Communication and voting in heterogeneous committees: An
experimental study
Mark T. Le Quement and Isabel Marcin∗
September 20, 2015
Abstract
We experimentally investigate the Condorcet Jury Model in three-person committees
featuring heterogeneous preference types in which agents communicate via straw vote before
voting by simple majority rule. We study the extent to which individuals engage in strategic
communication in the presence of preference heterogeneity, while checking for the possible
presence of social preferences and cognitive constraints. To that end, we apply a 2 × 2 design
varying the information provision protocol (exogenous group signals vs public messaging)
and the group composition (homogeneous or heterogeneous preferences). Results clearly
contradict the predictions of the standard model which emphasizes lying incentives as well as
the predictions a simple model of social preferences which assumes that subjects maximize joint
payoffs. In contrast, results are in line with the predictions of a simple model of cognitively
heterogeneous agents. While approximately 75% of (naive) subjects truth-tell and tend to
follow the majority of announced signals in their voting decision, approximately 25% of
(sophisticated) agents apply a decision rule significantly closer to their type-specific optimal
decision rule and consistently lie in a way that is payoff-improving.
Preliminary Working Draft
Please do not cite without the authors permission.
∗
We thank Christoph Engel and seminar participants at the Max Planck Institute for the study of Collective
Goods for many helpful comments. We also wish to thank Sebastian Goerg, Jens Grosser, Oliver Kirchkamp,
Sebastian Kube and Dimitri Landa for helpful suggestions.
1
1
Introduction
Much collective decision-making brings together individuals exhibiting publicly known and heterogeneous preferences. Individuals exchange information before deciding. There will be agreement on
the optimal decision after information exchange if the pooled information clearly indicates the state
of the world (whether a defendant is guilty or innocent, whether a reform will lower unemployment,
whether a job candidate is competent). Individuals will instead disagree if aggregate information
is less clear-cut and leaves high uncertainty about the state.
Consider the case of a faculty deciding whether to hire a candidate who is a microeconomist
and who may be either good or bad. The hiring committee is composed of macroeconomists and
microeconomists. All members want to hire her if she is good and prefer to turn her down is she is
bad, but payoffs from errors differ: it is common knowledge that microeconomists derive a higher
loss from a mistaken rejection than from a mistaken hire and vice versa for macroeconomists. In
such a setting, disagreement arises if available information is weakly favourable, in which case only
microeconomists favour hiring. Committee members hold relevant information in a decentralized
fashion. One member might know current colleagues of the candidate. Another may be well-versed
in her literature and have studied her publication list. Yet another may know the department
from which she graduated. Individual pieces of information, once pooled, are likely to establish
unambiguously whether the candidate is good or bad. Information sharing is thus desirable. It is
however unfortunately difficult to achieve. If the voting rule gives veto power to macroeconomists,
microeconomists will have an incentive to omit information indicating weaknesses of the candidate.
If the voting rule instead allows microeconomists to overrule a veto by macroeconomists, the latter
will have incentives to omit facts indicating that the candidate is good. The core ingredients of
the above scenario match many other types of committees. Consider parliamentary committees
featuring members of different political parties or boards of directors featuring different types of
stakeholders (public and private stockholders, employees, etc).1
This paper offers a first experimental investigation of the Condorcet Jury Model with publicly
known heterogeneous preference types. We study three-person committees voting under simple
majority rule and featuring two preference types. Our primary objective is to check to which
extent agents communicate strategically in the presence of preference heterogeneity, by either
systematically misrepresenting contrary evidence or simply communicating uninformatively (babbling). The question is motivated by standard rational choice theory, which predicts incentives to
1
Different preferences may be rooted in political attitudes, demographic characteristics, personality traits, etc.
In the psychological literature on judicial decision-making there is a strand of literature that empirically studies
the effects of demographic and personal characteristics on jury deliberations (for reviews see e.g. MacCoun, 1989;
Pennington and Hastie, 1990; Devine et al., 2001; Sommers and Ellsworth, 2003).
2
deviate from truth-telling on the part of minority agents who recognize that their favored decision
given shared information is likely to be overruled at the voting stage.
In addressing these questions, we consider it likely that two types of behavioral biases lead
to a departure from the standard prediction. First, individuals may face cognitive constraints.
Communicating strategically and best responding to information is difficult and individuals may
differ in their ability to perform these tasks. Second, group composition may affect agents’ decision
rules as a consequence of a variety of possible group induced preferences (inequity aversion,
altruism, social norms, etc). For instance, individuals may maximize joint payoffs as opposed to
individual payoffs.
Our theoretical base corresponds to three models, mirroring the above: the standard model
(Model I), a model of joint payoff maximization (Model II) and a model of cognitive heterogeneity
(Model III). The prediction of the standard model is that individuals in minority babble, individuals
in majority truthtell and all individuals apply their type specific optimal decision rule to the
available evidence. Our model of joint payoff maximization makes individuals de facto homogeneous
despite potentially different underlying payoff parameters. In the equilibrium of this second model,
individuals truthfully share their information and subsequently apply the joint payoff maximizing
decision rule. In our third model assuming cognitive heterogeneity, the numerically dominant
heuristic agents truth-tell and apply the majority heuristic while reasoning agents exploit this by
lying strategically, their ability to do so increasing in their level of cognitive sophistication.
In order to test these models, we use a 2 × 2 experimental design in which we vary the
composition of committees as well as the information protocol. The first dimension varies
the committee composition from heterogeneous to homogeneous while fixing the information
protocol. The second dimension shifts the information protocol from exogenous information
(signals are publicly shown to all three jurors) to endogenous information (simultaneous cheaptalk communication of private signals). Comparing treatments with communication featuring
either homogeneous or heterogeneous committees allows us to observe whether heterogeneity
triggers strategic communication as predicted by Models I and III. In so-doing, treatments with
exogenous information help us assesss to which extent individuals apply their optimal type specific
decision rules. Comparing treatments with exogenous information featuring either homogeneous or
heterogeneous committees allows us to test Model II, which implies that the decision rule applied
by jurors ought to vary with the committee composition.
Our results are as follows. At the aggregate level, behavior appears very simple both at the
communication and voting stages. The average lying rate is very low and each preference type’s
average decision rule is heavily skewed towards the majority heuristic (vote with the majority of
signals). A fraction of subjects (between 15% and 25%) behaves very differently throughout the
3
game. These roughly apply the optimal rule and consistently lie in a way that allows them to
favorably bend the implemented decision rule. These furthermore profitably lie also when part of
the majority group, which is advantageous due to the suboptimal decision rule applied by other
majority subjects. While results clearly contradict the prediction that we derive for the standard
model, these are in line with the prediction derived from model III which features heterogeneous
cognitive types. Group composition in contrast does not significantly affect jurors’ decision rules.
Related literature Building on De Condorcet’s (1785) seminal essays on voting, a formal
literature that models voting as information aggregation has blossomed over the last two decades.
Early contributions (Austen-Smith and Banks, 1996; Feddersen and Pesendorfer, 1998) study
private voting (see Gerardi, 2000; Martinelli, 2006; Meirowitz, 2007, for further refinements of
early insights). Key findings for this case have been confirmed and qualified experimentally in
Guarnaschelli et al. (2000) and Esponda and Vespa (2014).
A set of newer contributions study the realistic case of voting preceded by communication
and have focused on the so-called truthful communication and sincere voting outcome (TS
equilibrium). A milestone is the negative result obtained by Coughlan (2000) for the case of known
heterogeneous preference types. If full truth-telling leads to disagreement with positive probability,
there exists no equilibrium featuring truthful deliberation followed by sincere voting. Le Quement
and Yokeeswaran (2015), assuming two preference types (doves and hawks) and voting under
unanimity, finds a unique responsive (symmetric and pure strategy) equilibrium prediction. Jurors
of the type endowed with veto power (doves) truthfully announce their signals, while remaining
jurors (hawks) babble and all agents vote sincerely (simply using available information). A main
theme has since been how uncertainty about preference types allows the TS equilibrium to exist
despite possible disagreement at the voting stage (Austen-Smith and Feddersen, 2006; Meirowitz,
2007; Van Weelden, 2008; Le Quement and Yokeeswaran, 2015).
Voting with communication has also been examined experimentally. Guarnaschelli et al. (2000)
study a homogeneous jury and find, in contradiction with theory, a small lying rate (around 5%)
and skepticism towards information provided by others. Goeree and Yariv (2011)(GY in what
follows) study the case of privately known (and potentially heterogeneous) preference types and
find that agents on average follow a simple heuristic: truth-tell and vote with the majority of
signals. Our results, in our control treatment with homogeneous preferences as well as in our main
treatment with heterogeneous preferences, echo these findings.
Our experiment complements GY on several dimensions. First, theoretical predictions for
the cases considered by our experiments (known vs unknown preferences) differ: guaranteed
non-existence of the TS equilibrium in our setup, (mostly) possible existence in GY’s experiment.2
2
In two of the treatments in GY, the TS equilibrium does not exist, i.e. in the treatment with unanimity rule
4
Second, our objectives differ. GY aim at testing a hypothesis formulated in Gerardi and Yariv
(2007), namely that unrestricted communication renders all voting rules equivalent. Our focus
is instead on understanding how the game is played given a fixed voting rule (majority) and a
particular communication protocol (straw vote). GY accordingly examine free form communication
while we study straw votes. In our view, straw votes present the double advantage of making
strategic incentives in communication relatively easy to identify and of minimizing the likelihood
that subjects develop social preferences through rich interaction. These two aspects increase the
potential relevance of the standard model predictions offered in Coughlan (2000) and Le Quement
and Yokeeswaran (2015). A last difference between GY and our analysis is that we explicitly
explore the potential role of cognitive heterogeneity and preferences relating to group composition.
Structure of the paper Section 2 introduces the Condorcet Jury Game as well as Models
I and II. Section 3 presents our experimental design. Section 4 presents our findings regarding
average behavior. Section 5 presents our model of cognitive heterogeneity (Model III) and section
6 presents our findings regarding individual level behavior. Section 7 concludes. The appendix
contains among others discussions of further results from our post experimental tests.
2
The Condorcet Jury game
In what follows, we introduce the basic Condorcet jury game as well as two specific models that
build on the basic setup. These both assume fully rational agents but differ as to the preferences
assumed. In the first, agents maximize individual payoffs, whereas in the second, they maximize
joint payoffs. For both models, we shall examine equilibria in symmetric strategies, i.e. such that
all jurors of a given preference type use the same strategy.
The classical model: Model I We start by introducing the standard Condorcet Jury Model
(Model I) proposed by Austen-Smith and Banks (1996) and Feddersen and Pesendorfer (1998).
After recalling the negative result obtained by Coughlan (2000), we present our equilibrium
prediction for the model.
Suppose a jury composed of n = 3 members. A defendant is being judged and is either guilty
(red state) or innocent (blue state) with equal prior probability. The jury must decide whether to
convict (red decision) or acquit (blue decision) him. Each juror casts a vote in favour of either
red or blue. The voting rule is given by an integer R = 2 s.t. the defendant is convicted if and
only if at least R votes are cast in favour of red.
and in the treatment with partisan agents who favour a given outcome regardless of the realized state.
5
Each juror receives a single private signal prior to the vote. A signal s ∈ {red, blue} indicates
either the red or the blue state. A signal is "correct" with probability p = .7, i.e. P (s = red |
red) = P (s = blue | blue) = p, while P (s = blue | red) = P (s = red | blue) = 1 − p. Juror signals
are i.i.d.
The timing of the game is as follows. In stage 0, jurors receive private signals. In stage 1, jurors
communicate via a round of simultaneous public cheap talk. In stage 2, jurors simultaneously cast
a vote. In stage 3, the defendant is convicted if and only if R conviction votes were cast, where R
is the voting rule.
Each jury member j’s preferences are determined by a commonly known parameter q j ∈ (0, 1) .
Juror j’s payoff function is given as follows: −q j is the utility obtained when the defendant is
convicted despite being innocent, and −(1 − q j ) is the utility obtained when the defendant is
acquitted but guilty. The utility derived from acquittal of an innocent or conviction of a guilty
defendant is normalized to 0. Given these payoffs, a juror j prefers conviction over acquittal
whenever his posterior probability that the defendant is guilty exceeds q j . The parameter q j thus
measures the juror’s degree of aversion to wrongful conviction.
Juror preferences fall into two homogeneous categories. The jury contains nblue blue-biased
jurors with preferences qblue =
5
6
and nred red-biased jurors with preferences qred = 16 , where
nblue + nred = n. We occasionally simply speak of red types and blue types. A majority (minority)
type agent is a subject whose preference type is held by a majority (minority) of committee
members.
Given j ∈ {red, blue}, we call a j-signal held by a type-j juror a conform signal and a −j-signal
held by a type-j juror a contrary signal. Similarly, we call decision j (−j) the conform (contrary)
decision for a given type-j juror. We call j-decision rule a decision rule that specifies the likelihood
of voting for decision j as a function of the number of observed signals of color j. We call
threshold j-decision rule a deterministic decision rule that specifies a minimal number of signals of
color j above (below) which a juror votes for decision j with probability one (zero). Given our
assumptions, the optimal decision rule of each preference type given three signals is to choose the
conform decision when at least one conform signal is available. Let σ(r) = 0 denote the probability
of voting for a conform decision given r conform signals in three. We denote the decision rule
(σ(0), σ(1) = x, σ(2) = 1, σ(3) = 1) by Λ(x). Λ(1) is thus the optimal decision rule of each type.
Finally, we say that a subject votes sincerely if he casts a vote in favour of the decision that he
favours given the information available to him.
We start by recalling the impossibility result of Coughlan (2000). The latter states that if
nblue ≥ 1 and nred ≥ 1, there exists no equilibrium in which jurors truth-tell and vote sincerely.
To understand the result, assume that the committee contains a simple majority of blue-biased
6
subjects. The decision rule applied in the above putative equilibrium is the optimal decision rule
of blue-biased subjects, i.e. choose red after at least three red signals. The unique red-biased
agent, being rational, acts under the assumption that his announcement is pivotal (i.e. affects
the final outcome) and thus infers that two other jurors hold a red signal, which implies that he
favours a red decision (as he favours a red decision after at least one red signal). If he holds a blue
signal, he thus has an incentive to lie. Arguments appearing in Le Quement and Yokeeswaran
(2015) lead to the following simple equilibrium prediction for this committee.
Proposition 1. Let nblue ≥ 1 and nred ≥ 1. There exists an equilibrium in which majority type
subjects truth-tell while the minority subject babbles and all subjects vote sincerely.
Note simply that on the equilibrium path, a subject of the majority preference type implements
his favored decision rule conditional on public information. Sincere voting implies that majority
types condition their vote on messages sent by majority types and minority types condition their
votes on messages sent by majority types and their own signal.
Social preferences: Model II An alternative assumption is that jurors have social preferences and maximize joint payoffs. There are various explanations in behavioral economics for
group induced preferences (e.g. social preferences, altruism, social norms) as well as different
approaches to model these preferences.3 We base our modeling choice on GY who find group
choices to be consistent with the welfare maximizing decisions given available aggregate information. Maximizing joint payoffs is equivalent to jurors behaving as if they all shared the same
preference parameter given by the average preference parameter. In a committee with two (1)
blue-biased agents and one (2) red-biased agent, this would mean that all jurors act as if having
the preference parameter (5/6 + 5/6 + 1/6)/3 ≈ 0.61 (≈ .39). In any of the two above cases, the
optimal decision rule conditional on 3 signals is to vote for the decision supported by a majority
of signals. The committee would de facto be a homogeneous committee despite the different
underlying preference parameters of different jurors. For such a committee, we may state the
following equilibrium prediction.
Proposition 2. Assume that jurors maximize joint payoffs. Let nblue ≥ 1 and nred ≥ 1. There
exists an equilibrium in which all jurors truth-tell and subsequently vote for the decision supported
by the majority of signals. If instead all jurors share the same preference parameter (i.e. either
nblue = 3 or nred = 3), the only equilibrium featuring truth-telling by all agents is s.t. jurors
subsequently apply the optimal decision rule.
3
For instance, in the literature on social preferences there are outcome-based model that focus on inequity
aversion or taste for efficiency, as well as intention-based models that highlight the role of reciprocity, kindness, etc.
(e.g., Rabin, 1993; Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000; Charness and Rabin, 2002).
7
While Model II thus predicts truth-telling for any jury composition, it predicts that outcomes
will differ depending on the jury composition. While heterogeneous committees vote with the
majority of signals, homogenous committees implement their type-specific decision rules.
Given our focus on symmetric strategies, checking whether predictions are consistent with our
experimental data simply requires examining average behavior of different preference types.
3
Experimental design
Setup Our experimental setup replicates the standard Condorcet Jury Model and uses the setup
presented in Guarnaschelli et al. (2000) as well as Goeree and Yariv (2011). At stage 0, Nature
selects a jar, which is either blue or red. A blue jar contains 7 blue balls and 3 red balls while a
red jar contains 7 red balls and 3 blue balls. In the next stage, subjects are exposed to information
whose nature depends on the treatment (endogenous or exogenous) that they participate in. In
the endogenous treatment, each committee member draws a ball (i.e. a signal) with replacement
from the selected jar and subsequently chooses a simultaneously observed public message in the
set {red, blue}. In the exogenous treatment, subjects all observe a public profile of three balls
drawn with replacement from the picked jar. In the next stage, committee members each cast
a vote taken from the set {red, blue} and the decision having received a majority of votes is
implemented. In the final stage, committee members observe the number of votes cast for each
jar, the jar selected by nature as well as their payoffs.
Each committee has three members. The preference type of each subject is either red or blue.
The payoff matrix corresponding to each preference type appears in Table 1 below. These payoffs
are formally equivalent to thresholds qblue =
5
6
for blue-biased types and qred =
1
6
for red-biased
types in the classical Condorcet Jury Model. The game is played repeatedly over 20 rounds with
random rematching within matching groups of six subjects.
Table 1: Payoff structure
Blue Jar
Red Jar
Vote for Red Jar
10
40
Vote for Blue Jar
160
10
Blue Type
Blue Jar
Red Jar
Vote for Red Jar
10
160
Vote for Blue Jar
40
10
Red Type
Treatments We run a main treatment and three control treatments which correspond to the
four entries of a simple two by two matrix in a between-subjects design. The first dimension
8
determines whether information is endogenous or endogenous (see above). In the endogenous
case, messages are presented in a semi-anonymous way: each is shown only with an indication
of the preference type of its emitter. The second dimension determines whether the committee
features homogeneous preference types or is instead heterogeneous. The assignment process in
the heterogeneous treatments is as follows. Individuals are throughout the experiment part of a
fixed matching group containing three blue and three red types. Each period, two committees are
randomly formed within each such matching group, each with a different majority color (blue or
red types). Thereby homogeneous committees are by contruction not possible in the heterogeneous
treatment. The jury composition is commonly known at the beginning of the game. In the
heterogeneous treatments, it follows that each juror is very likely to experience multiple rounds as
a minority type and as a majority type. We thus run the following four treatments, treatment
number 2 constituting our main treatment featuring deliberation and voting in a heterogeneous
group.
Table 2: Overview of treatments
Preferences
Information
homogeneous
heterogeneous
endogenous
1: Endo-Hom
2: Endo-Het
exogenous
3: Exo-Hom
4: Exo-Het
Post-experiment tests At the end of the session, in all treatments subjects execute a set
of individual tests. The strategic communication test (SCT) aims at identifying whether agents
understand the advantages of lying. There are two stages: communication and voting (by majority).
An agent faces two computers and is replaced by a computer at the voting stage. His only decision
is whether to truthfully announce his signal. Other (computerized) subjects truthfully announce
their signals and vote sincerely under the assumption of truth-telling by all committee members.
The agent’s computerized alter-ego at the voting stage votes sincerely on the basis of his true
signal as well as other committee members’ truthfully announced signals. Payoffs obtained by
the two computers who replace other committee members are randomly allocated to two other
subjects in the session.
Subjects who participated in the Endo-Het treatment face four scenarios, corresponding to the
subject being either a majority or a minority preference type and holding either a contrary or a
conform signal. Of these scenarios, the only one providing an incentive to lie is that in which the
agent is a minority type and holds a contrary signal. Subjects who participated in the Endo-Hom
treatment face only two scenarios, both of which provide no lying incentive. In each of these,
9
subjects are part of a homogeneous committee and hold either a conform or a contrary signal.
The SCT is only taken by subjects who participate in endogenous treatments. The next three
tasks are executed after all treatments.
The second task is the individual decision test (IDT). An agent receives three signals and
chooses a jar alone. The objective here is to test whether subjects are able to decide optimally on
their own on the basis of multiple signals.4 For each treatment, we use behavior in the IDT to
assign subjects to each of four categories. Any subject is characterized by a preference type j and
a minimal number x for which he picked decision j in the test. Such a subject is classified as using
the threshold j-decision rule with threshold x and assigned to category x. The four categories
thus merge red and blue preference types. The third task is a classical lying aversion test Gneezy
et al. (2013), described in more detail in our discussion of results in the Appendix. The final task
is a social value slider aimed at measuring social preferences (Murphy et al., 2011).
Experimental procedure The experiment was conducted in the BonnEconLab in February
and March 2015. It was programmed and conducted with the software z-Tree (Fischbacher, 2007)
and organized with the software hroot (Bock et al., 2014). A total of 384 University of Bonn
students from various disciplines (15% with an economics major) participated in the experiment. In
each treatment 96 subjects (yielding 16 independent matchings groups per treatment) participated.
Subjects received written instructions which were read out loud by the experimenter. To familiarize
subjects with the game and check that subjects understood it, we asked control questions that
had to be answered correctly before subjects were allowed to proceed to the actual game. Subjects
were given the opportunity to privately ask questions. The amounts earned from the experiment
were exchanged at a rate of 150 ECU = 1 Euro. Subjects received the payment from all 20
rounds, which averaged 10.50 Euros and ranged from 5.50 Euros to 16.50 Euros. Subjects earned
additionally an average of 4.68 Euros in the post-experimental tests. On average, one session
lasted 65 minutes (40 minutes jury experiment and 25 min post-tests). 58.6 % of subjects were
female and average age was 22.6 years.
4
For this test we use the strategy method. Subjects make a decision for each of the four possible signal profiles.
The individual threshold decision rule is calculated by 4 minus the total number of decisions j, where j ∈ {red, blue}
is the preference type of the subject. For example, if a subject of type red chooses the red jar three times, he is
assigned the decision rule “vote j when holding at least one j-signal”. Note that the method bases on the assumption
that subjects’ decision rule is monotonic, i.e. that the likelihood of voting for a given jar increases in the number of
signals indicating this jar. Another caveat is that the method does not allow us to observe whether individuals’
decision rule is stochastic or deterministic.
10
4
Average behavior
4.1
Decision stage results
We start with results for the decision stage and then proceed to results for the communication
stage (for endogenous treatments only). Main findings for the decision stage are as follows.
Each preference type’s average decision rule is skewed towards the majority heuristic. Different
preference types’ average rules nevertheless significantly differ. Information provided by others is
discounted. Finally, group composition plays no role.
Reversal to the middle The average decision rule of both types exhibits a reversal to the
middle as compared to the optimal decision rule (assuming truthtelling in Endo treatments).
Figures 1a and 1b show the average decision rule of each type across the four treatments. The
decision rule is given as a function of the public profile of messages in the Endo treatments and as
a function of signals in the Exo treatments (though we simply write signals on the x-axis). The
optimal decision rule is to choose the conform decision as soon as there is at least one conform
signal in three. Subjects of each preference type on average choose the conform decision with
probability .23 − .42 after one conform item (signal or message), zero after no conform item and
one after two or three conform items.5 Assuming truthtelling in the Endo treatments, we may
generally conclude that across all treatments red-biased types act less red-biased and blue-biased
types less blue-biased than warranted by their optimal decision rules. Subjects thus act in a more
moderate way than predicted by optimal behavior. For each preference type, we henceforth call
5
To compare decision rules across preference types, we use a permutation test. First, for each treatment i,
preference type j, matching group k and number x of conform items, we retrieve the frequency f (i, j, k, x) of votes
for the conform decision. For each i, j, x, we then take the average over the 16 matching groups of f (i, j, k, x). Call
this number f (i, j, x). When comparing two given preference types j and j 0 for a given treatment i, we calculate for
each x the difference f (i, j, x) − f (i, j 0 , x) and multiply this difference by a number T (i, j, j 0 , x) which simply sums
the number of times that x was observed by preference type j in i and the number of times that x was observed
P
0
0
by preference type j 0 in i. Finally, we compute x=3
x=0 (f (i, j, x) − f (i, j , x))T (i, j, j , x). This is the test statistic
to which we apply the permutation test. The advantage of the permutation test is that it takes into account the
number of observations for each possible value of x in a given treatment i. As a more conservative test, we also run
a Mann-Whitney-Wilcoxon test. For every treatment i, preference type j and matching group k, we retrieve the
frequency fe(i, j, k) of votes for the conform decision given at least one conform item. Our reason for conditioning on
at least one conform item is that we want to compare the voting decisions only for those information profiles where
the rational decision is to vote for the conform jar (assuming largely predominant truth-telling in Endo treatments).
The permutation test confirms that the average red-decision rule of red-biased types is not significantly different
from the average blue-decision rule of blue-biased types in each treatment. The Mann-Whitney-Wilcoxon does
not find significant differences between decision rules of the different preference types in all treatments except in
Endo-Het (16 obs. per type, p-value= 0.0275).
11
the applied decision rule the moderated version of their optimal decision rule.
1
vote for conform jar in %
.4
.6
.8
.2
0
0
.2
vote for conform jar in %
.6
.8
.4
1
Fig. 1: Individual decision to vote for conform jar (in %)
0
1
2
number of conform signals per group
optimal rule
het exo blue
het-endo blue
3
0
hom-exo blue
hom-endo blue
1
2
number of conform signals per group
optimal rule
het exo red
het-endo red
(a) Blue types
3
hom-exo red
hom-endo red
(b) Red types
Heterogeneity across preference types Figure 2 shows the two preference types’ average
decision rules pooled over treatments (assuming truthtelling in Endo treatments). The rules
clearly differ in spite of the observed reversal to the middle. Though both types vote blue with
probability one after no red signal and vote red with probability one after three red signals, their
behavior differs clearly for one and two red signals. For one red signal, red-biased types already
vote red with probability .38 while blue-biased types vote red with probability .05. For two red
signals, red-biased types vote red with probability close to one (.93) while blue-biased types vote
red with probability .62. The average decision rule of red-biased types is thus unambiguously
more red-biased than that of blue types though the difference is attenuated with regards to the
optimal behavior scenario.
Skepticism towards others’ messages The comparison of average decision rules across
Exo and Endo treatments shows evidence that subjects discount information obtained through
others’ messages, which echoes the findings of Guarnaschelli et al. (2000). Figure 1 shows that
when transiting from the Exo to the Endo treatments, the average likelihood of a conform decision
moves downwards by approximately .15 (0.08) conditional on one (zero) conform signal.
We estimate a discrete choice model on each individual’s decision to vote for the conform
12
0
.2
vote for red jar in %
.6
.4
.8
1
Fig. 2: Individual decision to vote for red jar (in %) across types (pooled over treatments)
0
1
2
number of red signals per group
Blue types
3
Red types
jar as a function of several explanatory variables.6 In addition to dummy variables Endo and
Het corresponding to the two treatment dimensions Endo and Het and their interaction term
Endo*Het, we consider several additional variables: the threshold (category) obtained from the
IDT (threshold ), dummy variables for the number of conform messages (signals in Exo treatment)
per group (conform messages), a dummy for an own conform signal for Endo treatments (own
conform signal ), its interaction term with treatment Het (own conform signal*Het), and dummy
variables for number of conform messages sent by other types in Endo-Het (conform messages
other ). Table 3 contains the marginal effects from the fixed model component that correspond to
our estimations.
Regression (1) and (2) reveal that subjects in Endo are less likely to vote for the conform
jar compared to Exo treatments. Regression (3) also reveals that in the Endo treatments, a
subject’s own signal has a significant impact on his decision after controlling for aggregate available
information. This effect is in addition stronger in heterogeneous than in homogeneous committees
(see regression (4)), which is intuitive given the intensified trust issue arising in heterogeneous
groups. Regression (5) reveals that in Endo-Het, subjects discount information from own-preference
type subjects less than information revealed by subjects of the other preference type. As expected,
the dummy variables for the number of conform signals/messages in the group significantly increase
the probability to vote for the conform jar, as shown in regression (2) for all treatments and (3)
for both Endo treatments.
6
We run a mixed effects logistic regression model for choices nested in individuals nested in matching groups.
The random effect for the group is nested within subjects, which means that it takes on a different value for each
combination of group and subject.
13
Table 3: Logit estimations on conform decision
Het
Endo
(1)
(2)
(3)
(4)
(5)
All
All
Endo
Endo
Endo-Het
0.01
0.01
0.01
-0.01
(0.02)
(0.02)
(0.01)
(0.02)
-0.06***
-0.06***
(0.02)
(0.02)
-0.11***
-0.07***
-0.07***
-0.07***
(0.01)
(0.01)
(0.01)
(0.02)
0.25***
0.21***
0.20***
0.21***
(0.01)
(0.02)
(0.02)
(0.03)
0.92***
0.88***
0.88***
0.92***
(0.01)
(0.02)
(0.02)
(0.01)
0.94***
0.92***
0.92***
0.96***
(0.01)
(0.02)
(0.02)
(0.01)
0.10***
0.07***
(0.01)
(0.01)
Endo*het
0.01
(0.02)
IDT
1 conform message
2 conform messages
3 conform messages
Own signal
Own signal*Het
0.05***
(0.02)
1 conform message other type
-0.05***
(0.01)
2 conform message other type
0.05
(0.10)
Observations
7,680
7,680
3,840
3,840
1,920
# of Ind.
384
384
192
192
96
# of Groups
64
64
32
32
16
-255.48
-128.87
-48.76
-1001.47
-591.65
Log likelihood
The table contains marginal effects from the fixed components of the logistic mixed effects model.
Standard errors in parentheses.
*** p<0.01, ** p<0.05, * p<0.1
Joint payoff maximization Assuming truth-telling at the communication stage, figure 1
reveals no differences in average decision rules between the Exo-Het and the Exo-Hom treatments,
which demonstrates that group composition does not affect the decision rule applied by subjects.
This contradicts our prediction for Model II, according to which heterogeneous committees ought
14
to follow the majority heuristic while homogeneous juries implement the type-specific optimal
decision rule. We use a permutation test to test the statistical significance of the differences
between voting rules applied in the respective treatments.7 Comparing Exo-Hom and Exo-Het,
the p-value of 0.7 of our permutation test indicates no statistically significant difference.8
Beyond exogenous social preferences triggered simply by heterogeneous group composition, one
could envisage endogenous social preferences triggered by the conjunction of heterogeneous group
composition and communication. We find a statistically significant difference between Exo-Het
and Endo-Het (p-value=.0001 in the permutation test, p-value of 0.0011 in the Mann-WhitneyWilcoxon test), but this difference is replicated in the comparison of Exo-Hom and Endo-Hom
(p-value=.0003 in the permutation test, p-value of 0.062 in the Mann-Whitney-Wilcoxon test).
While deliberation thus does have an effect on applied decision rules, the effect does not reflect
the fact that deliberation generates or activates social preferences (endogenous social preferences).
Had this been the case, the difference between Exo-Het and Endo-Het would not be replicated
between Exo-Hom and Endo-Hom.
Majority juror voting and minority juror annoucements We need to test specifically
the behavior of minority and majority agents. Minority jurors ought to condition their vote on
three signals and apply the decision rule Λ(1). Majority jurors ought to use only two signals, their
own and that of the other majority subject, and vote for the conform decision as long as at least
one signal in two is conform.
In order to test Model II, we examine voting behavior by majority types in heterogeneous
committees.Table 4 below examines to which extent majority jurors condition their voting decision
on the announcement of the minority juror. Model I predicts that this should not be the case
given that the minority juror bables in equilibrium. The table shows a majority juror’s likelihood
of voting for the conform decision as a function of his own signal (a conform signal takes the
value of 1 and a contrary signal takes the value of 0) and the announcement of the two remaining
jurors, one majority juror and one minority juror. For each possible signal held the majority
juror, we examine three possible scenarios: Two involving a single conform signal held by others
(either the majority or the minority juror), and one involving no conform signal held by others.
Comparing these scenarios shows that that the information provided by the minority juror is not
7
The statistic to which the test is applied is computed in a similar way to the statistic used in the permutation
test described in footnote 5. The difference is that we now compare decision rules across treatments rather than
preference types.
8
As in footnote 5, we also run a Mann-Whitney-Wilcoxon test comparing the decision rule across Exo-Hom
and Exo-Het conditional on at least one conform signal/message. The test yields a p-value of 0.5976 which is not
significant.
15
only influential but furthermore approximately as influential as that provided by a majority juror.
Consider the scenario where a majority juror holds a contrary signal and exactly one other
juror announces a conform signal, the latter being either a majority juror or a minority juror. A
Wilcoxon signed-rank test confirms that the majority jurors’ decision does not significantly differ
between these two cases (p-value=0.393). Consider the scenario where a majority juror holds a
conform signal and exactly one other juror announces a conform signal, the latter being either a
majority juror or a minority juror. A Wilcoxon signed-rank test shows that the majority juror is
more likely to vote for the conform decision in the second case, i.e. if the minority juror announced
a conform signal (p-value=0.046). Our equilibrium prediction for Model I is thus clearly violated.
Table 4: Voting behavior by majority types in Endo-Het
Own conform
Conform message sent by
Vote for
Wilcoxon
signed-rank test
signal
own type
other type
conform decision
0
0
0
4.0%
0
1
0
16.7%
0
0
1
12.4%
1
0
0
44.4%
1
1
0
97.2%
1
0
1
100.0%
p=0.393
p= 0.046
The unit of independent observation is the matching group.
4.2
Communication stage results
Table 5 shows average lying rates based on individual averages in the Endo treatments; unconditional and conditional on the signal received and on being either a majority or a minority
type in the Endo-Het treatment. Lying rates are generally very low, in a way that is roughly
compatible with the prediction of Model II. We find little evidence for babbling by minority
subjects as predicted by Model I. In Endo-Het, the lying rate of minority types is only 11.4%,
while babbling would correspond to 50%. In Endo-Het, lying rates conditional on a contrary signal
are substantially larger than for conform signals (16.8% vs. 8.5%). A Mann-Whitney-Wilcoxon
test finds marginally significant difference between lying rates conditional on a contrary signal
in Endo-Hom and Endo-Het (based on matching group averages, p-value= 0.097) and an even
stronger effect comparing these lying rates in Endo-Hom and Endo-Het minority (p-value of 0.049).
16
Table 5: Lying rates in Endo treatments
Endo-Hom
Endo-Het
5.7
8.5
contrary signal
10.2
16.8
conform signal
0.7
0.5
general lying rate
5
lying rate in minority
11.4
contrary signal
21.9
conform signal
1.0
lying rate in majority
7.7
contrary signal
14.9
conform signal
0.5
A model of cognitive heterogeneity
Average behavior is thus not in line with the predictions of the two models that we have examined,
both of which assume no heterogeneity among agents of a given preference type. We now develop
a model that assumes cognitive constraints and heterogeneity across agents along that dimension.
A popular model that addresses cognitive contraints is the Quantal Response model proposed in
McKelvey and Palfrey (1995, 1998), but the latter does not feature heterogeneity in that regard.
Though the heterogeneity aspect could in principle be added to the Quantal-Response model, this
would likely make it unsolvable, the latter being already very difficult to solve in the absence of this
extra dimension. See for example Guarnaschelli et al. (2000) for a discussion of problems arising
in applying the model to a communication and voting game involving jurors with homogeneous
preferences.
We shall assume three general categories of agents: random agents, naive agents and reasoning
agents. Random agents act purely randomly. Heuristic agents obey a simple heuristic rather
than optimizing, and correspond to the subjects described in the experiment of Goeree and Yariv
(2011). Their behavior rule is truth-telling and vote with the majority of messages announced.
Reasoning agents optimize by best-responding, under constraints, to correct beliefs about others’
strategies. They however act noisily and the set of strategies they are able to envisage is limited.
Both of these constraints are however relaxed as their degree of cognitive sophistication increases.
An agent thus now has a preference type and a cognitive type (random, heuristic or reasoning),
the cognitive type being privately known.
In the simplest special case of the model, any reasoning agent is perfectly rational. It is easily
17
seen that if the likelihood of being a random or a rational agent is low enough, there exists an
equilibrium in which reasoning subjects always announce a conform signal and vote sincerely. In
such a case, a reasoning agent can be confident that he faces two heuristic agents. There are are
thus two rational lying scenarios in the above equilibrium. A lie by a minority type holding a
contrary signal allows him to beneficially bend the decision rule applied by heuristic majority
types. A lie by a majority type holding a contrary signal allows him to manipulate other majority
subjects who wrongfully apply Λ(0) instead of Λ(1).
The above special case neglects the diversity in cognitive ability across reasoning agents.
Consider the equilibrium described above. In order to identify the profitable lying opportunity, a
majority type subject must first wish to implement his optimal decision rule Λ(1) or one close
to the latter. Second, he needs to acknowledge that the average decision rule applied by other
agents of the majority type is Λ(0). Third, he must understand when to lie so as to favorably
bend the decision rule. The requirements are simpler for a minority type subject. He needs to
wish to implement his optimal decision rule Λ(1), but he has an incentive to lie whether or not he
believes the other preference type’s average decision rule to be Λ(1) or Λ(0). Lying in majority is
thus more complex than lying in minority. If agents differ in their cognitive sophistication, some
subjects might perceive lying incentives in both scenarios, others only in the simplest scenario
(minority), yet others in none.
We now formally introduce our model of cognitive heterogeneity (Model III). There are
three cognitive types: random agents, heuristic agents and reasoning agents, with respective
probabilities {, 1 − α − , α}. Cognitive types are privately known. Nature draws a publicly
observed heterogeneous committee composition specifying whether there are two blue-biased types
or two red-biased types in the committee (call this the heterogeneous treatment). The game is
then played according to the timing described earlier.
A reasoning agent i has a privately known sophistication level s(i) drawn from a distribution f
with full support over [0, 1]. A strategy σ for the game is composed of a communication strategy
conditional on being a majority type (σ(majority)), a communication strategy conditional on
being a minority type (σ(minority)) and a decision rule for the minority and majority cases. For
the sake of simplicity, we assume that only strategies involving decision rules Λ(1) or Λ(0) are
available. Let σ0 (τ ) specify truthtelling after a contrary and a conform signal given τ . Let σ1 (τ )
specify always announcing a conform signal given τ .
Each τ -communication strategy σ(τ ), for τ ∈ {majority, minority} has a complexity level k(σ(τ )) ∈ [0, 1] and each decision rule Λ(x) has a complexity level k(Λ(x)).
The complexity of a given strategy σ = (σ(majority), σ(minority), Λ(x)) is given by
max {k(σ(majority)), k(σ(minority)), k(Λ(x))}. A reasoning or heuristic agent picks one strategy
18
σ and plays it with a consistency level given by probability c > .5, while he otherwise randomizes
with equal probability among all available strategies. A so-called noisy strategy for such an agent
is a picked strategy σ and a consistency level c. A heuristic agent always picks the strategy
truth-tell and apply Λ(0). A random agent randomizes with equal probability among all available
strategies.
We now present an equilibrium concept that remains conceptually identical to the classical
Perfect Bayesian equilibrium concept and simply adds a restriction on the strategies that can be
played by a reasoning agent. A symmetric equilibrium E is given by a set of reasoning agents’
noisy strategies {(σs , cs ) |s ∈ [0, 1]} , s.t. for each s ∈ [0, 1]:
1) k(σs ) ≤ s.
2) For a reasoning agent of sophistication level s, it holds true that at every information set,
the noisy strategy (σs , cs ) weakly payoff dominates any other noisy strategy (σ 0 , cs ) satisfying
k(σ 0 ) ≤ s, given beliefs derived by Bayes’ rule (whenever possible) and assuming that others play
their assigned equilibrium strategy.
3) cs = g(s) ∈ (0, 1), where g is strictly increasing in s, with g(0) ≥ .5.
Condition 1) states that the agent’s picked strategy must belong to the set of strategies that
he is able to consider given his sophistication level. Condition 2) states that at every information
set at which the agent is called upon to act, there is no advantageous deviation within the set of
strategies that he can consider given his sophistication level. Condition 3) pins down the level
of consistency with which an agent plays his picked strategy. The function g is a parameter of
the model. It is increasing in s so as to capture the notion that the higher s, conditional on 1)
and 2) being satisfied, the more convinced the agent is about the strategy that he picks. The
function g also captures the fact that agents make errors. A similar assumption underlies the
well known Quantal-Response model McKelvey and Palfrey (1995, 1998) as well as the simpler
Noisy Nash Model considered in Guarnaschelli et al. (2000). As the Quantal Reponse model,
our model incorporates a rational expectation dimension, as individuals best respond to correct
beliefs about others’ (noisy) strategies. As in the Noisy Nash Model, the likelihood of an error
and the distribution over strategies conditional on an error are however independent of the payoffs
attached to different strategies.
e for the above model, assuming and α to be small,
We now propose an equilibrium scenario E
so that a reasoning agent is essentially sure that he faces two heuristic agents. First, let the picked
strategy of all reasoning agents specify truth-telling after a conform signal. Let there be three
ordered intervals of sophistication levels defined by two thresholds 1 > t2 > t1 > 0. The picked
strategy of reasoning agents belonging to the highest interval specifies lying after a contrary signal
19
in both majority and minority and applying Λ(1). The picked strategy of agents in the second
most sophisticated group involves lying after a contrary signal only in minority. This second group
furthermore divides into two subgroups, where the more sophisticated subgroup applies Λ(1) and
the less sophisticated applies Λ(0). The picked strategy of the third group of reasoning agents
specifies truth-telling after a contrary signal in both minority and majority and application of
Λ(0).
Consider now the case of a homogeneous committee featuring either only red-biased agents
or only blue-biased agents (call this the heterogeneous treatment). Lying in such a committee
seems even less intuitive than lying as a majority type in a heterogeneous committee. Consider an
b of the following form, given α, low. There is a threshold t3 ∈ (0, 1) s.t. reasoning
equilibrium E
agents with a sophistication level above t3 pick the strategy lie after a contrary signal, truth-tell
after a conform signal, apply Λ(1), while remaining reasoning agents pick a strategy involving
always truth-telling. The group of agents with a sophistication below t3 divides into two subgroups,
where the more sophisticated subgroup applies Λ(1) and the less sophisticated applies Λ(0). Let
σB be the communication strategy specifying lying after a contrarian signal and truth-telling after
a conform signal. Let σA specify truth-telling after both types of signals.
We may now state the following existence result for the two types of committees considered
above (heterogeneous and homogeneous).
e is an equilibrium
Proposition 3. 1. Assume the heterogeneous treatment. For α, small enough, E
scenario if k(σ0 (minority)) = k(σ0 (majority)) = k(Λ(0)) = 0 and 0 < k(σ1 (minority)) <
k(Λ(1)) < k(σ1 (majority)) < 1. 2. Assume the homogeneous treatment. For α, small enough,
b is an equilibrium scenario if the complexity order assumed in Proposition 3 is satisfied and
E
furthermore, 1 > k(σB ) > k(σ1 (majority)).
We omit a proof of the above results, the latter being trivial given that a reasoning agent is
sure to face two heuristic agents for 1 − α large enough. Our ranking of complexities reflects the
fact that we consider lying in minority very intuitive and lying in majority very unintuitive. As to
the optimal decision rule, we find its complexity intermediate. Figure 6 in the appendix gives
a graphical illustration of the scenario described in Point 1. Figure 7 in the appendix gives a
graphical illustration of Point 2.
Given the law of large numbers, the lying rate of agents should roughly coincide with the
consistency level specified by their noisy strategy. The lying behavior of individuals should thus be
predictive of their degree of sophistication. Agents who lie consistently in majority and minority
are almost exclusively highly sophisticated reasoning agents. Agents who lie only in minority are
almost exclusively moderately sophisticated agents. Finally, agents who lie only in minority are
almost exclusively random agents. Also, given the assumptions made in Points 1 and 2 regarding
20
b are on average more
the complexity of strategies, agents who pick the strategy (σB , Λ(1)) in E
e
sophisticated than agents who pick the strategy (σ1 (majority), σ1 (minority), Λ(1)) in E.
Given the consistency function g, expected lying rates (and thus actual lying rates) across
categories of consistent liers should thus exhibit a particular ordinal structure. First, the expected
lying rate of agents who lie in majority and minority in heterogeneous committees is higher
than that of those who lie only in minority. Second, the expected lying rate in minority of
agents who lie only in minority is higher than the expected lying rate in majority of agents
who lie only in majority. Furthermore, the expected lying rate of agents who lie consistently
in homogeneous committees is higher than that of agents who lie consistently in majority and
majority in homogeneous committees.
Proposition 3 also has implications for voting behavior as a function of communication behavior.
Point 2 implies that subjects who lie consistently in the homogeneous treatment should have a lower
threshold than subjects who lie consistently in both majority and minority in the heterogeneous
treatment. Also, in the heterogeneous treatment, subjects who lie consistently in majority and
minority should have a lower threshold than remaining subjects (who lie only in minority or only
in majority or neither in majority nor minority). Finally, subjects who lie only in minority should
have a lower threshold than subjects who never lie consistently. However, if we assume that
k(Λ(1)) is very close to k(σ1 (majority)), both categories may have the same threshold in that all
members of both categories pick the decision rule Λ(0).
6
Individual behavior
Our empirical strategy for testing Model III is twofold. We first study voting behavior and show
that choices in the decision stage as well as in the communication stage reflect heterogeneous
behavior. We then study communication behavior and show that lying on average allowed subjects
to increase their payoffs, as predicted. A more detailed analysis of lying rates reveals that their
ordering roughly replicates the predictions of Model III. We finally examine the relation between
communication behavior in the treatment and voting behavior in the treatment as well as in the
individual decision test.
6.1
Decision stage results
Our main finding here is that a small but not insignificant fraction of agents applies a decision
rule that is significantly closer to the optimal one.
Figure 3 summarizes voting behavior in the experiment across all four treatments. We focus
on voting behavior given one conform signal as this is the information set showing the highest
21
discrepancy between theoretical prediction and actual experimental behavior. The figure shows
the empirical cumulative distribution of the frequency of choosing the conform decision conditional
on one conform signal. The figure is constructed as follows. Any subject has an average frequency
x of choosing the conform decision after observing exactly one conform signal. In the Endo
treatments, we use the subject’s own signal and assume truthful messages by the remaining two
committee members. For any such frequency x, the figure indicates the proportion of individuals
across all treatments whose frequency is lower than x. The figure reveals that 32.3 % of subjects
vote for the conform jar with an average frequency of at least 50%, when exposed to exactly one
conform signal.
Fig. 3: Empirical cumulative distribution of frequency of conform vote conditional on one conform
0
empirical cumulative distribution
.6
.2
.4
.8
1
signal/message (pooled over treatments)
0
.2
.4
.6
.8
1
Average individual decision to vote for conform jar given one conform signal
The first column of Table 6 presents results of the individual decision test (IDT). Merging
subjects across treatments, it indicates the proportion of individuals applying each possible
threshold j-decision rule (0,1,2, or 3). Approximately 70 % of subjects share the same suboptimal
threshold of 2, i.e. require at least two conform balls to choose the conform decision. The remaining
30 % mostly have a threshold of 1, corresponding to the optimal decision rule. Results are pooled
over treatments since the proportion of individuals applying each IDT threshold does not differ
significantly between treatments (chi2-test, p-value of 0.19).
In Table 6, the columns named # of conform messages 0-3 show for each IDT threshold (from
0 to 3) the average frequency of voting for the conform jar for each possible signal profile, merged
across treatments.9 The table reveals that behavior in the IDT significantly relates to behavior in
the treatments. Subjects with an IDT threshold of 1 vote for the conform jar after one conform
signal much more frequently than subjects with an IDE threshold of 2 (63% vs 24%). Subjects
9
As done for Figure 3 for the Endo treatments, we add up the number of conform messages sent by others and
the signal held by the subject.
22
with an IDT threshold of 3 exhibit idiosyncratic behavior, but they are very few (only 6).
Table 6: Proportion of conform votes in % conditional on threshold from IDT and signal profile
# of conform signals
IDT threshold
proportion
0
1
2
3
0
0.78
66.7
54.2
100.0
100.0
1
27.34
11.91
63.2
97.8
99.81
2
70.31
7.92
24.2
97.7
99.52
3
1.56
36.03
38.4
69.1
72.2
The superscripts indicate that these cases feature fewer
observations since some subjects were never exposed to
the concerned signal profile. The numbers of subjects in
1, 2 and 3 are respectively 102, 262 and 5.
Finally, Regression (5) in Table 3 corroborates the above insights. The significant negative
coefficient of the IDT threshold in regression (1)and (2) shows that the test is predictive of voting
behavior in the treatment. The higher a subject’s IDT-threshold, the less likely he is to vote for
the conform jar.
6.2
Communication stage results
Our main finding here is that the low average lying rate reflects heterogeneous behavior distinguishing consistent truth-tellers and consistent liars. We find evidence that consistent liars are
cognitively sophisticated subjects whose decision rule is closer to the optimal decision rule than
average and who understand the advantage of lying.
The low average lying rate might reflect either homogeneous or heterogeneous individual
behavior. Either almost all subjects lie rarely in line with the average rate or a small but
significant fraction of subjects consistently lies while remaining agents consistently truth-tell. We
show that the first answer applies to the case of conform signals while the second applies to the
case of contrary signals.
In Table 7, the columns denoted by subjects lying ≥ 1 give the average lying rate for subjects
who lied at least once within the same category. We first focus on the lying rate conditional on a
contrary signal. In the Endo-Hom treatment, the unconditional lying rate is 10.2% and increases to
44.55% conditional on having lied at least once (abbreviated l ≥ 1 in what follows). In Endo-Het
minority, the rate increases from 21.9% to 63.04% conditional on l ≥ 1. In Endo-Het majority,
23
it increases from 14.9% to 44.71% conditional on l ≥ 1. A significant proportion of individuals
lies at least once in all three treatment groups (22.92% in the Endo-Hom treatment, 33.33% in
Endo-Het minority and 33.33% in Endo-Het majority). Figure 4 provides further detail regarding
lying behavior after contrary signals. We define consistent liars at a particular information set
as subjects who lie at least 50% of the time. The fraction of such agents is 8.3% in Endo-Hom,
14.06% in Endo-Het majority and 24.5% in Endo-Het minority.
Table 7: Heterogeneous lying rates in Endo treatments in %
Endo-Hom
Endo-Het
all
lying ≥ 1*
share
all
lying ≥ 1*
share
general lying rate
5.7
23.9
24.0
8.5
18.2
46.9
contrary signal
10.2
44.5
22.9
16.8
37.5
44.8
conform signal
0.7
22.3
3.1
0.5
10.5
5.2
lying rate in minority
11.4
31.9
35.4
contrary signal
21.9
63.0
33.3
conform signal
1.0
29.8
3.1
lying rate in majority
7.7
22.4
34.4
contrary signal
14.9
44.7
33.3
conform signal
0.5
15.9
3.1
* The subset contains in each category only subjects that lie at least once.
0
empirical cumulative distribution
.2
.8
.6
.4
1
Fig. 4: Empirical cumulative distribution of lying rates in Endo treatments
0
.2
.4
.6
average lying rate
Endo-Hom
Endo-Het majority
.8
1
Endo-Het minority
The profile of lying rates is very different after conform signals. In all three considered
treatment groups, a very low proportion of individuals lies at least once after a conform signal
24
(3.1% in all three cases). There is thus a marked difference in the profile of individual lying
behavior conditional on either contrary or conform signals. The low aggregate lying rate after
contrary signals reflects heterogeneous behavior to the extent that in both Endo treatments a
significant fraction of subjects lies frequently given such a signal. The low aggregate lying rate
after conform signals instead reflects homogeneous behavior, the fraction of frequent liars being
negligible. We conjecture that the observed lying behavior reflects the fact that a small fraction of
sophisticated subjects voluntarily seizes the profitable lying opportunity that it correctly identifies.
We provide some evidence in support of this in what follows.
Lying and expected payoffs Table 8 indicates average payoffs of subjects conditional on
holding a contrary signal and either lying or not lying. We distinguish between Endo-Hom,
Endo-Het majority and Endo-Het minority subjects. The table reveals that conditional on a
contrary signal, subjects who lie earn more on average than subjects who do not.
Table 8: Average profits given a contrary signal
Endo-Hom
Endo-Het majority
Endo-Het minority
Lie
59.04
59.58
57.14
No Lie
51.11
46.83
46.78
15.50%
27.21%
22.15%
Difference
Lying implies a coordination problem. If two subjects of the same preference type and both
holding a contrary signal lie simultaneously, the triggered shift in the decision rule will be excessive.
To study this problem empirically, we identify all Endo treatment aggregate signal realizations in
which two subjects of the same preference type (the majority type) hold a contrary signal. For this
subset of rounds, we compare the average group profits of majority types for the cases featuring
respectively no lie, one lie and two lies. Table 9 confirms that profits decrease significantly when
going from one lie to two lies. Crucially, however, two simultaneous lies only happen extremely
rarely, i.e. in less than 2% of cases, this being a trivial consequence of the low aggregate lying rate
and of the small committee size. For all practical purposes, a subject lying at the communication
stage can thus legitimately assume to be the only one lying, as in the unilateral deviation scenario
in the putative truthful-sincere equilibrium studied in Coughlan (2000).
25
Table 9: Profits of majority types per number of lies by majority types after a contrary signal
Endo-Hom
Endo-Het
No lie
1 lie
2 lies
Average profit
36.17
43.55
35.00
Share of obs.
76.97%
21.35%
1.69%
Average profit
35.54
39.41
20.00
Share of obs.
69.14%
29.14%
1.71%
Ordinal structure of lying rates Recall that our equilibrium for Model III (Point 1,
heterogeneous treatment) predicts a specific ordering of lying rates across subjects who lied
consistently either both in minority and majority, or only in minority or only in majority. The
first group ought to have a higher lying rate in minority than the second and the second ought
to have a higher lying rate in minority than the second in majority. The intuition being that
these groups are ordered in a decreasing order of cognitive sophistication. Point 2 of Proposition 3
furthermore predicts that consistent liers in majority and minority in the heterogeneous treatment
should have a lower lying rate in each case than consistent liers in the homogeneous treatment.
The intuition here is again that the second group is more sophisticated on average than the first.
Table 10: Lying shares and voting decision by lying types in Endo treatments
Category & theory equivalent
Het
Obs
Minority
Majority
Vote conform
IDT
C1 (s ≥ t2 )
11
80.2%
77.6%
66.9%
1.6
C2 (s ∈ [t1 , t2 ])
12
71.5%
10.4%
49.5%
1.8
3
6.7%
55.7%
42.6%
1.67
66
3.9%
3.5%
23.5%
1.80
C3 (random)
C4 (heuristic and s < t1 )
Hom
5: s ≥ t3
9
85.6%
70.3%
1.1
6: s < t3
87
2.4%
25.1%
1.8
Note: In Endo-Het, we could not categorize 4 subjects because they did not receive a
contrary signal in minority.
In order to test the predictions of the model, we classify Endo-Het treatment subjects into four
categories depending on whether they consistently lied after a contrary signal when respectively
in minority and in majority, consistent lying being defined as lying more than 50% of the time.
Category 1 subjects lied consistently in both scenarios. Category 2 and 3 lied consistently
respectively only in minority or majority. Category 4 did not lie consistently in any of the two
26
scenarios. Similarly, we classify Endo-Hom treatment subjects into two categories depending on
whether they consistently lied after a contrary signal. C5 subjects lied consistently, C6 subjects
did not. We abbreviate category x by Cx. In what follows, we shall see that the structure of lying
rates across these categories echoes the predictions of Model II, as appearing in Propositions 4
and 5.
Table 10 gives lying shares in Endo-Het and Endo-Hom after a contrary signal. For Endo-Het,
the table indicates lying shares in respectively minority and majority for each of the four categories
of subjects C1 to C4. For Endo-Hom, it indicates lying rates for categories C5 and C6.
Results for Endo-Het replicate the ordering of lying rates predicted by Proposition 3, Point
1. First, the lying rate of C1 subjects in minority (80.2%) is higher than that of C2 subjects in
minority (71,5%). Second, C2 subjects’ lying rate in minority is significantly higher than that
of C3 subjects in majority (55.7%). The low number of C3 subjects is also compatible with our
assumption that random agents have a low ex ante probability. Finally, C4 agents have a very low
average lying rate in both minority and majority, which is compatible with these being mostly
heuristic agents who almost always truth-tell.
Results for Endo-Hom replicate the prediction of Proposition 3, Point 2. Subjects who
lie consistently in Endo-Hom have a higher lying rate than C1 Endo-Het subjects (85.6% vs
(80.2%, 77.6%)).
Lying and the Strategic Communication Test After the Endo treatments, subjects took
a strategic communication test (SCT) aimed at checking their understanding of strategic lying.
We wish to establish how behavior in the SCT relates to communication behavior in the treatment.
Our conjecture is that better performance in the SCT correlates positively with lying more in the
treatment given contrary signals. The rationale is that subjects who understand strategic lying
well ought to be in a good position to seize the lying opportunities arising in the game.
Recall that in the SCT, it was made clear that the decision rule applied by subjects’ automatized
alter egos was their optimal decision rule. In the SCT taken by Endo-Het treatment subjects
(which we call the Endo-Het SCT), lying after a contrary signal in minority was optimal, truthtelling being optimal in all remaining cases. In the SCT taken by Endo-Hom subjects (Endo-Hom
SCT), truth-telling was always optimal.
Table 11 shows SCT results. Almost all subjects truth-tell after a conform signal in both the
Endo-Het SCT and the Endo-Hom SCT, which is optimal and reveals a certain comprehension of
the task. A significant proportion of minority subjects in the Endo-Het SCT lies after a contrary
signal (17%). Surprisingly, a larger proportion of majority subjects in the Endo-Het SCT lies
after a contrary signal (31%) though this is suboptimal. The latter lying rate is not only large but
also larger than that of minority subjects. In the Endo-Hom SCT, a non-negligible proportion of
27
subjects lies after a contrary signal (12,5%), which is suboptimal just as in the majority Endo-Het
SCT. These two last results are puzzling.
An immediate caveat is the possible presence of an order-effect: the SCT was done after the
treatment, so that treatment experience may have affected behavior in the test. In other words,
treatment experience may have partially caused behavior in the SCT rather than both purely
being caused by the subject’s exogenous degree of cognitive sophistication. Recall that in the
Endo-Het treatment, lying in minority and majority was optimal. The SCT having been taken
after the treatment, relatively sophisticated subjects may thus have naively continued acting as
in the treatment. We consider it likely that a substantial proportion of sophisticated subjects
did not fully understand the SCT setup, in particular the fact that all subjects were replaced
by automatized alter egos behaving differently (i.e. better) than their real counterparts. For an
extensive discussion of SCT behavior in relation to Model III see appendix A.2.
Table 11: Lying shares in strategic communication test
Endo-hom
Endo-het
contrary signal (minority)
.
17.7%
conform signal (minority)
.
1.0%
contrary signal (majority)
12.5%
31.3%
conform signal (majority)
1.0%
0.0%
*Specifications in brackets only apply to Endo-Het.
6.3
Relation between communication behavior and voting behavior
We now want to examine the relation between communication behavior and voting behavior.
Model III indeed predicts a positive correlation between lying after a contrary signal and applying
a decision rule closer to the optimal decision rule. Both indeed correlate positively with the
underlying factor of cognitive sophistication.
Lying and the Individual Decision Test The column IDT in Table 10 examines IDT
behavior as a function of lying behavior in the treatments, i.e. for each of the six categories of
agents (C1 to C6). Results are in line with the predictions derived for Model III (Proposition 3).
C5 Endo-Hom subjects have a lower threshold than C1 Endo-Het subjects. C1 subjects have a
lower threshold than C2 and C4 subjects. Categories C2 and C4 have almost identical thresholds,
which is consistent with Proposition 3 if we assume that k(Λ(1)) is very close to k(σ1 (majority)),
implying that C2 and C4 reasoning subjects all pick a strategy involving decision rule Λ(0).
28
Finally, C3 subjects have a surprisingly low threshold, which is not in line with our prediction but
rather uninformative. According to Proposition 3 these are random agents. They are furthermore
very few (3 subjects) in line with our assumption that random agents are very rare.
Lying and treatment voting behavior We also examine treatment voting behavior as a
function of lying behavior. The column Vote for conform jar in Table 10 gives the frequency of
votes for the conform jar in Endo treatments after one conform signal10 for each of the categories
of agents C1 to C6. As for the IDT column, results are compatible with the predictions of
Proposition 3 for Model III. C1 subjects vote for the conform jar with a frequency of 67%. This
frequency decreases to roughly 40% for subjects in C2 and C3 and to 23% for C4 subjects. C5
subjects vote for the conform jar with a frequency of 70%. This frequency decreases to roughly
25% for C6 subjects.
7
Conclusion
We report results from a 2x2 experimental design aimed at understanding how deliberative
committees featuring known heterogeneous preference types decide, with a particular focus on the
potential role of social preferences and cognitive heterogeneity.
We find simple average behavior reflecting heterogeneous individual behavior consistent with
heterogeneous sophistication levels. Naive agents truth-tell and apply a moderated version of
their type specific optimal decision rule. Sophisticated agents, essentially always alone in a
given committee, lie in a way that allows them to favorably affect the implemented decision rule.
Observed play thus echoes the profitable unilateral deviation scenario arising in the hypothetical
truthful-sincere equilibrium analyzed by Coughlan (2000). Social preferences in contrast play no
role.
Our experiment has focused on straw votes and simple majority rule, both of which are
restrictive. Future experiments ought to examine different deliberation protocols (e.g. sequential,
repeated, subgroup-based) as well as different voting rules (e.g. qualified majority or unanimity).
The optimal protocol in real environments is likely to differ from that predicted by rational
choice models due to the presence of a large share of naive agents. For example, while Subgroup
Deliberation is found by Le Quement and Yokeeswaran (2015) to outperform Plenary Deliberation
in theory, this may not be true in practice because Plenary Deliberation implies more information
sharing than predicted by theory while the proposed theory of Subgroup Deliberation assumes
too complex behavior. Finally, though this experiment finds no role for social preferences, richer
deliberation processes (face to face, rich language-based, prolonged) are likely to affect preferences.
10
To count the total number of conform signals, we use the subject’s own signal and assume that others truth-tell.
29
Debate might in some cases stimulate empathy, solidarity and common identity while it may in
other cases reinforce in vs outgroup dichotomies and cause preference polarization.
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A
Additional analysis
A.1
Lying aversion
Lying and the Lying Aversion Test The post-experiment lying aversion test is a two-player
deception game where the sender’s decision to lie increases own payment independent of the
counterpart’s decision (for more details please refer to the original paper). In contrast to Gneezy
et al. (2013), any subject is assigned twice to a two-persons matching group and plays the game
once as a sender and once as a receiver. We only use the decision made by subjects when assigned
to the sender role. Another difference w.r.t. the original study is that we only let subjects play
the game once in any given role(sender or receiver). Our results of the lying aversion test are very
close to the original paper (see figure 5 for an overview of lying rates).
6
Fig. 5: Lying Aversion test
.458
.448
.427
.432
.891
.021
.06
.039
.042
.484
.021
.049
.065
.081
.466
.029
.034
.044
.029
.404
.036
.01
.018
.031
.375
.013
.013
.029
.01
.367
.013
.016
.016
.016
.026
1
2
3
4
5
6
1
2
Message sent by A
3
4
5
.487
Number
Lying behavior in the jury experiment may have been affected by agents’ lying aversion,
which was measured in the post-treatment lying aversion test. We run three different regressions.
32
Regression (1) is a discrete choice model with the dummy variable lie given a contrary signal as
dependent variable. In regressions (2) and (3), we use a linear regression and take as dependent
variable the number of lies during the 20 periods. In regression (2) we include all subjects, while in
regression (3) we only include subjects who lied at least once. Regressions (1) and (2) allow us to
test whether the independent variables influence respectively the probability to lie or the frequency
of lying over the 20 rounds. We in addition use the restriction at least one lie for regression (3), as
we conjecture that subjects who lied at least once were more likely to identify the lying incentive.
We use as independent variables the treatment dummy Het, Period to control for learning
effects, the dummy variable SCT 11 to check for comprehension of lying incentives, IDT treshold,
lying aversion and SVO.
We find that lying aversion exclusively influenced the behavior of those subjects who lied at
least once. The variable lying aversion has no significant influence in either regression (1) or
regression (2). As soon as we drop all non-liars from the regression in (3), we find that lying
aversion exerts a negative impact on the number of lies.
The regressions provide a few other insights. Regression (1) shows that consistent with our
non-parametric results, preference heterogeneity in the Endo treatment significantly increases the
probability to lie. We also find that lying increases over the course of the experiment (significant
coefficient of Period ). The variable with the largest impact is SCT. In addition, the higher the
threshold in the IDT, the less likely a subject is to lie, which is also consistent with findings
presented earlier. Social value orientation (SVO), as measured in the fourth post-experimental
task, has no significant influence.
The regressions provide a few other insights. Regression (1) shows that consistent with our
non-parametric results, preference heterogeneity in the Endo treatment significantly increases the
probability to lie. We also find that lying increases over the course of the experiment (significant
coefficient of Period). The variable with the largest impact is the SCT. In addition, the higher
the threshold in the IDT, the less likely a subject is to lie, which is also consistent with findings
presented earlier.
11
We here use the answer from our first question in the SCT. Recall that it was rational to lie in Endo-Het, but
not in Endo-Hom. The test is useful as a proxy for comprehension of lying incentives.
33
Table 12: Estimation on lying respectively number of lies
Dependent variable
Model
(1)
(2)
(3)
lie contrary signal
# of lies
# of lies
Marginal effects, logit
Linear reg
Linear reg
only positive # of lie
Het
0.0472**
0.302
-0.965*
(0.0194)
(0.304)
(0.555)
0.175***
4.477***
3.424***
(0.0181)
(0.696)
(0.773)
-0.0384*
-0.682
-1.276**
(0.0205)
(0.420)
(0.592)
-0.00186
-0.0293
-0.114**
(0.00154)
(0.0180)
(0.0415)
-0.00110
-0.0157
-0.00571
(0.000721)
(0.00942)
(0.0225)
2.317**
6.712***
(0.892)
(1.565)
1,978
192
65
# of groups (cluster)
32
32
27
# of individuals
192
192
65
0.450
0.464
Period
0.00199***
(0.000742)
SCT
IDT threshold
Lying Aversion
SVO
Constant
Obs.
Log likelihood
-433.14712
R-squared
Standard errors in parentheses
*** p<0.01, ** p<0.05, * p<0.1
Regression (1) includes random effects on the group and individual level.
In Regression (2) and (3) standard errors are clustered on matching group level.
A.2
Strategic Communication Test
One may define four cognitive classes of subjects based on performance in the SCT. Sophisticated
agents (Group I) are those who lie only in minority in the SCT. Agents demonstrating an
34
intermediate level of sophistication come in two variants. Group II lies in minority and majority
in the SCT. Group III never lies. Unambiguously unsophisticated agents (Group IV) lie only
in majority in the SCT. Lying only in minority means that an agent behaved optimally both
in minority and majority, which we take as evidence that the person understands the why and
how of strategic lying. In contrast, always or never lying implies that a subject made a mistake
in one scenario and acted optimally in the other, which we take as inconclusive evidence of
his cognitive sophistication. These agents may be heuristic subjects who simply follow a rule,
which we consider very likely in the case of systematic truth-tellers. Alternatively, they might
be moderately sophisticated agents who understood well the lying incentives in one scenario and
misunderstood them in the other. Lying only in majority means that an agent acted suboptimally
both in minority and majority, which we take as clear evidence that the person did not understand
strategic lying.
Table 13 provides lying rates in the Endo-Het treatment for each of the four classes defined on
the basis of behavior in the SCT.
Table 13: Endo-Het lying rates by responses in SCT
Group
I
II
III
IV
min & maj
min
maj
never
13
4
17
62
32.31%
16.25%
10.59%
2.50%
contrary signal
60.00%
41.74%
21.94%
4.69%
lying rate in minority
36.38%
37.50%
9.34%
4.96%
contrary signal
67.73%
66.67%
21.89%
9.03%
lying rate in majority
30.14%
8.93%
11.80%
1.80%
contrary signal
57.05%
25.00%
21.29%
3.66%
SCT lying
# of subjects
general lying rate
A preliminary observation is that for Groups I, II and III, behavior in Endo-Het replicates
behavior in the SCT: these subjects lie consistently exactly in those cases where they lied
consistently in the SCT. Group IV agents stand out. While they lied consistently (only) in
majority in the SCT, they lied consistently neither in minority nor in majority in the treatment.
Results for Group IV imply that the subgroup of individuals who lie consistently in the
treatment does not contain these clearly unsophisticated subjects. This subtracts a set of
particularly unsophisticated agents from the group of consistent liars in Endo-Het, thereby offering
indirect support for our cognitive hierarchy explanation of lying behavior.
Results for Group I provide additional evidence in favour of our cognitive hierarchy conjecture
35
by indicating that a fraction of consistent liars in Endo-Het minority is constituted by sophisticated
agents who understand the logic of lying. The fact that these agents were very few (only 4 subjects
in 96) however represents a significant caveat.
Results for Group II challenge our cognitive hierarchy conjecture if we assume that these
individuals are heuristic agents who always lie for reasons orthogonal to expected payoffs. In
such a case, we must conclude that a substantial proportion of consistent liars in the treatment
consists of heuristic (as opposed to cognitively sophisticated) agents. If one instead agnostically
defines Group II subjects as moderately sophisticated subjects, results for this group are better
reconcilable with our cognitive hierarchy conjecture. Note again that the order effect potentially
exhibited by the data clearly plays against our cognitive hierarchy conjecture here because it may
induce us to underestimate the sophistication of Group II subjects. In the Endo-Het treatment,
lying in minority and majority was optimal. The SCT having been taken after the treatment,
relatively sophisticated subjects may thus have naively continued acting as in the treatment. We
consider it likely that a substantial proportion of sophisticated subjects did not fully understand
the SCT setup, in particular the fact that all subjects were replaced by automatized alter egos
behaving differently (i.e. better) than their real counterparts.
An important caveat is the possible presence of an order-effect: the SCT was done after the
treatment, so that treatment experience may have affected behavior in the test. In other words,
treatment experience may have partially caused behavior in the SCT rather than both purely
being caused by the subject’s exogenous degree of cognitive sophistication. Recall that in the
Endo-Het treatment, lying in minority and majority was optimal. The SCT having been taken
after the treatment, relatively sophisticated subjects may thus have naively continued acting as
in the treatment. We consider it likely that a substantial proportion of sophisticated subjects
did not fully understand the SCT setup, in particular the fact that all subjects were replaced by
automatized alter egos behaving differently (i.e. better) than their real counterparts.
We do not repeat the above exercise for Endo-Hom subjects as difficulties arise in disentangling
cognitive types on the basis of the SCT.
36
B
Additional Tables and Figures
e in Model II
Fig. 6: Equilibrium E
0
k(σ1 (min))
0
t1
k(λ(1))
k(σ1 (maj))
1
t2
1
σ0 (min),
σ1 (min),
σ1 (min),
σ1 (min),
σ0 (maj),
σ0 (maj),
σ0 (maj),
σ1 (maj),
λ(0)
λ(0)
λ(1)
k
s
λ(1)
b in Model II
Fig. 7: Equilibrium E
0
k(λ(1))
0
C
k(σB )
1
t
1
σA ,
σA ,
σB ,
λ(0)
λ(1)
λ(1)
k
s
Instructions
We print instructions for the Exo-Hom blue-biased type (B.1) and for the Endo-Hom blue-biased
type (B.2) treatments. Aspects where the instructions differ for red-biased types are indicated in
round brackets. Aspects where the instructions differ for heterogeneous groups are indicated in
square brackets. Instructions for post-experimental tests are available upon request.
37
C.1
Instructions Exo-Hom blue-biased type
General explanations for the participants
You are taking part in an economic experiment. Please read the following instructions carefully. You can
earn money in this experiment. Your payment will depend on your decisions and on the decisions of the
other participants.
During the experiment communication is prohibited. Failure to comply will result in exclusion from the
experiment and loss of earnings. Should you have any questions, please address them to us: hold your
hand out of the cabin and one of the experimenters will come to your seat.
At the end of the experiment, all sums of money will be paid to you in cash. During the experiment
monetary amounts do not correspond to Euro, but to points. In the end, the total point earnings that you
obtained during the experiment will be converted into Euro, where: 150 points = 1 Euro.
The study consists of four parts:
Part 1. Control Questions: you are asked to answer control questions to check comprehension.
Part 2. Experiment: The experiment consists of several parts. Your earning from all parts will be paid.
(1) The instructions for Part 1 can be found below.
(2) You will receive the instructions for the other parts later.
Part 3. End: After the experiment you will receive a questionnaire with general questions. Please fill
this out carefully.
Part 4. Payment: You will receive the payment privately. The other participants will not know the
amount of your payment.
Instructions Experiment Part 1
Part 1 of the experiment consists of 20 rounds. [At the beginning of the experiment, you will be
randomly assigned to a type, type A or type B. The type allocation is maintained throughout the
experiment.] In each round, all participants will be divided into groups of 3 participants randomly. [Per
group there are either two Type A-participants and a Type B-participant or a Type A-participant and two
type B-participants. You will be informed about the group composition at the beginning of each round.]
The group allocation is renewed at the beginning of each round. Therefore the group composition
changes in each round.
In the experiment you have the task to vote for one of two jars. There are two possible jars, which we
call the Red and the Blue Jar. The Red Jar contains 7 red balls and 3 blue balls. The Blue Jar contains 7
blue balls and 3 red balls.
At the beginning of the game one of the two jars will be selected for your group at random. The
probability that the Red Jar is selected is 50%. The probability that the
38
Blue Jar is selected is also 50%. You will not be told which Jar was selected. In Figure 1 you see the Red
and the Blue Jar. Figure 2 displays the image of the unknown jar.
Figure 1: Red and Blue Jar
Figure 2: Unknown Jar
As information you will receive the color of three randomly drawn balls from the jar (see Figure 3). In
three drawings one ball will be randomly drawn from the jar each time. Each drawing is carried out in
two steps:
1. A ball is drawn from the jar.
2. The color is written down and the ball is immediately thrown back into the jar.
The number of balls in the jar thus remains the same at each draw. There are three drawings to obtain
three balls. Each participant in your group receives the same three balls as information.
Differently colored balls may be drawn from the jar. However, all the balls are drawn from the same jar.


When the Red Jar is selected for your group, each time a ball is drawn from a jar that contains 7 red
balls and 3 blue balls.
When the Blue Jar is selected for your group, each time a ball is drawn from a jar that contains 7
blue balls and 3 red balls.
Figure 3: Example for ball draw
After the ball draw the vote takes place. The vote is governed by the following rules:
39


If the majority of participants votes for the Red Jar, your group decision is the Red Jar. If there are 2
to 3 votes for the Red Jar and 0 to 1 votes for the Blue Jar, the group decision is therefore the Red
Jar.
If the majority of participants votes for the Blue Jar, your group decision is the Blue Jar. If there are 0
to 1 votes for the Red Jar and 2 to 3 votes for the Blue Jar, the group decision is therefore the Blue
Jar.
The payment you receive for the group decision depends on the accuracy of your group decision and of
the actual jar.



If your group decision corresponds to the selected Jar and the actual jar is the Red Jar, then you will
receive 40 (160) points.
If your group decision corresponds to the selected Jar and the actual jar is the Blue Jar, then you will
receive 160 (40) points.
If your group decision does not correspond to the selected jar, then you will receive 10 points.
Table 1: Payments
Number of votes for
Red Jar
Number of votes for
Blue Jar
Group Decision
Actual Jar
Payment
[Type A]
[Payment
Type B]
2 or 3
0 or 1
Red Jar
Red Jar
40 (160)
[160]
2 or 3
0 or 1
Red Jar
Blue Jar
10
[10]
0 or 1
2 or 3
Blue Jar
Red Jar
10
[10]
0 or 1
2 or 3
Blue Jar
Blue Jar
160 (40)
[40]
After all participants have voted, the votes will be counted and you will be informed about the outcome
of the vote, i.e. votes for Red Jar, votes for Blue Jar, group decision, actual color of the jar and your
payment. After the end of the round you will be assigned into new randomly selected groups and the
next round begins.
You will receive the payments from all 20 rounds.
If you have questions about the experiment, please contact us now.
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C.2
Instructions Endo-Hom blue-biased type
General explanations for the participants
You are taking part in an economic experiment. Please read the following instructions carefully. You can
earn money in this experiment. Your payment will depend on your decisions and on the decisions of the
other participants.
During the experiment communication is prohibited. Failure to comply will result in exclusion from the
experiment and loss of earnings. Should you have any questions, please address them to us: hold your
hand out of the cabin and one of the experimenters will come to your seat.
At the end of the experiment, all sums of money will be paid to you in cash. During the experiment
monetary amounts do not correspond to Euro, but to points. In the end, the total point earnings that you
obtained during the experiment will be converted into Euro, where: 150 points = 1 Euro.
The study consists of four parts:
Part 1. Control Questions: you are asked to answer control questions to check comprehension.
Part 2. Experiment: The experiment consists of several parts. Your earning from all parts will be paid.
(1) The instructions for Part 1 can be found below.
(2) You will receive the instructions for the other parts later.
Part 3. End: After the experiment you will receive a questionnaire with general questions. Please fill
this out carefully.
Part 4. Payment: You will receive the payment privately. The other participants will not know the
amount of your payment.
Instructions Experiment Part 1
Part 1 of the experiment consists of 20 rounds. [At the beginning of the experiment, you will be
randomly assigned to a type, type A or type B. The type allocation is maintained throughout the
experiment.] In each round, all participants will be divided into groups of 3 participants randomly. [Per
group there are either two Type A-participants and a Type B-participant or a Type A-participant and two
type B-participants. You will be informed about the group composition at the beginning of each round.]
The group allocation is renewed at the beginning of each round. Therefore the group composition
changes in each round.
In the experiment you have the task to vote for one of two jars. There are two possible jars, which we
call the Red and the Blue Jar. The Red Jar contains 7 red balls and 3 blue balls. The Blue Jar contains 7
blue balls and 3 red balls.
At the beginning of the game one of the two jars will be selected for your group at random. The
probability that the Red Jar is selected is 50%. The probability that the
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Blue Jar is selected is also 50%. You will not be told which Jar was selected. In Figure 1 you see the Red
and the Blue Jar. Figure 2 displays the image of the unknown jar.
Figure 1: Red and Blue Jar
Figure 2: Unknown Jar
As information you will receive the color of three randomly drawn balls from the jar (see Figure 3). In
three drawings one ball will be randomly drawn from the jar each time. Each drawing is carried out in
two steps:
1. A ball is drawn from the jar.
2. The color is written down and the ball is immediately thrown back into the jar.
The number of balls in the jar thus remains the same at each draw.
Figure 3 shows that two cases can occur. You either will be shown a red ball (case 1) or a blue ball (case
2).
Figure 3: Example for randomly drawn ball
Case 1) A red ball was drawn
Case 2) A blue ball was drawn
Differently colored balls may be drawn from the jar to the participants of the same group. However, all
the balls are drawn from the same jar.


When the Red Jar is selected for your group, each time a ball is drawn from a jar that contains 7 red
balls and 3 blue balls.
When the Blue Jar is selected for your group, each time a ball is drawn from a jar that contains 7
blue balls and 3 red balls.
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Now there is an information stage. You will send a message about the color of the ball that was shown to
you to the other participants in your group. You can choose the content of the message independently of
the actual color of the ball (see figure 4).
Figure 4: Message information stage
After you have sent the message, you receive the message of all the other participants of your group
(see figure 5). In total, you see 3 messages, the messages of the other two participants and your own
message.
Figure 5: Example for results of information stage
Row with types only in
heterogeneous
treatment
After the information stage the vote takes place. The vote is governed by the following rules:


If the majority of participants votes for the Red Jar, your group decision is the Red Jar. If there are 2
to 3 votes for the Red Jar and 0 to 1 votes for the Blue Jar, the group decision is therefore the Red
Jar.
If the majority of participants votes for the Blue Jar, your group decision is the Blue Jar. If there are 0
to 1 votes for the Red Jar and 2 to 3 votes for the Blue Jar, the group decision is therefore the Blue
Jar.
The payment you receive for the group decision depends on the accuracy of your group decision and of
the actual jar.

If your group decision corresponds to the selected Jar and the actual jar is the Red Jar, then you will
receive 40 (160) points.
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

If your group decision corresponds to the selected Jar and the actual jar is the Blue Jar, then you will
receive 160 (40) points.
If your group decision does not correspond to the selected jar, then you will receive 10 points.
Table 1: Payments
Number of votes for
Red Jar
Number of votes for
Blue Jar
Group Decision
Actual Jar
Payment
[Type A]
[Payment
Type B]
2 or 3
0 or 1
Red Jar
Red Jar
40 (160)
[160]
2 or 3
0 or 1
Red Jar
Blue Jar
10
[10]
0 or 1
2 or 3
Blue Jar
Red Jar
10
[10]
0 or 1
2 or 3
Blue Jar
Blue Jar
160 (40)
[40]
After all participants have voted, the votes will be counted and you will be informed about the outcome
of the vote, i.e. votes for Red Jar, votes for Blue Jar, group decision, actual color of the jar and your
payment. After the end of the round you will be assigned into new randomly selected groups and the
next round begins.
You will receive the payments from all 20 rounds.
If you have questions about the experiment, please contact us now.
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