Competition, Preference Uncertainty, and Jamming:A Strategic Communication Experiment

William Minozzi∗ Jonathan Woon†

July 28, 2015

Abstract

We conduct a game-theoretic laboratory experiment to investigate the nature of in-formation transmission in a communication environment featuring competition andinformation asymmetry. Two senders have private information about their preferencesand simultaneously send messages to a receiver in a one-dimensional space with a largenumber of states, actions, and messages. We find that equilibrium predictions farepoorly and that senders overcommunicate by consistently exaggerating their messages.Over time, exaggeration increases and communication unravels. Our analysis suggeststhat exaggeration and unraveling can only be partially explained by bounded rational-ity models of iterated reasoning or belief learning. Instead, behavior is consistent witha naive form of exaggeration in which senders know they must exaggerate, but theydo so in an understated way that is less responsive to their private information or toopponents’ past behavior than would be fully optimal.

Keywords: Strategic information transmission; Sender-receiver games; Multiple senders;Laboratory experiment; Bounded rationality; Level-k; Behavioral models

JEL Classifications: C72, D82, D83

∗Associate Professor, Department of Political Science, 2137 Derby Hall, Ohio State University, Columbus,OH 43210, Email: minozzi.1@osu.edu†Corresponding author: Associate Professor, Department of Political Science, Department of Economics

(secondary), and Pittsburgh Experimental Economics Laboratory, 4814 Wesley W. Posvar Hall, Universityof Pittsburgh, Pittsburgh, PA 15260 Phone: 1-412-648-7266, Email: woon@pitt.edu

1 Introduction

Throughout daily life, people are confronted with conflicting messages from informed sources.

In politics, for example, candidates offer competing visions for national policy (Banks 1990),

legislators and lobbyists make conflicting claims for and against legislation (Gilligan and

Krehbiel 1989; Austen-Smith 1990, 1993), and courts rely on adversarial advocates to in-

form their judgments (Dewatripont and Tirole 1999). Each situation features well-informed,

interested actors whose preferences remain partially private. Those actors send competing

messages to decisionmakers who cannot verify the content of those messages. The set of

messages is limited to a single salient dimension, sometimes explicitly by germaneness rules,

yet within that dimension of disagreement, the number of possible messages is large. In this

paper, we describe a laboratory experiment on information transmission in this context.

Based on existing theory, we have contradictory expectations for information revelation in

such settings. Preference differences limit the information that can be conveyed in strategic

environments (Crawford and Sobel 1982), which can be exacerbated if the decisionmaker

is uncertain about information-providers’ preferences (Sobel 1985; Lupia and McCubbins

1998). Yet competition between information providers can also provide mechanisms for the

truth to emerge. Indeed, the “marketplace of ideas” is often invoked to justify the importance

of protecting free speech (Mill 1859). Competition among elites and the news media can

yield a more informed public (Page and Shapiro 1992; Gentzkow and Shapiro 2008). Yet

mixing competition and preference uncertainty may only provide further opportunities for

strategic obfuscation (Milgrom and Roberts 1986; Minozzi 2011).

To the best of our knowledge, ours is the first experiment to examine an environment

that incorporates competition and preference uncertainty in a unidimensional state space

with large message and action spaces. Studies in the extant experimental literature on

communication tend to focus on simple environments that feature either a single sender,

commonly known preferences, a small number of messages and actions, or a combination of

only a few of these features (Blume et al. 1998, 2001; Cai and Wang 2006; Dickhaut, McCabe,

1

and Mukherji 1995; Gneezy 2005; Hurkens and Kartik 2009; Peeters, Vorsatz, and Walzl 2008;

Sanchez-Pages and Vorsatz 2007). Studies that investigate environments with two senders are

rare. Lai, Lim, and Wang (2015) and Vespa and Wilson (Forthcoming) test for Battaglini’s

(2002) fully-revealing equilibrium, which requires a two-dimensional setting.1 In contrast, the

one-dimensional environment in our experiment makes information-revelation theoretically

more difficult. Unlike previous studies, in which there is typically a small set of discrete

states and messages, our setting allows for much more varied communication strategies. Our

setting is also spatial, meaning that our findings have straightforward applications to many

fundamental formal models of politics.

We present evidence for three main findings. First, there is overcommunication rela-

tive to equilibrium expectations, consistent with previous research on single-sender games

(Blume et al. 1998; Cai and Wang 2006). Senders select messages best modeled as additive

exaggerations in the direction of their hidden biases. In the competitive setting, this sort

of overcommunication means that receivers come very close to learning the hidden state

information simply by splitting the difference in messages.

Second, we find that there is unraveling. Specifically, we find that senders’ exaggerations

tend to increase with repeated play. As senders exaggerate more and more, their messages

eventually hit the boundaries of the message space. This unraveling suggests the possibility

that senders learn to exaggerate, in contrast with previous sender-receiver experiments.

Finally, we apply two models of limited strategic sophistication: a level-k model and a

belief learning model. Intriguingly, each model explains some of our empirical findings, but

neither model dominates the other in terms of predictive accuracy. Moreover, both models

over-predict exaggeration. We conclude that senders use a naive strategy consistent with

strategic cognition but significantly more limited in sophistication than in typical models of

bounded rationality.

1Boudreau and McCubbins (2008) conduct an experiment with competition and a version of preference un-certainty, but their experiment involves a decisionmaker who solves SAT math problems with the help of“experts.” Their setup departs from standard sender-receiver games because their receivers have heteroge-neous (unobserved and uncontrolled) beliefs about the true state.

2

2 Theoretical Model and Equilibrium Predictions

Consider a communication environment with two senders and one receiver.2 Both senders

observe a state variable, which we call the target T , uniformly distributed over the interval

[−100, 100]. Each sender i also privately observes his shift Si, which represents his prefer-

ence divergence from the receiver. The left sender has shift SL distributed uniformly over

[−50, 0], and the right sender has SR distributed uniformly over [0, 50]. These distributions

are common knowledge. The senders simultaneously select messages mi from the real line

to send to the receiver, who chooses action c.3 The receiver prefers c to be close to T while

sender i prefers c to be close to T + Si. Formally, the receiver’s payoff is 100− |c− T | and

the senders’ payoff is 100− |c− (T + Si)|. In terms familiar from the spatial voting model,

T is the receiver’s ideal point while T +Si is sender i’s ideal point. Thus, the receiver knows

that senders are opposed, but she is uncertain whose ideal point is closer to hers. We focus

on perfect Bayesian equilibria.

Cheap talk and signaling games typically have many equilibria. The purpose of our equi-

librium analysis is therefore not to make unique predictions, but to provide a framework that

organizes our experimental analysis and guides our expectations about the kinds of behavior

that are consistent with fully rational, strategic play (Schotter 2006). While previous studies

focus on the most informative equilibria (e.g., Cai and Wang 2006), we focus on equilibria

that are simple to implement and heuristically plausible, yet differentiated from each other

in structure and informativeness. The three equilibrium classes we consider are babbling,

partition, and jamming.4 Each of these equilibria involves a simplistic message strategy, and

2For a detailed formal analysis of this game, see Minozzi (2011). We use male pronouns to refer to sendersand female pronouns to refer to the receiver.

3In the experiment, messages are integers in [−150, 150]. As is typical in cheap-talk games, equilibriumoutcomes, which are mappings from states to actions, do not depend on the literal meanings of messages.For ease of exposition and to be consistent with the experimental design, we nevertheless present ourequilibrium predictions assuming that literal messages are used.

4In addition to these equilibria, there are other types that we do not focus on. First, hybrids of thesethree types can also be equilibria. For example, an equilibrium can be constructed in which a centralsubinterval of the state space is relegated to babbling while the remainder is treated as jamming. Weignore this sort of equilibrium because it relies on intensive coordination to construct and would be difficult

3

straightforward on-the-path and off-the-path beliefs. Moreover, these equilibria feature very

different predictions about informativeness and manipulability of outcomes by senders. Gen-

erally, babbling involves the least information transmission and thus opportunity for senders

to affect the receiver’s decisions, and jamming involves the most information transmission

and thus most opportunity for senders to change what receivers do. The equilibria also differ

in the particular ways that senders hold back what they know, so our experimental analysis

provides us with an opportunity to investigate strategies empirically. For each equilibrium,

we make predictions about the average value of observed messages and actions. Proofs and

complete statements of equilibria for these predictions are presented in the supplemental

appendix.

First, in a babbling equilibrium, messages are unrelated to the target. Thus the receiver

can learn nothing about the target, and she consequently ignores both messages. For com-

parison, Figure 1 illustrates the three equilibrium message strategies for a left sender with

shift SL = −25; the dashed line depicts the average message in a babbling equilibrium.

Babbling Predictions. There is an equilibrium with mL,mR, and c ∼ U[−100, 100]. Inthis equilibrium, the average message is E[mi] = 0, and the average action is E[c] = 0.

Second, in a partition equilbrium, senders transmit limited information by revealing that

the target is included in a subinterval of the state space (Crawford and Sobel 1982). The

simplest possibility is one in which each sender only reveals whether the target is high or low,

using a cutoff K and communicating that T is either in [K, 100] or in [−100, K]. Because

the senders are competitive, each prefers to partition the message space differently. Given

to differentiate from the “pure” types we consider here. Second, there are fully revealing equilibria thatrequire punishment strategies based on off-the-path beliefs to induce revelation (Krishna and Morgan 2001).We ignore these equilibria because such off-the-path beliefs are typically implausible (Battaglini 2002) andwould also require intensive coordination to construct. Third, any relabeling on the message space wouldyield equivalent equilibrium outcomes with, for example, the signs of the messages flipped. Again we focuson equilibria with literal messages and ignore the relabeled equilibria because the latter are difficult toimplement.

4

Figure 1: Example of left sender’s equilibrium message strategies for SL = −25

-150

-100

-50

050

100

150

Mes

sage

-100 -50 0 50 100Target

Babbling Prediction Partition PredictionJamming Prediction

K > 0, the left sender prefers to signal that the target is low unless his own ideal point

T + SL is above K. The right sender prefers instead to signal that the target is high, unless

his ideal point T +SR is below −K. Because shifts are private information, the exact cutoffs

used by each sender are unknown. Posterior beliefs therefore depend on marginalizing over

the hidden shifts. These partition strategies imply three equilibrium actions: c = 2K if both

senders send high messages, c = −2K if both send low messages, and c = 0 otherwise. Given

these actions and cutoffs, K∗ is the endogenous value such that posterior expectations for the

target match equilibrium actions. Here, K∗ = 506

(7−√

5)≈ 39.7. (See the Supplemental

Appendix for details). The dotted line in Figure 1 depicts the average message sent in a

partition equilibrium.

5

Partition Predictions. There is a partition equilibrium with strategies given by

mL(T, SL) ∼

{U[K∗, 100] if T + SL > K∗

U[−100, K∗] if T + SL < K∗

mR(T, SR) ∼

{U[−K∗, 100] if T + SR > −K∗

U[−100,−K∗] if T + SR < −K∗

c(mL,mR) =

2K∗ if mL > K∗ and mR > −K∗

−2K∗ if mL < K∗ and mR < −K∗

0 otherwise,

with K∗ = 506

(7−√

5)≈ 39.7. In this equilibrium, E[mL] = K∗+100

2if T + SL > K∗

and −100+K∗

2, and E[mR] = −100−K∗

2if T +SR < −K∗ and E[mR] = −K∗+100

2otherwise.

Actions are point identified by messages.

Third, in a jamming equilibrium (Minozzi 2011), senders sometimes tell the truth by

sending mi = T but also have incentives to jam the truthful messages from the opposing

sender by sending mi = −T . If the messages agree (mL = mR), the receiver infers the true

target T and chooses c = T . However, if the messages disagree (mL = −mR), the receiver

infers that one sender must have jammed, but she also remains uncertain which sender lied

and thus which message signals the true target. In that case, the receiver chooses a “default”

action. We focus on the equilibrium in which the default action is c = 0, which is the optimal

action based on prior beliefs.5

Senders’ message strategies also specify when to jam. Each sender recognizes that the

only actions on the equilibrium path are c = T and c = 0. Truthtelling might therefore lead

the receiver to choose c = T (if the other sender also tells the truth), but jamming will always

lead to conflicting messages and to the action c = 0. The sender’s best response is to engage

in conditional truthtelling: tell the truth when T + Si is closer to T than to 0 but otherwise

jam. It follows from this calculation that each sender has a jamming region—a set of targets

5Technically, other default actions (i.e., c 6= 0) can support other jamming equilibria, but the default ofc = 0 is a natural focal point that is supported by very simple off-the-path beliefs (e.g., where the receiverignores both messages). Alternatively, c = 0 is the optimal action in a perturbed game where there is asmall probability that both messages are purely random; in this case every possible message pair occurs onthe path of play and Bayes’ Rule always applies.

6

for which the sender will jam—and that the size and location of these jamming regions

depend on the sender’s shift. The left sender’s jamming region is the interval [0,−2SL]

and the right sender’s jamming region is the interval [−2SR, 0]. Because the receiver does

not observe the senders’ biases, her posterior beliefs about the target will be based on her

beliefs about which of the senders must have lied (i.e., that one of the sender’s shifts must

be sufficiently far away to make jamming optimal for that sender).

When a sender jams, he must jam in such a way to ensure that when the messages

disagree, the receiver will choose the default c = 0. To construct this message, first note

that the jamming regions described above imply that at least one of the senders will always

send a truthful message mi = T . When the receiver observes conflicting messages, she knows

that one of them must be the true state. The sender who wishes to lie must then exploit

the receiver’s preference uncertainty and construct a message that ensures the receiver will

remain completely uncertain about which of the senders actually jammed. Thus, jamming

messages must be countervailing. They must be on the opposite side of the default action

from the true target, and the more extreme the target, the more extreme the jamming

message. More specifically, when the default action is c = 0, the jamming message is

mi = −T . The solid line in Figure 1 illustrates the message strategy for a left sender with

SL = −25.

Jamming Predictions. There is a jamming equilibrium with strategies given by

mL(T, SL) =

{−T if T ∈ [0,−2SL]

T otherwise(1)

mR(T, SR) =

{−T if T ∈ [−2SR, 0]

T otherwise(2)

c(mL,mR) =

{mL if mL = mR

0 otherwise.(3)

(4)

In this equilibrium, messages mi are point identified by T and Si, and actions are pointidentified by messages.

7

3 Experimental Procedures

We conducted our experiments at the Pittsburgh Experimental Economics Laboratory using

subjects recruited through the lab’s centralized database. Most subjects were undergraduates

at the University of Pittsburgh, and no subjects were recruited from the authors’ classes.

Each subject participated in only one session.

Upon arriving at the lab, subjects gave informed consent and were seated at separate

computer terminals. All interactions between subjects took place anonymously through the

networked computers using software programmed and conducted using z-tree (Fischbacher

2007). Subjects received strict instructions not to communicate with one another in any

other way throughout the session. A full set of instructions is available in the appendix.

The instructions were presented on their computer screens and read aloud in an effort to

induce common knowledge among the participants. Subjects received printed copies of the

instructions, to which they were encouraged to refer as often as they needed, and were given a

quiz about the instructions in order to ensure comprehension. The quizzes were administered

through the computers so that subjects privately received immediate feedback about whether

or not they answered questions correctly (as well as explanations of the correct answers).

Consistent with the lab’s governance policy, no deception or false feedback was used in the

experiment.

After the instructions and quiz, the software randomly assigned subjects to roles: A (left

sender), B (right sender), or C (receiver). Roles were fixed throughout the session. The

instructions referred only to roles “A,” “B,” or “C,” and made no reference to “senders,”

“receivers,” “left,” or “right.” In our discussion, however, we continue to use these terms.

At the beginning of a round, subjects were randomly assigned to groups of three, with one

subject in each role in each group. Assignments were conducted with replacement so that it

was possible to be matched with the same group in different rounds. To preclude reputation

effects, subjects never knew the identities of the other subjects in their group. Targets T

and shifts SL and SR were then drawn independently for each group. In all conditions of the

8

Figure 2: Screenshot of sender’s graphical interface

experiment, T was drawn uniformly from the integers between −100 and 100, SL was drawn

uniformly from the integers between −50 and 0, and SR was drawn uniformly from integers

between 0 and 50. In the instructions and throughout the experiment, we referred to each

player’s ideal action as a “target.” That is, T is referred to as “C’s target,” T + SL is “A’s

target,” and T + SR is “B’s target.”6 The size of these spaces is in contrast with previous

experiments on cheap talk games: for example 3 states in Blume et al. (2001), 4 in Dickhaut,

McCabe, and Mukherji (1995) and 5 states in Cai and Wang (2006). The relatively large

state and action spaces afford the opportunity to conduct a more detailed investigation of

communication strategies than previous studies.

In each round, each sender simultaneously observed the receiver’s target and his own

target (but not the other sender’s target), and chose a message. The experimental interface

6In our presentation, we continue to refer to “targets” and “shifts.” When we do so, the “target” is under-stood to be C’s target.

9

presented information textually as well as graphically. As shown in Figure 2, possible mes-

sages and actions are displayed on a horizontal axis. To send a message, senders used the

mouse to drag a slider along this axis to a position that corresponded to a message (any

integer from −150 to 150).7 The interface also displayed the range of possible actions for the

receiver, the realized target, the range of possible targets for the sender and for the opposing

sender, and the sender’s own target, all of which is also presented textually at the top of the

screen. The sender’s interface also featured a payoff calculator (manipulated via a separate

slider) that shows the sender’s and receiver’s payoffs for each possible action the receiver

might choose. The receiver observed messages simultaneously after both senders finished,

again both graphically and textually. The receiver then dragged a slider to select an action,

any integer from −150 to 150.

At the end of every round, subjects were informed of all of the results from the round

for their group: both messages, the action, every player’s target, and every player’s payoff.

Subjects also observed the results from all previous rounds they played, but they never

observed the results for groups to which they did not belong. Payoffs for each round were

denominated in “points,” with 100 points being the maximum possible points a player could

earn in a round (if the receiver’s action matched their own target exactly). In terms of

points, the receiver’s payoff function was 100 − |c − T | and a sender’s payoff function was

100− |c− (T + Si)|. Subjects played either 24 or 32 rounds of the game.8

At the end of the experimental session, total points were converted to cash at the rate of

$1 for every 150 points.9 Subjects were paid the sum of their earnings plus a $7 participation

7We allow messages outside the target space so that senders could choose messages corresponding to their owntargets. Such messages also help differentiate between behavioral explanations. First, messages outside thisrange are inconsistent with our equilibrium predictions, modulo a relabeling of the message spaces. Thatis, if subjects send messages outside of the integers from −100 to 100, they clearly indicate to receiversthat either they are not using the literal meanings of messages, or they are lying about the target and theirmessages should be discounted or completely ignored. If, however, subjects exaggerate, then we shouldexpect more extreme messages.

8We chose 32 rounds as our baseline length of play because pretesting indicated that this was the maximumnumber of rounds that was feasible to implement given the time constraint of our sessions. In sessions withadditional tasks, the maximum feasible number of rounds was 24.

9In principle, the feasible range of earnings is large and likely to be meaningful. The maximum possible

10

payment, and their earnings (including the show-up fee) ranged from $16.70 to $23.80, with

an average of $19.90. We conducted four sessions of the experiment (69 subjects). Each

session involved between 12 and 18 subjects (4 to 6 groups).

4 Evaluation of Equilibrium Predictions

Result 1. For receivers, we find a substantial correlation between actions and targets, whichis inconsistent with the Babbling Predictions and slightly more consistent with the Jam-ming Predictions than the Partition Predictions. The average payoff is closer to theJamming Predictions than the other predictions, and the mean squared prediction erroris also smallest for the Jamming Prediction. However, we also find that the correla-tion between actions and targets persists even under conditions in which equilibriumanalysis predicts it should diminish.

The first two rows of Table 1 present aggregate measures of information transmission.

Each row compares the observed amount to the amount predicted by each of the equilibria

we discussed in the theoretical section.10 On average, the receiver’s payoff is 78.65, which is

much higher than the expected average from the babbling equilibrium prediction and close to

midway between the expected averages of the partition equilibrium and jamming equilibrium

predictions. We also find a high correlation of 0.867 between targets and actions. This is

much higher than the babbling equilibrium (which predicts no correlation), higher than the

partition equilibrium prediction, and very close to the jamming prediction.

In addition to these aggregate measures, we also quantify how well each of the theoretical

equilibria predicts the receiver’s actions by computing the mean squared prediction error

(from the expected value of the sender’s message). These results, found in the third row of

Table 1, show that the jamming equilibrium produces the smallest prediction errors, at a

earning in a session was 3200 ÷ 150 + 7 = $23.33 while the minimum possible (given the exceedingly rareevent in which the action is at one extreme of the space and the sender’s ideal point is at the other extreme)was −6400 ÷ 150 + 7 = −$35.67. Subjects had negative earnings in only 17 rounds out of 1280, and nosubject had negative earnings aggregated over all rounds in a session.

10Note that our numerical predictions are based on discrete distributions since the values in the experimentare integers.

11

Table 1: Information transmission

PredictedObserved

Babbling Partition Jamming

Receiver Average Payoff 49.75 73.91 83.25 78.65

Target-Action Correlation 0.000 0.820 0.865 0.867√

Mean squared prediction error 51.48 34.88 32.50 –

level approximately 37% lower than the babbling equilibrium and 7% lower than the partition

equilibrium. Thus, the aggregate results seem to provide better support for the jamming

equilibrium predictions than the alternatives.

Before turning to the senders’ messages, we examine the relationship between actions and

targets more closely. The scatterplot in Figure 3 shows visually the strong correspondence

between receivers’ actions and the true targets. The regression line in Figure 3, which is

reported in the first column of Table 2, has an estimated slope coefficient of 0.75. The slope

is also significantly less than 1, which suggests that receivers trust the senders’ messages less

for extreme targets and respond by choosing actions closer to the midpoint. The regressions

reported in the second and third columns in Table 2 test whether actions depend on the

target in ways consistent with the partition and jamming equilibria, respectively.

To test whether there is evidence for the partition equilibrium that is not apparent from

the visual inspection of the data, the regression model in the second column of Table 2

includes indicator variables for whether the target is high for left senders (T > K∗ + |SL|)

or low for right senders (T < K∗ − SR) as well as the interactions between these indicators

and the target. The partition equilibrium predicts that the coefficient for target and the

interactions should be 0 and the indicators should be about 2K∗ ≈ 79.4 and −2K∗ ≈ −79.4.

We find that the main coefficient for target remains positive and statistically significant,

while the interaction terms are close to 0, and the intercepts are far smaller in magnitude

than expected. Thus, even when we allow for the relationship between actions and targets

12

Figure 3: Targets and receivers’ actions

-100

-50

050

100

Actio

n

-100 -50 0 50 100Target

Full Revelation Fitted Line

to vary in a way that the partition equilibrium predicts, we find evidence that actions are

related to targets within each region of the partition.

The specification in the third column tests whether actions correspond to the jamming

equilibrium by including an interaction between target and an indicator for the jamming

region. Jamming predicts that the main coefficient should be 1, and the interaction term

should be −1, which would indicate actions are unrelated to targets only when the target is

within one of the senders’ jamming regions. This is not what we find. The results in the third

column of Table 2 show that while the slope decreases in magnitude in the jamming region

(the interaction coefficient is negative and significant), actions remain positively related to

targets as we can reject the hypothesis that the coefficient for the interaction term is −1 (or

that the sum of the two coefficients equals 0).

Two additional model specifications explore the effects of preference divergence on the

receiver’s actions. The model in the fourth column of Table 2 adds the senders’ shifts to the

13

Table 2: Regression analysis of receiver actions

(1) (2) (3) (4) (5)

Target 0.75** 0.66** 0.81** 0.78** 0.92**

(0.03) (0.04) (0.03) (0.03) (0.05)

Target × Left High -0.16

(0.12)

Target × Right Low -0.05

(0.14)

Target × Jam -0.22** -0.11* -0.01

(0.04) (0.04) (0.05)

Left High 23.08*

(9.05)

Right Low -14.81

(12.28)

Left Shift 0.34** 0.37**

(0.08) (0.08)

Right Shift 0.33** 0.39**

(0.06) (0.06)

Target × Left Shift 0.004**

(0.001)

Target × Right Shift -0.002

(0.001)

Constant -1.97 -2.10 -2.11 -1.75 -2.31

(1.51) (1.57) (1.52) (2.68) (2.71)

R2 0.75 0.76 0.76 0.78 0.79

N 640 640 640 640 640

* p < 0.05, ** p < 0.01

OLS with standard errors clustered by subject (in parentheses)

14

jamming specification. The coefficients for these variables are positive and significant, which

suggests that senders who are more extreme exert greater influence on the receiver’s actions

than senders who are more moderate. The responsivness of receivers to senders’ shifts is

in contrast to a fully-revealing equilibrium (in which receivers should only be responsive to

changes in the target). The diminished magnitude of the coefficient for the interaction term

in the fourth specification also suggests that the attenuated relationship between actions and

targets within jamming regions is due more to the influence of senders than it is to receivers’

distrust. The model in the final column provides further support for this interpretation.

When we add additional interactions between the senders’ shift and target, we find that the

main coefficient for target increases (so that it is closer to 1) while the interaction between

target and the jamming region becomes closer to 0 and is no longer statistically significant.

Thus, we find that actions are highly correlated with targets but the correlation does not

diminish in any way predicted by equilibrium analysis. Instead, the correlation is decreasing

only in the senders’ biases.

Result 2. For senders, we find that the Jamming Predictions fit the data better than ei-ther the Babbling or Partition Predictions. Although messages are less correlated withtargets within jamming regions, they are not countervailing. Moreover, we find sys-tematic patterns in the direction of the prediction errors. Left senders’ messages areconsistently exaggerated to the left of the equilibrium predictions, while right senders’messages are consistently exaggerated to the right. Regression analysis further showsthat exaggeration is increasing in the senders’ degree of bias.

To compare predictions, Table 3 presents both the (signed) mean prediction errors and

the root mean squared prediction errors for each equilibrium by type of sender (using the

expected values of senders’ messages as the Babbling and Partition Predictions). We retain

the sign of the error for senders so that we can examine their direction. Doing so also allows

us to assess how well each equilibrium theory predicts messages by testing the hypothesis that

the errors have mean 0. We see from Table 3 that the ranking of the equilibrium predictions is

15

consistent with what we found for the receivers’ actions: the jamming prediction outperforms

the partition prediction, which outperforms the babbling prediction. This is true if we look

at the mean error as well as the root mean squared error, and these differences appear to be

more pronounced for senders than they are for receivers. However, none of the equilibrium

theories predicts senders’ behavior very well. Not only are the magnitudes of the prediction

errors for senders larger than they are for receivers, but we can also reject the hypothesis

that the errors have mean 0 for all three equilibria.

The patterns in the data suggest that senders exaggerate their messages in the direction

of their biases. Left senders’ messages are consistently to the left of the predictions of each

equilibrium theory, while right senders’ messages are consistently to the right. We obtain a

clearer picture of sender behavior by plotting messages against targets in Figure 4. Because

the predicted messages depend on both the target and shift, we disaggregate and plot the

data for different ranges of shift values. For example, the plot in the upper left corresponds

to SL between −10 and 0 while the plot in the lower right corresponds to SR between 40

and 50. Each plot also includes the predicted jamming equilibrium message strategy (for the

mean of the specified range of shift values).

Two characteristics of behavior are evident from Figure 4. First, messages are highly

correlated with targets no matter what the value of the shift parameter Si. This correlation

is consistent with the predictions of the jamming theory, in sharp contrast with both the

babbling equilibrium and partition equilibrium predictions. Visual inspection of the scat-

Table 3: Sender deviations from equilibrium predictions

Babbling Partition Jamming

Mean prediction errorLeft senders -60.13 -50.76 -44.44

Right Senders 58.93 45.81 38.45

√Mean squared prediction error

Left senders 85.97 69.28 62.01

Right senders 88.54 71.27 63.58

16

Figure 4: Messages and targets, by range of senders’ shifts

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Left Shift ∈ [-10, 0]-1

000

100

Mes

sage

-100 -50 0 50 100Target

Left Shift ∈ [-20, -10]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Left Shift ∈ [-30, -20]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Left Shift ∈ [-40, -30]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Left Shift ∈ [-50, -40]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Right Shift ∈ [0, 10]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Right Shift ∈ [10, 20]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Right Shift ∈ [20, 30]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Right Shift ∈ [30, 40]

-100

010

0M

essa

ge

-100 -50 0 50 100Target

Right Shift ∈ [40, 50]

17

terplots also strongly suggests the absence of any partition structure in which messages are

uncorrelated with targets for high or low subsets of the target parameter.11 Second, messages

are rarely, if ever, truthful. In fact, only 12 out of 1,280 messages are equal to the target.12

Instead, the disaggregated results reinforce the finding of exaggeration by senders we noted

above. Left senders reliably send messages well below the target, while right senders reliably

send messages well above the target. Furthermore, senders also select messages outside of the

target space, some of which are at the extreme boundaries of m = −150 and m = 150. This

fact also indicates that senders systematically exaggerate relative to equilibrium predictions.

Table 4 presents estimates of two regression models for each range of shift values presented

in Figure 4. The first model is simply a regression of message on target. Not surprisingly,

we can clearly reject the babbling equilibrium, which implies that the slope should be 0. In

fact, the slope is statistically indistinguishable from 1 for many values of SL and SR. The

results also show that as the magnitude of Si increases, the slope tends to decrease while the

magnitude of the intercept tends to increase.

The second model adds an interaction term to test the jamming theory’s prediction that

messages will be countervailing (inversely related to the target) when the target is within a

sender’s jamming region. The interaction is between target and an indicator for the jamming

region (as defined in the theoretical section). This allows the slope of the message function to

differ inside and outside of the jamming region.13 While the jamming theory implies that the

coefficient for the interaction should be −2, we can reject the hypothesis that the coefficient

is −2 for all but one set of shift values. There is one other region for which the interaction is

in the correct direction, but its magnitude implies something closer to babbling within the

jamming region than countervailing messages: for SL ∈ [20, 30], the confidence interval for

11This visual inspection is supported by a series of piecewise linear models that partition the set of targetsaccording to the partition equilibrium.

12We consider a “truthful” message to be one that literally matches the true target. Although the exactmeaning of messages is endogenous to players’ choices and their perceptions of the game, it seems naturalto interpret a numerical message m as a report that means “the target is m” or “you should pick m.”

13Including only this interaction effectively constrains the piecewise linear function to be continuous at T = 0while allowing for a discontinuity at T = −2SL for left senders and at T = −2SR for right senders.

18

Tab

le4:

Reg

ress

ion

anal

ysi

sof

mes

sage

s,by

shif

t

Shif

t:[-

10,0

][-

20,-

10]

[-30

,-20

][-

40,-

30]

[-50

,-40

]

Tar

get

0.97

**0.

97**

0.98

**0.

98**

0.90

**0.

92**

0.79

**0.

79**

0.73

**0.

72**

(0.0

3)(0

.03)

(0.0

3)(0

.03)

(0.0

5)(0

.05)

(0.0

5)(0

.05)

(0.0

6)(0

.07)

Tar

get×

Jam

-0.3

90.

20-0

.48*

0.03

0.02

(1.5

3)(0

.54)

(0.1

7)(0

.22)

(0.1

1)

Con

stan

t-4

3.34

**-4

3.24

**-5

3.54

**-5

3.88

**-5

9.58

**-5

7.09

**-7

6.19

**-7

6.61

**-7

6.90

**-7

7.35

**

(3.5

5)(3

.54)

(3.6

4)(3

.87)

(4.1

4)(4

.14)

(3.9

4)(3

.70)

(5.9

2)(6

.52)

R2

0.82

0.82

0.85

0.85

0.78

0.79

0.69

0.69

0.59

0.59

N12

512

514

214

210

710

711

711

714

314

3

Shif

t:[0

,10]

[10,

20]

[20,

30]

[30,

40]

[40,

50]

Tar

get

0.94

**0.

94**

0.91

**0.

92**

0.87

**0.

85**

0.86

**0.

84**

0.82

**0.

86**

(0.0

6)(0

.06)

(0.0

5)(0

.04)

(0.0

6)(0

.06)

(0.0

8)(0

.09)

(0.0

5)(0

.11)

Tar

get×

Jam

-0.7

6-1

.02*

*0.

270.

15-0

.11

(0.6

5)(0

.29)

(0.2

5)(0

.21)

(0.2

2)

Con

stan

t40

.78*

*40

.31*

*50

.17*

*48

.29*

*56

.62*

*58

.54*

*64

.44*

*66

.17*

*74

.93*

*72

.35*

*

(4.3

6)(4

.39)

(6.1

2)(6

.18)

(6.7

6)(6

.74)

(6.7

3)(6

.85)

(7.2

0)(9

.65)

R2

0.65

0.65

0.74

0.75

0.70

0.70

0.55

0.55

0.62

0.62

N14

214

212

012

011

711

712

712

713

013

0

*p<

0.05

,**

p<

0.01

,O

LS

wit

hst

and

ard

erro

rscl

ust

ered

by

sub

ject

(in

pare

nth

eses

)

19

the interaction is [−0.94,−0.01]. In all other cases, we cannot reject the hypothesis that the

interaction term is 0, meaning that we cannot statistically distinguish between the slope for

the message function inside and outside of the jamming region.

Result 3. Receivers use a “split the difference” strategy that is consistent with a best re-sponse to observed sender behavior. Specifically, regression analysis reveals a consistentrelationship between actions and the average message sent by the two senders.

Rather than following equilibrium predictions, receivers appear to behave rationally given

the message strategies senders actually used. That is, if senders engage in symmetric ex-

aggeration strategies consistent with the findings in Table 4, the message functions can be

written approximately as additive functions of the target, shift, and an additional constant:

mi = T + Si + Ei, where EL < 0 for left senders and ER > 0 for right senders. Given any

pair of messages, the symmetry of the shift distributions implies that, conditional on these

message strategies, the receiver’s best response is to simply take the average of the two mes-

sages.14 This strategy does not depend on the the receiver’s beliefs about the magnitudes of

the senders’ exaggeration, only their symmetry.

The regressions reported in Table 5 support this interpretation of receivers’ behavior.

Comparing models in the first two columns, we see that the coefficient on average message

is above 0.80. Adding target, left shift, and right shift to the model does not add any

additional explanatory power. The absence of differences between specifications suggests that

information about the target is transmitted through the behavioral mechanism of averaging

sender messages. The model in the third column shows that this strategy does not change

much over the course of an experimental session. Since the receivers appear to use a “split

the difference” strategy that is simple and intuitive given the symmetry of the senders’ shift,

we assume in the remainder of the analysis that senders correctly anticipate that receivers

14When one of the messages is at the boundary of the message space, receivers should discount the messageand put greater weight on the interior message. However, the numerical effect of the boundaries is quitesmall and so the average remains a good approximation to the best response function.

20

Table 5: Regression analysis of receivers’ responses to messages

(1) (2) (3)

Average Message 0.85** 0.82** 0.84**

(0.02) (0.05) (0.05)

Target 0.04 0.04

(0.04) (0.04)

Left Shift 0.01 0.01

(0.06) (0.06)

Right Shift 0.01 0.01

(0.05) (0.05)

Avg. Message × Round -0.002

(0.003)

Round -0.04

(0.11)

Constant 0.33 0.38 0.94

(0.98) (1.66) (1.54)

R2 0.89 0.89 0.89

N 640 640 640

* p < 0.05, ** p < 0.01

OLS with standard errors clustered by subject (in parentheses)

follow this strategy.

Although the amount of overall information transmission matches some features of the

predictions of equilibrium theory, we can thoroughly reject the possibility that this is because

senders strategically limit the information they reveal. Consistent with previous experiments

on cheap-talk games, we find substantial evidence of overcommunication: messages are highly

correlated with the underlying state of the world. In our experiment, this overcommunication

takes the form of systematic exaggeration, in the form of adding a constant to the target.

Such exaggeration is decodable because, if the receiver knows the constant that has been

added, she can simply deduct that constant to infer the true target. Senders with biases to

the right of the receiver inflate their messages to the right, while senders with biases to the

21

left of the receiver deflate their messages to the left. The magnitude of exaggeration is also

increasing in the senders’ biases. We next explore whether subjects’ behavior can instead be

explained by a form of strategic bounded rationality.

5 Limited Strategic Sophistication

The fundamental idea underlying equilibrium analysis is the mutual consistency of beliefs

and actions. Each player is assumed to choose the best response given her beliefs about

what others will do, and those beliefs are also assumed to be consistent with what others

actually do. But given the multiplicity of equilibria in signaling games as well as the subtle

logic required to construct equilibrium strategies (especially off the path of play), the level

of sophistication required for subjects to engage in equilibrium play is too demanding for

many individuals. Nevertheless, it is quite plausible that subjects engage in behavior that

exhibits limited strategic sophistication. We consider whether forms of bounded rationality

that relax the mutual consistency assumption help to explain the patterns of information

transmission and exaggeration that we find.

Specifically, we take two approaches to modeling how beliefs might be formed. First,

we apply a level-k model (e.g. Camerer, Ho, and Chong 2004; Costa-Gomes, Crawford, and

Broseta 2001; Crawford 2003; Nagel 1995; Stahl and Wilson 1995) in which subjects form

expectations based on iterated reasoning. Some subjects are naive, minimally sophisticated

subjects best respond to naive behavior, more sophisticated subjects best respond to min-

imally sophisticated behavior, and so on. Such models provide a useful starting point for

several reasons. First, they have been used to explain overcommunication in cheap-talk

games (e.g. Cai and Wang 2006; Kawagoe and Takizawa 2009; Wang, Spezio, and Camerer

2010), so our application of level-k facilitates comparisons to previous work. Second, the

models are useful for generating predictions about subjects who confront the game for the

first time, especially in early rounds of play when previous experience provides an insufficient

22

guide to behavior. Third, the level-k model accounts for heterogeneity in subjects’ exagger-

ation as a function of differences in cognitive ability or strategic sophistication. Although

we cannot observe cognitive differences directly, they can be inferred from behavior.

In our second approach, we develop a model of “experiential best response,” in which

beliefs are based not on a reasoning process, but on past observations. Our simple model is

in the spirit of Cournot best response dynamics or fictitious play (e.g. Camerer and Ho 1999;

Cheung and Friedman 1997; Fudenberg and Levine 1998). In this model, we posit that a

sender expects his opponent’s message will be the average of the messages he has observed

his opponents send previously, and so he will best respond to opponents’ past exaggeration.

The latter model helps to understand how individuals might adapt their behavior over time

and to gauge how sophisticated they are in doing so.

Finally, we pit these two approaches head-to-head in a test of out-of-sample predictive

power. Here, we fit models of each form of limited strategic sophistication to data from the

first several rounds of play. Then we estimate mean and root mean square predictive errors

for a test set of future rounds. Intriguingly, we find that neither model dominates the other

in terms of out-of-sample predictive power.

5.1 Level-k Reasoning

In the level-k framework, k denotes a subject’s degree of sophistication, in terms of the

number of steps of iterated reasoning. Level-0 players are non-strategic and use naive decision

rules. Level-1 players believe their opponents are level-0 and choose the appropriate best

response. In general, level-k players best respond given the belief that their opponents are

level-(k − 1).

Applying the level-k framework to our game poses two potential complications. First, in

contrast to previous applications, our game involves players in three roles.15 Thus, we must

15In previous cheap-talk experiments, there are only two roles, senders and receivers. While there are moreplayers in other games, such as the beauty contest, these feature only one role (players have the sameactions and utility functions). The assumption that a level-k player believes all other players to be k − 1

23

be able to pin down a sender’s beliefs about not just the other sender but the receiver as well.

Because a level-k receiver believes that both senders are level-(k − 1), it follows from the

symmetry of the senders’ levels that the receiver’s best response is to choose an action equal

to the average message. This holds for any k for the receiver, and as noted above, receivers

indeed chose actions close to the average message. Thus, in the analysis that follows, we

focus attention exclusively on the sophistication of senders.

The second complication is that we must choose an appropriate level-0 type with which

to “anchor” the analysis. The key property is that level-0 behavior must be naive and non-

strategic. There are two plausible level-0 strategies that senders might employ. Senders

might be naive truthful types who report the truthful message mt-0 = T , which we denote

as t-0. This is the assumption that Cai and Wang (2006) make.16 Alternatively, senders

might be naive selfish types, which we denote by s-0, who instead report their own targets,

ms-0 = T + Sj. The latter type of sender might be thought of as attempting to maximize

utility but is non-strategic because he fails to consider either how the receiver interprets the

message or how his opponent’s strategy affects the receiver’s action.

To derive the form of each type’s message function, suppose first that the naive truthful

type t-0 anchors the iterated reasoning process. Type t-1 denotes the level-1 subject who

believes he is playing a truthful opponent. A subject of this type believes that the receiver

will choose c = 12(T +mt-1), and so his best response is to choose mt-1 = T + 2S. To see this,

note that the sender wants to induce the receiver to choose an action equal to his own target,

T +S. Thus, c = 12(T +mt-1) = T +S if and only if mt-1 = T +2S. (This argument is equally

valid for left and right senders, regardless of the sign of S.) At the next level of sophistication,

type t-2 believes he faces a type t-1 opponent. Type t-2 believes that c = 12(T +2Sopp+mt-2),

where Sopp is the opponent’s shift. Although he does not know his opponent’s shift, each

and the structure of the beauty contest game allows beliefs to be summarized by a single parameter (theaverage guess).

16Applications of level-k to symmetric beauty contest games usually assume that level-0 players chooserandomly, but anchoring level-k on such behavior implies that senders’ messages will sometimes be on theopposite side of the target from their shift. This is clearly rejected by our previous analysis.

24

Table 6: Level-k message strategies

Type Left Sender Message Right Sender Message

t-0 T T

t-1 T + 2S T + 2S

t-2 T + 2S − 50 T + 2S + 50

t-3 T + 2S − 100 T + 2S + 100

s-0 T + S T + S

s-1 T + 2S − 25 T + 2S + 25

s-2 T + 2S − 75 T + 2S + 75

sender does know its sign and distribution. The best response is to choose the message

that will ensure E(c) = T + S, which implies mt-2 = T + 2S − 2E(Sopp). For example, the

left sender knows that Sopp is distributed uniformly between 0 and 50; therefore, his best

response is mt-2 = T + 2SL − 50. Similarly, the t-2 type right sender’s best response is

mt-2 = T + 2SR + 50. Continuing this pattern of reasoning, the message functions for t-3

types are mt-3 = T + 2SL − 100 for left senders and mt-3 = T + 2SR + 100 for right senders.

Anchoring the analysis on the naive selfish type s-0 rather than t-0 yields a similar set of

strategies that differ only by a constant for types k > 1. Message functions are summarized

in Table 6. Messages that reflect levels k > 1 take the general form m = T + 2S + α, where

α(k) = 50(k− 1) if the anchor is the naive truthful t-0 type and α(k) = 50(k− 1) + 25 if the

anchor is the naive selfish s-0 type. In contrast, naive strategies are either less responsive or

unresponsive to the shift parameter and do not involve a constant term.

Our level-k analysis yields best response functions characterized by levels of exaggeration

and that are continuous in the target and shift parameters. This is in contrast to equilibrium

strategies, which involve discontinuous categorical or piecewise functions. We also find a clear

pattern between the level of sophistication k and a sender’s best response. Naive, level-0,

senders ignore strategic considerations and report the truth or their own ideal points. Level-

1 (and above) senders realize that their goal is to send a message such that the average

25

(of their own message and the opposing player’s message) is equal to their own ideal point.

This implies that senders at least as sophisticated as level-1 types will exaggerate in the

direction of their own shift: left senders exaggerate to the left in order to pull the average

message to their own ideal point on the left and right senders exaggerate to the right to

pull the average to their ideal point on the right. The degree of exaggeration also depends

on the sender’s degree of bias, and higher level senders will exaggerate even further so as

to counteract the exaggeration of lower level senders. Thus, a model of bounded rationality

potentially explains the patterns of exaggeration that we observe in the data.

Result 4. Most senders can be classified with some level of limited strategic sophistication,with 75% falling between one and two levels of strategic reasoning. However, regressionanalysis suggests that senders under-exaggerate relative to the predictions of level-kreasoning. Messages are less responsive to changes in shifts than would be optimal inthe level-k model but also unravel in the sense that naive exaggeration increases overtime.

We use two methods to empirically assess the level-k framework. First, we classify each

subject’s level of sophistication using a two-step process following the method of Costa-

Gomes, Crawford, and Broseta (2001) and Cai and Wang (2006). In step one, we compare

observed messages to each type’s predicted message (for t types as well as s types) and

classify each message as being consistent with a type if the distance between the prediction

and message is within an error band of ±12.17 In step two, we then classify a subject as

being a particular type based on the modal classification of the subject’s messages. If there

is a tie among the modal message type, we classify a subject according to the lower level. If

the two types are both level k but are based on different anchors, we consider t-k as being

a lower type than s-k.

17This error band is 8% of the total message space and was chosen to maximize the number of messagesuniquely classified. Due to the boundedness of the message space, we account for potential censoring inthe predicted messages (i.e., predicted messages greater than 150 are set to 150 and predicted messagesless than −150 are set to −150). Overall, 91.6% of messages fit at least one classification, and 67.3% ofmessages are uniquely classified.

26

The classification analysis summarized in Table 7 reveals heterogeneity in the distribution

of subjects’ levels of strategic sophistication. Most subjects send messages that are relatively

consistent: 71% of subjects send a majority of messages that belong to the same classification.

And most (78%) of these possess some degree of strategic sophistication beyond naivete (i.e.,

k > 0). Overall, 24% are classified as one of the level-0 naive types, 45% are classified level-1,

and 30% as level-2. No senders are classified as having sophistication greater than level-2.

The level-k framework therefore appears to organize the data reasonably well.18

Table 7: Level-k classifications

Minimum Classification Success

Type > 30% > 40% > 50% > 60% > 70% > 80% Pct. Type

t-0 0 1 0 0 1 0 4%

s-0 2 1 2 1 1 2 20%

t-1 0 2 2 1 2 0 15%

s-1 1 2 8 2 0 1 30%

t-2 0 4 4 4 0 0 26%

s-2 1 0 0 1 0 0 4%

Pct. Min. Success 9% 22% 35% 20% 9% 7%

The second method we use to assess the level-k framework is to regress the message on

target, shift, and separate intercepts for the left and right sender. As noted above, our level-k

analysis implies that the level of exaggeration (after controlling for the true target) depends

on the value of shift and a constant. Specifically, the coefficient on target should be 1, the

coefficient on shift should be 2, and the intercepts should be a multiple of 25.

The results, reported in the first column of Table 8, provide additional but qualified

support to our interpretation that bounded rationality—in the form of limited strategic

sophistication—helps to explain exaggeration in messages. The coefficient on target is close

18We also compared behavior in the competitive communication game with a one-shot Beauty Contest gamebut found that classifications in the two games differed markedly, likely because of significant differences intheir strategic contexts. Whereas the Beauty Contest game implicates few obvious norms of good behavior,communicating immediately raises the prospect of truthtelling and lying.

27

Table 8: Regression analysis of senders’ messages and level-k

All Levels Excluding Level 0

(1) (2) (3) (4)

Target 0.87** 0.98** 0.85** 1.00**

(0.02) (0.03) (0.02) (0.03)

Shift 0.88** 0.85** 1.00** 0.97**

(0.08) (0.12) (0.07) (0.13)

Left Sender -40.22** -19.62** -41.34** -21.31**

(3.55) (4.41) (3.22) (5.34)

Right Sender 34.85** 19.23** 44.64** 22.43**

(4.75) (5.29) (3.36) (6.26)

Target × Round -0.01* -0.01**

(0.00) (0.00)

Shift × Round 0.00 -0.00

(0.01) (0.01)

Left Sender × Round -1.53** -1.53**

(0.36) (0.39)

Right Sender × Round 1.17** 1.66**

(0.42) (0.44)

R2 0.84 0.86 0.88 0.90

N 1280 1280 976 976

* p < 0.05, ** p < 0.01

OLS with standard errors clustered by subject (in parentheses)

to (albeit significantly less than) 1, and we see that the magnitudes of the left and right

sender intercepts are between 25 and 50.19 The regression estimates therefore suggest that

the average subject’s level of sophistication is around s-1 or t-2, which is somewhat higher

than what we found in the classification analysis.

When we look at the regression coefficient for senders’ shifts, the evidence for level-k

thinking becomes murkier. The coefficient on shift is close to, but significantly less than,

19These results persist if we drop the subjects we code as level-0, for whom messages should not depend onshift and the constant (see models (3) and (4) of Table 8).

28

1. Messages are therefore unconditional functions of shifts, consistent with the direction

predicted by the level-k framework. But since the coefficient is much less than 2, messages

are much less responsive to senders’ biases than the framework predicts. Thus, it seems that,

relative to payoff-maximizing levels of exaggeration established by the level-k framework,

there is systematic understatement in exaggerations.20 Indeed, a coefficient on target of 1

and a coefficient on shift of 1 is consistent with behavior in which senders tack on a constant

amount of exaggeration to their own ideal points. Such behavior is even less sophisticated

than that in the level-k framework. It is consistent instead with a characterization of subjects’

behavior in which they hold a naive (yet perhaps intuitive) belief that some amount of

exaggeration is necessary to pull the receiver’s actions toward their own target but fail

to recognize that the optimal way to do so is to induce the (expected) midpoint between

messages to be equal to their own target.

Both of these analyses assume that sophistication is fixed for each subject throughout the

session. To allow for the possibility that subjects might adapt their behavior and eventually

learn to play more sophisticatedly over time, we conduct two more analyses. First, we

estimate a model including interactions with the round of play (see model (2) from Table

8).21 Here we are interested in whether the coefficient on shift increases with repeated

rounds of play. If subjects learn to play level-k best responses over time, we would expect

to see the coefficient for the interaction between shift and round to be positive. Instead, we

find no evidence whatsoever that messages become more responsive to senders’ shifts over

time. Rather, based on the interaction of round and the left and right sender intercepts, it

20The level-k framework suggests that the messages of sophisticated senders would hit the boundaries fre-quently. For example, the messages of t-2 types would be censored 25% of the time on average. However,we observe very little censoring in the data. Overall, only 7.3% of messages are censored, although we dofind that the rate of censoring increases over time (from 2.7% in the first 8 rounds to 14% in the last 8rounds of sessions with 32 rounds). The conclusion that messages are understated does not change whenwe estimate a Tobit model that allows for censoring at the boundaries of the message space, as there isvery little difference between the OLS and Tobit coefficients (see the Appendix).

21We code round to begin at 0 so that we can interpret the uninteracted coefficients as the coefficients inthe first round of play.

29

Figure 5: Exaggeration and unraveling: Predicted sender intercepts over time

-100

-50

050

100

Pred

icte

d Ex

agge

ratio

n

1 5 9 13 17 21 25 29Period

Left Sender Right Sender

appears that a form of naive exaggeration increases over time.22 Figure 5 plots the predicted

intercepts over time (with 95% confidence intervals, based on model (2) from Table 8). While

the intercepts begin near −19 and 19, for the left and right senders respectively, by the 32nd

round of play the estimated levels of exaggeration end near −67 and 55. If we use these

estimates to project out of sample by assuming that exaggeration increases at the same

rate, then play would approach an uninformative equilibrium (i.e., all messages reach the

boundaries of the space with certainty) by around the 100th round of play. Messages unravel,

but not quickly enough for communication to break down completely before the end of each

experimental session.

As a second investigation of increasing sophistication with repeated play, we split the

sessions in half to see whether subjects have higher sophistication levels in the second half.

22Both lying aversion or some form of social preference (e.g. Hurkens and Kartik 2009) as well as quantalresponse equilibrium (McKelvey and Palfrey 1998) predict stable patterns of behavior, so the increases inexaggeration over time that we observe in the data suggest against these two alternative interpretations.

30

In Table 9, we compare level-k classifications using messages from the first half of each

session (first 12 periods in sessions with 24 periods and first 16 in session with 32) against

the classifications using the second half. Counting s-k types as higher levels than t-k types

(e.g., s-1 is higher than t-1), we classify 28 subjects as having a “higher” level in the second

half of each session than in the first half while 17 subjects’ levels remain the same, and

only 1 subject can be classified as having a “lower” level (originally t-2 but then later s-1).

Moreover, this comparison shows that 15 subjects can be classified as level-0 in the first half

while only four subjects remain in the level-0 classification in the second half.

Table 9: Change in level-k classifications

Second half

t-0 s-0 t-1 s-1 t-2 s-2 Total

First half

t-0 2 0 0 0 1 0 3

s-0 0 2 2 4 3 1 12

t-1 0 0 4 9 4 0 17

s-1 0 0 0 6 2 0 8

t-2 0 0 0 1 3 1 5

Total 2 2 6 20 13 2 45

Although the level-k framework provides a better explanation for behavior than equilib-

rium analysis, this explanation remains incomplete. Senders appear to engage in systematic

exaggeration in a way that is only partly consistent with limited strategic sophistication. If,

in fact, senders are engaged in best response play based on mutually inconsistent beliefs,

then senders appear to do so in an understated manner—senders do not fully incorporate

their shifts into their message strategies. While messages appear to be consistent with level-k

types, the regression analysis suggests instead that individuals attempt to be strategic by

exaggerating, but they neither engage in iterated reasoning nor recognize that the best re-

sponse function must take into account the receivers’ averaging strategy. But we also find

that experience matters, as messages become more exaggerated over time, and players are

classified at higher levels of sophistication later in the session. It is still possible that sub-

31

jects play a form of best response, but the level-k analysis misspecifies the belief formation

process. Our next framework focuses explicitly on the role of experience in the formation of

beliefs.

5.2 Experiential Best Responses

In the level-k framework, we assumed that subjects form beliefs about what others will

do through a process of iterated reasoning, anchored by intuitive conjectures about naive

behavior. But we also saw that as individuals play the game, they exaggerrate more and

may even become more sophisticated. An alternative explanation for this behavior is that

senders continually adjust their beliefs to incorporate their observations about opponents’

behavior and thus play the best response to the empirical distribution of messages they have

experienced.

The “experiential best response” framework is a form of belief learning or fictitious play

that generalizes the insight of the level-k analysis. Like level-k, we assume players choose

best responses to their (non-equilibrium) beliefs. But rather than assume that beliefs depend

on iterated reasoning, we instead assume that beliefs depend on each subject’s empirical

observation. This is plausible for at least two reasons. First, subjects are reminded of their

history at the end of each round; thus, they may simply be acting on the information we

offer them. Second, we found above that senders exaggerate more over time. Rather than

becoming more sophisticated (as a level-k interpretation might suggest), it is possible that

senders are instead simply responding to a self-perpetuating trend in the messages they

observe. By focusing on the alternative framework, we attempt to disentangle these two

explanations.

We now assume each sender believes his opponent to send a message equal to the oppo-

nent’s ideal point plus the average exaggeration he has observed. If the sample average of

the opposing sender’s past exaggeration is et =∑t−1

τ=1(mτopp−T τ ), then the expected value of

an opposing sender’s current message is E(mopp) = T + et. The logic involved in deriving the

32

experiential best response that follows from this belief is virtually identical to the derivation

of best responses in the level-k framework, and as in the level-k framework, we assume that

receivers choose the average of the senders’ messages. Given the expectation et, a sender’s

best response in round t is to choose message m = T + 2S − et. The more a sender has

observed his opponents exaggerate in the past, the more the sender will himself exaggerate

in order to pull the average message (i.e., what the sender expects to be the receiver’s ac-

tion) to his own ideal point. The experiential best response framework implies that messages

will be partly functions of target and shift, as the level-k framework does. Unlike the level-k

framework, however, it predicts that the remaining extent of exaggeration will be completely

determined by opponents’ past exaggeration rather than by the distribution of opponents’

shifts or by the level of strategic sophistication. That is, once the target, shift, and oppo-

nents’ exaggeration are taken into account, there should be no additional exaggeration by

either left or right senders.

Result 5. Senders’ messages are responsive to the average of opponents’ past exaggeration.However, senders also under-exaggerate relative to the predicted experiential best re-sponse, as their messages remain less responsive to changes in shifts than predicted bythe framework. The level of under-exaggeration decreases over time and is much closerto the predicted messages by the last round of play.

To apply this framework to our data, we regress message on target, shift, indicators

for left and right sender, and opponents’ past average exaggeration. Because senders may

have short or long memories, we used two different measures of past exaggeration. First, we

measure exaggeration over the entire history a sender experienced, from round 1 up to the

previous round. Second, we measure exaggeration as a moving average over the most recent

five rounds. In each case, we expect the coefficient on past exaggeration to be −1, on shift to

be 2, on target to be 1, and on left sender and right sender both to be 0. As in the previous

section, we also estimate a specification that includes interactions with round to account for

changes in behavior over time that are not already accounted for by exaggeration.

33

Table 10: Regression analysis of senders’ messages and experiential best responses

(1) (2) (3) (4)

Target 0.88** 0.99** 0.88** 1.00**

(0.02) (0.03) (0.02) (0.03)

Shift 0.87** 0.93** 0.85** 0.94**

(0.08) (0.14) (0.08) (0.14)

Exaggeration (last 5) -0.56** -0.33*

(0.07) (0.13)

Exaggeration (all) -0.68** -0.24

(0.14) (0.18)

Left Sender -12.19* -10.45 -11.25 -14.92

(5.30) (6.24) (7.71) (7.49)

Right Sender 3.61 9.83 3.51 15.30*

(5.67) (6.58) (6.67) (7.38)

Target × Round -0.01** -0.01**

(0.003) (0.003)

Shift × Round -0.01 -0.01

(0.01) (0.01)

Exag. (last 5) × Round -0.01

(0.01)

Exag. (all) × Round -0.02

(0.01)

Left Sender × Round -0.46 0.03

(0.50) (0.65)

Right Sender × Round -0.08 -0.63

(0.66) (0.80)

R2 0.87 0.88 0.86 0.87

N 1234 1234 1234 1234

* p < 0.05, ** p < 0.01

OLS with standard errors clustered by subject (in parentheses)

34

The regression coefficients reported in Table 10 suggest that taking into account oppos-

ing senders’ past behavior provides an improved account of exaggeration but that senders

continue to under-exaggerate. In the models without round interactions, the coefficient for

opponents’ past exaggeration is −0.56 when we use the last 5 rounds of play, and it is −0.68

when we use the entire history of play. Both coefficients are significantly smaller in magni-

tude than the experiential best response framework predicts and we can reject the hypothesis

that −1 lies in the confidence interval for the coefficients. However, when we allow the rela-

tionship between opponents’ past exaggeration and messages to vary over time by including

an interaction with the round of play, we see that the coefficients for the interaction term for

both versions of past exaggeration are negative (but not statistically significant). By the 32nd

round, messages are as responsive to opponents’ past exaggeration as the experiential best

response framework predicts; the predicted coefficient using all rounds of opponents’ past

exaggeration is −0.98, with the hypothesized value of −1 within the confidence interval.23

Nevertheless, two discrepancies between the experiential best response framework and

senders’ observed messages appear to remain. First, we see that the intercepts for left sender

(in all models) and right sender (in the last column) remain statistically significant, which

suggests that subjects continue to add a constant amount of exaggeration that is unrelated

to opponents’ past exaggeration or to their own shifts. The interaction model estimates,

however, suggest that this extra exaggeration decreases as subjects gain experience. The

second discrepancy is that we also continue to find that the coefficient on shift remains far

less than 2. Thus, even when we account for experience, senders are less responsive to the

level of their biases than either the level-k or experiential best response frameworks predict.

Naive understatement in exaggeration persists.

23Based on model (4) from Table 10, the coefficient of Exaggeration (all) in the 32nd round is −0.24 +32(−0.02) = −0.98 with 95% confidence interval [−1.61,−0.35]. In the sessions with only 24 rounds,exaggeration in the 24th round is −0.24 + 24(−0.02) = −0.79 with 95% confidence interval [−1.26,−0.32].Separating these sessions from each other yields similar results. Details appear in the SupplementalAppendix.

35

5.3 Comparison of Limited Strategic Sophistication Models

Result 6. Neither model of limited strategic sophistication dominates the other at predict-ing data out-of-sample. Level-k predictions are better than experiential best responsepredictions for right senders, and vice versa for left senders.

To help determine which model of limited strategic sophistication better predicts behav-

ior, we performed an out-of-sample comparison. We first generate level-k predictions using

observed messages in periods 1–19 to classify each sender’s type according to the procedures

described above. We then generate predicted messages for five rounds, 20–24. We can think

of this procedure as treating periods 1–19 as the estimation sample and periods 20–24 as

out-of-sample predictions.24 Comparing periods 20–24 puts the models on an even playing

field, since the level-k predictions are based on a classification (estimation) procedure while

there are no parameter estimates involved in generating the experiential best response pre-

dictions. The level-k predictions are based on each subject’s type classification (bounded by

the range of possible messages). Given our finding that sophistication increases over time

(Table 9), we also used more recent periods as an alternative estimation sample (periods

11–19).

For the experiential best response framework, we compute the predicted messages for

subject i in round t as mit = Tit + 2Sit − eit, where eit is the sample average of opponents’

past exaggeration (as used in Table 10) over either the five most recent rounds or all previ-

ous rounds. Table 11 shows the prediction errors from the limited strategic sophistication

frameworks.

The results in Table 11 provide mixed evidence with respect to which model of limited

strategic sophistication is better. For left senders, the experiential best response model based

on all previous rounds of play provides the best out-of-sample predictions, as measured in

terms of both mean prediction error and root mean squared prediction error. In contrast, for

24Splitting estimation and test sets in this way also allows us to pool together sessions with 24 and 32 rounds.

36

Table 11: Sender deviations from limited strategic sophistication predictions

Level-k Experiential Best Response

(1–19) (11–19) (Past 5) (All)

Mean prediction errorLeft Senders 25.62 14.47 -21.51 -13.78

Right Senders -16.28 -8.26 32.65 25.81

√Mean sq. pred. error

Left Senders 38.38 26.69 30.41 24.88

Right Senders 32.62 26.60 47.63 43.78

right senders, the level-k predictions are superior. When we base the level-k classifications

on later periods (11–19), the prediction errors decrease in magnitude, but the relative com-

parisons do not change. We are left without clear evidence that either framework dominates

the other.

6 Conclusion

In this paper, we have presented three main findings from a sender-receiver experiment in

an environment with competition, preference uncertainty, a unidimensional state space, and

large message and action spaces. First, there is scant evidence of behavior that comports

with equilibrium predictions from the game. Senders’ messages do not accord well with

predictions from babbling, partition, or jamming equilibria. Instead, they are best modeled

as additive exaggerations from the target (the hidden state information). Consequently,

receivers can excel at matching their actions to the target simply by splitting the difference

between the two messages. Moreover, such receiver behavior is conditionally rational given

the message strategies that senders use.

Second, we find evidence of substantial unraveling in the messages that senders select.

As game play proceeds, senders gradually increase the exaggeration that they add to the

target. This process resembles phenomena like grade inflation and similar kinds of rating

37

systems in which messages continue to convey information despite differing from their literal

meanings.

Third, we apply two models of limited strategic sophistication to explain the first two

findings. A level-k model accounts well for variation in strategic sophistication. Moreover,

this model suggests that senders become more sophisticated over time, as classification into

levels within the first several rounds diverges from that based on the last several. In contrast,

a model of “experiential best response” provides an alternative account of these putative

increases in sophistication. In this model, senders do not become more sophisticated. Rather,

they simply condition their messages on past observations of their opponents’ behavior.

While both models account for sender behavior better than equilibrium predictions, neither

model dominates the other in terms of minimizing prediction errors. Moreover, both models

predict much more exaggeration than we observe. Determining the precise cause of this

exaggeration is a subject we leave for future research.

In sum, we find reasons to be cautiously optimistic about information transmission via

the “marketplace of ideas.” Competitive communication causes the proliferation of polarized

messages that depart from the truth, but nevertheless allows receivers to infer what the truth

is. This optimism may only be justified in the short term, however, perhaps notably for new

policy issues or social cleavages. Further research is needed to determine whether competition

achieves sustainable improvements in information transmission and the extent to which those

improvements diminish as an issue matures.

38

Acknowledgements

We gratefully acknowledge comments and advice from two anonymous reviewers, Dan Butler,

Mark Fey, Sean Gailmard, Becky Morton, Michael Neblo, Laura Paler, John Patty, Maria

Petrova, Joel Sobel, Craig Volden, Alistair Wilson, Rick Wilson, and seminar audiences

at Duke University and Washington University in St. Louis. Previous versions of this

paper were presented at the Experiments Mini-Conference at the 2011 Annual Meeting

of the Southern Political Science Association, the 2011 Annual Meeting of the Midwest

Political Science Association, and the 2011 Annual Meeting of the American Political Science

Association.

39

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42

Supplemental Appendix for“Competition, Preference Uncertainty, and Jamming:

A Strategic Communication Experiment”

William Minozzi∗ Jonathan Woon†

July 13, 2015

A1 Proofs.

Game Setup. There are three players, senders i ∈ {L,R} and receiver. Each sender

privately observes his shift Si, where SL ∼ Uniform[−50, 0] and SR ∼ Uniform[0, 50]. Both

senders observe the target T ∼ Uniform[−100, 100]. After observing Si and T , the senders

simultaneously choose messages mi ∈ R. After observing these messages, the receiver chooses

an action c. Payoffs are given by u(c, T, Si) = 100 − |c − (T + Si)|, i ∈ {L,R}, where the

receiver has S = 0. We use the notation mi(T, Si) and c(mL,mR) to denote strategies, and

density g(T |mL,mR) to denote R’s posterior beliefs. We focus on perfect Bayesian equilibria.

Proposition 1. Babbling Predictions. There is a babbling equilibrium such that E[mi] = 0

and E[c] = 0.

Proof. Suppose mL,mR, and c ∼ Uniform[−100, 100]. According to Bayes rule, if the

senders use these strategies, the receiver’s posterior beliefs equals her prior, g(T |mL,mR) =

Uniform[−100, 100]. No player can profitably deviate from his strategy, and the expectations

are straightforward to calculate based on strategies.

∗Associate Professor, Department of Political Science, 2137 Derby Hall, Ohio State University, Columbus,OH 43210 Phone: 1-614-247-7017, Email: minozzi.1@osu.edu†Corresponding author: Associate Professor, Department of Political Science and Faculty, Pittsburgh

Experimental Economics Laboratory, 4814 Wesley W. Posvar Hall, University of Pittsburgh, Pittsburgh, PA15260 Phone: 1-412-648-7266, Email: woon@pitt.edu

A1

Definition. A partition strategy is a message strategy such that the state space is partitioned

−100 < K1 < K2 < ... < Kn < 100, and mi(T, Si) maps to a uniform distribution over one of

the partitions. A partition equilibrium is one in which both senders use partition strategies

(with different partitions).

Proposition 2. Partition Predictions. There is a K and a partition equilibrium in which

E[mL] = K+1002

if T + SL > K and −100+K2

, and E[mR] = −100−K2

if T + SR < −K and

E[mR] = −K∗+1002

otherwise.

Proof. Consider the following strategies and on-the-path beliefs:

mL(T, SL) ∼

U[K, 100] if T + SL > K

U[−100, K] if T + SL < K

(A1)

mR(T, SR) ∼

U[−K, 100] if T + SR > −K

U[−100,−K] if T + SR < −K(A2)

c(mL,mR) =

2K if mL > K and mR > −K

−2K if mL < K and mR < −K

0 otherwise

(A3)

g(T |mL > K,mR > −K) =

T−K

50(75−K)if T ∈ [K, 50 +K]

175−K

if T ∈ [50 +K, 100]

(A4)

g(T |mL < K,mR < −K) =

−K−T

50(75−K)if T ∈ [−K − 50,−K]

175−K

if T ∈ [−100,−K − 50]

(A5)

g(T |mL < K,mR > −K) =

150+2K

(1− T−K50

) if T ∈ [K, 50 +K]

150+2K

if T ∈ [−K,K]

150+2K

(1− −T−K50

) if T ∈ [−K − 50,−K]

(A6)

A2

and suppose that off-the-path beliefs given by any posterior distribution with expected value

0. If these strategies and beliefs constitute an equilibrium, then the predictions stated in the

text clearly follow. We show that this is an equilibrium, first by focusing on the receiver’s

beliefs and strategy, and then turning to sender behavior.

Given the message strategies in (A1) and (A2), the receiver’s on-the-path posterior

beliefs are calculated by marginalizing over shifts. For example, consider the beliefs given in

A4. When mL > K and mR > −K, the receiver infers that T > K −SL and T > −K −SR.

Because SL < 0 < SR, T > K − SL implies that T > −K − SR, so the latter inequality

provides no additional information. Focusing on T > K − SL and marginalizing over SL,

there are two cases: (a) K < 50, so g(T |mL,mR) = T−K50(75−K)

for T ∈ [K, 50 + K] and

g(T |mL,mR) = 175−K

for T ∈ [50 + K, 100], or (b) K > 50, so g(T |mL,mR) = 2(T−K)(100−K)2

for

T ∈ [K, 100]. In either case, K is identified by setting the posterior expected value equal to

2K. Doing so for case (a)

∫ 50+K

K

T −K50(75−K)

TdT +

∫ 100

50+K

1

75−KTdT = 2K

which identifies K = 253

(7−√

5) ≈ 39.7, which is less than 50, as supposed in (a). For case

(b), this equation has no solution. Thus the choice strategy given in (A3) is optimal for its

first case given this K.

A symmetrical argument applies for mL < K and mR < −K, resulting in posterior

beliefs given in equation (A5). Similarly, the choice strategy given in (A3) is optimal for its

second case.

The final on-the-path case is when mL < K and mR > −K. Based on these message

strategies, the receiver infers that T ∈ [−50−K, 50+K]. Again subdividing into subintervals

for T and marginalizing over SL and SR yields the beliefs given in equation (A6). The set

of off-the-path messages includes all pairs such that mL > K and mR < −K. Because of

symmetry, neither sender is more likely than the other to have sent an off-the-path message.

A3

Therefore, we make the anodyne assumption that beliefs in this case are given by any dis-

tribution that has expected value 0. Given these on-the-path and off-the-path beliefs, and

the choice strategy given in (A3) is optimal for its third case.

Turning to sender behavior, each sender knows that there are three intervals to con-

sider: T < −K, T > K, and T ∈ [−K,K]. When T < −K, sender R’s message has the

effect of selecting c = 0 or c = −2K, and this depends on whether u(c = −2K,T, SR) >

u(c = 0, T, SR), which is equivalent to T + SR < −K, the cutoff used in (A2). Sender L

can only deviate by sending a message that would increase c (with some probability), and as

he prefers c to be smaller, this deviation is not profitable. A symmetrical argument applies

when T > K and results in the cutoff used in (A1). Finally, when T ∈ [−K,K], both senders

infer the other’s strategy (Uniform[−100, K] for L and Uniform[−K, 100] for R), meaning

that both know that the outcome will be c = 0 unless they deviate. But if either deviates,

he will move the action further from his preferred ideal point. Thus there are no profitable

deviations from these strategies.

Proposition 3. Jamming Predictions. There is an equilibrium with strategies and on-the-

path beliefs given by

mL(T, SL) =

−T if T ∈ [0,−2SL]

T otherwise

(A7)

mR(T, SR) =

−T if T ∈ [−2SR, 0]

T otherwise

(A8)

c(mL,mR) =

mL if mL = mR

0 otherwise

(A9)

g(T |mL,mR) =

1 if mR = mLandT = mR

12

if mR = −mLandT ∈ mR,mL

(A10)

A4

and off-the-path beliefs given by any posterior distribution with expected value 0.

Proof. First consider the receiver’s beliefs and actions. Given these message strategies, both

senders only send the same message only when it is truthful. Therefore, the probability

that their message equals T is 1, and the optimal choice is c = T . Now consider messages

such that mR = −mL. In this case the receiver’s on-the-path beliefs are proportional to the

probabilities of two events: (a) T = mR and SL < −12mR, and (b) T = mL and SR > −1

2mL.

Based on the prior distributions of the shifts, the probability of event (a) is 150

(50 − 12mR),

and the probability of event (a) is 150

(50 + 12mL). Since mR = −mL, these probabilities are

equivalent, and so the posterior probability that T = mR is 12, and similarly so for T = mL.

Given that these values are symmetric around 0, the optimal choice for c = 0. Finally,

the off-the-path beliefs are selected so that c = 0 for any pairs of messages that are not of

identical magnitude.

Turning now to sender behavior, each sender effectively chooses between c = T and

c = 0. For example, straightforward substitution and algebra show that u(c = 0, T, SR) >

u(c = T, T, SR) if and only if T ∈ [0,−2SL]. This identifies the interval for whichmL(T, SL) =

−T , and a symmetrical argument identifies T ∈ [−2SR, 0] as the interval for whichmR(T, SR) =

−T . Because off-the-path beliefs result in the same choice for c as sending mi = −T , there

are no profitable deviations.

A2 Additional Analysis of Experimental Data

The analysis of limited strategic sophistication reported in the main text does not account

for the censoring of messages due to the boundaries of the message space. Here, we show

that our findings and interpretations are robust to alternative specifictions of Tables 8 and

10 using Tobit regressions that allows for censoring, as we find that the values of the Tobit

coefficient estimates are very similar to the OLS estimates reported in the text.

A5

Table A1: Tobit version of Table 8 (senders’ messages and level-k)

All Levels Excluding Level 0

(1) (2) (3) (4)

Target 0.91** 0.99** 0.90** 1.00**

(0.02) (0.03) (0.02) (0.03)

Shift 0.93** 0.85** 1.08** 0.97**

(0.08) (0.12) (0.08) (0.13)

Left Sender -40.96** -19.34** -41.83** -21.03**

(3.87) (4.39) (3.60) (5.37)

Right Sender 34.77** 19.88** 44.91** 22.82**

(4.95) (5.35) (3.61) (5.70)

Target × Round -0.01 -0.01**

(0.00) (0.00)

Shift × Round 0.01 0.01

(0.01) (0.01)

Left Sender × Round -1.60** -1.57**

(0.37) (0.40)

Right Sender × Round 1.10* 1.64**

(0.43) (0.39)

σ 36.44** 33.93** 34.03** 30.14**

(2.69) (3.04) (3.26) (0.59)

Log likelihood -6033.91 -5944.95 -4467.80 -4359.20

N 1280 1280 976 976

* p < 0.05, ** p < 0.01

Tobit regression with standard errors clustered by subject (in parentheses)

A6

Table A2: Tobit version of Table 10 (senders’ messages and experiential best responses)

(1) (2) (3) (4)

Target 0.93** 1.00** 0.92** 1.00**

(0.02) (0.03) (0.02) (0.04)

Shift 0.93** 0.93** 0.91** 0.94**

(0.09) (0.14) (0.09) (0.14)

Exaggeration (last 5) -0.62** -0.34**

(0.07) (0.13)

Exaggeration (all) -0.74** -0.24

(0.15) (0.19)

Left Sender -10.04 -10.22 -9.12 -15.02

(5.48) (6.74) (8.14) (7.80)

Right Sender 0.02 11.07 0.05 17.02*

(5.88) (7.22) (7.17) (7.79)

Target × Round -0.01* -0.01*

(0.00) (0.00)

Shift × Round -0.00 -0.00

(0.01) (0.01)

Exag. (last 5) × Round -0.01

(0.01)

Exag. (all) × Round -0.03*

(0.01)

Left Sender × Round -0.35 0.21

(0.55) (0.69)

Right Sender × Round -0.40 -1.03

(0.77) (0.88)

σ 33.03** 32.38** 34.04** 33.00**

(3.21) (0.86) (3.25) (3.29)

Log likelihood -5681.62 -5662.61 -5719.06 -5682.56

N 1234 1234 1234 1234

* p < 0.05, ** p < 0.01

Tobit regression with standard errors (in parentheses) clustered by subject

A7

In our analysis of senders’ behavior over time (models with interaction terms in Table

10), we refer to the responsiveness of senders’ messages to the game parameters (i.e., target,

shift) “by the last period of play.” To clarify what we mean by this, we can write model

with time-varying coefficients, where the coefficients are a function of the period t:

Messageit = α + (β1 + γ1t)Target + (β2 + γ2t)Shift + (β3 + γ3t)Experience

+(β4 + γ4t)Left Sender + (β5 + γ5t)Right Sender + εit

Our hypotheses about the last period of play concern the values of the coefficients where

t = 24 or t = 32. That is, our hypotheses are about the linear combinations of the parameters

for tL ∈ {24, 32}:

H1 : β1 + γ1tL = 1

H2 : β2 + γ2tL = 2

H3 : β3 + γ3tL = −1

H4 : β4 + γ4tL = 0

H5 : β5 + γ5tL = 0

The estimated parameter combinations are presented in Table A3 (for exaggeration based on

the last 5 periods) and Table A4 (for exaggeration based on all previous periods). The first

two sets of estimates are based on the parameter estimates reported in Table 10 (pooling

across sessions) for period tL = 24 and for period tL = 32. The last two columns report

estimates separately by the number of periods in a session (i.e., the estimates for tL = 24

are based on regression using only sessions with 24 periods, and likewise for tL = 32). The

results do not vary much in terms of whether we use tL = 24 or tL = 32 as the last period or

whether we estimate separate regressions for sessions with different numbers of periods. By

the last period of play, the coefficient for Target remains less than 1 and the coefficient for

A8

Shift remains less than 2—this is consistent with underexaggeration or underresponsiveness

to these variables. The coefficient for exaggeration is statistically distinguishable from 0

regardless of which measure of past exaggeration we use, and the confidence interval tends

to exlude the hypothesized value of −1 when we use the most recent 5 periods, while it

includes −1 when we use the entire history of play. The estimated intercepts vary by sender

and by the measure of past opponent exaggeration. When we use the most recent 5 periods,

the confidence intervals for the left sender intercept sometimes exludes 0. When we use all

previous periods, the confidence intervals include 0 for both left and right senders.

A9

Tab

leA

3:C

oeffi

cien

tsin

last

per

iod

ofpla

y(e

xp

erie

nce

last

5p

erio

ds)

All

Ses

sion

s24

Per

iod

Ses

sion

s32

Per

iod

Ses

sion

s

Var

iable

Par

amet

ers

t L=

24t L

=32

t L=

24t L

=32

Tar

get

β1

+γ1t L

0.82

(0.0

4)0.

76(0

.06)

0.90

(0.0

3)0.

72(0

.08)

[0.7

4,0.

89]

[0.6

3,0.

88]

[0.8

3,0.

96]

[0.5

5,0.

88]

Shif

tβ2

+γ2t L

0.82

(0.1

3)0.

78(0

.20)

0.79

(0.0

8)0.

79(0

.26)

[0.5

5,1.

09]

[0.3

7,1.

18]

[0.6

2,0.

96]

[0.2

4,1.

33]

Exag

gera

tion

β3

+γ3t L

−0.

53(0

.13)

−0.

60(0

.19)

−0.

51(0

.16)

−0.

62(0

.22)

[−0.

79,−

0.28

][−

0.98,−

0.22

][−

0.84,−

0.18

][−

1.07,−

0.17

]

Lef

tSen

der

β4

+γ4t L

−21.0

0(9

.29)

−24.6

7(1

2.85

)−

33.7

5(1

0.94

)−

17.6

7(1

4.74

)

[−39.7

2,−

2.29

][−

50.5

4,1.

20]

[−56.3

8,−

11.1

2][−

48.3

2,12.9

8]

Rig

ht

Sen

der

β5

+γ5t L

7.88

(11.

86)

7.20

(16.

80)

21.7

8(1

1.62

)0.

55(1

9.85

)

[−16.0

2,31.7

8][−

26.6

3,41.0

3][−

2.27,4

5.8]

[−40.7

4,41.8

3]

Coeffi

cien

t(s

tan

dar

der

ror)

,[9

5%co

nfi

den

cein

terv

al]

A10

Tab

leA

4:C

oeffi

cien

tsin

last

per

iod

ofpla

y(e

xp

erie

nce

all

pre

vio

us

per

iods)

All

Ses

sion

s24

Per

iod

Ses

sion

s32

Per

iod

Ses

sion

s

Var

iable

Par

amet

ers

t L=

24t L

=32

t L=

24t L

=32

Tar

get

β1

+γ1t L

0.80

(0.0

4)0.

74(0

.06)

0.89

(0.0

3)0.

69(0

.08)

[0.7

3,0.

88]

[0.6

2,0.

85]

[0.8

3,0.

96]

[0.5

3,0.

85]

Shif

tβ2

+γ2t L

0.77

(0.1

3)0.

71(0

.19)

0.79

(0.0

9)0.

65(0

.26)

[0.5

1,1.

03]

[0.3

2,1.

10]

[0.6

1,0.

98]

[0.1

2,1.

19]

Exag

gera

tion

β3

+γ3t L

−0.

79(0

.23)

−0.

98(0

.32)

−0.

84(0

.37)

−1.

22(0

.34)

[−1.

26,−

0.32

][−

1.61,−

0.35

][−

1.62,−

0.07

][−

1.92,−

0.51

]

Lef

tSen

der

β4

+γ4t L

−14.1

5(1

3.94

)−

13.8

9(1

8.64

)−

22.4

6(1

9.55

)4.

05(1

8.57

)

[−42.2

2,13.9

1][−

51.4

3,23.6

5][−

62.9

0,17.9

8][−

34.5

8,42.6

7]

Rig

ht

Sen

der

β5

+γ5t L

0.78

(15.

37)

−4.

27(2

1.41

)12

.58

(21.

00)

−24.9

6(2

5.38

)

[−30.1

7,31.7

2][−

47.3

9,38.8

5][−

30.8

6,56.0

1][−

77.7

5,27.8

2]

Coeffi

cien

t(s

tan

dar

der

ror)

,[9

5%co

nfi

den

cein

terv

al]

A11

1

Instructions General Information This is an experiment in communication. The University of Pittsburgh has provided funds for this research. If you follow the instructions closely and make appropriate decisions, you may make a considerable amount of money. In addition to the $7 participation payment, these earnings will be paid to you, in cash, at the end of the experiment. During the experiment, all earnings will be denominated in points, which will be converted to cash at the rate of $1 per 150 points. The exact amount you receive will be determined during the experiment and will depend on your decisions and the decisions of others. You will be paid your earnings privately, meaning that no other participant will find out how much you earn. Also, each participant has a printed copy of these instructions. You may refer to your printed instructions at any time during the experiment. If you have any questions during the experiment, please raise your hand and wait for an experimenter to come to you. Please do not talk, exclaim, or try to communicate with other participants during the experiment. Also, please ensure that your cell phones are turned off and put away for the duration of the experiment. Participants intentionally violating the rules will be asked to leave the experiment and may not be paid. Roles, Rounds, and Matching Each participant will be assigned to one of three roles: A, B, or C. Your role will be assigned before the first round and will remain fixed throughout the experiment. In this experiment you will make decisions in a series of rounds, and there are a total of 32 rounds. Each round is a separate decision task. Before every round, you will be randomly matched with two other participants. In every group of three participants there will be one player in each role (one A player, one B player, and one C player). You will not know the identity of the other participants you are matched with in any round, and your earnings for each round depend only on your action in that round and the actions of the participants you are matched with in that round.

2

Targets At the beginning of every round, the computer will randomly select a target for each player. Player C’s target will be a number between -100 and 100. Each number is equally likely to be C’s target. Player A’s target will be less than Player C’s target. The difference between A’s target and C’s target will be some amount between 0 and 50 units. Each amount is equally likely, and the exact amount will be selected at random in every round. Player B’s target will be greater than Player C’s target by some amount between 0 and 50 units and each amount of difference is equally likely. For example, suppose that the computer selects 25 as Player C’s target. For Player A’s target, the computer will randomly select a number from -25 to 25. Likewise, Player B’s target will be a randomly selected number from 25 to 75. It is important to note that Player A’s target and Player B’s target are randomly selected by the computer independently. That is, the value of Player A’s target does not affect the value of Player B’s target and vice versa. Similarly, the computer will randomly determine each player’s target at the beginning of the round so that the targets in one round are selected independently of the targets in another round. Sequence of Decisions The sequence of decisions in every round is as follows:

1. Players A and B each find out the value of Player C’s target and the value of their own target. (Note that Player A does not see Player B’s target, nor does Player B see Player A’s target.) Independently and simultaneously, Players A and B each select a message to send to Player C.

2. Player C sees the messages sent by Player A and Player B. Player C then chooses an action (any number between -150 and 150). (Note that Player C sees both messages but none of the targets.)

3

Payoffs Each player’s payoff depends only on how close Player C’s action is to his or her own target. More specifically, a player earns 100 points if the action is equal to his or her own target and 1 point less for each unit of difference between the action and the target. This is described by the following formula (where the straight lines indicate absolute value):

Player’s Payoff = 100 – |Player’s Target – C’s Action| Note that the messages sent by Player A and Player B are not part of the payoff formula. To illustrate, consider a few examples. Suppose you are Player A, your target is 10 and Player C chooses the action 40. The difference between your target and the action is 30, so your payoff would be 70. If Player C’s target is 25, then the difference between C’s target and the action is 15, so C’s payoff would be 85. Now suppose instead that Player C chooses the action -40. If Player A’s target is 20, then the difference between A’s target and the action is 60 and A’s payoff would be 40. If Player B’s target is 80, then the difference between B’s target and the action is 120, so B’s payoff would be -20. If Player C’s target is 45, then the difference between C’s target and the action is 85, so C’s payoff would be 15. (Note that it is possible for payoffs to be negative.) Sample Screens We will now see what the screens look like for each type of player during the experiment. This is the screen that will be seen only by Player A. There is a brief set of instructions in the upper left-hand corner. A description of the payoff formula is also shown on the left side of the screen. The top of the screen shows several values: C’s actual target, A’s target (which is labeled “your target”), and the range of possible targets for B. The targets are indicated graphically in the figure in the middle of the screen, which also indicates the possible range of values for each player’s target. Player A chooses a message by dragging the white tab to any position along the horizontal black line. After moving the tab, it will indicate the value of the selected message. Note that there is also a section on the left marked “payoff calculator.” Click on the “Show” button to reveal an orange tab that can be used to calculate hypothetical payoffs for each possible action that Player C can take. If you move the orange tab to different positions, the bold text at the bottom of the screen changes to indicate what Player A’s payoff and player C’s payoff would be. Note that the payoff calculator does not show B’s hypothetical payoff because you do not know the value of B’s target. Note also that you can hide the payoff calculator by clicking on the “hide” button.

4

When Player A is ready to send the message, he or she will click on the “Send Message” button in the lower right-hand corner of the screen. Feel free to move the message tab and try out the payoff calculator. When you are ready to continue, click on the “Send Message” button. This is the screen that only Player B will see. B players see this screen at the same time that the A players see their screens. It is pretty much the same as Player A’s screen except that B’s target is known while A’s is not. When you are done looking at this screen, click on the “Send Message” button to continue. After Player A and Player B send their messages, Player C will see this screen. In the upper-left corner there is again a brief set of instructions. The top of the screen shows the numerical values of the messages. The messages are also indicated graphically in the middle of the screen. To select an action, Player C moves the red tab to the desired location. As with the other tabs, it shows the numerical value of its location after it is moved. Note that Player C does not have a payoff calculator because the actual values of the targets are not known. Try moving the “Action” tab and the click on “Choose Action” button when you are ready to continue. At the end of every round, you will see this screen, which shows you the results from the round—including the actual targets of every player, both messages, the action chosen by Player C, and the payoffs earned by every player in your group. At the bottom of the screen, it will show the results of every previous round that you played.

5

QUIZ INSTRUCTIONS. To check your understanding of the decision tasks, please answer the questions below as best you can. Note that your quiz answers do not affect your earnings, and you may refer to your printed instructions as often as you like. When you are finished, feedback about the correct answers will be shown on the screen. You must attempt to answer all of the questions. If you have any further questions at this time, please raise your hand and the experimenter will come to you.

1. C’s target can be any number from: [0 to 10, 0 to 100, -100 to 100, -150 to 150] 2. If C’s target is -40, then A’s target can be any number from: [-100 to 0, -90 to -40, -40 to 10, 40 to 90] 3. If C’s target is 30, then B’s target can be any number from: [-20 to 30, 0 to 50, 30 to 80, 50 to 100] 4. If you are Player C, your target is 85, and you choose the action 45, how many points will you receive? [15, 40, 60, 85] 5. If you are Player A, your target is -70, and Player C chooses the action 50, how many points will you receive? [-70, -20, 30, 50] 6. Suppose that you are Player B, your target is 10 and Player C’s target is -15. If you send the message 10 and Player C chooses the action 0, how many points will you receive? [10, 15, 85, 90] 7. Suppose that you are Player C. Player A sent you the message -50 while Player B sent you the message 50. If you choose the action 30 and your actual target was 50, how many points will you receive? [20, 30, 70, 80] 8. In every round, will you be matched with same participants? [Yes, No]