Overcommunication in Strategic Information

Transmission Games

Hongbin Cai ∗, Joseph Tao-Yi Wang

Department of Economics, UCLA, Los Angeles, CA 90095, USA

Abstract

Since Crawford and Sobel (1982), the theory of strategic information transmissionhas found a wide range of applications and has become increasingly important inthe literature. In this paper we conduct laboratory experiments to test this theory.Our experimental results strongly support the basic insight of the theory, namely,that less information is transmitted when preferences of the sender and the receiverdiverge. Moreover, the average payoffs for the senders, the receivers, and the overallsubject population are very close to those predicted by the most informative equi-librium. However, the evidence shows that subjects consistently overcommunicatein that the senders’ messages are more informative about the true states of theworld and that the receivers rely more on the senders’ messages in choosing actions,compared with what the theory allows in the most informative equilibrium. To un-derstand the overcommunication phenomenon, we use two popular approaches ofbounded rationality: behavior type analysis and quantal response equilibrium, toanalyze subjects’ behavior in our experiment data.

JEL classifications: C72; C92; D82

Keywords: Overcommunication; Communication Games; Bounded Rationality

1 Introduction

In many real world situations decision makers have to rely on others forinformation needed to make good decisions; these informed others often havepreferences different from the decision makers’ and hence may act strategicallyin communicating their information. Following the seminal work of Crawfordand Sobel (1982), the literature on strategic information transmission has

∗ Corresponding author.Email address: cai@econ.ucla.edu (Hongbin Cai).URL: http://www.econ.ucla.edu/cai/ (Hongbin Cai).

Preprint submitted to Elsevier Science 10 March 2005

grown rapidly, with a wide range of applications in economics and relatedfields such as political science (e.g., congress committee decisions), finance(e.g., stock analysts), and organization theory (e.g., managers versus subor-dinates or consultants). Given its increasing importance in the literature, itis natural to ask what extent the theoretical insights of Crawford and Sobel(1982) and the subsequent works hold empirically. However, aside from thestandard limitations of field data (e.g., no controlled environments, too muchnoise), this theory is very difficult to test with field data, because its key vari-ables (e.g., communication, preference differences) are inherently unobserv-able. Experiments are well suited to test the theory by providing controlledenvironments and by allowing experimenters choose the variables unobservablein field data (e.g., preference differences).

In this paper, we conduct laboratory experiments to test the theory ofstrategic information transmission. We are interested in what extent the maininsights of the theory are supported by experimental data. In the model ofCrawford and Sobel (1982), one player (the sender) communicates his privateinformation about the state of the world to a decision maker (the receiver).They show that in all equilibria the state space is partitioned so that thereceiver takes the same action within each partition element. Moreover, inthe most informative equilibrium, less information can be transmitted as thepreference difference between the two players increases. With non-trivial pref-erence differences, little information can be communicated even in the mostinformative equilibrium, that is, the finest partition in equilibrium is coarse. 1

We design a relatively simple game of strategic information transmission withdiscrete states, messages and actions. The most informative equilibrium of thisgame ranges from the completely informative equilibrium (truth-telling by thesender and fully-trusting by the receiver), to partially informative equilibria,to the completely uninformative equilibrium (“babbling” equilibrium), as thepreference difference between the sender and the receiver increases. We runa series of experiments with varying preference differences, and find evidencethat strongly supports the theoretical comparative statics predictions. Specif-ically, the experimental results clearly show that as the preference differenceincreases, less information is transmitted from the senders and used by thereceivers, in the sense that correlations between states and messages, betweenmessages and actions, and between states and actions all decrease in prefer-ence difference. Moreover, the average payoffs for the senders, the receivers,and the overall subject population are very close to the equilibrium payoffspredicted by the theory. These findings suggest that the subjects not only un-derstand the game and behave strategically (and hence become less “trusting”when their partner’s preference is more different from theirs), but also seem

1 In the canonical example in which the state is uniformly distributed on [0, 1] andutility functions are quadratic, there can be at most two partitions (less than orgreater than 1/4) when the preference difference between the two players is only1/8.

2

to be playing according to the most informative equilibrium.A closer look at the subjects’ behavior in the data, however, clearly rejects

the conclusion that the subjects as a whole play according to equilibrium. Infact, the data show that there is a robust tendency for the subject populationto communicate more than what the most informative equilibrium predicts,the upper bound of communication the theory allows. In the case of large pref-erence differences, correlations between states and messages, between messagesand actions, and between states and actions are all significantly positive, whilethe theory predicts that they should all be zero. We then examine distribu-tions of states conditional on messages, distributions of actions conditionalon messages, and distributions of actions conditional on states. In all cases,the experimental data clearly show that these conditional distributions (fre-quencies) are not consistent with theory predictions: messages are informativeabout the states, and actions rely on messages and hence are correlated withstates. For example, in our game with the state space of {1, 3, 5, 7, 9}, in thecase of large preference difference in which the babbling equilibrium is theunique equilibrium of the game, for the message of 5, the empirical distrib-ution of states is: s = 1 (43.75%), s = 3 (16.25%), s = 5 (28.75%), s = 7(6.25%), s = 9 (5.00%), with an average state of 3.25. Compared with theuniform distribution with an expected state of 5 predicted by the theory, thisempirical distribution contains a substantial amount of information about thetrue state of the world.

These findings raise two questions. 2 First, why are the average payoffs soclose to the equilibrium predictions while at the same time subjects’ individualbehavior is clearly different from equilibrium strategies? Second, what are thepossible explanations for the subjects’ behavior? In particular, what are thereasons behind the overcommunication phenomenon? Two approaches to mod-eling bounded rationality, behavior type analysis and quantal response equilib-rium, are commonly used in the existing literature to explain non-equilibriumphenomena in experimental data. To understand the overcommunication phe-nomenon in our experiment, we apply both approaches to analyze our data.

The behavior type analysis approach is developed by a number of scholars,e.g., Nagel (1995), Stahl and Wilson (1995), Ho et al. (1998), Costa-Gomeset al. (2001, 2004), and Camerer et al. (2004). They demonstrate that sub-jects in their experiments consistently behave in a specific bounded rationalway: subjects of different levels of sophistication have the non-equilibrium be-lief that other subjects are one level lower than themselves in sophisticationand best response to that belief. 3 In many games it is typical to start byspecifying the lowest level of sophistication to behave completely randomly

2 These findings are robust to certain variations of payoff parameters and noisysignals. In addition, we do not find any clear learning effect in our data. See Section7 for details.3 See Crawford (1997) and Camerer (2003) for comprehensive surveys and furtherreferences.

3

(choosing every available strategy with equal probability). However, completerandomization is the sender’s equilibrium strategy in the babbling equilibriumof communication games (which always exists). Thus, the system of behaviortypes in communication games cannot be anchored on complete randomiza-tion. Similarly, it is not straightforward to apply the cognitive hierarchy theoryof Camerer et al. (2004) to communication games. To apply this approach incommunication games, Crawford (2003) cites earlier experiment evidence (e.g.,Blume et al. 2001) and argues that the system of subject types should be an-chored on the “truster” type of senders and the “believer” type of receivers.In our data, there are indeed senders who always tell the truth (the trustertype) and receivers who always believe the senders (the believer type). So wefollow Crawford (2003) to define the system of behavior types for our gameby starting with these simple types. Assuming receivers are of the believertype, the best response of the next level of sophistication senders is to sendmessages identical to their most preferred actions. These senders are calledthe liar type. Then, believing senders are of the liar type, receivers’ best re-sponse is to invert the messages they receive by taking actions equal to themessages minus the preference difference. The system of behavior types canbe constructed in this iterative way.

We then follow the classification methodology of Costa-Gomes et al. (2001,2004) to classify subjects whose actions can be consistently identified as fittinginto one of the behavior types. About 75% of the subjects in our experimentscan be classified. We calculate the expected payoffs and correlations generatedby the distribution of behavior types combined with some random noise. Theresults match the actual data quite well. This suggests the following inter-pretation of our experimental results. Subjects of lower levels of sophistica-tion overcommunicate, resulting in more communication than what the theorypredicts. However, since the subjects are randomly paired, a sender of a cer-tain type can be matched with possibly many types of receivers in a givenround. Some of these matches are equilibrium-alike, others (“mismatches”)give rise to outcomes quite different from equilibrium. Combined with somenoise, these mismatches tend to offset the payoff gains and losses (relative tothe equilibrium predictions), leading to average payoffs close to the equilibriumpredictions.

An alternative approach to bounded rationality is the “Quantal ResponseEquilibrium” (QRE) proposed by McKelvey and Palfrey (1995, 1998), in whichplayers have correct beliefs about their opponents but do not maximize theirpayoffs perfectly given the beliefs. One advantage of this approach is that itexplicitly takes into account noisy behavior of subjects. To apply this approachto our experiment data, we follow McKelvey and Palfrey (1998) to solve for thelogit-agent quantal response equilibrium (logit-AQRE) for our game, and dothe maximum likelihood estimation using our data. The results suggest thatthe logit-AQRE explains the communication patterns in our experimental datapretty well.

In the existing literature there are very few experimental works testing the

4

theory of strategic information transmission. Closest to our paper is Dick-haut et al. (1995), who pioneered direct experimental testing of the theoryof strategic information transmission. Their experimental results also confirmthe comparative statics predictions of the theory. Our experimental design issimilar to theirs, but differ in several aspects. First, our experimental designallows sharper tests of the theory. Dickhaut et al., use a game in which boththe state space and the action space are {1, 2, 3, 4}. In this game the receiver isoften indifferent between two actions, which makes outcomes of different equi-libria not sharply distinguishable. 4 Moreover, we control for repeated gameeffects while Dickhaut et al., let each pair of subjects play three rounds of thegame in sessions with 4 pairs of subjects. Despite the differences in design,it is comforting that their findings are broadly confirmed in our experiments.However, we go beyond confirming the comparative statics predictions of thetheory. We find similarities between average payoffs and equilibrium-predictedpayoffs, and more importantly, the overcommunication phenomenon. We es-tablish the robustness of these findings along several dimensions. Lastly, weanalyze the possible reasons behind the non-equilibrium behavior of subjectsand what results in payoffs being so close to the equilibrium predicted payoffs.

In two other closely related papers, Blume et al. (1998, 2001) conduct a seriesof experiments on strategic information transmission games. Their experimen-tal results clearly indicate an overcommunication tendency by the subjects,see, e.g., Tables V-VII of Blume et al. (2001). They also do a quite thorough in-vestigation into whether such non-equilibrium phenomenon is consistent withseveral different equilibrium solution concepts. Since they are mainly inter-ested in the evolution of meanings of messages, their games and experimentaldesigns are different from ours. Our paper focuses on overcommunication anduses the bounded rationality approaches to shed light on this phenomenon,thus complements their studies.

There is a very active literature, both theoretical and experimental, on pre-game communication when players try to communicate to their opponentstheir intentions to play the game in a particular way, e.g., see Farrell andRabin (1996). 5 Due to space limit, we will not discuss the relationships be-tween such pre-game communication and strategic information transmission(where players communicate their private information to their opponents).Crawford (1998) offers an excellent overview of the theoretical connections

4 For example, in the babbling equilibrium the receiver is supposed to choose either2 or 3 and the sender should purely randomize among 1, 2, 3 or 4. If the receiver’stie-breaking leans toward matching the sender (i.e., choosing 2 for message 2, and3 for message 3), this babbling equilibrium is indistinguishable with the completelyinformative equilibrium 50% of time.5 Palfrey and Rosenthal (1991) study a model in which players have private infor-mation about their costs of contributing to a public project and also can announcetheir intentions (whether to contribute) before the contribution stage. They obtainmixed experimental results regarding communication in the cheap talk stage.

5

and distinctions between these two types of cheap talk games as well as therelated experimental literature.

Several theoretical models, e.g., Eyster and Rabin (2000), Ottaviani andSquintani (2002), and Crawford (2003), predict that under certain condi-tions players communicate their private information more than the standardBayesian Nash equilibrium allows. We will discuss the relations of these paperswith our study in Section 5.

The rest of the paper is organized as follows. Section 2 presents the theoret-ical model and its predictions, and Section 3 discusses our experiment design.Then we present the experimental results of the baseline sessions in Section 4.The next two sections use two approaches of bounded rationality to interpretour data. Section 5 analyzes the data using the behavior type analysis ap-proach mentioned above, and Section 6 applies quantal response equilibriumto the data. In Section 7 we examine robustness of our experimental resultsand possible learning effects. Concluding remarks are contained in Section 8.

2 Theoretical Model and Predictions

In our experiments, subjects are paired to play the following specific gameof strategic information transmission. One player in each pair is the sender andthe other is the receiver. The sender is informed about the state of the world,which is a number uniformly drawn from the state space S = {1, 3, 5, 7, 9}. Thereceiver knows the distribution of s, but not its realization. The sender thenchooses to send a message to the receiver, where the feasible message space isany subset of M = {1, 3, 5, 7, 9}. After receiving a message from the sender, thereceiver chooses an action from the action space of A = {1, 2, 3, 4, 5, 6, 7, 8, 9}.The true state of the world and the receiver’s action determine the two players’payoffs in points according to the following pre-specified formula

uR = 110 − 10 · |s − a|k

uS = 110 − 10 · |s + d − a|k

where uR and uS are the payoffs for the receiver and the sender, respectively,s is the true state of the world, a is the receiver’s action, d is the preferencedifference between the sender and the receiver, and k is a positive parameter.Thus, the receiver’s ideal action is to match the true state of the world, whilethe sender’s is the true state of the world plus d.

In games of strategic information transmission, it is well known that thereexists a babbling equilibrium in which the sender always sends a purely un-informative message (message not correlated with his information about thetrue state of the world) and the receiver always ignores the sender’s messageand makes decisions based on her own prior knowledge about the state of theworld. In the babbling equilibrium, no information is transmitted from theinformed sender to the receiver. Naturally the focus of research is on the most

6

informative equilibrium, that is, how much information can be possibly trans-mitted in any equilibrium. Informativeness can be measured by the correlationbetween actions and the true states of the world. The correlation is zero in thebabbling equilibrium, and takes the maximum value of one if actions perfectlymatch the states of the world. In addition, the informativeness of the sender’smessages can be measured by the correlation between the true states of theworld and the messages he sends; and how “trusting” the receiver is can bemeasured by the correlation between the messages she receives and the actionsshe takes.

While theoretically messages do not have to confirm with their natural lan-guage meanings in equilibria, we assume that subjects use messages for theirnatural language meanings in our experiments. There are two reasons for thisassumption. First, as the results of Blume et al. (2001) show, equilibriummessages tend to be consistent with their natural language meanings. Second,we used a highly structured communication protocol in our experiments thatspecifically asked senders to report the states they observed. It is highly un-likely that subjects in our experiments would coordinate on meanings of themessages other than their natural language meanings.

When the most informative equilibrium involves pooling or partial pooling,there exist many different equilibrium messages that lead to the same state-action partitions. For example, in a partial pooling of states 5, 7 and 9, senderscan randomize uniformly over {5, 7, 9}, or choose to send a single message of“The number I received was 9” (or other messages) in all three states. Weuse the randomization pooling in our analysis, for the following reasons. First,for a large part of our analysis we focus on the case of d = 4 in which thebabbling equilibrium is the unique equilibrium, in which messages must beun-informative about the states and hence the correlation between messagesand states should be zero. In this case our choice of the randomization poolingdoes not have any effect. Secondly, our main points about overcommunicationare not affected if we use other messages in equilibria with partial pooling. Itcan be verified that the correlations between messages and states and betweenmessages and actions are not very different qualitatively if, for example, weuse a message of 9 for the states {5, 7, 9}. In addition to correlation analysis,we further examine the empirical conditional frequencies of states given mes-sages, actions given messages, and actions given states, which provides strongevidence for overcommunication. Thirdly, the randomization pooling allowsthe most flexibility in interpreting the data from the theoretical point of view,thus strengthening our main points about overcommunication relative to thetheoretical predictions.

In the game used in our experiments, as the preference difference varies, themost informative equilibria can be characterized as follows.Proposition 1 For k ≥ 1, the most informative equilibria of the game (fordifferent d’s) are(1) the separating (completely informative) equilibrium if d ≤ 1, in

which for every state of the world s, the sender always tells the truth, and

7

the receiver always chooses the action according to the (truthful) message;(2) the partial pooling equilibrium if 1 < d ≤ 1.5, in which the sender

sends a same message for states 1 and 3, and another message for states5, 7, and 9, and the receiver chooses 2 or 7 ;

(3) the partial pooling equilibrium if 1.5 < d ≤ 2.5, in which the senderchooses m(s = 1) = 1 and pools for states 3, 5, 7, and 9, while the receiverchooses 1 if m = 1 and 6 otherwise;

(4) the babbling equilibrium if d > 2.5, in which the sender pools forstates 1, 3, 5, 7, and 9, and the receiver always chooses 5 no matter whatmessage she receives.

Our experiments have four treatments with d1 = 0.5, d2 = 1.2, d3 = 2, andd4 = 4, corresponding to the four cases in Proposition 1. We choose a valueof 1.4 for parameter k in our baseline sessions, so that payoffs are sensitive tothe choices subjects make (the receiver’s payoffs range from 110 to -73.79 andthe sender’s earning can range from 110 to - 214.23 when d = 4), but not toohigh that subjects may suffer from a loss in one round too large to recover inthe experiment (for example, for k = 2 and d = 4, the sender could get payoffsof -1330 in one round). Besides, it is not desirable for one or few abnormalrounds to have too large an impact on the payoffs and to potentially affectsubjects’ behavior in later rounds. In our experiments with k = 1.4 and d = 4,subjects sometimes got negative point payoffs in some rounds but all recoveredtheir losses and got positive total payoffs by the end of the experiments. Forrobustness analysis, we also conduct experiments with different values of k,which will be discussed later in Section 7.

The properties of the most informative equilibria are presented in Table 1below.

The main insight of Crawford and Sobel (1982) is clearly shown in Ta-ble 1. As the preference difference d between the sender and the receiverincreases, less information can be transmitted in the most informative equilib-rium. Specifically, as d increases, the sender’s message becomes less informa-tive as he pools more. As a result, as d increases from 0.5, to 1.2, to 2, and to4, the correlation between the true state and the sender’s message decreasesfrom 1, to 0.75, to 0.5, all the way down to 0. Correspondingly, as d increases,the receiver trusts the sender less as her actions are less correlated with hismessages. For d = 0.5, 1.2, 2, 4, the correlation between the sender’s messagesand the receiver’s actions decreases from 1, to 0.866, to 0.707, all the waydown to 0. Consequently, as a measure of how much information is reflectedin the receiver’s decisions, the correlation between actions and the true statesdecreases as d increases. 6 Accordingly, both the sender’s and the receiver’s

6 Note that the correlation between messages and actions is identical to the cor-relation between states and actions. Since we treat pooling of states as an explicitmixing of these states (done by the computer program), the distribution of the real-ized messages is the same as the distribution of states. For example, when d = 1.2,the sender will mix between 1 and 3 equally for states s = 1 and s = 3 (each with

8

Tab

le1.

Pro

pert

ies

ofth

eM

ost

Info

rmat

ive

Equ

ilibr

ia

dE

quilib

rium

Equil.

Sen

der

s’R

ecei

vers

’O

vera

llC

orr.

Cor

r.C

orr.

Mes

sage

sA

ctio

ns

Pay

offs

Pay

offs

Pay

offs†

(S,M

)‡(M

,A)

(S,A

)

0.5

{1},{3},{5},

{7},{9}

1,3,

5,7,

910

6.21

(0.

00)

110.

00(

0.00

)10

8.11

(1.

89)

1.00

01.

000

1.00

0

1.2

{13}

,{57

9}2,

789

.52

(18.

06)

95.4

4(1

0.33

)92

.48

(15.

01)

0.75

00.

866

0.86

6

2{1},{3

579}

1,6

72.3

7(3

1.77

)87

.38

(19.

88)

79.8

8(2

7.54

)0.

500

0.70

70.

707

4{1

3579}

529

.46

(66.

32)

71.5

9(2

7.26

)50

.52

(54.

90)

0.00

00.

000

0.00

0

Not

es:N

umbe

rsin

side

the

pare

nthe

sis

are

the

stan

dard

devi

atio

nsof

the

corr

espo

ndin

gpa

yoffs

.†

Thi

sis

the

aver

age

ofth

ese

nder

’san

dre

ceiv

er’s

payo

ffs.

Thi

sm

easu

reis

usef

ulbe

caus

eou

rex

peri

men

tsra

ndom

lyas

sign

asu

bje

ct’s

role

asth

ese

nder

orre

ceiv

erat

the

begi

nnin

gof

each

roun

d.‡

Whe

nca

lcul

atin

gth

eco

rrel

atio

nsin

volv

ing

mes

sage

s,w

ein

terp

ret

the

mes

sage

sas

mix

ing

acro

ssth

epo

ssib

lest

ates

.In

othe

rw

ords

,{1

3}w

ould

bem

ixin

gbe

twee

n1

and

3,an

dso

on.

9

payoffs decrease in d, with the sender’s payoff suffering a much greater reduc-tion than the receiver’s as d increases from 0.5 to 4. The following hypothesessummarize the theoretical predictions of the model.Hypothesis 1 As the preferences of the sender and the receiver diverge, lessinformation is transmitted by the sender and utilized by the receiver: the cor-relations between states and messages, between messages and actions, and be-tween states and actions all decrease.Hypothesis 2 As the preferences of the sender and the receiver diverge, boththe sender’s and the receiver’s payoffs decrease.

3 Experiment Design

The experiment was conducted in the California Social Science Experi-mental Laboratory (CASSEL) located at UCLA, and programmed with thesoftware z-Tree (Fischbacher 1999). Typically, sessions lasted from 1.5 to 2.5hours, and subjects were predominantly UCLA undergraduate students, withfew graduate students. Subjects interacted with each other only through com-puter terminals during the experiments. Subjects are paid a $5 show-up feeplus whatever they earn from playing the games; their dollar earnings areconverted from point payoffs using a pre-specified exchange rate. 7

In each experimental session, 2N subjects were matched into N pairs. Pair-ing was done in such a way that two subjects would play against each otherat most once; and this is made known and clear to all subjects to avoid prob-lems of repeated interactions. 8 In each round, within each pair one player wasrandomly chosen to be the sender and the other to be the receiver. For eachpair, the computer program generated a number uniformly from {1, 3, 5, 7, 9},and revealed the number to the sender. After knowing this number, the senderchose a message to send to the receiver. Besides one of the five possible states,the sender could also voluntarily randomize over the state space. For example,the sender could instruct the computer to send “3 or 5 or 7”, and the computerprogram would generate a number uniformly from the set the sender speci-fied and sent it to the receiver. Once the receiver got the sender’s message,she chooses an action from {1, 2, 3, 4, 5, 6, 7, 8, 9}. The even numbers were in-cluded in the action space so that the receiver could make better decisions ifshe tried to maximize her expect payoff and her beliefs led to the expectedstate being one of the even numbers (e.g., the state being either 1 or 3 with

probability of 15 ·

12 + 1

5 ·12 = 0.2), leading to the realized message of m = 1 or m = 3

each with a probability of 0.2. Moreover, the receiver will take an action of 2 whenseeing messages 1 or 3, which corresponds to states 1 or 3. Therefore, the correlationof actions with states is the same with messages.7 Excluding the $5 show-up fee, average dollar earning ranges from about $10.30to $32.11 from the lowest payoff session to the highest payoff session.8 A sample of instructions for one of the sessions can be found athttp://www.econ.ucla.edu/cai/.

10

Fig. 1. Sender’s (Member A’s) Screen.

equal probabilities). Payoff functions and parameters were publicly announcedto the subjects, while possible values of payoffs are presented in tables to thesubjects so they do not have to do the calculations. For example, when thetrue state of the world is 7, the sender’s and the receiver’s payoffs in pointsfor the four treatments d = 0.5, 1.2, 2, 4, are as shown in Table 8.

In the experiments, the tables of payoffs as functions of the true state of theworld and the receiver’s action were presented to the subjects as shown in thefollowing screen shots. The preference difference d was shown in a box in theupper right corner of each player’s screen (in the example, d = 4). The payofftable was presented on the left side of the screen for each player. The firstcolumn was titled “Secret number”, which represents the state of the world inthe experiments. Subjects needed to click on the button of a number in thefirst column to view the payoffs for both players (which depend on memberB’s actions) if the true value of the secret number was that number. In theexample, member A clicked on buttons of “1”, “3”, “5”, and not (yet) on“7” or “9”, while member B clicked all the buttons except “7”. On the rightside of member A’s screen, member A could find the true value of the secretnumber (“5” in this example) and was asked to choose a message in the boxbelow. In this example, member A chose “5 or 7 or 9”, which meant thatthe computer program would generate a number from among the three withequal probability. The number turned out to be “9” in this case, which thecomputer sent to member B. The right side box of member B’s screen toldher that member A sent a message “The number I received is 9”, where she

11

Tab

le2.

Pay

offs

Whe

nTru

eSt

ate

s=

7(k

=1.

4)

Act

ions

acti

on1

acti

on2

acti

on3

acti

on4

acti

on5

acti

on6

acti

on7

action

8ac

tion

9

mem

ber

B-1

2.86

14.8

240

.36

63.4

483

.61

100.

0011

0.00

100.

0083

.61

mem

ber

A(d

=0.

5)-2

7.43

1.23

27.8

752

.23

73.9

392

.36

106.

2110

6.21

92.3

6

mem

ber

A(d

=1.

2)-4

8.59

-18.

639.

4435

.43

59.0

479

.84

97.0

910

8.95

102.

68

mem

ber

A(d

=2

)-7

3.79

-42.

45-1

2.86

14.8

240

.36

63.4

483

.61

100.

0011

0.00

mem

ber

A(d

=4

)-1

41.1

9-1

06.7

4-7

3.79

-42.

45-1

2.86

14.8

240

.36

63.4

483

.61

Not

es:T

hese

nder

isca

lled

“mem

ber

A”,

and

the

rece

iver

“mem

ber

B”

inth

eex

peri

men

ts.T

here

ceiv

er’s

payo

ffde

pend

son

the

stat

ean

dhe

rac

tion

,an

dis

inva

rian

tto

d.

12

Fig. 2. Receiver’s (Member B’s) Screen.

was also reminded that “This message could come from a random draw”. 9

Member B was then asked to choose an action from the action space. At theend of each round, a summary table revealed all the relevant information toboth players, including the true state of the world, the sender’s message, thereceiver’s action, and each player’s payoff.

We ran three sessions using our base design with the payoff parameter kset at 1.4. Session 1 was conducted with 28 subjects and a total of 21 rounds,five rounds each for d = 0.5, 1.2, 2 and six rounds for d = 4, resulting in 70observations for each of the first three cases and 84 for the last. Session 2 wasrun with 32 subjects and a total of 31 rounds, all with d = 4, resulting in 496observations. Session 3 was run with 32 subjects and a total of 20 rounds, allwith d = 2, resulting in 320 observations. The results from these three sessionsturned out to be very close, so we grouped them together as a single sample.

4 Experimental Results

In this section we present the experimental results of the three baselinesessions.Result 1 The experimental outcomes are strongly supportive of Hypothesis 1:less information is transmitted by the sender and utilized by the receiver aspreferences between the sender and the receiver diverge. (i.e. as d is larger.)

9 We restrict the language protocol to “The number I received is” to ensure thatboth subjects communicate in a common language. We do not deal with issues ofevolution of meanings of messages that Blume et al., (1998, 2001) study.

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Specifically, the correlations between states and messages, between messagesand actions, and between states and actions all decrease as d increases.

Table 3 presents the correlations calculated from our experimental data.

Table 3Actual Information Transmission

Number of k d Corr. Corr. Corr.Observations (S,M) (M,A) (S,A)

70 1.4 0.5 0.916 0.965 0.876

70 1.4 1.2 0.896 0.924 0.830

390 1.4 2 0.734 0.794 0.618

580 1.4 4 0.391 0.542 0.207

As d increases from 0.5 to 4, the senders’ messages become less and lessinformative, measured by the correlation between states and messages. 10

While states and messages are highly correlated for the cases d = 0.5 andd = 1.2, the correlation between states and messages decreases significantlywhen d = 2 (corr.(S, M) = 0.734) and drops much further when d = 4(corr.(S, M) = 0.391). The receivers show a similar pattern, with the correla-tion between messages and actions decreasing from 0.965, to 0.924, to 0.794,and then to 0.542 as d increases from 0.5, to 1.2, to 2, and then to 4. As ameasure of how much information is actually used in the receivers’ actions, thecorrelation between states and actions also exhibits a clear comparative staticpattern predicted by theory: it decreases from 0.876, to 0.830, to 0.618, andthen to 0.207 as d increases from 0.5, to 1.2, to 2, and then to 4. Therefore,our experimental outcomes are strongly supportive of the theoretical predic-tion of Hypothesis 1. This indicates that subjects in our experiments as awhole understand the basic strategic aspects of the game and are responsiveto incentives presented in the experiment design.Result 2 The experimental outcomes are strongly supportive of Hypothesis2: both the senders’ and the receivers’ average payoffs decrease as the prefer-ence difference increases. Moreover, the average payoffs for the senders, thereceivers, and the subject population are very close to those predicted by themost informative equilibrium.

We calculate the average payoffs for the senders, the receivers, and the wholesubject population, for each value of preference difference d = 0.5, 1.2, 2, 4.Table 4 presents the results, as well as the equilibrium predicted payoffs (fromTable 1) for comparison. It is clear from Table 4 that the actual payoffs exhibitstrong monotonic patterns as preferences between the senders and the receivers

10 In 10 − 15% of the observations senders sent mixing messages such as “7 or 9”.For those observations, we used the realized messages generated by the computer tocalculate the correlation between messages and states. Our results are not affectedmuch by this.

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diverge, just as theory predicts. In particular, as d increases from 0.5 to 4,average payoffs for both the senders and receivers fall. These results lendstrong support for the comparative static prediction of the Crawford and Sobel(1982).

Table 4Theoretical vs. Actual Payoffs

# of Senders’ Payoffs Receivers’ Payoffs Averageobs. d Predicted Actual Predicted Actual Predicted Actual

(s.d.) (s.d.) (s.d.) (s.d.) (s.d.) (s.d.)

70 0.5 106.21 99.08∗ 110.00 101.79∗∗ 108.11 100.44∗

( 0.00) (24.16) ( 0.00) (25.82) ( 1.89) (24.95)

70 1.2 89.52 88.76 95.44 93.54 92.48 91.15(18.06) (18.10) (10.33) (19.97) (15.01) (19.14)

390 2 72.37 75.03 87.38 83.69∗ 79.88 79.36(31.77) (37.28) (19.88) (32.69) (27.54) (35.30)

580 4 29.46 36.89∗∗ 71.59 65.84∗∗ 50.52 51.37(66.32) (68.38) (27.26) (42.72) (54.90) (58.80)

∗ T-test shows actual payoffs differ from equilibrium payoffs significantly at the 5-percent level of confi-dence.∗∗ T-test shows actual payoffs differ from equilibrium payoffs significantly at the 1-percent level of confi-dence.

Another finding from Table 4 is that the actual payoffs are very close to theircorresponding payoffs predicted by the most informative equilibrium, givenhow noisy experimental data usually are. For d = 0.5, the most informativeequilibrium is the separating (completely informative) outcome in which boththe sender and the receiver get deterministic payoffs. In this case one expectsthat some subjects in the experiments make errors sometimes or just wantto try other strategies (e.g., trying to learn), thus leading to somewhat lowerpayoffs. From Table 4, the senders and the receivers on average get 7.13 and8.21 points less, respectively, than their equilibrium payoffs when d = 0.5.For the cases of d = 1.2 and d = 2, actual payoffs and equilibrium payoffsare very close for both the senders and the receivers and for the populationaverage. In most instances, they are almost identical, e.g., 89.52 (equilibrium)and 88.76(actual), 95.44 (equilibrium) and 93.54 (actual), 79.88 (equilibrium)and 79.36 (actual). For d = 4, the actual payoffs for the senders and thereceivers are still quite close to the corresponding equilibrium payoffs, butnot as close as in the cases of d = 1.2 and d = 2, and are significantlydifferent. The actual average population payoff (51.37) for d = 4 is very closeto the equilibrium payoff (50.52), since the senders’ gain relative to equilibrium(36.89− 29.46 = 7.43 points) is largely offset by the receivers’ loss relative toequilibrium (71.59 − 65.84 = 5.75). Note also that the standard deviationscalculated from our experimental data also come very close to those predicted

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by the theory, with the case of d = 0.5 being the obvious exception.Finally, we perform t-statistic tests to determine whether actual payoffs

differ from equilibrium payoffs in a statistically significant way. For the casesof d = 1.2 and d = 2, actual payoffs are not statistically different from thecorresponding equilibrium payoffs at the 5-percent level of confidence, exceptfor the receivers’ average payoff when d = 2. Since the sample size is large ford = 4, it is easy to get significant values of the t-statistics and reject the nullhypothesis that the actual payoff equals the equilibrium payoff. However, evenwhen the t-test is significant, all it says is that statistically the actual payoffis not equal to the equilibrium payoff. The differences between theoretical andactual payoffs in those cases are still small given how noisy experimental datausually are. In sum, the evidence in Table 4 indicates that the actual payoffsin our experiments are very close to those predicted by theory.

Does this mean that subjects play according to the most informative equi-librium? The results on payoffs suggest a positive answer. However, after ex-amining the data in more detail, we find that this is not the case.Result 3 Subjects behave differently from the equilibrium strategies predictedby theory. In particular, except for the full revelation case, the senders tend tocommunicate information and the receivers tend to “trust” the senders morethan the most informative equilibrium.

To understand this result, in Table 5 below we compare correlations fromthe actual experimental data (from Table 3) and theoretical correlations inthe most informative equilibrium (from Table 1).

Table 5Theoretical vs. Actual Information Transmission

# of d Correlation(S,M) Correlation(M,A) Correlation(S,A)Obs. Predicted Actual Predicted Actual Predicted Actual

70 0.5 1.000 0.916 1.000 0.965 1.000 0.876∗

70 1.2 0.750 0.896∗∗ 0.866 0.924 0.866 0.830

390 2 0.500 0.734∗∗ 0.707 0.794∗∗ 0.707 0.618∗

580 4 0.000 0.391∗∗ 0.000 0.542∗∗ 0.000 0.207∗∗

∗ T-test shows actual payoffs differ from equilibrium payoffs significantly at the 5-percent level of confi-dence.∗∗ T-test shows actual payoffs differ from equilibrium payoffs significantly at the 1-percent level of confi-dence.

Again, the full revelation case (d = 0.5) is special since the theoretical cor-relations are all equal to one. The actual correlations are necessarily smallerthan but are still close to one. What is interesting is that in all other cases(d = 1.2, 2, 4), the correlations between states and messages from the ex-periments are all greater than those predicted by theory. In particular, ford = 4, the actual correlation is almost 0.4 while the theoretical correlationis zero. This clearly suggests that the senders communicate more informationabout the true states of the world in their messages than the theory allows

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in the most informative equilibrium. Similarly, in all three cases d = 1.2, 2, 4,the correlations between messages and actions from the experiments are allgreater than those predicted by theory, especially when d = 4 (the actualcorrelation is 0.542, compared to zero for the theoretical correlation). At leastin the case of a large preference difference (d = 4), the evidence stronglysuggests that the receivers tend to trust the senders much more than theoryallows. Compared with equilibrium strategies, both senders and receivers inour experiments show an overcommunication tendency for cases with largepreference differences. 11

Surprisingly, the overcommunication by both senders and receivers doesnot necessarily imply that information actually used in the decisions is morethan what theory predicts. Table 5 shows that the correlation between statesand actions from the experiments is actually lower than that predicted bytheory (except for d = 4). When d = 1.2 and d = 2, the actual correlationsbetween states and actions are lower than the equilibrium predictions: 0.830(actual) versus 0.866 (equilibrium) for d = 1.2, and 0.618 (actual) versus0.707 (equilibrium) for d = 2. For the case of d = 4, even though the actualcorrelation between states and actions (0.207) is still significantly greater thanthe equilibrium correlation of zero, its magnitude is much smaller comparedwith the strong overcommunication tendency of senders and receivers.

We perform statistical tests to see if the actual correlations are significantlydifferent from the predicted ones. 12 Consistent with our casual observationabove, for the correlation between states and messages, the t-tests are signifi-cant for all but the case of d = 0.5; for the correlation between messages andactions, the t-tests are significant for d = 2 and d = 4, but not significant ford = 0.5 and d = 1.2; and for the correlation between states and actions, thet-tests are all significant except for the case of d = 1.2.

While the correlations of the aggregate data in Table 5 clearly show an over-communication tendency by both senders and receivers, to gain more insightsabout this phenomenon, a more detailed analysis of the experimental data isnecessary. We focus on the observations for the case of d = 4, because the ac-tual correlations in this case differ from equilibrium predictions the most andalso because we have the most observations for this case. We first examinethe frequencies of actions taken by the receivers conditional on the true statesof the world. For d = 4, the only equilibrium is the babbling equilibrium, in

11 Note that since senders and receivers communicate more than what is allowed inthe most informative equilibrium, their behavior is even further away from the lessinformative equilibria of the game.12 Specifically, we run the following regression: Y = α + (rXY + β)X + ε whererXY = sY

sX· σXY , sX , sY are the sample standard deviation of X and Y , and σXY

is the correlation between X and Y predicted by theory. The t-test on β would tellus if Corr(X, Y ) (the actual correlation between X and Y ) is statistically differentfrom σXY (the theoretical correlation). We run the regressions with (Y, X) beingone of the three pairs (Message, State), (Action, Message) and (Action, State).

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which the receiver should choose action 5 regardless of the messages she re-ceives and hence regardless of the true states of the world. The experimentalresults are presented in the upper panel of Table 6. For simplicity, we groupactions into three groups: “actions 1 to 3”, “actions 4 to 6”, and “actions 7to 9”. For the 580 observations we have on d = 4, the frequencies of the fivestates (s = 1, 3, 5, 7, 9) are very close to uniform.

Several points are immediate from the experimental results. First, for eachof the 5 states, only about half of the time the actions 4, 5 6——actions consid-ered broadly consistent with equilibrium——are actually chosen. Secondly, asthe state becomes larger, the frequency weights switch from the small actionsto the large actions. Specifically, as the state increases, while the frequenciesof actions 4 to 6 remain more or less constant, the frequencies of small actions1 to 3 decreases (s = 7 poses a minor exception) 13 and the frequencies oflarge actions 7 to 9 increases. Thirdly, the average action clearly exhibits anincreasing trend as the state becomes larger.

We use χ2 tests to further examine if the empirical distribution of actionsconditional on each state is consistent with the babbling equilibrium. If thetheory prediction is interpreted literally, namely, the receivers should alwayschoose action 5, the test is trivially rejected by our data. We use a stronger testby postulating that subjects “tremble hands” while playing the equilibriumstrategy so that the observed choices of action 5 are equilibrium plays while allother actions result from erroneous plays with equal probabilities. Since action5 is observed less than 50% of the time, our test is very strong in that it allowsnoisy plays more than 50% of the time. The last column of the upper panelin Table 6 presents the χ2 test results. For every state s = 1, 3, 5, 7, 9, the testresults strongly reject the null hypothesis of equilibrium play with errors atthe 2-percent level of confidence. Therefore, with very high probabilities theobserved frequencies of actions conditional on each state are not consistentwith the babbling equilibrium hypothesis, even if a large amount of errors areallowed.

For the senders’ behavior, the frequencies of the true states conditional onthe senders’ messages are presented in the middle panel of Table 6. The em-pirical frequencies of states conditional on messages indicate how informativeeach message (1, 3, 5, 7, 9) is about the true state of the world. In the babblingequilibrium, the sender’s message should be completely uninformative so thatthe distribution of states conditional on any message should be uniform over{1, 3, 5, 7, 9}. In the 580 observations with d = 4, the frequencies of messagesare very skewed: message 1 was sent only 31 times while message 9 was sent332 times. The fact that messages are concentrated on 9 does not itself re-ject the hypothesis that messages are uninformative, since senders in differentstates could all pool at a same message. However, the frequencies of statesconditional on messages clearly rejects the hypothesis that messages are un-

13 This is because some receivers simply subtract 4 from 7, resulting in many 3’s.See Section 5 for further explanations.

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Table 6Frequencies of A conditional on S, S on M, and A on M (d=4)

Actions State 1 State 3 State 5 State 7 State 9 Total

1,2,3 34 24 16 19 12 105

4,5,6 59 58 61 47 52 277

7,8,9 22 40 41 47 48 198

Total 115 122 118 113 112 580

aver. A 4.670 5.516 5.746 5.973 6.071

χ2(8) 19.48 28.38 38.76 46.10 48.29(p-value†) (0.012) (0.000) (0.000) (0.000) (0.000)

State Message 1 Message 3 Message 5 Message 7 Message 9 Total

1 19 14 35 22 25 115

3 4 21 13 25 59 122

5 2 6 23 15 72 118

7 3 8 5 14 83 113

9 3 4 4 8 93 112

Total 31 53 80 84 332 580

aver. S 2.871 3.755 3.250 4.071 5.964

χ2(4) 33.35 18.04 42.75 10.88 41.92(p-value‡) (0.000) (0.001) (0.000) (0.028) (0.000)

Action Message 1 Message 3 Message 5 Message 7 Message 9 Total

1,2,3 20 22 23 31 9 105

4,5,6 11 25 49 36 156 277

7,8,9 0 6 8 17 167 198

Total 31 53 80 84 332 580

aver. A 2.774 4.094 4.288 4.786 6.611

χ2(8) 44.13 28.90 20.07 81.83 257.13(p-value†) (0.000) (0.000) (0.010) (0.000) (0.000)

† The confidence level (p-value) is reported for the χ2 test against the null hy-pothesis that Action 5 is the intended equilibrium play while all other actions areerrors made with equal probability. To do so, we adjust the error probabilities sothat the predicted frequency of Action 5 equals the actual frequency observed.‡ The confidence level (p-value) is reported for the χ2 test against the null (bab-bling equilibrium) hypothesis that all states are possible with equal probabilityconditional on any message.

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informative. For message 1, with more than 60% probability the state is one,and with more than 80% probability the state is less than or equal to 5. Ifa receiver is shown Table 6, her optimal action upon receiving a message of1 is to choose the average state conditional on message 1, which is 2.87, farfrom the equilibrium action 5. Similar patterns are observed for messages 3and 5, where the frequency weights are predominantly placed on small states1, 3, and 5, and the conditional average state is much smaller than the equilib-rium prediction of 5. 14 The conditional frequencies of states is closest to theuniform distribution for message 7, except that the frequency weights are sur-prisingly still skewed towards small states, 15 leading to a conditional averagestate of only about 4. On the other hand, the conditional frequencies of statesfor message 9 are clearly skewed towards large states 5, 7, and 9, resulting ina conditional average state of almost 6. The last row of the middle panel inTable 6 presents the results of χ2 tests for each message s = 1, 3, 5, 7, 9. Thetest results strongly reject the uniform distribution of state hypothesis at the0.1-percent level of confidence for all messages except for m = 7, in which casethe null hypothesis is rejected at the 3-percent level of confidence. To sum up,the evidence here strongly indicates that the senders’ messages are informativeabout the true states of the world, and that they overcommunicate comparedwith the theoretical prediction.

The receivers’ overcommunication tendency can be seen from the lowerpanel of Table 6, which presents the frequencies of actions conditional on mes-sages. Again, in the babbling equilibrium, the receiver should choose action5 regardless of the message she receives. This is clearly not the case. Actions4 to 6 are taken less than 50% of the time except for message 5. Receiversare far more likely to take small actions 1 to 3 for small messages 1 or 3 thanlarge messages 5, 7, and (especially) 9, and are also far more likely to takelarge actions 7 to 9 for large messages 7 or 9 than small messages 1,3, and 5.The average action exhibits a clear monotonic trend as the message increases.From the χ2 test results in the last row of the lower panel in Table 6, the bab-bling equilibrium hypothesis is rejected at the 1-percent level of confidence forall messages. The evidence clearly indicates that receivers’s actions vary withthe senders’ messages, hence they “over-trust” the senders compared with thebabbling equilibrium prediction.

14 Interestingly, the average state when the message is 5 is lower than that when themessage is 3. This is because some receivers simply subtract 4 from 5, resulting ina probability of 43.75% that Action 1 is chosen when the message is 5. See Section5 for further explanations.15 This can be attributed to the fact that some receivers would subtract 4 from 7,resulting in a probability of 29.76% that Action 3 is taken when the message is 7.

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5 Bounded Rationality: Behavior Type Analysis

Our experimental results show that on the aggregate level subjects under-stand the basic strategic elements of the game (hence the comparative staticshypotheses 1 and 2 are strongly confirmed), but both senders and receiverstend to overcommunicate relative to what theory allows in the most informa-tive equilibrium. So the following questions naturally arise. What accounts forthe non-equilibrium behavior? Why are the actual average payoffs so close tothe equilibrium predictions? In order to answer these questions, we need toexamine the subjects’ individual behavior to figure out what kinds of non-equilibrium behavior the subjects exhibit. In this section, we apply one ap-proach of bounded rationality, behavior type analysis, to our experiment datato understand the overcommunication phenomenon. To identify behavior pat-terns, we need a non-trivial number of plays of the same game for each subject.For this reason, we use the data from Session 2 of the base design, in whichthere are 31 rounds of play with preference difference d = 4. With 31 roundsof play, each subject on average plays 15-16 rounds as the sender and 15-16rounds as the receiver.

Table 7Type Classification Definition(d=4)

Type Sender’s Message (given S) Receiver’s Action (given M)

Name S=1 S=3 S=5 S=7 S=9 M=1 M=3 M=5 M=7 M=9

L0 1 3 5 7 9 1 3 5 7 9

L1 5 7 9 9 9 1 1 1 3 7

L2 9 9 9 9 9 5 5 5 5 5

Eq any any any any any 5 5 5 5 5

Soph 7 9 9 9 9 2 3 3 4 6

Using earlier experiment evidence (e.g., Blume et al., 2001) on communi-cation games, Crawford (2003) defines the system of behavior types that isanchored on the “truster” type of senders and the “believer” type of receivers.We follow Crawford (2003) to define the system of behavior types for our gamewith d = 4. The definitions are listed in Table 7. A sender of the lowest level ofsophistication, type L0 (the truster type), simply sends the truthful messageall the time. In response to the type L0 sender, the type L0 ( the believer type)receiver completely trusts the sender and always chooses the action identicalto the sender’s message. The next level sender, type L1 (the liar type), be-lieves that the receiver is of the L0 type, and hence incorporates his policybias (d = 4) into his messages. In response to the type L1 sender, the type L1(the inverted type) receiver will choose action 1 for a message of 5, action 3 fora message of 7, and action 7 for a message 9 (since it is equally likely from an

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L1 sender in states 5, 7, 9). Messages 1 and 3 would be “out-of-equilibrium”for an L1 sender; we assign the belief of state 1 for the L1 receiver. 16 An L2sender best responds to an L1 receiver and will always send a message of 9regardless of the true state. 17 An L2 receiver best responds to an L2 senderand hence will always choose action 5. In equilibrium the sender could sendany message (or even randomize), and hence all message will be consistent,so it is not meaningful to identify an equilibrium type sender. On the otherhand, an equilibrium type receiver will always choose action 5 regardless ofthe message she receives, which coincides with the L2 type. Finally, a sophis-ticated type sender maximizes his payoffs given the empirical distribution ofthe receivers’ behavior, while a sophisticated type receiver best responses tothe empirical distribution of the senders’ behavior. Their optimal strategiesare given in the last row of Table 7.

Table 8Type Classifications Results: Senders

Type Count Subject Number (% of consistency)

L0 2 18(80%), 22(90.9%)

L1 8 1(83.3%), 2(66.7%), 12(71.4%), 14(69.2%), 16(60%),23(76.9%), 29(63.2%), 32(64.7%)

L2 10 3(78.9%), 4(100%), 6(100%), 9(62.5%), 11(85.7%),17(86.7%), 20(82.4%), 25(81.3%), 27(72.2%), 31(64.7%)

Soph 4 5(94.4%), 8(88.2%), 19(100%), 30(100%)

N/A 8 7, 10, 13, 15, 21, 24, 26, 28

We then follow the classification methodology of Costa-Gomes et al., (2001,2004) to classify subjects whose actions can be consistently identified as fittinginto one of the behavior types. In applying the type definitions to classify the32 subjects, we allow for errors of plus or minus 1. 18 Then we count how manytimes a subject’s choices are consistent with one of the types as a sender andas a receiver separately. Subjects are classified into a particular type if theirbehavior is consistent with that type for more than 60% of the time (e.g., at

16 There are two reasons for this choice. The first reason is for monotonicity, sincean L1 type receiver believes that the state is 1 upon observing a message of 5. Thesecond reason is that it does not affect the classification of higher types that much aslong as an L1 type receiver does not choose an action greater than 3 when observinga message of 1 or 3. The L2 type senders’s best responses to the L1 type receiversare always M=9.17 In state 1, an L2 sender is indifferent between sending a message of 7 and amessage of 9 to an L1 receiver, and we break the tie towards 9. The results are notsignificantly affected by this tie-breaking choice.18 Note that this has no effect on classification of the senders, since their messagesdiffer from each other by 2.

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least 10 out of 16 times). 19 If a subject is consistent with more than one typeby the 60% rule, we classify him or her as the type he or she is most consistentwith. If there is a tie between two or more types, we classify that subject tothe lowest type of sophistication. 20 Table 8 presents the classification resultsfor the senders.

Among the 32 subjects, 24 can be classified as one of the behavior types.Two can be classified as the L0 type, 8 the L1 type, 10 the L2 type, and 4 thesophisticated type.

The classification results for the receivers is presented in Table 9. Amongthe 32 subjects, 26 can be classified as one of the behavior types. Three canbe classified as the L0 type, 3 the L1 type, 11 the L2 or Equilibrium type, and9 the sophisticated type.

Table 9Type Classifications Results: Receivers

Type Count Subject Number (% of consistency)

L0 3 18(62.5%), 22(95%), 25(66.7%)

L1 3 5(69.2%), 8(92.9%), 19(86.7%)

L2/Eq 11 1(68.4%), 4(100%), 6(77.8%), 9(80%), 12(82.4%), 15(70.6%),21(66.7%), 24(88.2%), 27(61.5%), 30(80%), 32(100%)

Soph 9 2(78.9%), 7(62.5%), 10(60%), 11(70.6%), 14(77.8%),17(81.3%), 23(72.2%), 26(73.3%), 28(71.4%)

N/A 6 3, 13, 16, 20, 29, 31

The fact that about 75% of the subject population fits into the behaviortypes seems to indicate that this approach of bounded rationality is reasonablysuccessful in describing the behavior of a majority of subjects. Moreover, it isinteresting to note that some subjects, such as 18 and 22, are highly consistentas the truster type L0 senders and believer type L0 receivers; and quite a manysubjects are highly consistent as either the liar type L1 senders or the invertertype L1 receivers. 21

19 In around 12% of the observations in this session, senders sent mixing messagessuch as “7 or 9”. For those observations, we used the realized messages generatedby the computer in the analysis of sender behavior. Our classification result is notaffected much by this.20 There are three subjects (20, 29, 31) who are tied for two or more types as sendersand five subjects (5, 8, 15, 19, 24) who are tied for two or more types as receivers.Using alternative tie-breaking rules does not have much effect on the analysis below.21 Examining these low type subjects’ individual behavior, we do not find indicationof learning over time, that is, they do not become more rational over time. Thisis quite remarkable for the L0 types, since the potential gain from becoming morerational is rather significant. By our calculation, given the empirical distribution ofthe receivers’ actions, the expected payoffs for the sender’s types are 22.77, 46.06,

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This suggests roughly the following interpretation of our data. Those sub-jects of lower levels of sophistication overcommunicate, resulting in the greater-than-equilibrium amount of information transmitted from the senders to thereceivers. However, since the subjects are randomly paired, types of sendersand types of receivers are sometimes matched “correctly” to produce equilibrium-like outcomes (e.g., an L2 sender and an L2 receiver), but some other timesmatched “incorrectly” according to the equilibrium. In different matches, thereceivers get more than their equilibrium payoffs in some cases (e.g., an L0receiver versus an L0 sender), but in other cases get less than their equilibriumpayoffs (e.g., a L0 receiver versus a L1 sender). Combined with some noise,these mismatches tend to offset the payoff gains and losses (relative to theequilibrium predictions), leading to average payoffs close to the equilibriumpredictions.

To make the above interpretation more concrete, we calculate the payoffsand correlations implied by the distribution of types, and compare it with theactual ones. To proceed, however, we must specify behavior patterns for thesubjects who cannot be classified by our criterion. There is no a prior guidanceof how re-classification should be done, since unclassified subjects by definitiondo not show strong behavior patterns. We consider two scenarios as illustrativeexamples. In scenario 1, we suppose these subjects on average act like L0 typesand L1 types half of the time each (and hence cannot be classified by our 60%consistency requirement). In scenario 2, we suppose they on average act likeL0 types half of the time and act in a completely random way in the other halfof the time. In each scenario, we use the type distributions to calculate theimplied sender’s and receiver’s expected payoffs and the implied correlationsbetween states, messages, and actions. Table 10 reports the results as well asthe actual data and the equilibrium predictions for comparison.

Table 10Estimation Results of Nash Equilibrium, Type Analysis and AQRE (d = 4)

Actual Nash Type Type Crawford AQRE

Scenario 1 Scenario 2 Equilibrium (λ = 2.00)

Senders’ uS 37.37 29.46 40.54 35.82 50.24 23.49

Receivers’uR 65.67 71.59 72.86 67.85 79.75 66.20

Corr(S,M) 0.376 0.000 0.531 0.388 0.630 0.326

Corr(M,A) 0.534 0.000 0.631 0.517 0.829 0.400

Corr(S,A) 0.183 0.000 0.361 0.222 0.608 0.178

48.68 and 50.28, for L0, L1, L2, and Sophisticated types, respectively. Given the em-pirical distribution of the senders’ messages, the expected payoffs for the receiver’stypes are 58.16, 71.85, 71.59 and 76.56, for L0, L1, L2, and Sophisticated types,respectively.

24

From Table 10, we can see that scenario 1 with unclassified subjects as halfL0 types and half L1 types overestimates communication (all three correla-tions are greater than those of actual data) and the senders’ and receivers’expected payoffs. On the other hand, scenario 2 (that all unclassified sub-jects are supposed to act randomly half of the time and act according to L0types) does a very good job matching up with the actual data in terms ofboth correlations and expected payoffs. This shows that at least with somereasonable adjustments the behavior type analysis can explain our experimen-tal data quite well, predicting both the degree of overcommunication and theplayers’ expected payoffs. Having said that, we offer a word of caution. Themain issue is that subjects of higher levels of sophistication in our games arenot very distinguishable from each other, for example, L2 and sophisticatedsenders only differ in state 1, while L2 and equilibrium senders and receiversare not distinguishable at all (Table 7). Moreover, several subjects are tiedwith two or three higher types on both the sender and receiver sides. Eventhough the tie-breaking rule does not have important effects on our result, itis possible that the type classification may not be robust in other somewhatdifferent environments. Thus, the results from the behavior type analysis inthis section are not conclusive. Models and experimental designs that allowsharper distinction of subject types are needed for further investigation. 22

A related, but slightly different way to interpret our experiment data is toexplain the data as some equilibrium in a model in which some subjects arerational and others are of some behavior types. Crawford (2003) studies acheap talk model with a zero-sum underlying game in which players can berational or the L0 and L1 types (i.e., truster and liar senders, believer andinverter receivers). In equilibrium, rational type players best response to theiropponents, taking into account that their opponents can be one of the threetypes. He characterizes the equilibrium of the game and shows how it changesas the initial type distribution changes. Even though the underlying game inour study is quite different from that in Crawford (2003), it is easy to apply hismethod here, because subjects of L2 and Equilibrium types in our games arenot very distinguishable from the sophisticated types (Table 7). As a result,in the classification quite a many subjects classified as L2 types are equally(or very close to be) consistent with sophisticated types. More precisely, if L2type is ruled out, 8 out of 10 L2 type senders in Table 8 can be re-classifiedas sophisticated type, and 5 out of 11 L2 type receivers in Table 9 can bere-classified as sophisticated type. Reclassifying those unclassified subjects sothat they on average act like L0 type half of the time and L1 type in theother half of the time, we arrive at a type distribution roughly as follows: the

22 For example, in sequential bargaining games, Camerer et al. (1993) and Johnsonet al. (2002) use “lookups” by subjects to infer whether or not they use backwardinduction. In simultaneous guessing games, Costa-Gomes et al. (2001) and Costa-Gomes and Crawford (2004) use subjects’ patterns of searching for information toprovide sharper distinction between behavior types.

25

senders are L0, L1 and sophisticated with proportions (20%, 40%, 40%), andthe receivers are L0, L1 and sophisticated with proportions (30%, 30%, 40%).With these type distributions, it can be verified that in equilibrium the sophis-ticated type sender sends a message 7 when the state is 1, and 9 in all otherstates, which coincides with the behavior of the sophisticated sender who bestresponses to the empirical distribution of the receivers’ actions (Table 7). Inequilibrium, the sophisticated type receiver will choose 1 in state 1, choose 3in states 3, 5, 7, and choose 7 in state 9. Compared with the behavior of thesophisticated receiver who best responses to the empirical distribution of thesenders’ actions (Table 7), the theoretically sophisticated receiver’s strategy isvery close but not identical in states 1, 7, 9. We compute the correlations andexpected payoffs in this equilibrium. The results are in Table 10, under thecolumn titled “Crawford Equilibrium”. Evidently, the Crawford equilibriumsubstantially overestimates communication and expected payoffs. The reasonfor the overestimation by the Crawford equilibrium and by the non-equilibriumtype simulation of scenario 1 above (with half L0 types and half L1 types forunclassified subjects) is the same. In our experiment data, there is much noisein subjects’ behavior, which is not taken into account by either interpretation.The second scenario (with half L0 types and half complete randomization)matches the data very well, because it introduces complete randomness intothe subject population.

6 Bounded Rationality: Quantal Response Equilibrium

An alternative explanation for non-equilibrium behavior in experimentaldata is the concept of Quantal Response Equilibrium (QRE) introduced inMcKelvey and Palfrey (1995, 1998). In a Quantal Response Equilibrium, play-ers have correct beliefs about their opponents’ behavior, but they either makemistakes while trying to maximizing their payoffs or their preferences overdifferent strategies are stochastic. Unlike the rational approach and the be-havior type analysis, the QRE explicitly takes into account noisy behavior ofsubjects. McKelvey and Palfrey (1995, 1998) demonstrate that the QRE canbe useful in organizing and interpreting experimental data for a variety ofexperiments. In this section we apply the concept of QRE to our experimentdata to shed light on the overcommunication phenomenon.

Specifically, following McKelvey and Palfrey (1998), we solve for the logitagent quantal response equilibrium (logit-AQRE) for our game. 23 Let u(a, s)and v(a, s) be the utilities of the sender and receiver, respectively, when thestate is s and the receiver chooses an action a. Then in the logit-AQRE witha parameter λ, the sender in state s sends a message m ∈ {1, 3, 5, 7, 9} with

23 For details of the solution concept and its applications, see McKelvey and Palfrey(1998).

26

probability psm = Pr(m|s):

psm =eλus(m)∑5,7,9

j=1,3, eλus(j,)

where us(m) =∑9

a=1 qmau(a, s) is the sender’s expected payoff from sendinga message m in state s; the receiver chooses an action a upon receiving amessage m with probability qma = Pr(a|m):

qma =eλvm(a)∑9

j=1 eλvm(j)

where vm(a) =∑5,7,9

s=1,3 r(s|m)v(a, s) is the receiver’s expected payoff fromchoosing an action a upon the receipt of message m and r(s|m) = psm/

∑9i=1 pim

is her posterior belief about the states.In this specification, in each state the sender assesses the expected payoff

of sending a message m ∈ {1, 3, 5, 7, 9}, given the receiver’s strategy. Thenhe sends a message with a probability based on its expected payoff relativeto sending other messages, according to the exponential form given above.The better a message strategy is against the receiver’s strategy, the greaterthe probability the sender is going to use it. The parameter λ measures howclosely the sender’s mixed strategy follows the expected payoffs. When λ = 0,the sender randomizes over all messages, irrespective of their relative expectedpayoffs. When λ goes to infinity, the sender’s strategy converges to his bestresponse. The receiver’s strategies can be interpreted similarly.

Hence, P = (psm)5×5 and Q = (qma)5×9 represent the sender’s and receiver’smixed strategies, respectively, in the logit-AQRE. For any given λ, to solvefor the seventy variables from the system of seventy equations, we iterateby starting with an initial P0 and obtaining the new Q1 and P1 using theabove equations. We perform this procedure iteratively until P and Q convergeto a fixed point. We then calculate the likelihood for this λ and obtain themaximum likelihood estimator by maximizing the likelihood function. 24

Just as choosing the right anchor for the behavior type analysis is inportant,choosing a proper initial point P0 is important in applying the logit-AQRE tostrategic information transmission games. In these games, since the sender’sstrategies (namely, their messages) do not directly affect payoffs (thus the term“cheap talk”), there are many payoff-equivalent equilibria but with differentequilibrium strategies. For example, in the d = 4 case, while babbling is theonly equilibrium outcome, there are in fact many babbling equilibria, in eachof which the sender sends a particular message in all states. So if we choose acomplete random strategy for the sender as the initial P0, the iteration processquickly settles on a babbling equilibrium. To avoid this problem, we use the

24 We use the golden section method in the estimation since λ is one dimension andL(λ) is unimodal according to our grid search.

27

truth-telling strategy for the senders as the initial P0 (i.e., identity matrix). 25

For the ease of comparison, the result of the logit-AQRE estimation is dis-played in the last column of Table 10. As shown in Table 10, the maximumlikelihood estimator of λ is 2.00. In the estimated AQRE, the correlationbetween state and action is very close to the actual data, the correlationsbetween state and message and between message and action are fairly closeto the actual data. Hence the logit-AQRE provides a good interpretation forthe communication patterns in our experimental data. In terms of payoffs,the receiver’s expected payoff in the estimated AQRE is very close to thatof actual data, while the sender’s is somewhat lower. This may suggest thatthe logit-AQRE incorporates all kinds of noisy actions, while subjects in thedata do not make mistakes (especially the very bad mistakes) to the extentthe logit-AQRE postulates. 26

7 Robustness Analysis

To test how robust the results from our baseline sessions are, we ran twosessions of experiments with somewhat different designs but with equilibriumpredictions essentially identical to the base design. By changing the exper-imental design while holding the theoretical predictions fixed, we can testwhether the experimental results are sensitive to the design details.

For the first robustness test, we set the payoff parameter k at 1.2, so thatpayoffs are less sensitive to the difference between action and state comparedwith the baseline sessions where k is 1.4. Moreover, we used a different designwhen subjects choose messages such as “1 or 3”. Specifically, instead of doingthe mixing automatically for the senders in the baseline sessions, senders mustsend a unique message “1 or 3”. We ran a session with 32 subjects and a total31 rounds, 5 rounds of d = 0.5, 13 rounds of d = 1.2, 5 rounds of d = 2, and8 rounds of d = 4.

For the second robustness test, we ran another session in which the payoffparameter k is further lowered to 1, making payoffs even less sensitive to thedifference between action and state. The message space is the same as the pre-vious robustness session. An additional modification introduced in this sessionis that we gave senders noisy signals instead of perfect ones. Specifically, atthe beginning of each round, senders were not perfectly informed about thetrue state of the world, but were given a noisy signal. With 90% chance thesignal was equal to the true state of the world, and with 10% chance the signal

25 We obtain the same result if using the lying strategy of the L1 type senders asthe initial P0.26 To be sure, the AQRE requires that players make worse mistakes less often. Inour game, the sender suffers more than the receiver in their respective worst payoffcases: when the state is 1 and the action is 9, the loss to the sender is 214.23 and theloss to the receiver is 73.79. So this may explain why the sender’s expected payoffis under-estimated by the logit-AQRE.

28

was randomly drawn by the computer program uniformly from {1, 3, 5, 7, 9}.The reason for introducing noisy signals is the following: One possible expla-nation for overcommunication is that the receivers tend to trust the senders“too much” because they believe that the senders are more “powerful” dueto their knowledge about the state of the world (in applications the sendersare usually experts of some sort). If this is a force behind overcommunication,one expects its effect will diminish with noisy signals since the senders are lessknowledgable and hence less powerful. We ran the session with 48 subjectsand a total 20 rounds, 5 rounds for each of the four cases d = 0.5, 1.2, 2, 4.Since in this session the senders only get the signals but do not observe thestates, we use the correlations with respect to signals, for they reflect moreclosely how much information is communicated than correlations with respectto states. The equilibrium predictions about the correlations with respect tosignals are exactly the same as before, making comparisons with the previousresults easier.

As in the baseline sessions (Table 4), our experimental results show thatthe average payoffs for the senders, the receivers, and the subject populationare close to those predicted by the most informative equilibrium in each ofthe robustness test sessions. Because of space limits, these results are notpresented here, but can be found in Cai and Wang (2004).

Table 11 presents the correlations between states (or signals) and messages,between messages and actions, and between states (or signals) and actions forthe two robustness test sessions. 27 The upper panel shows the results for thefirst robustness test session. Clearly Hypothesis 1 is supported in this case,as all correlations decrease as the preference difference increases. As in thebaseline sessions, here both senders and receivers exhibit a strong overcom-munication tendency for relatively large preference differences, since the actualcorrelations between states and messages and between messages and actionsare both greater than their counterparts in the most informative equilibrium.In particular, when d = 4, the correlation between states and messages is0.391 and the correlation between messages and actions is 0.642, while theequilibrium prediction is zero for both correlations. Also, the correlation be-tween states and actions is 0.312, indicating that substantial information isactually used in the receivers’ decisions. Comparing Table 11 with Table 5 alsoreveals that the correlations are quite close in the two cases, especially for thecorrelation between states and messages, which differs only in the third orfourth digit (not shown) for d = 0.5, 1.2,and 4. Therefore, the evidence fromthe first robustness test shows that the main results of the baseline sessionsare robust for a smaller k and a somewhat different message space.

27 In both sessions, senders sometimes used messages such as “1 or 3”. In computingthe correlations between states (or signals) and messages and between messagesand actions, we use a computer program to do the randomization and record therealizations as the messages for correlation calculations. This gives us a basis tocompare these correlations with those of the baseline sessions.

29

Tab

le11

.T

heor

etic

alvs

.A

ctua

lIn

form

atio

nTra

nsm

issi

onfo

rR

obus

tnes

sTes

ts

Num

Pro

b.

Cor

rela

tion

(S,M

)C

orre

lati

on(M

,A)

Cor

rela

tion

(S,A

)

ofC

orre

ctk

d

obs.

Sig

nal

Pre

dic

ted

Act

ual

Pre

dic

ted

Act

ual

Pre

dic

ted

Act

ual

801

1.2

0.5

1.00

00.

916

1.00

00.

955

1.00

00.

923

208

11.

21.

20.

750

0.89

70.

866

0.91

20.

866

0.89

5

801

1.2

20.

500

0.83

70.

707

0.85

00.

707

0.75

5

128

11.

24

0.00

00.

391

0.00

00.

642

0.00

00.

312

120

0.9

1.0

0.5

1.00

00.

868

1.00

00.

924

1.00

00.

870

120

0.9

1.0

1.2

0.75

00.

887

0.86

60.

904

0.86

60.

832

120

0.9

1.0

20.

500

0.85

80.

707

0.86

20.

707

0.76

9

120

0.9

1.0

40.

000

0.35

40.

000

0.45

70.

000

0.25

9

30

The lower panel of Table 11 shows the results for the second robustness testsession. The correlations for d = 0.5 look somewhat lower than before, perhapsbecause some subjects were still learning about the game in the first few rounds(in which d = 0.5 was used) and the game is somewhat more complicatedthan in the other designs due to the noisy signals. However, the results inthis session are still consistent with those in the baseline sessions. For d =1.2, 2, 4, correlations all decrease in d, confirming Hypothesis 1. Moreover, bothsenders and receivers overcommunicate as the correlations between signalsand messages and between messages and actions are greater than those in themost informative equilibrium. In fact, the correlations for d = 1.2, 2, 4 hereare reasonably close to those in Table 5, without any clear dominance pattern.Therefore, we conclude that noisy signals do not seem to have any effect on theexperimental outcomes. It suggests that the “trusting the expert” explanationof overcommunication does not play an important role.

Another possible explanation for subjects’ overcommunication tendency ob-served in our experiments is insufficient learning. The game used in our ex-periments is not a trivial one for people without game theory training, so itmay take a while for subjects to learn to play the equilibrium strategies. 28

To examine this learning hypothesis, we ran a session using our base design(Session 2 as described in the end of Section 3) in which 32 subjects played31 rounds of the game with preference difference d = 4. By holding the design(including preference difference) fixed throughout the session, we can compareoutcomes in early rounds with those in later rounds to see if subjects behavedifferently after gaining experience playing the game. We focus on the case ofd = 4 to test the learning effect for two reasons. One reason is that amongthe four difference preferences, deviations from equilibrium predictions are thelargest for d = 4. Hence, if learning leads to convergence to equilibrium, thenits effect will be strongest for d = 4. Another reason we focus on d = 4 is thatin this case subjects should have the strongest incentives to learn to play theequilibrium strategies, because it is most costly for subjects to be outguessedby their opponent. In trying to increase the subjects’ incentives to learn, weuse a generous exchange rate of points to dollars ($1=50 points) in this ses-sion. The session lasted less than 2.5 hours, the highest payoff was $45.1 andthe lowest was $12.7 (excluding the $5 show-up fee).

Our analysis of the average payoffs over time does not find any learningeffect, see Cai and Wang (2004) for details. Examining the table of correlationsreveals the same picture. From Table 12, casual observation reveals that noneof the three correlations exhibit a clear time trend, let alone convergence toequilibrium. For example, the correlation between states and messages andthe correlation between messages and actions both are greater in the last 6rounds than in the first 5 rounds after some ups and downs in between. We runregressions to formally test for any time trend in the correlations. Our results

28 In all of our experiments, subjects play several practice rounds before they playthe real rounds to learn about the experimental design.

31

show that none of the three correlations in any group of 5 earlier roundsis significantly different from rounds 26-31, except the correlation betweenmessages and states of the first 5 rounds. Overall, the results here indicatethat there is no clear learning effect. 29

Examining subjects’ individual behavior reveals a similar picture. Due tospace limitation, we only present how the average of the true state conditionalon seeing each message changes across time in Table 12. There is no cleartrend for Messages 3, 5, 7 and 9, where the average state in the last 6 roundsis higher than the first 5 rounds for Message 3 and 9, while lower for Message5 and 7. The average state conditional on Message 1 experiences a significantdrop after round 10, which might suggest that senders who send Message 1 aremore truthful later on. However, this is highly unreliable since observations ofMessage 1 are very few (each 5 rounds have no more than 7 observations).

8 Conclusion

In this paper we conduct laboratory experiments to test the theory of strate-gic information transmission originated by Crawford and Sobel (1982). Ourmain findings are the following. First, our experimental results strongly sup-port the basic insight of the theory, namely, less information is transmittedwhen preferences of the sender and the receiver diverge. Secondly, the evi-dence shows that subjects consistently overcommunicate in that the senders’messages are more informative about the true states of the world and thereceivers use the senders’ messages more in choosing actions, compared withwhat the theory allows in the most informative equilibrium. Thirdly, theseresults are robust to certain variations of payoff parameters and noisy signals,and are robust to subjects’ learning. Fourthly, we use both the behavior typeanalysis and quantal response equilibrium to analyze subjects’ behavior inour experiment data. Both approaches have some success in interpreting ourdata. In particular, with each of the two approaches, we generate the kinds

29 Of course, this is not to say that learning definitely will not lead to convergenceto equilibrium. It is possible that it takes much longer than 31 rounds for subjectsto learn to play equilibrium strategies. But we have to leave it to future researchto investigate this possibility. It is interesting to note that, in contrast to our resulthere, Blume et al. (2001) report evidence indicating significant learning effects:senders became more “sophisticated” (e.g., revealing their types less often) andreceivers became less gullible. Because Blume et al. are interested in the evolution ofmessages, their experimental design facilitates possible subject learning much morethan ours. In their experiments, there were 60 rounds of play and subjects wererandomly paired with 6 opponents and thus interacted with an opponent about 10periods. Moreover, summary history about other subject pairs was revealed to allsubjects at the end of each period. In our experiments, opportunities for subjectsto learn are more limited since they played 31 rounds, interacted with any othersubject only once, and only knew the outcomes of the games they played in.

32

Tab

le12

.A

ctua

lIn

form

atio

nTra

nsm

issi

onan

dA

vera

geSt

ate

for

each

Mes

sage

:Lea

rnin

gE

ffect

Rou

nds

kd

Cor

r.C

orr.

Cor

r.A

vera

geSta

tefo

rea

chM

essa

ge

(S,M

)(M

,A)

(S,A

)M

=1

M=

3M

=5

M=

7M

=9

1–5

1.4

40.

244∗

∗0.

500

0.13

95.

000

3.85

73.

714

5.66

75.

533

6–10

1.4

40.

351

0.53

00.

091

5.00

03.

444

2.60

03.

462

5.97

8

11–1

51.

44

0.43

40.

449

0.32

92.

200

4.33

33.

250

4.00

06.

111

16–2

01.

44

0.47

00.

566

0.17

51.

000

3.50

03.

600

4.77

86.

171

21–2

51.

44

0.43

90.

627

0.24

52.

000

4.50

02.

200

4.77

86.

107

26–3

11.

44

0.34

40.

557

0.13

31.

000

3.88

93.

000

3.61

55.

567

∗Si

gnifi

cant

lydi

ffere

ntfr

omth

ela

st6

roun

ds(r

ound

26-3

1)at

the

5-pe

rcen

tle

velof

confi

denc

e.∗∗

Sign

ifica

ntly

diffe

rent

from

the

last

6ro

unds

(rou

nd26

-31)

atth

e1-

perc

ent

leve

lofco

nfide

nce.

33

of correlations (overcommunication) and average payoffs (close to equilibriumpayoffs) observed in the experimental data.

While we have some success in explaining our experimental results (inparticular, overcommunication and near equilibrium payoffs) using the twobounded rationality approaches, the test may not be very “powerful” in thesense that for large preference differences, messages and actions tend to clustertoward the highest state (state 9), as the sender’s bias is in one direction. Infuture work, it is desirable to develop models such that biases are symmetricso that clustering does not arise. Questions about what the underlying motivesfor the L0 types are naturally arise. Subjects 18 and 22 are highly consistentas L0 senders and L0 receivers in that they send truthful messages and chooseactions blindly following the messages. Are they driven by social preferences(e.g., trust)? Where do their beliefs come from? These questions all await forfurther investigation.

Acknowledgement

We are grateful to David Levine for his encouragement, support and advice.We also thank Vincent Crawford, Thomas Palfrey, Leeat Yariv, the associateeditor and an anonymous referee for very helpful comments. All remainingerrors are our own.

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35