intended message mλi ,

so that the asymmetric information regarding senders not knowing receivers’ language

types is eliminated. This is analogous to what happens in many tourist attractions, where

information is written in different languages, and tourists from different countries just

jump to the bit written in a language they understand to retrieve the information. Thus,

messages with multiple dimensions do more than just increase the size of the message

space. To see this, consider the following example, with one sender and one receiver who

15

share a common interest. Suppose that

T = A = fα, β, γg

uS (t, a) = uR (t, a) =

8<: 1 i f a = t

0 i f a 6= t

M = Z; ΛS = fλSg ; ΛR =�

λ�R , λ+R

λS = f�100,�99, ..., 0, ..., 99, 100g ; λ+R = f1, 2, ...g ; , λ�R = f�1,�2, ...g

Assume that each t 2 T has positive probability but that the true realization is the

sender’s private information whereas both λ�R and λ+R also have positive probability, but

their realization is the receiver’s private information. Since sender and receiver have iden-

tical preferences, the only issue is how the sender can communicate her information to the

receiver. Clearly, without language barriers, the efficient outcome (i.e., perfect communi-

cation) is an equilibrium. However, it is not an equilibrium for 1-dimensional communi-

cation under these language barriers: to achieve efficiency, λ+R must be able to distinguish

between m (β), m (γ) and m (δ) , the three equilibrium messages from the sender in the

three states. Hence, at least two of the three messages must be in λ+R , and as a result,

λ�R cannot distinguish these two messages, since λ+R \ λ�R = ?, i.e., efficiency cannot be

achieved in an equilibrium. It is also easy to see that even if we increased the number of

messages available to the sender up to the point where λS = M, the same difficulty would

remain. On the other hand, we could achieve full communication even if we restricted

λS to the set f�2,�1, 0, 1, 2g , but allowed 2-dimensional messages. This would allow the

sender to produce a message (m� (t) , m+ (t)) where m� and m+ are the strings that de-

scribe the payoff relevant information for each possible type of receiver.11 Note that in this

example the number of messages is not the issue as when we make λS = f�2,�1, 0, 1, 2g,

even with 2-dimensional messages we have actually reduced the number of messages

available to the sender compared to the case λS = f�100,�99, ..., 0, ..., 99, 100g. Yet, effi-

cient communication can now be guaranteed.

11For instance, there is an equilibrium where the strings m� and m+ are described by

m+ (α) = 0, m+ (β) = 1, m+ (γ) = 2,

m� (α) = 0, m� (β) = �1, m� (γ) = �2,

i.e., λ+R can distinguish m+ (α), m+ (β) and m+ (γ), and type λ�R can distinguish m� (α), m� (β) and m� (γ).

16

The second problem is that, without knowing the senders’ language types, the re-

ceivers do not know how to interpret the senders’ messages. Hence, to achieve effective

communication, the senders must reveal their language types. Fix any N such that

N � 3������[i2I

Λi

����� . (5)

Ignoring incentive compatibility, the following lemma shows that there is a procedure

such that every sender is able to reveal his language type. The proof of Lemma 1 can be

found in Appendix A.1.

Lemma 1 For every (i, j, λ) 2 I� I�Λ with i 6= j, there exists a function Υ(i,λj): Λi �! MN

such that

Υ(i,λj) [λi] 2 (λi)

N , 8λi 2 Λi,

and λi 6= λ0i =) Υ(i,λj) [λi] �λj Υ(i,λj)

�λ0i�

. (6)

Given i 6= j, suppose agent i follows Υ(i,λj)to reveal his language type to agent j

whose type is λj: if agent i is of type λi, he sends Υ(i,λj) [λi] 2 (λi)

N to type λj. For

any two distinct language types of i, λ0i and λ00i , because of (6), λj can distinguish the

message sent by λ0i (i.e., Υ(i,λj)�λ0i�) from the message sent by λ00i (i.e., Υ(i,λj)

�λ00i�). Thus,

the asymmetric information due to receivers not knowing the senders’ language type is

eliminated.

Once again, the N-dimensional nature of messages is crucial. Consider the following

sender-receiver common-interest example where now it is the sender that has language

barriers:

T = A = fα, βg

uS (t, a) = uR (t, a) =

8<: 1 i f a = t

0 i f a 6= t

M = Z; ΛR = fλRg ; ΛS =�

λ0S, λ00S , λ000S

λR = f1, 2g ; λ0S = f1, 2g ; λ00S = f1, 3g ; λ000S = f2, 3g

17

Suppose all language types and all payoff states have positive probability. Clearly, with-

out language barriers, the efficient outcome (i.e., perfect communication) is an equilib-

rium. However, it is not an equilibrium for 1-dimensional communication for these par-

ticular language barriers. To see this, suppose otherwise. Then, to achieve efficiency for

λ0S, states α and β must be truthfully revealed by messages 1 and 2. Without loss of gen-

erality, λR plays α and β upon receiving messages 1 and 2, respectively. As a result, if λR

plays α upon receiving messages 3, then efficiency is not achieved for λ00S ; if λR plays β

upon receiving messages 3, then efficiency is not achieved for λ000S —we get a contradiction.

Nevertheless, because of Lemma 1, an equilibrium with multi-dimensional mes-

sages�mλ, mt (λ)

�which guarantees full communication exists. In such equilibria, the

first component, mλ identifies the sender’s language type while the second component

identifies the payoff type. As in the previous example, giving arbitrary additional mes-

sages to each sender type would not work because the receiver would not be able to

understand such messages.

The last obstacle is technical and amounts to making sure that, once asymmetries

of information about language types are resolved, there are still enough dimensions to

convey the payoff relevant information. For a given hΛ, πi , some sender’s language type

λi may have fewer messages than needed to replicate (σ�, ρ�), i.e., jλij <���E (σ�,ρ�)

i

���, where

E (σ�,ρ�)

i denotes the set of messages player i sends under (σ�, ρ�). We show this too can

be overcome by multiple dimensions via the following lemma and its proof can be found

in Appendix A.2. Fix any positive integer bN such that

bN � maxn���E (σ�,ρ�)

i

��� : i 2 Io

. (7)

Lemma 2 For every (i, j, λ) 2 I� I�Λ with i 6= j, there exists a function Γ(λi,λj): E (σ

�,ρ�)i �!

(λi)bN such that for any m, m0 2 E (σ

�,ρ�)i ,

m 6= m0 =) Γ(λi,λj) (m) �λj Γ(λi,λj)

�m0� . (8)

By the previous two steps, both senders’ and receivers’ language types can be truth-

fully revealed. Given this and i 6= j, suppose sender λi follows Γ(λi,λj)to send messages

18

to receiver λj, and Γ(λi,λj)translates equilibrium messages in E (σ

�,ρ�)i to i’s endowed mes-

sages in (λi)bN. Then, for any two distinct messages m0, m00 in E (σ

�,ρ�)i , because of (8),

receiver λj can distinguish the two translated messages, Γ(λi,λj) (m0) and Γ(λi,λj) (

m00),

i.e., equilibrium messages are effectively transmitted. In this case, N-dimensional mes-

sages have the role of increasing the number of messages at the sender’s disposal, and

they achieve this using components that the receiver can understand.

We now proceed to integrate these observations together in the more general frame-

work where incentive compatibility matters, thus proving Theorem 1.

4.2.2 Proof of Theorem 1

Fix any game G� without language barriers and any finite-message equilibrium (σ�, ρ�)

in G�, which are listed as follows.

G� =D

I, M, T, Λ�, π

�, A, (ui)i2I , N

�= 1

E;

(σ�, ρ�) =h(σ�i : Ti ! M)i2I ,

�ρ�i : Ti � MI ! Ai

�i2I

i.

Consider N =�

N + bN�� �����[i2I

Λi

�����, where N and bN are defined in (5) and (7), respectively.

For notationl convenience, for every i 2 I and every�λi, λ0i

�2 Λi �Λi, fix any two

functions:

Υ(i,λ0i) : Λi �! MN;

Γ(λi,λ0i)

: E (σ�,ρ�)

i �! (λi)bN .

These two functions will not play any essential role in our equilibrium (see footnote 14).

Senders’ strategies: let m(i,λj)denote the message intended from sender i to receiver

j of language type λj. For every player i 2 I and every (ti, λi) 2 Ti �Λi, define

σi (ti, λi) =�

m(i,λj)

�j2I, λj2Λj

=�

Υ(i,λj) (λi) , Γ(λi,λj) [

σ�i (ti)]�

j2I, λj2Λj2 MN. (9)

I.e., given j 6= i, sender λi tells receiver λj about i’s true language type via Υ(i,λj) (λi) as

19

described in Lemma 1 and the equilibrium message σ�i (ti) under (σ�, ρ�) via Γ(λi,λj)�σ�i (ti)

�as described in Lemma 2.12

Fix any eti 2 Ti.

Receivers’ strategies: upon receiving the intended message m(i,λj)from sender i ( 6= j),

receiver λj uses the following function to translate it back to an equilibrium message un-

der (σ�, ρ�) via the following function:

Σ(λj,i)

hm(i,λj)

i=

8>>><>>>:σ�i (ti) ,

if there exists (ti, λi) 2 Ti �Λi such that

m(i,λj)=�

Υ(i,λj) (λi) , Γ(λi,λj)

�σ�i (ti)

��;

σ�i�eti�

, otherwise.

(10)

Note that, by Lemmas 1 and 2, if there exist multiple (ti, λi) 2 Ti �Λi such that m(i,λj)=�

Υ(i,λj) (λi) , Γ(λi,λj)

�σ�i (ti)

��, then λi must be unique, and all (ti, λi) have the same

equilibrium message σ�i (ti), and hence, Σ(λj,i)

hm(i,λj)

iis well-defined. We are ready

to define ρj for every j 2 I as follows.

ρj

�tj, λj,

�m

λji , bmλj

i

�i2I

�= ρ�j

�tj,�

σ�j�tj�

, Σ(λj,i)

hm(i,λj)

i�i2I�fjg

�.

That is, under (σ, ρ) and any given (t, λ) 2 T � Λ, each sender’s type λi follows

σ�i (ti) by sending two pieces of information,�

Υ(i,λj) (λi) , Γ(λi,λj)

�σ�i (ti)

��, to each re-

ceiver’s type λj, where the former truthfully reveals λi, and the latter is the coded message

of σ�i (ti) by using the endowed message of λi. Upon receiving the message, each receiver

λj decodes it back to σ�i (ti), and plays the action ρ�j

htj,�σ�i (ti)

�i2I

i. As a result,

ρ�i

�ti,hσ�j�tj�i

j2I

�= ρi

hti, λi,

�σj�tj, λj

��j2I

i, 8 (i, t, λ) 2 I � T �Λ,

i.e., (σ, ρ) and (σ�, ρ�) are outcome-equivalent.

12For i = j, the messages�

Υ(i,λ0i) (λi) , Γ(λi ,λ0i)�σ�i (ti)

��is never used in our equilibrium — it is the

message intended from λi to λ0i, but λi knows herself/himself is the language type λi not λ0i, and as a result,

such messages are redundant. We include these redundent messages purely for notational convenience.

20

Finally, we show the incentive compatibility of both senders and players. First, for

any receiver j under (σ�, ρ�), he forms a posterior belief on t�j upon receiving the mes-

sages�σ�i (ti)

�i2I , and chooses the best strategy ρ�j

htj,�σ�i (ti)

�i2I

i. For receiver j under

(σ, ρ), he receives two pieces of information, i.e., λ in addition to�σ�i (ti)

�i2I . Since T and

Λ are independent by Assumption 1, receiver j forms the same posterior belief on t�j as

that under (σ�, ρ�). And hence, the same strategy ρ�j

htj,�σ�i (ti)

�i2I

iis a best reply for

j. Second, for any sender i under (σ�, ρ�), sending σ�i (ti) is a best strategy given the true

payoff state ti, i.e., sending σ�i (ti) is weakly better than sending σ�i�t0i�

for any t0i 2 Ti.

Note that under (σ, ρ), the equilibrium message of sender i with the true payoff state ti

will be interpreted as σ�i (ti) by the receivers. Furthermore, any message from sender i

would be interpreted as σ�i�t0i�

for some t0i 2 Ti (see (10)). Since sending σ�i (ti) is weakly

better than sending σ�i�t0i�

for any t0i 2 Ti, it is a best strategy for sender i to send the

equilibrium message under (σ, ρ).�

4.2.3 Implications of Theorem 1

One immediate implication of Theorem 1 is that any language barriers in the canonical

Crawford and Sobel (1982) cheap-talk model can be overcome. In that model, there ex-

ists a maximally-revealing equilibrium, in which finite messages are transmitted. Hence,

all equilibria in the model are finite-message equilibria, and Theorem 1 immediately im-

plies that all of them can be replicated whatever language barriers there are, if multi-

dimensional communication is allowed.

A second, less immediate, implication focuses on the setting studied by Blume and

Board (2013), which is that of a common-interest sender-receiver game. Specifically, we

assume the following:

Assumption 2 (common-interest sender-receiver game)

I = f1, 2g , jA1j = jT2j = 1, jT1j > 1, jA2j > 1, jMj < ∞,

u1 � u2 � u (i.e., common interest) is continuous and T1 and A2 are compact metric spaces.

I.e., player 1 is the sender and player 2 is the receiver; A1 and T2 are degenerate; we

21

use u to denote the common utility function for both players. In this setting Blume and

Board (2013) prove that indeterminacies of meaning are inevitably induced by language

barriers under 1-dimensional communication. As previously discussed, this means that

there will not be efficient equilibria.13

However, given no language barrier, Proposition 1 below shows that approximate

efficiency can always be achieved if there are sufficiently many, albeit finite, messages. Its

proof can be found in Appendix A.3.

Proposition 1 For any ε > 0 and any game with 1-dimensional communication and no language

barriers G� =D

I, M, T, Λ�, π

�, A, (ui)i2I , N

�= 1

Ewhich satisfies Assumption 2, there exists

a positive integer K such that

jMj � K =)

������ sup(σ,ρ)2ΣG�

U (σ, ρ)�Z

t2T

�maxa2A

u (t, a)�

πT (dt)

������ � ε,

where ΣG� denotes the set of equilibria of G�.

Note thatZ

t2T

[maxa2A u (t, a)]πT (dt) is the maximal utility that players can possibly

get. We say an equilibrium (σ, ρ) achieves ε-efficiency if and only if������U (σ, ρ)�Z

t2T

�maxa2A

u (t, a)�

πT (dt)

������ � ε.

Then, Theorem 1 and Proposition 1 together immediately imply the following corollary:

Corollary 1 For any ε > 0 and any game satisfying Assumptions 1 and 2, ε-efficiency can be

achieved in equilibria of similar games G =

I, M, T, Λ, π, A, (ui)i2I , N�

for sufficiently large

N.

That it, multi-dimensional communication not only eliminates indeterminacies of

meaning caused by language barriers, but also achieves almost-efficiency.14

13Indeterminancies of meaning imply inefficiency, or equivalently, efficiency implies determinancy of

meaning.14Theorem 1 assumes that language types and payoff states are independently distributed. For common-

22

5 Main Results: 1-dimensional Communication

In the previous section, we showed that for any language barriers, if multi-dimensional

communication is allowed, we can always replicate outcomes that would obtain in the

absence of such language barriers. In this sense, in the presence of language barriers

multi-dimensional communication allows us to do no worse than if such language bar-

riers did not exist. In this section, we change quantifier and focus on one-dimensional

communication to study whether there exist language barriers that allow us to do “better”

than what we can achieve without them.

In particular, we follow the Goltsman, Horner, Pavlov, and Squintani (2009) strat-

egy of studying several modified versions of cheap-talk communication games, although

the games studied here generalizes theirs over two dimensions: we consider any arbi-

trary distributions and utility functions, while Goltsman, Horner, Pavlov, and Squintani

(2009) focus on the uniform distribution and the quadratic utility function.15 In Section

5.2, we define arbitration and mediation equilibria (Goltsman, Horner, Pavlov, and Squin-

tani (2009)), noisy-talk equilibria (Blume, Board, and Kawamura (2007)), and language-

barrier equilibria, all of which may Pareto dominate cheap-talk equilibria. In Section 5.3,

we provide a linear ranking regarding the maximal welfare induced by all the equilibria

except for arbitration equilibria. In Sections 5.4 and 5.5, we further clarify the relation-

ship between arbitration equilibria and language-barrier equilibria on the one hand and

language-barrier equilibria and noisy-talk equilibria on the other.

We begin in Section 5.1 by showing, through an example, that language barriers can

(ex-ante) Pareto improve outcomes strictly.

interest games, however, in previous versions of this paper we showed that even in the absence of indepen-

dence, with a N-dimensional protocol there exist ε-equilibria that achieve almost-efficiency.15(cite BB WP version) also undertook an exercise similar to ous, by adding language-barrier equilibria

to the Goltsman, Horner, Pavlov, and Squintani (2009) analysis. However, they focused on the Goltsman,

Horner, Pavlov, and Squintani (2009) class of games, so that our results, which consider more general set-

tings, differ.

23

5.1 An example of language barriers strictly improving over the cheap

talk

Example 1 Consider a canonical Crawford and Sobel (1982) one sender- one receiver game, i.e.,

I = fS, Rg; jASj = jTRj = 1,

TS = M = f0, 1g and AR = (�∞, ∞) ,

uS = ��

ar � tS �58

�2

and uR = � (ar � tS)2 , 8 (ar, tS) 2 AR � TS,

Consider two scenarios:

1. no language barriers:

ΛS = ΛR = fMg ,

and the prior on T�ΛS �ΛR has a uniform distribution;

2. language barriers for the sender:

bΛS =n

λS = f0g , bλS = f0, 1go

,

ΛR = fMg ,

and the prior on T� bΛS �ΛR has a uniform distribution.

Without language barriers, with one-dimensional communication, only the pooling

equilibrium exists, because the bias between the sender and receiver is too large (i.e., 58 >

12 ). In the pooling equilibrium, the receiver takes the action 1

2 , and the ex-ante expected

utilities of the two agents are EuS = �4164 and EuR = �1

4 . However, with the language

barriers for the sender specified above, it is easy to check that the following strategy

profile is an equilibrium: type λS always sends message 0, type bλSsends the message 1

if t = 1, and the message 0 if t = 0,. Finally, the receiver plays action 1 if he gets the

message 1 and action 13 if he gets message 0.16 The agents’ ex-ante expected utilities in

16It is straightforward to see that the receiver and λS are choosing best replies in equilibrium. To check

the incentive compatibility of bλS, note that the receiver plays only two actions: 1 and 1

3 . Furthermore, the

24

this equilibrium are:

EuS =12� EuλS

+12� EubλS

= �321576

> �4164

uR = �12��

13� 0

�2

� 14��

13� 1

�2

� 14� (1� 1)2 = �1

6> �1

4

Therefore, the equilibrium with these language barriers (ex-ante) Pareto dominates the

equilibrium with no language barriers.

5.2 Cheap talk communication devices

The welfare improvement in Example 1 is not entirely surprising, since the literature has

already pointed out that while faulty communication may reduce message precision, it

may also weaken the sender’s incentive compatibility constraints in such a way to more

than compensate for the lesser precision. In particular, Blume, Board, and Kawamura

(2007) show this for “noisy talk” in a sender-receiver game, where talk is “noisy” in the

sense that there is an exogenous probability that the Receiver will not hear the intended

message, but one randomly chosen by nature. Furthermore, Goltsman, Horner, Pavlov,

and Squintani (2009) show that if an an unbiased mediator is introduced in the standard

Crawford and Sobel (1982) model, this mediator may improve communication by sending

noisy messages to the receiver.

Example 1 shows that language barriers can also achieve welfare improvements, so

the question we wish to answer is: what is the relationship regarding welfare between

language barriers and cheap talk or generalized versions of cheap talk such as noisy talk

and mediated communication mentioned above? To answer this question, we follow the

ideal point for the sender is 58 when t = 0, and����13 � 5

8

���� = 724<

924=

����1� 58

���� .

As a result, it is a best reply for bλSto send the message 0 when t = 0. Similarly, the ideal point for the

sender is 138 when t = 1, and ����13 � 13

8

���� = 3124>

1524=

����1� 138

���� .

As a result, it is a best reply for bλSto send the message 1 when t = 1.

25

Goltsman, Horner, Pavlov, and Squintani (2009) strategy in studying optimal equilibria

under language barriers, noisy talk, mediated communication, and arbitrated communi-

cation, and compare their welfare properties.17

Recall that a communication game is defined by a tupleI, M, T, Λ, π, A, (ui : T � A �! R)i2I , N

�.

From now on, we fix any primitive (excluding language barriers),I, M, T, πT, A, (ui : T � A �! R)i2I , N = 1

�,

so as to make comparisons meaningful. In particular, we fix πT but allow Λ and the

marginal distribution on it to change. For simplicity, we assume

I = f1, 2g , jA1j = jT2j = 1,

i.e., we focus on the standard one sender-one receiver game, and 1 and 2 are the sender

and the receiver, respectively.

Note that jA1j = jT2j = 1 means that the sender (i.e., player 1) takes a degenerate

action, and the receiver (i.e., player 2) observes a degenerate payoff type — this is com-

mon knowledge. Hence, it is with loss of generality for us to omit A1 and T2 and to use

A and T to denote A2 and T1, respectively. However, we still consider Λ = Λ1 �Λ2, i.e.,

we allow for the possibility that both the sender and the receiver have language barriers.

Finally, for simplicity, we assume

M = A = R.

5.2.1 Mediation and arbitration equilibria

First, we define arbitration and mediation equilibria.

17Of course, this means comparing the best available equilibrium under the different “faulty” devices.

For instance, we say language barriers strictly improve welfare over noisy talk if and only if the optimal

equilibrium under some language barriers has a strictly larger welfare than the optimal equilibrium under

all noisy talk.

26

Definition 4 [p : T �! 4 (A)] is an arbitration equilibrium with adverse selection ifZa2A

u1 [t, a] p (t) (da) �Z

a2A

u1 [t, a] p�t0�(da) , 8t, t0 2 T. (11)

[p : T �! 4 (A)] is an arbitration equilibrium with moral hazard if

8ι : A �! A, (12)ZT

24 Za2A

u2 [t, a] p (t) (da)

35πT [dt] �ZT

24 Za2A

u2 [t, ι (a)] p (t) (da)

35πT [dt] .

[p : T �! 4 (A)] is a mediation equilibrium if both (11) and (12) are satisfied.

We say [p : T �! 4 (A)] is an arbitration equilibrium if it is either an arbitration

equilibrium with adverse selection or an arbitration equilibrium with moral hazard. Clearly,

a mediation equilibrium is an arbitration equilibrium. In particular, we share the same

definition of mediation equilibrium with Goltsman, Horner, Pavlov, and Squintani (2009).

Our notion of arbitration equilibrium with adverse selection is the same as the origi-

nal ”arbitration equilibrium” defined in Goltsman, Horner, Pavlov, and Squintani (2009),

while ”arbitration equilibrium with moral harzard” is a new notion.18

Our terminology is inspired by Myerson (1991)’s notions of moral hazard and ad-

verse selection in communication games. Suppose there is a non-strategic middleman

(i.e., an arbitrator or a mediator) besides the players. The sender reports his private pay-

off type to the middleman; upon receiving t, the middleman commits to draw from a

lottery on A following the distribution p (t); given every realized value a of the lottery,

the receiver plays a. In the case of mediation, the sender is not committed to reporting

the true payoff type and condition (11) requires that truthful reporting be optimal for the

sender in an equilibrium. At the same time, the receiver is also not committed to follow

the action suggested by the middleman, and condition (12) describes the incentive com-

patibility condition for the receiver, where the function ι : A �! A in (12) represents the

18Presumably, ”arbitration equilibrium with adverse selection” is a more practically relevant one between

the two. However, our sole purpose to introduce the other notion is for conceptual clarification. More

precisely, it helps us compare language-barrier equilibria and arbitration equilibria (see Section 5.4).

27

receiver’s (possible) deviation from the recommended actions, i.e., when the mediator

recommends a, the receiver may deviate to play ι (a). The two forms of arbitration follow

when we impose just one of the two conditions. In arbitration with adverse selection,

the receiver must follow the arbitrator’s recommended action but the adverse selection

problem (i.e., the sender still needs to be incentivized to report her true payoff state) re-

mains. In arbitration with moral hazard, the sender must report her true payoff state, but

the moral hazard problem (i.e., the receiver still needs to be incentivized to follow the

arbitrator’s recommended action) remains.

5.2.2 Noisy-talk equilibria

A noisy-talk game is defined by a tuple (ε, ξ) 2 [0, 1] �4 (M), i.e., with probability ε,

the sender’s message is replaced by an exogenous and independent noise following the

distribution ξ. A potential candidate for a noisy-talk equilibrium is a strategy profile

([s : T �! 4 (M)] , [r : M �! 4 (A)]) .

Given [(ε, ξ) , s, r], type t of the sender follows s (t) 2 4 (M) to send a random mes-

sage; for any realized message m from the sender, with probability (1� ε), the receiver

observes m, and with probability ε, the receiver observes a random message generated by

the distribution ξ; finally, upon receiving a (possibly distorted) message m0, the receiver

takes a random action r (m0) 2 4 (A). We aggregate this process as follows.

p[(ε,ξ), s, r] : T �! 4 (A) , (13)

p[(ε,ξ), s, r] (t) [E] =ZM

24(1� ε)� r (m) [E] + ε�ZM

r (m) [E] ξ [dm]

35 s (t) [dm] , 8E � A,

i.e., p[(ε,ξ), s, r] (t) is the ex-post action distribution induced by the equilibrium, given t. We

now define noisy-talk equilibria.

Definition 5 ([s : T �! 4 (M)] , [r : M �! 4 (A)]) is a noisy-talk equilibrium if there ex-

28

ists (ε, ξ) 2 [0, 1]�4 (M) such that

8t 2 T, 8s0 : T �! 4 (M) , (14)Za2A

u1 (t, a) p[(ε,ξ), s, r] (t) (da) �Z

a2A

u1 (t, a) p[(ε,ξ), s0, r] (t) (da) ,

and 8r0 : M �! 4 (A) , (15)ZT

24 Za2A

u2 (t, a) p[(ε,ξ), s, r] (t) (da)

35πT [t] �ZT

24 Za2A

u2 (t, a) p[(ε,ξ), s, r0] (t) (da)

35πT [t] .

(14) and (15) in Definition 5 describe the incentive compatibility conditions for the

sender and the receiver, respectively.

5.2.3 Language-barriers equilibria

A valid language-barriers game is defined by a tuple [Λ = Λ1 �Λ2, π 2 4 (T �Λ)] such

that the marginal distribution of π on T matches the fixed πT and assumptions (3) and (4)

are satisfied. In game [Λ, π], a potential candidate for a language-barriers equilibrium is

a strategy profile

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] .

We say [σ, ρ] is a valid strategy profile if and only if σ and ρ are measurable with respect

to Λ1 and Λ2, respectively, where measurability is as defined in Section 3.

Given (t, λ1, λ2), the sender follows σ (t, λ1) 2 4 (M) to send a random message;

upon receiving a realized message m, the receiver follows ρ (λ2, m) 2 4 (A) to play a

random action. Abstractly, we use the function p(σ, ρ) defined below to aggregate this

process.

p(σ, ρ) : T �Λ1 �Λ2 ! 4 (A) , (16)

p(σ, ρ) (t, λ1, λ2) [E] =ZM

[ρ (λ2, m) [E]] σ (t, λ1) (dm) , 8E � A.

29

Definition 6 For any valid language-barriers game (Λ, π), we say a valid strategy profile

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] ,

is a language-barriers equilibrium if

8 (t, λ1) 2 T �Λ1, 8σ0 : T �Λ1 �! 4 (M) , (17)ZΛ2

0@ Za2A

u1 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u1 (t, a) p(σ0, ρ) (t, λ1, λ2) [da]

1Aπ [dλ2 j t, λ1] � 0,

and 8λ2 2 Λ2, 8ρ0 : Λ2 � M ! 4 (A) , (18)ZT�Λ1

0@ Za2A

u2 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u2 (t, a) p(σ, ρ0) (t, λ1, λ2) [da]

1Aπ [(dt, dλ1) j λ2] � 0.

Furthermore, we say it is an independent-language-barriers equilibrium, if T and Λ are indepen-

dent according to π.

Finally, to compare these different notions of equilibria, we define a notion of out-

come equivalence, as in Definition 3.

Definition 7 Consider any arbitration equilbrium [p : T �! 4 (A)], any noisy-talk equilib-

rium (s, r) under noise (ε, ξ) and any language-barriers equilibrium [σ, ρ], which, given t, in-

duce ex-post action distributions p (t), p[(ε,ξ), s, r] (t) as defined in (13), and P (σ, ρ) (t) as defined

in (16), respectively. Any two of the equilibria are outcome-equivalent if they induce the same

ex-post action distribution for any t 2 T.

5.3 Welfare comparison

In this section, we compare welfare induced by different equilibria. Goltsman, Horner,

Pavlov, and Squintani (2009) consider the canonical Crawford and Sobel (1982) with quadratic

utility, where, in any mediation equilibrium, the sender’s expected utility differs from the

receiver’s expected utility by a constant determined by the “bias.” In that setting, it is

30

without loss of generality to compare only the sender’s (or the receiver’s) expected utility

in different equilibria. However, in the general communication model as studied here,

this nice property no longer holds. We thus use any weakly-increasing social welfare

function Φ to aggregate players’ utility, i.e.,

Φ : RI �! R such that

xi � x0i , 8i 2 I =) Φ�(xi)i2I

�� Φ

h�x0i�

i2I

i.

That is, under an equilibrium, if every player i 2 I gets expected utility xi, we say this

equilibrium achieves social welfare of Φ�(xi)i2I

�. Then, given a social welfare func-

tion Φ, let ΦA�MH, ΦA�AS, ΦM, ΦN, ΦLB, ΦILB denote the supremum of the social

welfare achieved by equilibria in each of our possible protocols (arbitration with moral

hazard, arbitration with adverse selection, mediation, noisy-talk, language-barriers, and

independent-language-barriers, respectively). We now present the main result of this sec-

tion.

Theorem 2 For any weakly increasing social welfare function Φ, we have

ΦLB � ΦA�MH � ΦM � ΦILB � ΦN.

It is straightforward to see ΦA�MH � ΦM, because every mediation equilibrium

is an arbitration equilibrium with moral hazard. Given this, Theorem 2 is immediately

implied by the following three lemmas. The idea of the proofs is to show that equilibria

with language barriers, arbitration with moral hazard, mediation, independent language

barriers and noisy talk correspond to a series of increasingly restrictive incentive compat-

ibility conditions in that order. The proof of Lemmas 3, 4 and 5 are relegated to Appendix

A.5, 4 and 5.

Lemma 3 For any noisy talk equilibrium, there exists an outcome-equivalent independent-language-

barrier equilibrium.

Lemma 4 For any independent language barriers equilibrium, there exists an outcome-equivalent

mediation equilibrium.

31

Lemma 5 For any arbitration equilibrium with moral hazard, there exists an outcome-equivalent

language-barrier equilibrium.

It is straightforward to see ΦA�AS � ΦM, because every mediation equilibrium

is an arbitration equilibrium with adverse selection. One question remaining is how to

compare ΦA�AS and ΦLB (and ΦA�MH), which will be discussed in Section 5.4.

5.4 Arbitration equilibria and language-barrier equilibria

It is difficult to directly compare the maximal welfare of languag-barrier equilibrium and

arbitration equilibrium with adverse selection (i.e., the original ”arbitration equilibrium”

defined in Goltsman, Horner, Pavlov, and Squintani (2009)). However, it is easy to com-

pare the two forms of arbitration equilibrium, which is the reason that we introduce the

new arbitration notion. Furthemore, the comparision helps us clarify the relationship

between languag-barrier equilibrium and arbitration equilibrium with adverse selection.

The following example shows that neither ΦA�AS � ΦA�MH nor ΦA�MH � ΦA�AS

hold generally.

Example 2 Consider the standard cheap-talk model with quadratic utility such that

u1 (a, t) = ��

a� t� 34

�2

;

u2 (a, t) = � (a� t)2 .

and µT 2 ∆ (T) is defined as

µT (f0g) = µT (f1g) = 12

.

Consider [bp : T �! A] and [ep : T �! A] such that

bp (0) =34

and bp (1) = 74

;

ep (0) = 0 and ep (1) = 1.

That is, the sender and the receiver achieve the ideal actions (at both payoff states) in bpand ep, respectively. As a result, bp and ep are arbitration equilibrium with adverse selection

32

and arbitration equilibrium with moral hazard, respectively. Consider bΦ : R2 �! R andeΦ : R2 �! R defined as

bΦ [u1, u2] � u1 and eΦ [u1, u2] � u2.

Hence, bΦA�AS = eΦA�MH = 0. (19)

It is easy to show

bΦLB � � 916

, (20)

eΦA�AS � � 1128

. (21)

Then, (19), (20) imply bΦA�AS > bΦLB. Furthermore, (19), (21) and Lemma 5 imply eΦLB >eΦA�AS. Therefore, neither ΦLB � ΦA�AS nor ΦA�AS � ΦLB hold generally. The detailed

analysis can be found in Appendix A.8.

5.5 Independent-language barrier equilibria and noisy-talk equilibria

Example 3 Consider the sender-receiver game with I = fS, Rg; jASj = jTRj = 1,

TS = M = [0, 1] and AR = (�∞, ∞) ,

uS = ��

ar � tS �14

�2

and uR = � (ar � tS)2 , 8 (ar, tS) 2 AR � TS,

with the common prior µ (f0g) = µ

��3572

��= µ (f1g) = 1

3.

Then, there exists an independent-language-barriers equilibrium that strictly Pareto dominates

any noisy talk equilibria.

The proof is quite tedious and is relegated to Appendix A.9, but here we provide

some intuition. Given the prior µ described above, a mediated communication equilib-

rium can be constructed where the mediator proposes action zero when the type is zero,

action 12 when the type is 35

72 and mixes when the type is 1 by proposing action 1 with

probability 3536 and action 1

2 with complementary probability. The same outcome can be

33

obtained by an independent-language-barriers setting when one language type has three

messages (regardless of payoff type, this language type occurs with probability 3536) and

the other has only two of the three messages available. Then, there is an equilibrium

where one common message is used by both language types when the payoff state is zero

to communicate that the action that should be taken is zero, the second common message

is used by both language types when the payoff state is 3572 to indicate that the action that

should be taken is 12 and finally, if that payoff state is 1, then the remaining message is

used to communicate that the action that should be taken is one by the language type that

has that message available; the other language type, who only has two messages, uses the

second common message.

These equilibria cannot be replicated by a noisy-talk equilibrium. In mediated com-

munication it is the mediator that injects noise and can do so depending on the payoff

state: in this particular case, the mediator can make the receiver unsure of the sender’s

payoff state when she proposes action 12 , but when she proposes actions zero and one,

there is no uncertainty about the underlying payoff state. In the independent-language-

barriers case, this can be replicated because upon observing the second common message

the receiver is again uncertain about the sender’s payoff state, whereas there is no uncer-

tainty with the other two messages. In noisy-talk, this cannot be replicated because the

same noise distribution must apply for each payoff state.

6 Conclusions

At an intuitive level, “language barriers” bring obstacles to communication. However,

in this paper we show that they may not be if a different communication protocol (than

that in Crawford and Sobel (1982)) is allowed. In particular, with N-dimensional commu-

nication, (almost) efficiency can always be achieved in common-interest games, and any

equilibrium in the canonical cheap-talk game can be mimicked by an equilibrium under

any “language barriers.” As a result, players cannot be worse off under “language barri-

ers.” Of course in the real world, plently of examples of miscommunication exist, so our

results imply that miscommunication must arise from something outside of this setting.

A simple extension would be to incorporate in the model the cognitive cost of sending

34

and comprehending more complex messages thus reconciling our results with those of

Blume and Board (2013).19 More generally, miscommunication might also arise from the

fact that real-world messages have a semantic meaning and different agents might not

have the same vocabulary. In our model the meaning of messages is emergent in equi-

librium. We would argue that this makes the notion of equilibrium itself unsuitable for

studying this aspet of language, whereas a more promising approach would be based on

learning.

The second part of our paper shows that even if the original (1-dimensional) com-

munication in Crawford and Sobel (1982) is imposed, some language barriers can im-

prove upon equilibria obtainable in their absence. In particular, we show the optimal

independent-language-barrier equilibrium always weakly dominates—and sometimes,

strictly dominates— any generalized noisy talk equilibria, which includes the equilibria

in the canonical cheap-talk Crawford and Sobel (1982) model (without noise) as special

cases.20

We also believe that the Blume and Board (2013) framework with language types

utilized here is rich enough to accommodate both the standard cheap talk and the “per-

suasion games” literature (e.g.,Milgrom (1981)) as special cases. In cheap-talk, the sender

could send any possible messages, whereas in persuasion games, the privately informed

parties cannot lie about the payoff states. This no-lie assumption is equivalent to M =

2T� f?g and each payoff state t correponds to a language type who is endowed with a

set of subsets (of M) containing t. I.e., at t, the sender can send only a message E such

that t 2 E, meaning that only states in E are possibly true. Much of the literature on per-

suasion games has focused on the conditions on preferences necessary to guarantee full

communication of payoff type.21 It is easy to see that guaranteeing full communication

with arbitrary language barriers is easy but future research should focus on determining

the minimal conditions on language types necessary to guarantee full communication for

a given preference profile.

19See Garicano and Prat (2013) for a discussion of cognitive costs in communication in organisations.20One open question remains. We show that the optimal mediated communication is always weakly

better than communication under the optimal independent “language barriers.” Is the converse true?21See Seidmann and Winter (1997), Giovannoni and Seidmann (2007) and Jeanne Hagenbach and Perez-

Richet (2014) for details.

35

A Proofs

A.1 The proof of Lemma 1

Fix any (i, j, λ) 2 I � I � Λ with i 6= j. Recall N � 3� j[l2IΛlj, and hence, jΛij � N.

Label the elements in Λi as λ(1)i , λ

(2)i , ..., λ

(K)i , where K = jΛij � N.

For each λ(k)i 2 Λi with k � K, we have λj \ λ

(k)i 6= ? and

���λ(k)i

��� � 2, due to (3) and

(4). Thus, we fix some m(k) 2 λj \ λ(k)i , and some em(k) 2 λ

(k)i �

nm(k)

o, i.e., m(k) 6= em(k).

Note that

m(k) �λj em(k), (22)

when either em(k) 2 λj or em(k) /2 λj is true.

Then, define Υ(i,λj): Λi �! MN as follows. For each k 2 f1, 2, ..., Kg,

Υ(i,λj)

hλ(k)i

i= [ml]

Nl=1 2 MN such that ml =

8<: m(k), if l = k;em(k), otherwise.

That is, type λ(k)i uses m(k) to denote ”yes” and em(k) for ”no.” Furthermore, player i as-

sociates each of the first K dimensions of the message Υ(i,λj)

hλ(k)i

ito one element in Λi,

and player i reveals whether he is that element in the associated dimension. Precisely, λ(k)i

says ”yes” (i.e., m(k)) in the k-th dimension, and ”no” (i.e., em(k)) in all other dimensions.

For k 6= k0, we show Υ(i,λj)

hλ(k)i

i�λj Υ(i,λj)

hλ(k0)i

i, as needed in (6). By the defini-

tion of Υ(i,λj), we have

Υ(i,λj)

hλ(k)i

i= [ml]

Nl=1 =

�mk = m(k),

�ml = em(k)

�l 6=k

�;

Υ(i,λj)

hλ(k0)h

i= [ bml]

Nl=1 =

� bmk0 = m(k0),� bml = em(k0)

�l 6=k0

�.

Consider two cases: (1) m(k) 6= em(k0) and (2) m(k) = em(k0). In case (1), m(k) 6= em(k0) and

m(k) 2 λj implies

mk = m(k) �λj em(k0) = bmk,

i.e., in the k-th dimension, mk �λj bmk, which further implies Υ(i,λj)

hλ(k)i

i�λj Υ(i,λj)

hλ(k0)i

i.

36

In case (2), recall N � 3 by (5). Pick any k00 2�

1, ..., N� fk, k0g. Then, (22) implies

mk00 = em(k) �λj m(k) = em(k0) = bmk00 ,

i.e., in the k00-th dimension, mk00 �λj bmk00 , which further implies Υ(i,λj)

hλ(k)i

i�λj Υ(i,λj)

hλ(k0)h

i.�

A.2 The proof of Lemma 2

Fix any (i, j, λ) 2 I � I �Λ with i 6= j. Recall λi \ λj 6= ? and jλij � 2. Thus, we fix some

m 2 λi \ λj, and some em 2 λi� fmg, i.e., m 6= em. Note that

m �λj em, (23)

when either em 2 λj or em /2 λj is true.

Recall bN ����E (σ�,ρ�)

i

��� by (7). Label the elements in E (σ�,ρ�)

i as m(1), m(2), ..., m(K),

where K =���E (σ�,ρ�)

l

��� � bN. Then, define Γ(λi,λj): E (σ

�,ρ�)i �! (λi)

bN as follows. For each

k 2 f1, 2, ..., Kg,

Γ(λi,λj)

hm(k)

i= [ml]

bNl=1 2 M bN such that ml =

8<: m, if l = k;em, otherwise.

That is, type λi use m to denote ”yes” and em for ”no.” Furthermore, λi associates each

of the first K dimensions of the message Γ(λi,λj)

hm(k)

ito one element in E (σ

�,ρ�)i , and λi

reveals whether he intends to send that element in the associated dimension. Precisely,

to send the message m(k) 2 E (σ�,ρ�)

i , λi say ”yes” (i.e., m) in the k-th dimension, and ”no”

(i.e., em) in all other dimensions.

For k 6= k0, we show Γ(λi,λj)

hm(k)

i�λj Γ(λi,λj)

hm(k0)

i, as needed in (8). By the

definition of Γ(λi,λj), we have

Γ(λh,λi)

hm(k)

i= [ml]

bNl=1 =

hmk = m, (ml = em)l 6=k

i;

Γ(λh,λi)

hm(k0)

i= [ bml]

bNl=1 =

h bmk0 = m, ( bml = em)l 6=k0

i.

Since k 6= k0, (23) implies

mk = m �λi em = bmk,

i.e. in the k-th dimension, mk �λi bmk, which further implies Γ(λi,λj)

hm(k)

i�λj Γ(λi,λj)

hm(k0)

i.�

37

A.3 Proof of Proposition 1

We use the following two lemmas to prove Proposition 1, and the proofs can be found in

Appendix A.3.1 and A.3.2.

Lemma 6 Suppose Assumption 2 holds. For any ε > 0, there exists δ > 0 such that

8t, t0 2 T, d�t, t0�< δ =)

����maxa2A

u (t, a)� u [t, a�]���� < ε, 8a� 2 arg max

a2Au�t0, a

�. (24)

Lemma 7 For any game satisfying Assumption 2, there exists an optimal equilibrium (σ�, ρ�) in

G� =D

I, M, T, Λ�, π

�, A, (ui)i2I , N

�E

such that U (σ�, ρ�) � U (σ, ρ) for any strategy profile

(σ, ρ) in G�.

Proof of Proposition 1: Fix any game satisfying Assumption 2 and any ε > 0. By

Lemma 6, there exists δ > 0 such that

8t, t0 2 T, d�t, t0�< δ =)

����maxa2A

u (t, a)� u [t, a�]���� < ε, 8a� 2 arg max

a2Au�t0, a

�. (25)

Since T is compact, it is totally bounded. Hence, there exists a positive integer K, such

that T can be partitioned by fE1, ..., EKg and

t, t0 2 Ek =) d�t, t0�< δ , 8k 2 f1, ..., Kg . (26)

For each k 2 f1, ..., Kg, fix some tk 2 Ek and some ak 2 arg maxa2A u (tk, a). Then,

∑k2f1,...,Kg

Zt2Ek

u (t, ak)πT (dt) � ∑k2f1,...,Kg

Zt2Ek

�maxa2A

u (t, a)� ε

�πT (dt) (27)

=Z

t2T

�maxa2A

u (t, a)�

πT (dt)� ε,

where the inequality follows from (25) and (26).

Suppose jMj � K. Then, the expected utility ∑k2f1,...,Kg

Rt2Ek

u (t, ak)πT (dt) can be

achieved in a strategy profile, i.e., fix K messages, m1, ..., mK; the sender sends mk if and

38

only t 2 Ek; and the receiver plays ak if and only if he receives mk. By Lemma 7, an optimal

equilibrium exists, and denote it by (σ�, ρ�), and hence

U (σ�, ρ�) � ∑n2f1,...,Ng

Zt2En

u (t, an)πT (dt) . (28)

Furthermore, Zt2T

�maxa2A

u (t, a)�

πT (dt) � U (σ�, ρ�) . (29)

Thus, (27), (28) and (29) imply������U (σ�, ρ�)�Z

t2T

�maxa2A

u (t, a)�

πT (dt)

������ � ε,

which completes the proof of Proposition 1.�

A.3.1 Proof of Lemma 6

Since u is continuous and T, A are compact, u is uniformly continuous. Then, by Berg’s

Maximum Theorem, φ (t) � maxa2A u (t, a) is continuous on t 2 T. Since T is compact,

φ (t) is uniformly continuous, and hence,

8ε > 0, 9δ > 0, such that d�t, t0�< δ =)

����maxa2A

u (t, a)�maxa2A

u�t0, a

����� < ε

2, (30)

The uniform continuity of u implies

8ε > 0, 9δ > 0, such that (31)

d�t, t0�< δ =)

����u (t, a�)�maxa2A

u�t0, a

����� = ��u (t, a�)� u�t0, a�

��� < ε

2, 8a� 2 arg max

a2Au�t0, a

�.

Then, (30) and (31) imply

8ε > 0, 9δ > 0, such that d�t, t0�< δ =)

����maxa2A

u (t, a)� u (t, a�)���� < ε, 8a� 2 arg max

a2Au�t0, a

�.

This completes the proof of Lemma 6.�

39

A.3.2 Proof of Lemma 7

Suppose jMj = n. Define a function, ψ : An �! R as follows.

ψ (a1, ..., an) =Z

t2T

�max

a2fa1,...,angu (t, a)

�πT (dt) .

First, we show ψ is uniformly continuous, i.e.,

8ε > 0, 9δ > 0, such that (32)

jbak � eakj < δ, 8k 2 f1, 2, ..., ng =) jψ (ba1, ...,ban)� ψ (ea1, ...,ean)j < ε.

Consider any (ba1, ...,ban) and (ba1, ...,ban) such that maxk2f1,2,...,ng jbak � eakj < δ. For each

t 2 T, fix any k (t) 2 arg maxk2f1,...,ng u (t,bak). We thus have

ψ (ba1, ...,ban) =Z

t2T

hu�

t,bak(t)

�iπT (dt) . (33)

By uniform continuity of u,

8ε > 0, 9δ > 0, such that (34)

jbak � eakj < δ, 8k 2 f1, 2, ..., ng =)

������Z

t2T

hu�

t,bak(t)

�iπT (dt)�

Zt2T

hu�

t,eak(t)

�iπT (dt)

������ < ε.

Furthermore, by the definition of ψ (ea1, ...,ean), we have

ψ (ea1, ...,ean) �Z

t2T

hu�

t,eak(t)

�iπT (dt) . (35)

Then, (33), (34) and (35) imply

8ε > 0, 9δ > 0, such that (36)

jbak � eakj < δ, 8k 2 f1, 2, ..., ng =) ψ (ea1, ...,ean) � ψ (ba1, ...,ban)� ε.

If we change the roles of (ea1, ...,ean) and (ba1, ...,ban), and repeat the analysis, we get

8ε > 0, 9δ > 0, such that (37)

jbak � eakj < δ, 8k 2 f1, 2, ..., ng =) ψ (ba1, ...,ban) � ψ (ea1, ...,ean)� ε.

40

Therefore, (36) and (37) imply (32), i.e., ψ is uniformly continuous.

Second, there exists

(a�1 , ..., a�n) 2 arg max(a1,...,an)2An

ψ (a1, ..., an) , (38)

due to compactness of A and continuity of ψ, i.e.,Zt2T

"max

a2fa�1 ,...,a�ngu (t, a)

#πT (dt) �

Zt2T

�max

a2fa1,...,angu (t, a)

�πT (dt) , 8 (a1, ..., an) 2 An.

Third, recall that there are at most jMj = n messages. Label the elements in M as

m1, ..., mn, i.e., fm1, ..., mng. For any fixed strategy profile (σ, ρ), let ak 2 A denote the

action taken by the receiver upon getting mk under (σ, ρ). Then, the expected utility of

the players under (σ, ρ) is at mostZ

t2T

hmaxa2fa1,...,ang u (t, a)

iπT (dt).

Finally, (a�1 , ..., a�n) as defined in (38) corresponds to an equilibrium, denoted by

(σ�, ρ�), under which players’ expected utility isZ

t2T

hmaxa2fa�1 ,...,a�ng u (t, a)

iπT (dt). To

see this, define

Ek =

(t 2 T : a�k 2 arg max

a2fa�1 ,...,a�ngu (t, a)

), 8k 2 f1, 2, ..., ng .

Then, define

E1 = E1 and

Ek = Ek�h[k�1

l=1 Ek

i, 8k 2 f2, ..., ng .

As a result,�

E1, ..., En

is a partition of T, and each a�k is the optimal action for every

t 2 Ek. Thus, the following strategy profile is an equilibrium.24 sender’s strategy: send mk if and only if t 2 Ek, 8k 2 f1, 2, ..., ng .

receiver’s strategy: play a�k if and only if he receives mk, 8k 2 f1, 2, ..., ng .

35The incentive compatibility of the sender is implied by the definition of Ek and the incen-

tive compatibility of the receiver is implied by (a�1 , ..., a�n) 2 arg max(a1,...,an)2An ψ (a1, ..., an).

To sum, the last two ponts show the existence of an equilibrium (σ�, ρ�) such that

U (σ�, ρ�) � U (σ, ρ) for any strategy profile (σ, ρ). �

41

A.4 Weak-language-barrier equilibria

We introduce a notion of weak-language-barrier equilibria (resp. weak-independent-

language-barrier equilibria), which differ from language-barrier equilibria (resp. independent-

language-barrier equilibria) only on one assumption:

jλij � 2, 8 (i, λ) 2 I �Λ. (39)

That is, every language type must have at least two messages in any language-barrier

equilibrium, but language types in a weak-language-barrier equilibrium may be endowed

with just one single message.

Clearly, a language-barrier equilibrium is a weak-language-barrier equilibrium. Con-

versely, for any weak-language-barrier equilibrium, there is an outcome-equivalent language-

barrier equilibrium, which is summarized in the following lemma. Because of this, it is

without loss of generality to focus on weak-language-barrier equilibria.

Lemma 8 For any weak-language-barrier equilibrium, there exists an outcome-equivalent language-

barrier equilibrium. Furthermore, for any weak-independent-language-barrier equilibrium, there

exists an outcome-equivalent independent-language-barrier equilibrium.

Proof: Fix any valid language-barrier game (Λ, π), and any weak-language-barrier

equilibrium,

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] .

Recall M = R. Pick any disjoint M� (� M) and M�� (� M) which are both homeomor-

phic to M, e.g., M� =�

0, 13

�and M� =

�23 , 1�. Let

γ� : M �! M�,

� : M �! M��,

denote the homeomorphisms, and let �1 and ��1 denote the inverse function.

42

Define a new valid language-barrier game (Λ, π),

eΛ1 = fγ� (λ1) [ γ�� (λ1) : λ1 2 Λ1g ;eΛ2 = fγ� (λ2) [ γ�� (λ2) : λ2 2 Λ2g ;

eπ (E) = π (f[t, λ1, λ2] : [t, γ� (λ1) [ γ�� (λ1) , γ� (λ2) [ γ�� (λ2)] 2 Eg) , 8E � T � 2M � 2M,

i.e., each of the sender’s language type λ1 is transformed to a new type containing two

copies of the original type, with the first copy transformed from λ1 via γ� and the second

copy from λ1 via γ��; similar construction applies to the receiver’s language types; the

new prior eπ inheritates the distribution from the original prior π.

For any µ 2 4 (M), define γ� (µ) 2 4 (M�) as

γ� (µ) [E] = µ�

�1 [E]�

, 8E � M�,

i.e., for any random message generated by µ, we transform it to a message in M� via γ�,

and γ� (µ) is the the distribution of the transformed message from µ.

For each λ2 2 Λ2 such that λ2 $ M, fix any mλ2 2 M�λ2. Furthermore, if λ2 = M,

fix any mλ2 2 M. The sole purpose of construction of mλ2 is for the measurablility (with

respect to λ2) of eρ defined below.

We now define the outcome-equivalent language-barrier equilibrium.heσ : T � eΛ1 ! 4 (M) , eρ : eΛ2 � M ! 4 (A)i

,

eσ [t, γ� (λ1) [ γ�� (λ1)] = γ� (σ [t, λ1]) ,

eρ [γ� (λ2) [ γ�� (λ2) , m] =

8>><>>:ρ�λ2, γ��1 (m)

�, if m 2 M�;

ρ�λ2, γ���1 (m)

�, if m 2 M��;

ρ�λ2, mλ2

�, otherwise.

That is, a new sender’s type γ� (λ1) [ γ�� (λ1) follows the strategy σ [t, λ1] of the old

type λ1, but transform the (random) message to transform a message in M� via γ�; a new

receiver’s type γ� (λ2) [ γ�� (λ2) first decode the messages in M� and M�� via γ��1 and

γ���1, respectively, and then follows the strategies ρ�λ2, γ���1 (m)

�and ρ

�λ2, γ���1 (m)

�of the old type λ2.

43

First, with probability 1, the sender sends messages in M� and M��. Second, the

receiver treats M� and M�� as the transformed copies of the same set M (via γ� and

�, respectively). Hence, it is without loss of generality for the sender to send messages

only in M�.22 Given this, [eσ, eρ] just replicates [σ, ρ], and [eσ, eρ] inheritates the incen-

tive comptibility of the players from [σ, ρ]. Therefore, [eσ, eρ] is an outcome-equivalent

language-barrier equilibrium.

A similar argument applies to weak-independent-language-barrier equilibria.�

A.5 Proof of Lemma 3

In light of Lemma 8, it is without loss of generality for us to focus on weak-independent-

language-barrier equilibria. Fix any noisy-talk game (ε, ξ) 2 [0, 1] � 4 (M), and any

noisy-talk equilibrium ([s : T �! 4 (M)] , [r : M �! 4 (A)]) in the game. Define a language-

barrier game (Λ, π), such that T and Λ are independent under π, and

Λ1 = fMg [ ffmg : m 2 Mg ;

Λ2 = fMg ;

πΛ [fMg � fMg] = 1� ε;

πΛ [E� fMg] = ε� ξ [fm : fmg 2 Eg] , 8E 2 2M� fMg .

That is, the receiver understands all messages in M; with probability 1� ε, the sender

understand all messages in M, and with probability ε, the sender is endowed with a

single message; conditional on the probability-ε event, the distribution follows ξ, with

fmg replacing m.

Then, we define a weak-independent-language-barrier equilibrium

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] ,22Any message in M�� has a corresponding message M� which plays the same role.

44

such that for every (t, m) 2 T � M,

σ (t, λ1 = M) = s (t) ,

σ (t, λ1 = fmg) = δm,

ρ (λ2 = M, m) = r (m) ,

where δm denotes the Dirac measure on m. Clearly, incentive compatibility for every λ1 =

fmg is satisfied. Then, the incentive compatibility of the sender’s language type λ1 = M

and the receiver’s language type λ2 = M in [σ, ρ] inheritates the incentive compatibility

of the sender and the receiver in the noisy-talk equilibrium (s, r), respectively. I.e., [σ, ρ]

is an outcome-equivalent weak-independent-language-barrier equilibrium. Finally, by

Lemma 8, an outcome-equivalent independent-language-barrier equilibrium exists.�

A.6 Proof of Lemma 4

Fix any valid language-barrier game (Λ, π), and any independent-language-barrier equi-

librium

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] ,

Recall p(σ, ρ) : T �Λ1 �Λ2 ! 4 (A) defined in (16):

p(σ, ρ) (t, λ1, λ2) [E] =ZM

[ρ (λ2, m) [E]] σ (t, λ1) (dm) , 8E � A.

i.e., p(σ, ρ) (t, λ1, λ2) is the ex-post action distribution induced by [σ, ρ], given (t, λ1, λ2).

Then, define

P (σ, ρ) : T ! 4 (A) ,

P (σ, ρ) (t) [E] =ZΛ

hp(σ, ρ) (t, λ1, λ2) [E]

iπΛ [dλ1, dλ2] , 8E � A, (40)

i.e., P (σ, ρ) (t) is the ex-post action distribution induced by [σ, ρ], given t. We now show

P (σ, ρ) : T ! 4 (A) defined above is a mediation equilibrium. First, since [σ, ρ] is a

45

language-barrier equilibrium, (17) in Definition 6 implies

8 (t, λ1) 2 T �Λ1, 8σ0 : T �Λ1 �! 4 (M) ,ZΛ2

0@ Za2A

u1 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u1 (t, a) p(σ0, ρ) (t, λ1, λ2) [da]

1Aπ [dλ2 j t, λ1] � 0,

(41)

Recall Λ and π are indepdent, and hence (41) reduces to

ZΛ2

0@ Za2A

u1 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u1 (t, a) p(σ0, ρ) (t, λ1, λ2) [da]

1Aπ [dλ2 j λ1] � 0.

(42)

Given the definition of P (σ, ρ) defined in (40), if we integrate (42) over Λ1, we getZa2A

u1 [t, a]P (σ, ρ) (t) [da] �Z

a2A

u1 [t, a]P (σ0, ρ) (t) [da] , 8t, σ0. (43)

Finally, for every t0 2 T, consider σ0 (t) � σ (t0), and (43) becomesZa2A

u1 [t, a]P (σ, ρ) (t) [da] �Z

a2A

u1 [t, a]P (σ, ρ) (t) [da] , 8t, t0 2 T.

Second, since [σ, ρ] is a language-barrier equilibrium, (18) in Definition 6 implies

and 8λ2 2 Λ2, 8ρ0 : Λ2 � M ! 4 (A) ,ZT�Λ1

0@ Za2A

u2 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u2 (t, a) p(σ, ρ0) (t, λ1, λ2) [da]

1Aπ [(dt, dλ1) j λ2] � 0.

(44)

Given the definition of P (σ, ρ) defined in (40), if we intergrate (44) over Λ2, we get

ZT

24 Za2A

u2 [t, a]P (σ, ρ) (t) (da)

35πT [dt] �ZT

24 Za2A

u2 (t, a)P (σ, ρ0) (t) (da)

35πT [dt] , 8λ2, 8ρ0,

which further implies

8ι : A �! A,ZT

24 Za2A

u2 [t, a]P (σ, ρ) (da)

35πT [dt] �ZT

24 Za2A

u2 [t, ι (a)]P (σ, ρ) (da)

35πT [dt] .

Therefore, P (σ, ρ) : T ! 4 (A) defined above is a mediation equilibrium.�

46

A.7 Proof of Lemma 5

Fix any arbitration equilibrium with moral hazard [p : T �! 4 (A)], i.e.,

8ι : A �! A,ZT

24 Za2A

u2 [t, a] p (t) (da)

35πT [dt] �ZT

24 Za2A

u2 [t, ι (a)] p (t) (da)

35πT [dt] . (45)

In light of Lemma 8, it is without loss of generality for us to focus on weak-language-

barrier equilibria. Recall M = A = R. Define a language-barrier game (Λ, π), such

that

Λ1 = ffag : a 2 A = Mg ,

Λ2 = fMg ,

π [E] =ZT

p (t) [fa : (t, fag , M) 2 Eg]πT [dt] , 8E � T � 2M � 2M,

i.e., the receiver has a unique language type M, who understands all messages; the sender’s

language type has the form fag for a 2 A = M; conditional on payoff type t, π [λ1 = fag , λ2 = M j t]

inheritates the distribution from p (t) [a], with λ1 = fag replacing a.

Define [σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] as follows.

σ [t, λ1 = fag] = δa, 8a 2 A = M,

ρ [λ2 = M, m = a] = δa, 8a 2 A = M,

where δa is the Dirac measure on a. Clearly, incentive compatibility of each sender’s

language type fag is satsified. The incentive comptability of the receiver follows from

(45). More specifically, p(σ, ρ) : T �Λ1 �Λ2 ! 4 (A) defined in (16) has the value

p(σ, ρ) [t, λ1 = fag , λ2 = M] = δa.

And hence, (45) implies

8λ2 2 Λ2, 8ρ0 : Λ2 � M ! 4 (A) ,ZT�Λ1

0@ Za2A

u2 (t, a) p(σ, ρ) (t, λ1, λ2) [da]�Z

a2A

u2 (t, a) p(σ, ρ0) (t, λ1, λ2) [da]

1Aπ [(dt, dλ1) j λ2] � 0,

47

i.e., incentive comptability of the receiver is satisfied, and [σ, ρ] is an outcome-equivalent

weak-language-barrier equilibria. Finally, by Lemma 8, an outcome-equivalent independent-

language-barrier equilibrium exists.�

A.8 Analysis on Example 2

Recall

u1 (a, t) = ��

a� t� 34

�2

;

u2 (a, t) = � (a� t)2 ;

µT (f0g) = µT (f1g) = 12

.

First, we consider bΦ [u1, u2] � u1, and show bΦLB � � 916 . Fix any language-barrier

equilibrium,

[σ : T �Λ1 ! 4 (M) , ρ : Λ2 � M ! 4 (A)] .

Since the receiver has the strictly quadratic utility u2 (a, t) = � (a� t)2, his best reply is

to take the pure action a = Et, where the expection is taken over his posterier belief on t.

Hence,

ρ (λ2, σ (t, λ1)) = E [tjλ2, σ (t, λ1)] .

By the rule of iterative expection, we have

E(t,λ)�π [ρ (λ2, σ (t, λ1))] = Et�πT [t] ,

or equivalently,

E [aj (σ, ρ)] = Et�πT [t] , (46)

where E [aj (σ, ρ)] denotes the expected value of the equilibrium actions. Furthermore,

let E [u1 (a, t) j (σ, ρ)] and E [u2 (a, t) j (σ, ρ)] denote the expected utility of the two play-

ers. We thus have

E [u1 (a, t) j (σ, ρ)] = E

"��(a� t)� 3

4

�2

j (σ, ρ)

#= E

h� (a� t)2 j (σ, ρ)

i� 9

16

= E [u2 (a, t) j (σ, ρ)]� 916

,

48

where the second inequality follows from (46). Then E [u2 (a, t) j (σ, ρ)] � 0 implies

E [u1 (a, t) j (σ, ρ)] � � 916

.

Since (σ, ρ) is arbitrary, we therefore conclude bΦLB � � 916 .

Second, we consider eΦ [u1, u2] � u2, and prove eΦA�AS � � 1128 by contradiction.

Suppose otherwise. I.e., there exists an arbitration equilibrium with adverse selection

[p : T �! 4 (A)] such that

12� Ea�p(0)

h� (a� 0)2

i+

12� Ea�p(1)

h� (a� 1)2

i> � 1

128,

which impies

Ea�p(0)

h(a� 0)2

i<

164

(47)

and Ea�p(1)

h(a� 1)2

i<

164

. (48)

Note that

Ea�p(0)

h(a� 0)2

i= Ea�p(0)

�h�a� Ea�p(0) [a]

�+�

Ea�p(0) [a]� 0�i2

�(49)

= Ea�p(0)

��a� Ea�p(0) [a]

�2�+h�

Ea�p(0) [a]� 0�i2

.

Then, (47) and (49) imply ���Ea�p(0) [a]� 0��� � r 1

64. (50)

Similar argument shows ���Ea�p(1) [a]� 1��� � r 1

64. (51)

Now, consider payoff state t = 0. If the sender sends message 0, the receiver follows p (0).

As a result, the sender’s expected utility is

Ea�p(0)

"��

a� 0� 34

�2#= �Ea�p(0)

h(a� 0)2

i� 9

16+

32� Ea�p(0) [a] (52)

� � 916+

32� Ea�p(0) [a]

� � 916+

32�r

164

� �38

,

49

where the second inequality follows from (50). At payoff state t = 0, if the sender sends

message 1, the receiver follows p (1). As a result, the sender’s expected utility is

Ea�p(1)

"��

a� 0� 34

�2#= Ea�p(1)

"��

a� 1+14

�2#

(53)

= �Ea�p(1)

h(a� 1)2

i� 1

16+

12� Ea�p(1) [a� 1]

� � 164� 1

16+

12� Ea�p(1) [a� 1]

� � 116� 1

16� 1

2�r

164

� � 316

.

where the first inequality follows from (48) and the second inequality follows from (51).

Hence, (52) and (53) imply that the sender prefers message 1 to message 0 at t = 0,

contradicting [p : T �! 4 (A)] being an arbitration equilibrium with adverse selection.

A.9 Analysis on Example 3

Recall

uS = ��

ar � tS �14

�2

and uR = � (ar � tS)2 ,

µ (f0g) = µ

��3572

��= µ (f1g) = 1

3.

In what follows, we sometimes write z = 3572 to economize on notation. Consider language

barriers as described below, where T and Λ are independently distributed.

Λ =

�λ1 =

�0,

12

, 1�

, λ2 =

�0,

12

��;

πΛ (λ1) =3536

and πΛ (λ2) =1

36.

Consider the pure-strategy independent language-barrier equilibrium (h, g) defined as

50

follows. �[h (t, λ) 2 λ](t,λ)2T�Λ , [g (m) 2 A]m2M

�h (0, λ1) = 0; h (z, λ1) =

12

; h (1, λ1) = 1

h (0, λ2) = 0; h (z, λ2) =12

; h (1, λ2) =12

g (m) = m for m 2 M

It is easy to check that this is indeed an equilibrium according to Definition 6.23 Clearly,

in this equilibrium, payoff types are almost fully revealed and the expected utility of the

receiver is:

� 13��

12� 35

72

�2

� 136� 1

3

�1� 1

2

�2

= � 373� 72� 72

' �0.002379 (54)

Given quadratic utility, it is easy to see that, in any language-barrier equilibrium, the

sender’s expected utility differs from the receiver’s expected utility by a constant de-

termined by the “bias” (see the discussed in Section A.8). Furthermore, any noisy-talk

equilibrium can be transformed to an outcome equivalent language-barrier equilibrium

(see Lemma 3). Therefore, it is without of loss generality for us to compare only the the

receiver’s expected utility. In particular, we show the expected utility of the receiver in

any noisy-talk equilibrium is less that that of (h, g) constructed above.

Now, consider any noisy talk equilibrium associated with (ε, ξ) 2 [0, 1] �4 (M),i.e., with probability ε, the receiver, instead of getting the message from the sender, gets

an exogenously chosen noise which is independently (to the sender’s message) generated

according to the distribution ξ. Because of the independence, conditional on noise, payoff

types are uniformly distributed, with a mean z+13 = 107

3�72 <12 . I.e., conditional on noise,

the best strategy for the receiver is to take the action z+13 , and the welfare loss induced by

the noise is at least

13��

z+ 13

� 0�2

+13��

z+ 13

� z�2

+13��

z+ 13

� 1�2

=29�"�

z� 12

�2

+34

#>

16

.

Hence, the total welfare loss induced by the noise is larger than 16 � ε, which, together

23The ideal point of t = 0, 3572 , 1 are 1

4 , 3572 +

14 , 5

4 , and it is easy to check h (t, λ) is consistent with their

preference. Furthermore, E (tjm = 0) = 0, E (tjm = 1) = 1, E�

tjm = 12

�=

13�

3536+

136�

13�1

13+

136�

13

= 12 .

51

with (54), implies that (h, g) dominates the noisy talk equilibrium if

� 373� 72� 72

� �16� ε

() ε � 3772� 36

And, hence, we must have

ε <37

72� 36<

140

. (55)

By the revelation principle, a noisy-talk equilibrium can always be transformed to

one in which the sender takes a (mixed) strategy of recommending actions, and the re-

ceiver follows the recommended actions, and furthermore, it is a best reply for the sender

to recommend the designated actions, and conditional on receiving a recommended ac-

tion, it is a best reply for the receiver to follow it. We now prove our result by contra-

diction in 7 steps, i.e., we assume there is such an equilibrium, in which the receiver’s

expected utility is larger than � 373�72�72 (as calculated in (54)).

Given quadratic utility function, the receiver’s best reply in an equilibrium is the

expectation of his posterior belief on t upon receiving a message. Given the sender’s

quadratic utility function, each payoff type t has at most two best actions to recommend

in an equilibrium (i.e., one smaller than than his ideal point, and the other larger than it).

Step 1: we show that, with a positive probability, a sender of type t 2 f0, z, 1grecommend an action in the interval

�t� 1

11 , t+ 111

�. Suppose otherwise. Then, given

ε < 140 , the total welfare loss for the receiver (at payoff state t) is at least

1� ε

3��

111

�2

�1� 1

403

� 1121

=13

4840' 0.0026859,

which is larger than 373�72�72 ' 0.002379, i.e., the receiver’s welfare loss in (h, g) (see

(54))—a contradiction.

Furthermore, since the ideal point of the sender of type t is t+ 14 and t+ 1

11 < t+ 14 ,

type t must have a unique action in�

t� 18 , t+ 1

8

�to recommend in the equilibrium. Let

at 2�

t� 111 , t+ 1

11

�denote the action recommended by type t 2 f0, z, 1g.

Step 2: a sender with type t must recommend at with a probability larger than 1920 .

Suppose otherwise, i.e., type t recommend another action, denoted by bat, with at least

52

probability 120 . To make both actions best replies, they must have the same distance to

the ideal point t + 14 . Furthermore, since at < t + 1

11 , we conclude bat > 2��

t+ 14

���

t+ 111

�= t+

�12 �

111

�. Then the total welfare loss for the receiver due to type t recom-

mending bat is at least

120� 1� ε

3��

12� 1

11

�2

� 120�

1� 140

3� 81

484=

1053387200

' 0.0027195,

which is larger than 373�72�72 ' 0.002379, i.e., the receiver’s welfare loss in (h, g) (see

(54))—a contradiction.

Step 3: the sender of type t = 1 has an ideal point 54 > 1. As a result, she has a

unique best recommendation in the equilibrium, which is the largest action recommended

in the equilibrium — this is a1. Furthermore, let ba denote the second largest action recom-

mended in the equilibrium, i.e., ba < a1. Since type t = 1 never recommends ba, only type

t = z < 12 , type t = 0 and the noise (with mean z+1

3 < 12 ) may recommend ba. As a result,

ba = E (tjm = ba) < 12< z+

14

.

I.e., az � ba < z+ 14 , where z+ 1

4 is the ideal point for t = z. Therefore, ba = az.

Step 4: we calculate an upper bound for az = E (tjm = az). Note 0 < z < z+13

and that only the noise (with mean z+13 ) and types t = 0, t = z may recommend az. We

would increase the posterior expectation of t if type t = 0 is not allowed to recommend az.

Moreover, to further increase the expectation, we should reduce the probability of type

t = z and increase the probability of noise, due to z < z+13 . To sum, we have

az = E (tjm = az) (56)

�1920 �

1�ε3 � z+ ε� z+1

31920 �

1�ε3 + ε

�1920 �

1� 140

3 � 3572 +

140 �

3572+1

3

1920 �

1� 140

3 + 140

=2807557672

< 0.4869,

where the second inequality follows from z = 3572 and ε < 1

40 (see (55)).

53

Step 5: the sender of type t = 0 has an ideal point of 14 , and we have shown a0 <

14 < az. Hence, to make a0 a best recommendation for type t = 0, we must have

14� a0 � az � 1

4, (57)

i.e., a0 is more close to the ideal point 14 . Then, (56) and (57) imply

a0 > 0.0131. (58)

Step 6: let γ denote the ex-ante probability that noise generates the recommendation

a0, and we show γ < 1130 . Suppose otherwise. Recall a0 < 1

11 . Then, the total welfare loss

due to the noise recommending a0 is at least

γ�"

13��

111� z�2

+13��

111� 1

�2#� 1

130�"

13��

111� 35

72

�2

+13��

111� 1

�2#

=616369

244632960' 0.002519

which is larger than 373�72�72 ' 0.002379, i.e., the receiver’s welfare loss in (h, g) (see

(54))—a contradiction.

Step 7: since a0 < az < z+ 14 < a1 < 5

4 , where z+ 14 and 5

4 are the ideal points of

types t = z and t = 1 respectively. As a s result, types t = z and t = 1 do not recommend

a0, i.e., only type t = 0 and the noise may recommend a0. By Step 2 above, the sender of

type t = 0 recommend a0 with a probability larger than 1920 (which corresponds to ex-ante

probability 1920 �

1�ε3 ). Hence, we have

a0 = E�

tjm = a0��

1920 �

1�ε3 � 0+ γ� z+1

31920 �

1�ε3 + γ

�1

130 �3572+1

3

1920 �

1� 140

3 + 1130

=8560

710856' 0.01204,

(59)

where the second inequality follows from z = 3572 and γ < 1

130 . In particular, (58) contra-

dicts (59)).�

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