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Competition, Preference Uncertainty, and Jamming:A Strategic Communication Experiment
William Minozzi∗ Jonathan Woon†
March 10, 2014
Abstract
We conduct a game-theoretic laboratory experiment to investigate the nature ofinformation transmission in a complex communication environment featuring compe-tition and information asymmetry. Two senders have private information about theirpreferences and simultaneously send messages to a receiver in a one-dimensional spacewith a large number of states, actions, and messages. We find that equilibrium predic-tions fare poorly and that senders overcommunicate by consistently exaggerating theirmessages. Our analysis suggests that exaggeration can only be partially explained bybounded rationality models of iterated reasoning or belief learning. Instead, behavioris consistent with a naive form of exaggeration in which senders know they must ex-aggerate, but they do so in an understated way that is less responsive to their privateinformation or to opponents’ past behavior than would be fully optimal.
Keywords: Sender-receiver games; Strategic information transmission; Laboratoryexperiment; Bounded rationality; Behavioral models
JEL Classifications: C72, D82, D83
∗Assistant Professor, Department of Political Science, 2137 Derby Hall, Ohio State University, Columbus,OH 43210 Phone: 1-614-247-7017, Email: [email protected]†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: [email protected]
1 Introduction
Throughout daily life, people are confronted with conflicting messages from informed sources.
In politics, for example, candidates offer voters competing visions for national policy (Banks
1990), lobbyists and legislators construct conflicting arguments for and against legislation
(Austen-Smith 1990, 1993), and courts rely on adversarial advocates to inform their decisions
(Dewatripont and Tirole 1999). Although these situations differ substantially in their details,
they share several common elements. Each features well-informed, interested actors whose
preferences remain partially private. Those actors send rival messages to less-informed deci-
sionmakers who cannot verify the content of those messages. The set of messages is limited,
sometimes explicitly by germaneness rules, to a single salient dimension. Yet within that
dimension of disagreement, the number of potential messages is very large. We refer to such
settings as complex communication environments. In this paper, we describe a laboratory
experiment on the nature of information transmission in this context.
Despite the ubiquity of complex communication environments, our understanding of
them is quite limited. Consider the contradictory expectations one might have for infor-
mation revelation in such settings. Preference differences limit the information that can
be conveyed in strategic environments (Crawford and Sobel 1982). This problem is exacer-
bated if the decisionmaker is uncertain about information-providers’ preferences (Sobel 1985;
Lupia and McCubbins 1998). Yet competition between information providers may provide
mechanisms for the truth to emerge. Indeed, the notion of the “marketplace of ideas” is
often invoked to justify the importance of protecting free speech (Mill 1859). Similarly,
competition within heterogeneous legislative committees is thought to yield more credible
information to the entire chamber (Gilligan and Krehbiel 1989), competition among lobbyists
is thought to lead to better decisions by legislators (Austen-Smith and Wright 1992), and
competition among elites and the news media is thought to yield a more fully informed public
(Page and Shapiro 1992; Gentzkow and Shapiro 2008). Yet mixing competition and prefer-
ence uncertainty only provides further opportunities for strategic obfuscation (Milgrom and
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Roberts 1986; Minozzi 2011). And empirically, experimental treatments of sender-receiver
games reveal evidence of overcommunication, in which senders reveal more information than
equilibrium analysis predicts (Blume et al. 1998; Cai and Wang 2006).
To the best of our knowledge, ours is the first experiment to examine an environ-
ment that incorporates competition and preference uncertainty in a unidiminesional state
space with nearly continuous sets of messages and actions. Studies in the extant experimen-
tal literature on communication tend to focus on simple environments that feature either
a single sender, commonly known preferences, a small number of messages or actions, or
a combination of only a few these features (Blume et al. 1998, 2001; Cai and Wang 2006;
Dickhaut, McCabe, and Mukherji 1995; Gneezy 2005; Hurkens and Kartik 2009; Peeters,
Vorsatz, and Walzl 2008; Sanchez-Pages and Vorsatz 2007). A few studies investigate en-
vironments with two senders, but they are rare. Lai, Lim, and Wang (2011) and Vespa
and Wilson (2012) 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 a small set of discrete states and messages, our setting allows for much more var-
ied communication strategies. Our setting is also spatial, meaning that our findings have
straightforward applications to many fundamental formal models of politics.
Despite the complexity of the communicative environment, we find that receivers
come remarkably close to learning the hidden state information. Consistent with the find-
ings of single-sender experiments, senders overcommunicate in the sense that their messages
reveal more information than any of several equilibrium expectations would predict. But we
also find that senders persistently exaggerate in the direction of their biases, and receivers
can guess the state on average by employing a simple split-the-difference strategy. The end
result seems to be a kind of resurgence of the marketplace of ideas in which competition
1 Boudreau and McCubbins (2008) conduct an experiment with competition and a version of preferenceuncertainty, but their experiment involves a decisionmaker who solves SAT math problems with the helpof “experts.” Their setup departs from standard sender-receiver games because their receivers have het-erogeneous (unobserved and uncontrolled) beliefs about the true state.
2
between senders allows receivers to learn (or approximate) the truth in complex communi-
cation environments. In the long-run, however, the benefits of competition diminish: less
information is transmitted as senders exaggerate more and more over time.
Our analysis suggests that notions of strategic bounded rationality provide only in-
complete explanations for the patterns of exaggeration we observe. Models of iterated reason-
ing similar to those used in previous analyses to explain overcommunication can rationalize
exaggeration that is inconsistent with equilibrium analysis and potentially explains differ-
ences between subjects (Cai and Wang 2006), but we find that senders consistently under-
exaggerate relative to the predictions of this model. Although we can classify subjects’
behavior as being consistent with some levels of sophistication, their degree of exaggeration
is less responsive to senders’ biases than any possible best response function. In contrast to
previous experiments, we also find clear learning effects: senders learn to exaggerate more
and more over time. However, such adaptive behavior can only partially be explained by a
belief learning model, which still fails to account for the under-responsiveness of messages
to the level of preference divergence between senders and the receiver. We are left to con-
clude that subjects engage in a particularly naive form of exaggeration. Despite this naive
exaggeration, receivers remain remarkably capable of discerning the truth.
2 Theoretical Model and Equilibrium Predictions
Consider a simple political environment in which there can be communicative competition
and incomplete information about preferences: one with two senders and one receiver.2 At
the outset, both senders observe a state of the world, which we call the target T . The target
functions as the objective “truth” in the game. In our experiment, T is uniformly distributed
over the integers from −100 to 100. Each sender i also privately observes his shift Si, which
represents the direction and degree of preference divergence between i and the receiver. We
2 For 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.
3
designate one sender as the left sender, with shift SL uniformly distributed over the integers
from −50 to 0 and one sender as the right sender, with shift SR uniformly distributed over
the integers from 0 to 50. The distributions of T , SL, and SR are common knowledge. The
senders then simultaneously select messages mi to send to the receiver, who then chooses an
action c.
The receiver prefers c to be as close as possible to T while each sender i prefers
that c be as close as possible to T + Si. More specifically, the receiver’s payoff function is
UR = 100− |c− T | and the senders’ payoff function is USi= 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. Importantly, the receiver knows that the senders are opposed but is uncertain
whose ideal point is closer to hers.
Cheap talk and signaling games typically have many equilibria. The purpose of our
equilibrium analysis is therefore not to make unique predictions, but to provide a framework
that organizes our experimental analysis and that guides our expectations about the kinds
of behavior that may be consistent with fully rational, strategic play (Schotter 2006). While
previous studies have typically focused on the most informative equilibria (e.g. Cai and Wang
2006), we focus on three classes of equilibria that vary in their informativeness. We believe
these equilibria may be focal because of their relatively simple structure.
At one extreme, there is always a babbling or uninformative pooling equilibrium in
which the senders’ messages are unrelated to the target. In this equilbrium, no information is
transmitted. The receiver can learn nothing about the target; consequently, she ignores both
messages and chooses an action to maximize her ex ante expected utility. This reasoning
leads to the following predictions.
Babbling Predictions. Senders’ messages will be unrelated to targets, and the averagemessage will be E[mi] = 0. Receivers’ actions will be unrelated to messages and targets,and the average action will be E[c] = 0.
4
At the other extreme, there can exist fully informative equilibria (e.g., Krishna and
Morgan 2001). However, complete information transmission can only be supported by re-
ceiver strategies that are sufficiently severe off the path of play as to deter senders from
deviating from truthtelling. As Battaglini (2002) shows, however, such fully-revealing equi-
libria rely on implausible out-of-equilibrium beliefs: if out-of-equilibrium messages reveal too
much about the true state, then receivers actions will not be sufficiently severe to prevent
deviations and fully-revealing equilibria do not survive in two-sender games.3 We therefore
turn our attention to equilibria in which information transmission is only partial.
In the spirit of Crawford and Sobel (1982), one such equilibrium has a partition
structure. In a partition equilibrium, senders limit the information conveyed by revealing
only categorical information (that the target lies within a subset of the target space). The
simplest possibility is one in which each sender only reveals whether the target is high or
low. However, competition combined with preference uncertainty implies that each sender
would prefer to partition the target space differently. More specifically, the left sender
would prefer to claim that the target is low, unless his own ideal point is above K, for
some cutpoint K > 0; thus, if T + SL > K, the left sender randomizes uniformly over the
interval [K, 100] and otherwise randomizes uniformly over [−100, K]. Similarly, the right
sender would prefer to claim that the target is high, unless his ideal point is below −K; the
right sender randomizes over [−100,−K] if T + SR < −K and otherwise randomizes over
[−K, 100]. Figure 1 illustrates the equilibrium message strategies for a left sender with shift
SL = −25. The dotted line depicts the average message sent in a partition equilibrium while
the dashed line depicts the average message in a babbling equilibrium.
These strategies imply three possible equilibrium actions. The receiver chooses a high
action c = 2K if both senders reveal that the state is high, a low action c = −2K if both
3 Full revelation requires that receivers’ strategies essentially “punish” senders for not telling the truth offthe path-of-play by choosing an outcome that is far worse for the sender than the c = T , but such strategiesrequire beliefs to substantially diverge from the true state. This divergence of beliefs is limited when at leastone message reveals enough about the true state, thus limiting the severity of the punishment outcome,which will then be insufficient to prevent some senders from deviating from full revelation strategies.
5
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
senders reveal the state is low, and an intermediate action c = 0 if the senders disagree (the
left sender claims a low state and the right sender claims a high state).4 For the parameters
of our experiment, K∗ = 506
(7−√
5)≈ 39.7.
Partition Predictions. Within the intervals of their respective partitions, senders’ mes-sages will be unrelated to targets. If T + SL > K∗, left senders’ average message willbe E[mL] = K∗+100
2; otherwise, it will be E[mL] = −100+K∗
2. If T + SR < −K∗, right
senders’ average message will be E[mR] = −100−K∗
2; otherwise, it will be E[mR] =
−K∗+1002
.
4 Following the standard logic of partition equilibria, a left sender with a target of T + SL = −K isindifferent between the low outcome −2K and the middle outcome 0 while right sender with an ideal pointof T + SR = K is indifferent between the high outcome 2K and the middle outcome 0. The receiver’sposterior beliefs are distributed uniformly over the trapezoid defined by T ∈ [K−SL, 100] and SL ∈ [−50, 0]and the equilibrium value of K is the solution to∫ 0
−50
∫ 100
K−SL
T
(1
3750− 50K
)dTdSL = 2K
6
The third class of equilibrium we examine is the jamming equilibrium described by
Minozzi (2011). In this equilibrium, senders sometimes tell the truth but also have incentives
to jam the truthful messages from the opposing sender. If the messages agree, the receiver
infers the true target T and responds by choosing c = T . However, if the messages disagree,
the receiver infers that one sender must have jammed, but remains uncertain about which
sender lied and about the location of the true target. In that case, the receiver’s equilibrium
response is to choose a “default” action. We focus on the equilibrium in which this default
outcome is c = 0, which is the optimal action based only on her prior beliefs.5
Senders’ message strategies specify both when and how to lie. Given the receiver’s
strategy just specified, each sender recognizes that the only possible equilibrium path actions
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 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 lie in such a way to ensure that when the messages
disagree, the receiver will in fact 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
5 Technically, 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 occurson the path of play and Bayes’ Rule always applies.
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will always send a truthful message mi = T . Thus, 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 as the true target and the more extreme the target, the more extreme the
jamming message. More specifically, the jamming message function is mJ(T ) = −T (in the
equilibrium of interest where the default is c = 0).6 The solid line in Figure 1 illustrates the
message strategy for a left sender with SL = −25.
Jamming Predictions. Senders’ messages will reveal the target, mi = T , unless the targetis in the senders’ jamming region. The left sender’s jamming region is [0,−2SL],and the right sender’s jamming region is [−2SR, 0]. When the target is in a sender’sjamming region, the message will be countervailing, mi = −T . The receiver will choosean action equal to the target c = T when the senders’ messages agree; otherwise, thereceiver chooses the default action c = 0.
To summarize, in all three of the equilibria we consider, strategic incentives inhibit
the information senders transmit to receivers, consistent with the basic insight of Crawford
and Sobel (1982). However, the equilibria 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. In a babbling equilibrium, no information is transmitted because all
players have mutual expectations that messages will be uninformative and therefore ignored.
6 To see how the receiver remains uncertain, suppose the true state is θ so that if the left sender jams, thejamming message is −θ and the message pair is (−θ, θ). But this is the same message pair that wouldhave been observed if the true state were instead −θ and the right sender sends the jamming message−(−θ) = θ. The receiver’s marginal posterior beliefs are such that the true state is equally like to be θ asit is to be −θ. As a consequence, the optimal action is the midpoint c = 0, which is precisely the defaultaction. For other equilibria with a different default outcome c = D, message pairs that disagree will induceposterior beliefs for which the expected value of the target is equal to the default. That is, the jammingfunction is constructed so that a message pair m = (θ, θ′) implies the receiver’s posterior belief is such thatE[T |m] = Pr(θ|m)θ + Pr(θ′|m)θ′ = D. See Minozzi (2011) for the full characterization that includes theasymmetric case.
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In a partition equilibrium, senders are willing to reveal coarse but ambiguous information
about what they know. This results in some probability the receiver will learn that the
target is extreme (very high or very low) but will otherwise not learn its precise location.
Interestingly, the jamming equilibrium involves both greater honesty and greater deception.
It involves greater honesty because senders will often reveal the true target exactly. It
involves greater deception because when senders do not tell the truth, they instead make
countervailing claims that become further from the truth when targets are more extreme.
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. 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 and 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 one of the
roles in the game: A (left sender), B (right sender), or C (receiver). The instructions only
9
referred to the roles as “A,” “B,” or “C” and made no reference to “senders,” “receivers,”
“left,” or “right.” In our presentation and discussion, however, we continue to use these
terms. Subjects proceeded to play between 24 and 32 rounds of the game, with fixed roles
throughout the session.
At the beginning of every round, subjects were randomly matched into groups of
three, with one subject in each role in each group. Groups were selected 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 ID numbers of the other subjects in their group.
The targets T and shifts SL and SR were then drawn independently for each group. In all
conditions of the experiment, T was drawn uniformly from the integers between −100 and
100. The left sender’s shift was drawn uniformly from the integers between −50 and 0, and
the right sender’s shift 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.”7 Our use of nearly continuous distributions is in contrast with previous
experiments on cheap talk games that typically involve a small state and action space: for
example 3 states in Blume et al. (2001), 4 in Dickhaut, McCabe, and Mukherji (1995) and 5
states in Cai and Wang (2006). This innovation accurately conveys the notion of the spatial
model to subjects and affords us the opportunity to conduct a more detailed investigation
of communication strategies than previous studies.
The experimental interface we used presents information to subjects textually as well
as graphically as shown in Figure 2. The graphical display intuitively conveys the notion of
spatial distance inherent in the utility functions. We reasoned that this would allow subjects
to focus their cognitive resources on thinking strategically rather than on computing payoffs.
Although the instructions describe the set of targets and shifts as integers, our visual display
reinforces the notion that the distributions are to be treated as continuous and spatial.
7 In our presentation, we continue to refer to “targets” and “shifts.” When we do so, the “target” isunderstood to be C’s target.
10
Figure 2: Screenshot of sender’s graphical interface
In every round, each sender simultaneously observed the receiver’s target and his own
target (but not the other sender’s target), and then chose a message. As shown in Figure
2, possible messages and actions are displayed on a horizontal axis. To send a message,
senders use the mouse to drag a slider along this axis to a position that corresponds to the
desired message (any position between −150 and 150).8 The interface also displays the range
of possible targets 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 features a payoff calculator
(manipulated via a separate slider) that shows the sender’s and receiver’s payoffs for each
8 Although we chose to allow messages outside the target space so that senders could choose messagescorresponding to their own targets, we did not expect them to do so for two reasons. First, if subjects playaccording to equilibrium predictions, allowing such messages is irrelevant. Second, when subjects sendmessages outside of the interval [−100, 100], they clearly indicate to receivers that they are lying aboutthe target and their messages should be discounted or completely ignored.
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possible action the receiver might choose. The receiver observed messages simultaneously
after both senders had finished, and the interface displayed this information both graphically
and textually. The receiver then dragged a slider to select an action (any position between
−150 and 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)|.
At the end of the experimental session, total points were converted to cash at the
rate of $1 for every 150 points. Subjects were paid the sum of their earnings plus a $7
participation payment. We conducted four sessions of the experiment (69 subjects). Each
session involved between 12 and 18 subjects (4 to 6 groups), and each subject participated
in only one session.
4 Evaluation of Equilibrium Predictions
In our presentation of the results, we first summarize each main result and then present the
analysis that supports our findings.
Result 1. We find a substantial correlation between the receiver’s actions and targets thatis inconsistent with the Babbling Predictions and slightly more consistent with the Jam-ming Predictions than the Partition Predictions. The average payoff and correlationbetween target and action are closer to the Jamming Prediction than the other predic-tions, and the mean squared prediction error is also smallest for the Jamming Predic-tion. However, we also find that the correlation between actions and targets persistseven under conditions that the equilibrium theories predict it should diminish.
12
The first two rows of Table 1 present aggregate measures of information transmission
and compares the observed amount to the amount predicted by each of the equilibria we
discussed in the theoretical section.9 On average, the receiver’s payoff is 78.65, which is
much higher than 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.
Table 1: Information transmission
PredictedObserved
Babbling Partition Jamming
Receiver Average Payoff 49.75 73.91 83.00 78.65
Target-Action Correlation 0.000 0.820 0.864 0.867√
Mean squared prediction error 51.48 34.88 32.5 –
In addition to these aggregate measures, we also quantify how well each of the the-
oretical equilibria predict the receiver’s actions by computing the mean squared prediction
error. These results, found in the third row of Table 1, show that the jamming equilibrium
produces the smallest prediction errors, at a 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 correspon-
dence 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
9 Note that our numerical predictions are based on discrete distributions since the values in the experimentare integers.
13
Figure 3: Receivers’ actions
-100
-50
050
100
Actio
n
-100 -50 0 50 100Target
Full Revelation Fitted Line
slope is also significantly less than 1, which suggests that receivers trust the senders’ mes-
sages 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 may be some 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
14
magnitude than expected. Thus, even when we allow for the relationship between actions
and targets to vary in a way that the partition equilibrium predicts, we find no evidence
that actions are unrelated 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 while the main coefficient should be 1, 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 hypotheses 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
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. Note that this provides evidence against any fully-
revealing equilibrium because in any such equilibrium, actions will respond only to the target
and will not depend on the senders’ shifts. 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) and
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.
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Table 2: Regression analysis of receiver actions
Target 0.75 0.66 0.81 0.78 0.92
(0.02) (0.03) (0.02) (0.02) (0.04)
Target × Left High -0.16
(0.16)
Target × Right Low -0.05
(0.17)
Target × Jam -0.22 -0.11 -0.01
(0.04) (0.04) (0.05)
Left High 23.08
(12.63)
Right Low -14.81
(14.05)
Left Shift 0.34 0.37
(0.07) (0.07)
Right Shift 0.33 0.39
(0.07) (0.07)
Target × Left Shift 0.004
(0.001)
Target × Right Shift -0.002
(0.001)
Intercept -1.97 -2.10 -2.11 -1.75 -2.31
(1.01) (1.27) (0.99) (2.49) (2.47)
R2 0.75 0.76 0.76 0.78 0.79
N 640 640 640 640 640
16
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 errors. Left senders’ messages are consistentlyexaggerated to the left of the equilibrium predictions, while right senders’ messages areconsistently exaggerated to the right. Regression analysis further shows that exaggera-tion is increasing in the senders’ degree of bias.
Table 3 presents both the (signed) mean prediction errors and the root mean squared
prediction errors for each equilibrium prediction by type of sender. We retain the sign of the
error for senders so that we can examine their direction; this 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 consistent
with what we found for the receivers’ actions: the jamming prediction outperforms the
partition prediction, which outperforms the babbling equilibrium. 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 appear to predict 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.
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
17
The patterns in the data suggest that senders exaggerate their messages in the di-
rection 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-
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.10 Second, messages
are rarely, if ever, truthful. In fact, only 12 out of 1,280 messages are equal to the target.11
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 above the target. Furthermore, senders also surprisingly select messages out-
side of the target space, many of which are at the extreme boundaries of m = −150 and
m = 150.
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
10This visual inspection is supported by a series of piecewise linear models that partition the set of targetsaccording to the partition equilibrium.
11We consider a “truthful” message to be one that matches the true target. Although the exact meaning ofmessages is endogenous to players’ choices and their perceptions of the game, it seems natural to interpreta numerical message m as a report that means “the target is m” or “you should pick m.”
18
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]
19
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. Thus, it appears that
exaggeration is increasing in the sender’s degree of bias.
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.12 While the jamming theory
implies that the coefficient for the interaction should be −2, this turns out to be the case
for only one set of shift values: for SR ∈ [10, 20] the 95 percent confidence interval for the
interaction estimate is [−2.04,−0.003]. 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
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.
Although the amount of overall information transmission in our complex communi-
cation environment matches some features of the predictions of equilibrium theory, we can
thoroughly reject the possibility that this is because senders strategically limit the informa-
tion they reveal. Consistent with previous experiments on cheap-talk games, we find sub-
stantial evidence of overcommunication: messages are highly correlated with the underlying
state of the world. In our experiment, this overcommunication takes the form of exagger-
ation. Senders with biases to the right of the receiver inflate their messages to the right,
while senders with biases to the 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.
12Including 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.
20
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
4)(0
.04)
(0.0
3)(0
.04)
(0.0
5)(0
.05)
(0.0
5)(0
.06)
(0.0
5)(0
.07)
Tar
get×
Jam
-0.3
90.
20-0
.48
0.03
0.02
(1.6
8)(0
.40)
(0.2
3)(0
.15)
(0.1
5)
Inte
rcep
t-4
3.34
-43.
24-5
3.54
-53.
88-5
9.58
-57.
09-7
6.19
-76.
61-7
6.90
-77.
35
(2.3
9)(2
.44)
(2.1
4)(2
.25)
(2.8
4)(3
.05)
(2.7
8)(3
.47)
(3.0
8)(4
.32)
N12
512
514
214
210
710
711
711
714
314
3
R2
0.82
0.82
0.85
0.85
0.78
0.79
0.69
0.69
0.59
0.59
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
.05)
(0.0
5)(0
.06)
(0.0
7)(0
.08)
(0.0
6)(0
.09)
Tar
get×
Jam
-0.7
6-1
.02
0.27
0.15
-0.1
1
(1.2
4)(0
.51)
(0.2
6)(0
.23)
(0.1
8)
Inte
rcep
t40
.78
40.3
150
.17
48.2
956
.62
58.5
464
.44
66.1
774
.93
72.3
5
(3.2
7)(3
.37)
(3.1
5)(3
.26)
(3.3
4)(3
.80)
(3.9
2)(4
.80)
(3.4
0)(5
.37)
N14
214
212
012
011
711
712
712
713
013
0
R2
0.65
0.65
0.74
0.75
0.70
0.70
0.55
0.55
0.62
0.62
21
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 mini-
mally sophisticated behavior, and so on. Although such models are typically motivated by
situations in which players have no previous experience, they 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 as well as in early rounds of play when previous experience provides an insuffi-
cient to guide to behavior. Third, the level-K model is promising because it can explain
heterogeneity in subjects’ exaggeration as a function of differences in cognitive ability or
strategic sophistication. Although we cannot observe cognitive differences directly, they can
be inferred from behavior by applying the level-K model.
22
In our second approach, we develop a model of “experiential best responses,” in
which beliefs are based not on a reasoning process, but on past observations. Our simple
model is essentially a belief learning model 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 that his opponent’s message will
be the average of the messages he has observed his opponents send previously. Sophisticated
subjects will therefore 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.
We first analyze receivers’ behavior and then explore whether senders’ behavior might
be consistent with beliefs formed by either iterated reasoning or a learning process. Thus,
in the spirit of backward induction, we begin with receivers.
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.
If senders engage in symmetric exaggeration 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 Ei < 0 for left senders and
Ei > 0 for right senders. Given any pair of messages, the symmetry of the shift distributions
implies that the receiver’s best response is to simply take the average of the two messages.13
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
13When 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.
23
Table 5: Regression analysis of receivers’ responses to messages
Average Message 0.85 0.82 0.84
(0.01) (0.03) (0.04)
Target 0.04 0.04
(0.03) (0.03)
Left Shift 0.01 0.01
(0.05) (0.05)
Right Shift 0.01 0.01
(0.05) (0.05)
Avg. Message × Round -0.002
(0.001)
Round -0.04
(0.08)
Intercept 0.33 0.38 0.94
(0.68) (1.76) (2.05)
N 640 640 640
R2 0.89 0.89 0.89
above 0.80. Adding target, left shift, and right shift to the model does not add any additional
explanatory power. This absence of difference 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 follow this strategy.
24
5.1 Level-K Reasoning
In the level-K framework, K denotes the degree of sophistication a subject evinces. Level-0
senders are non-strategic and use naive decision rules. Level-1 senders believe their opponents
are level-0 and choose the appropriate best response. In general, level-K senders best respond
given the belief that their opponents are level-(K − 1). Thus, K refers to the number of
steps of iterated reasoning.
Applying the level-K framework to our game poses two complications. First, in
contrast to previous applications, our game involves players in three roles.14 Thus, we must
be able to pin down a sender’s beliefs about not just the other sender but the receiver as
well. As noted above, there is evidence that senders choose actions equal to the average
message, and it is therefore both intuitive and plausible that senders believe receivers follow
this strategy.
The second complication is that we must choose an appropriate level-0 type with
which to “anchor” the analysis. The key property, in our view, 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 mt0 = T ;
we denote these types as t0. This is the assumption that Cai and Wang (2006) make.15
Alternatively, senders might be naive selfish types, which we denote by s0, who instead
report their own targets, ms0 = 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.
14In most previous cheap-talk experiments, there are only two types, senders and receivers. Although thebeauty contest game allows for any number of players, the usual assumption is that a level-K playerbelieves all other players to be K − 1 and the structure of the game allows beliefs to be summarized by asingle parameter (the average guess).
15Applications 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.
25
Table 6: Level-K message strategies
Type Left Sender Message Right Sender Message
t0 T T
t1 T + 2S T + 2S
t2 T + 2S − 50 T + 2S + 50
t3 T + 2S − 100 T + 2S + 100
s0 T + S T + S
s1 T + 2S − 25 T + 2S + 25
s2 T + 2S − 75 T + 2S + 75
To derive the form of each type’s message function, suppose first that the naive
truthful type t0 anchors the iterated reasoning process. Type t1 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 + mt1), and so his best response is to choose mt1 = 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 + mt1) = T + S if and only if mt1 = 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 t2 believes he faces a type t1 opponent. Type t2 believes that
c = 12(T + 2Sopp + mt2), where Sopp is the opponent’s shift. Although he does not know his
opponent’s shift, each sender does know its sign and distribution. The best response is to
choose the message that will ensure E(c) = T + S, which implies mt2 = T + 2S − 2E(Sopp).
For example, the left sender knows that Sopp is distributed uniformly between 0 and 50;
therefore, his best response is mt2 = T + 2SL− 50. Similarly, the t2 type right sender’s best
response is mt2 = T + 2SR + 50. Continuing this pattern of reasoning, the message functions
for t3 types are mt3 = T + 2SL − 100 for left senders and mt3 = T + 2SR + 100 for right
senders.
26
Anchoring the analysis on the naive selfish type yields a similar set of strategies that
differ only by a constant for types K > 1. Type s1’s messages are ms1 = T + 2SL − 25 for
left senders and ms1 = T + 2SL + 25 for right senders. At the next level, s2, the functions
are ms2 = T +2SL−75 for left senders and ms2 = T +2SR+75 for right senders. In general,
level-K message strategies will be a linear combination of T , S, and a constant. 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 t0 type
and α(K) = 50(K − 1) + 25 if the anchor is the naive selfish s0 type. In contrast, naive
strategies are either less responsive or unresponsive to the shift parameter and do not involve
a constant term.
Our simple level-K analysis yields best response functions characterized by contin-
uous levels of exaggeration. 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 (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 rationalizes 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-K
27
reasoning. Specifically, messages are less responsive to changes in shifts than would beoptimal in the level-K model.
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 types t0-t3 and s0-s3) and classify
each message as being consistent with a type if the distance between the prediction and
message is within an error band of ±10.16 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 tK as being a lower
type than sK.
The classification analysis summarized in Table 7 reveals heterogeneity in the dis-
tribution of subjects’ levels of strategic sophistication. Most subjects send messages that
are relatively consistent: 69% of subjects send a majority of messages that belong to the
same classification. And most of these (76%) possess some degree of strategic sophistication
beyond naivete (i.e., K > 0). Specifically, 24% are classified as one of the level-0 naive types,
while 45% are classified level-1, and 30% as level-2. No senders are classified as having so-
phistication greater than level-2. The level-K framework therefore appears to organize the
data reasonably well.17
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)
16This error band is 6.7% of the total message space and was chosen to maximize the number of messagesuniquely classified.
17We 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.
28
Table 7: Level-K classifications
Minimum Classification Success
Type > 30% > 40% > 50% > 60% > 70% > 80% Pct. Type
t0 0 1 0 0 1 0 4%
s0 2 1 2 1 1 2 20%
t1 0 2 2 1 2 0 15%
s1 1 2 8 2 0 1 30%
t2 0 4 4 4 0 0 26%
s2 1 0 0 1 0 0 4%
Pct. Min. Success 9% 22% 35% 20% 9% 7%
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
to (albeit significantly less than) 1, and we see that the magnitudes of the left and right
Sender intercepts are between 25 and 50. The regression estimates therefore suggest that
the average subject’s level of sophistication is around s1 or t2, 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,
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. Indeed, a coefficient on target of 1 and a
29
Table 8: Regression analysis of senders’ messages and level-K
Target 0.87 0.98
(0.16) (0.07)
Shift 0.88 0.85
(0.03) (0.12)
Left Sender -40.22 -19.62
(2.14) (3.84)
Right Sender 34.85 19.23
(2.14) (3.73)
Target × Round -0.008
(0.002)
Shift × Round 0.001
(0.007)
Left Sender × Round -1.53
(0.24)
Right Sender × Round 1.17
(0.24)
N 1280 1280
R2 0.84 0.86
30
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
postulated by the level-K framework. It is consistent instead with a characterization of
subjects’ behavior in which they hold a naive (yet still 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.
To allow for the possibility that subjects might adapt their behavior and eventually
learn to play a level-K best response over time, we estimate an additional specification
that includes interactions with the round of play.18 If subjects learn to play best responses
over time, we would expect to see that the coefficient for the interaction between shift and
round to be positive (and the overall effect of shift by the last round should be 2). We
find no evidence whatsoever that messages become more responsive to senders’ shifts over
time. Instead, the second specification suggests that senders continue to engage in a naive
form of exaggeration and that this exaggeration increases over time. We can see this from
the fact that when we add the round interactions, the magnitude of the intercepts in the
second column diminish to about half the size of the estimates in the first column while
the interactions between sender intercepts and round are significant and in the appropriate
direction for each sender.19 While the intercepts begin near -20 and 19, for the left and right
sender respectively, by the 32nd round of play the estimated levels of exaggeration end near
-69 and 57. If we use these estimates to project out of sample by assuming that exaggeration
keeps increasing, then play would approach an uninformative equilibrium (i.e., all messages
would reach the boundaries of the space) by around the 100th round of play.
18We adjust Round to begin at 0 so that we can interpret the uninteracted coefficients as the coefficients inthe first round of play.
19Both 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.
31
The level-K framework has provided a better explanation for behavior in complex
communication environments than equilibrium analysis, although the 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 response function must take into
account the receivers’ averaging strategy. But we also find that experience matters, as
messages become more exaggerated over time. It is still possible that subjects play a form of
best response, but the level-K analysis misspecifies the process by which they form beliefs.
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 beliefs about what others will do are formed through
a process of iterated reasoning (anchored by intuitive conjectures about naive behavior).
But as individuals play the game, it is reasonable to assume instead that senders will 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. That is, we assume players choose
best responses to their out-of-equilibrium beliefs but replace beliefs formed through conjec-
ture and iterated reasoning with beliefs formed by 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-
32
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
opponent’s ideal point plus the average exaggeration he has observed. If the sample average
of the opposing sender’s past exaggeration et =∑t−1
τ=1(mτopp−T τ ), then the expeced value of
an opposing sender’s current message is E(mopp) = T + et. The logic involved in deriving the
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 that 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 action) to his own ideal point. The experiential best response framework implies,
like in the level-K framework, that messages will be functions of target and shift. 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 level of strategic sophistication. That is, once the target, shift, and
opponents’ 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 predictedby the frameowrk. We also observe that the level of under-exaggeration decreases overtime and is much closer to the predicted messages by the last round of play.
33
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 most 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 to both 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.
The regression coefficients reported in Table 9 suggest that taking into account op-
posing senders’ past behavior provides an improved account of exaggeration but that senders
continue to under-exaggerate. In the models without period interactions, the coefficient for
opponents’ past exaggeration is −0.56 when we use the last 5 periods of play, and it is −0.68
when we use the entire history of play. Both coefficients are significantly smaller in magni-
tude than 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 coefficient for the interaction term
for both versions of past exaggeration are negative and statistically significant. This implies
that senders’ messages become more responsive to opponents’ exaggeration as they gain more
experience. Indeed, by the last period of play, messages are as responsive to opponents’ past
exaggeration as the experiential best response framework predicts; the predicted coefficient
using all periods of opponents past exaggeration is −0.98, with the hypothesized value of −1
in the confidence interval.
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
34
Table 9: Regression analysis of senders’ messages and experiential best responses
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
(0.06) (0.12) (0.06) (0.12)
Exaggeration (last 5) -0.56 -0.33
(0.04) (0.08)
Exaggeration (all) -0.68 -0.24
(0.06) (0.09)
Left Sender -12.19 -10.45 -11.25 -14.92
(2.80) (4.90) (3.26) (5.24)
Right Sender 3.61 9.83 3.51 15.30
(2.96) (5.06) (3.42) (5.33)
Target × Round -0.008 -0.008
(0.002) (0.002)
Shift × Round -0.005 -0.007
(0.007) (0.008)
Exag. (last 5) × Round -0.009
(0.004)
Exag. (all) × Round -0.024
(0.006)
Left × Round -0.46 0.03
(0.30) (0.37)
Right × Round -0.08 -0.63
(0.32) (0.40)
N 1234 1234 1234 1234
R2 0.87 0.88 0.86 0.87
35
(in all models) and right sender (in the last column) remain statistically significant, which
suggests that subjects continue to add a constant amoung of exaggeration that is unrelated to
opponents’ past exaggeration or to their 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.
6 Conclusion
In the realm of politics, lobbyists, businesses, activists, and policy experts compete to in-
fluence legislation, regulatory rules, and court decisions. Parties, candidates, and public
intellectuals also clash in attempts to sway public opinion. Although many scholars have
suggested that competition can help resolve information transmission problems (Austen-
Smith and Wright 1992; Gilligan and Krehbiel 1989; Krehbiel 1991; Page and Shapiro 1992),
the theoretical analysis of incentives suggests a more cautious view might be warranted,
as the enthusiasm for the benefits of competition should be tempered by the realization
that there are potent strategic incentives for competing interests to engage in obfuscation
and misdirection (Milgrom and Roberts 1986; Minozzi 2011). Equilibrium analysis therefore
gives reasons to be concerned that competition in the “marketplace of ideas” may be replete
with manipulative and false advertising.
We find instead that a substantial amount of information is communicated in our
experiment involving a complex communicative environment that features competition, pref-
erence uncertainty, and a rich strategy space. This is because senders engage in very simple
communication strategies, exaggerating in the direction of their biases. While this results in
polarized messages that depart from the truth, competition then allows receivers to guess
36
the hidden state information by averaging the two messages. The “marketplace of ideas”
appears to work after all, albeit a bit noisily.
Our analysis of message strategies also suggests that in contrast to previous experi-
ments in single sender or simple environments, overcommunication cannot be fully explained
by models of strategic bounded rationality. Observed levels of exaggeration generally com-
port with some features of the best response functions implied by a model of iterated rea-
soning, but are also less responsive to variation in the degree of the senders’ preference
divergence than such models predict. A belief learning model partially accounts for in-
creases in exaggeration over the course of the experiment but similarly fails to account for
the insufficient responsiveness of senders’ messages to preference divergence. Senders in the
experiment therefore appear to use a naive communication strategy that reflects some de-
gree of strategic intuition but is even more limited in sophistication than models of bounded
rationality normally considered in the literature assume.
37
Acknowledgements
We gratefully acknowledge comments and advice from Dan Butler, Mark Fey, Sean Gail-
mard, 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 Southern Political Science Association meeting,
the 2011 Midwest Political Science Association Meeting, and the 2011 American Political
Science Association Meeting.
38
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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]