#  Communication in the Infinitely Repeated Prisoner’s Dilemma: Theory and
Experiments††thanks: I thank Lisa Bruttel, Marco Caliendo, Sebastian Fehrler,
Juri Nithammer, Vasilisa Petrishcheva, Tobias Werner as well as participants
at the Economic Seminar at the University of Potsdam for very helpful comments
and suggestions. Birte Prado Brand provided excellent research support. This
research is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – project number 389447910, which I gratefully
acknowledge. This project was preregistered at OSF before the data collection
had started, see
https://osf.io/xtmwv/?view_only=81a71835f0ca44019f4897a059825409.

Maximilian Andres Universität Potsdam. Email<EMAIL_ADDRESS>
###### Abstract

So far, the theory of equilibrium selection in the infinitely repeated
prisoner’s dilemma is insensitive to communication possibilities. To address
this issue, we incorporate the assumption that communication reduces—but does
not entirely eliminate—an agent’s uncertainty that the other agent follows a
cooperative strategy into the theory. Because of this, agents still worry
about the payoff from cooperating when the other one defects, i.e. the
sucker’s payoff $S$, and, games with communication are more conducive to
cooperation than games without communication. This theory is supported by data
from laboratory experiments, and by machine learning based evaluation of the
communication content.

JEL-codes: C73, C92, D83
Keywords: belief, cooperation, communication, infinitely repeated game,
machine learning

##  1 Introduction

Infinitely repeated prisoner’s dilemma games are useful to study cooperation
in social dilemmas.111This game has been used to investigate cooperation, for
example, in oligopolies (see Mailath and Samuelson, 2006), in environmental
policy (see Harstad et al., 2022), in international relations (see Powell,
1993), and in interpersonal relationships (see Wilson and Wu, 2017).
Theoretical studies examine changes in the critical discount factor
$\delta^{pd}$ to determine institutions that are more or less conducive to
cooperation (see Blonski et al., 2011).

Experimental studies, however, suggest that this yields misleading results
(see Dal Bó and Fréchette, 2018, and the literature therein). This is because
the critical discount factor $\delta^{pd}$ does not account for the payoff
from cooperating when the other one defects, i.e. the sucker’s payoff $S$. The
critical discount factor $\delta^{pd}$ is based on the assumption that agents
are certain that the other agent follows a cooperative strategy. Hence, agents
should ignore the payoff from cooperating when the other one defects ($S$).
Blonski et al. (2011), however, show that agents do consider the sucker’s
payoff $S$ by demonstrating that changes in the sucker’s payoff $S$ do affect
the level of cooperation in laboratory experiments. The authors explain their
finding by proposing that agents are maximally uncertain about the other
agent’s strategies and, hence, do account for the sucker’s payoff $S$.222 In a
similar vein, Dal Bó and Fréchette (2018) point out that exclusively focusing
on changes in the critical discount factor $\delta^{pd}$ yields inaccurate
predictions for the level of cooperation in laboratory experiments. In a meta
study, the authors investigate the determinants of cooperation. For this aim,
they include all the experimental papers they know of satisfying certain
conditions. The stage game is a two by two prisoner’s dilemma with perfect
monitoring. The game is either one-shot or repeated using a random
continuation rule which does not change during a supergame. Also the pairing
of participants stays the same for a supergame. Additionally, the authors
included only their own papers or studies published before 2014 if they had
access to the data. The studies are: Andreoni and Miller (1993); Aoyagi and
Fréchette (2009); Blonski et al. (2011); Bruttel and Kamecke (2012); Cooper et
al. (1996); Dal Bó (2005); Dal Bó et al. (2010); Dal Bó and Fréchette (2011,
2019); Dreber et al. (2008); Duffy and Ochs (2009); Fréchette and Yuksel
(2017); Fudenberg et al. (2012); Kagel and Schley (2013); Sherstyuk et al.
(2013). Only recently, in multi-agent continuous time laboratory experiments,
Martínez-Martínez and Normann (2022) alternate the sucker’s payoff $S$ to
disentangle the effects of strategic uncertainty and strategic incentives in
prisoner’s dilemma games. The authors find that changes in the sucker’s payoff
$S$ do affect cooperation, however, only in late supergames.

Yet, current literature neither aims to replicate their study nor looks at the
interaction of this effect with communication: even though this interaction
might lead agents to ignore the sucker’s payoff $S$. Experimental studies
indicate that communication fosters the certainty of subjects about the
cooperation of the other subjects (see Dvorak and Fehrler, 2021, and the
literature therein). Incorporating this fact into a model suggests that agents
will worry about the sucker’s payoff $S$ as long as communication reduces, not
eliminates, their uncertainty about the cooperative strategy of the other
agent. If, however, communication eliminates the uncertainty, the model
suggests that agents disregard the sucker’s payoff $S$. Thus, according to the
model, the effect of the sucker’s payoff $S$ on cooperation depends on the
effect of communication on uncertainty. This is why it is important to study
both in conjunction.333Similar in spirit, Dal Bó and Fréchette (2018) conclude
that it is of great interest to clarify the sufficiency of only looking at
changes in the critical discount factor $\delta^{pd}$ to determine the level
of cooperation once we allow communication.

This paper studies, in laboratory experiments, the effect of changes in the
sucker’s payoff $S$ on cooperation in infinitely repeated prisoner’s dilemma
games, both with and without communication. For this aim, the laboratory
experiments incorporate a two by two design that varies the sucker’s payoff
$S$ and whether subjects can communicate. Comparing treatments with a high and
a low sucker’s payoff $S$ in games without communication may first replicate
the results of Blonski et al. (2011). Additionally, to test whether this
effect persists once we allow for communication, we compare treatments with a
high and a low sucker’s payoff $S$ in games with communication. Finally,
eliciting subject’s beliefs and analyzing the content of the communication
might shed light on the effect of communication on cooperation.

Our results show that changes in the sucker’s payoff $S$ affect cooperation in
infinitely repeated prisoner’s dilemma games, both with and without
communication. First, in line with Blonski et al. (2011), an increase in the
sucker’s payoff $S$ increases the level of cooperation in games without
communication. Second, with communication, an increase in the sucker’s payoff
$S$ also increases the level of cooperation, but at an overall higher level of
cooperation. Similarly to the literature (see, e.g., Blonski et al., 2011; Dal
Bó and Fréchette, 2018), we conclude that theoretical studies that use
infinitely repeated prisoner’s dilemma games to determine the conduciveness of
institutions to cooperation and disregard the sucker’s payoff $S$, by only
examining changes in the critical discount factor $\delta^{pd}$, may yield
misleading results in settings, both, with and without communication.444In
contrast to previous laboratory experiments, this paper studies the
interaction of the role of the sucker’s payoff $S$ and the beliefs with
communication. Previous laboratory experiments focus on how much communication
is needed to sustain cooperation (see Dvorak and Fehrler, 2021) and the
cooperation enhancing effect of communication (see Kartal and Müller, 2022).
Note, in contrast to our paper, Kartal and Müller (2022) refer to a model of
endogenous preferences to explain the cooperation enhancing effect of
communication.

Explaining the results, we refer to our model. In our model, changes in the
sucker’s payoff $S$ affect cooperation in games without communication, and in
games with communication, but at an overall higher level of cooperation. This
is because, according to our model, communication reduces, but does not
eliminate, strategic uncertainty.555This explanation ties into a growing body
of literature (see, e.g. Aoyagi et al., 2022; Martínez-Martínez and Normann,
2022), that suggests strategic uncertainty as a prominent reason for
cooperation. That is in line with the data in two ways. First, using an
unsupervised machine learning algorithm to analyze the communication content,
we provide suggestive evidence that subjects are indeed uncertain about the
cooperation of the other subject before choosing an action. Second, the
analysis of beliefs suggests that communication reduces the uncertainty of
subjects about the cooperation of the other subjects.

The remainder of this paper is organized as follows. In Section 2, we present
the theory. In Section 3, we describe the experimental design, and in Section
4, we develop the hypotheses. In Section 5, we present the results before we
conclude in Section 6. An Appendix complements the article with the
theoretical background (A), the instructions (B), the comprehension questions
(C), the data (D) and the original tokens in their corresponding figures (E).

##  2 Theory

This section presents the model. Before turning to the model, we formalize the
stage game and the repetition procedure.

#### Stage game

In a standard prisoner’s dilemma $\Gamma$, two agents $i\in\\{X,Y\\}$
simultaneously face a choice $a$ between cooperation ($C$) and defection
($D$), $a_{i}\in\\{C,D\\}$. If both agents cooperate, each agent earns a
reward payoff $R$. If both agents defect, each agent earns a punishment payoff
$P$. If one agent defects while the other one cooperates, the defector earns a
temptation payoff $T$ and the cooperator a sucker’s payoff $S$. The stage game
payoffs in the prisoner’s dilemma $\Gamma$ are shown in Table
LABEL:table:stagegametheoryorginal.

| C | D
---|---|---
C | R | S
D | T | P
|  |
(a)

| C | D
---|---|---
C | $\frac{R-P}{R-P}=1$ | $\frac{S-P}{R-P}=-l$
D | $\frac{T-P}{R-P}=1+g$ | $\frac{P-P}{R-P}=0$
|  |
(b)

Table 1: Row agent stage game payoffs in the prisoner’s dilemma $\Gamma$.

Following Rapoport et al. (1965), a prisoner’s dilemma $\Gamma$ features the
conditions $T>R>P>S$ and $2\cdot R>T+S$. The condition $T>R>P>S$ ensures that
agents earn more from mutual cooperation than from mutual defection ($R>P$).
It also guarantees that cooperation entails a risk to earn less ($P>S$) as
each agent has an incentive to defect if the other agent chooses to cooperate
($T>R$). The condition $2\cdot R>T+S$ ensures that mutual cooperation is more
efficient than the asymmetric outcome.

A normalization of payoffs reduces the prisoner’s dilemma $\Gamma$ to two
parameters: gain $g$ from defection when the other agent cooperates and loss
$l$ from cooperation when the other agent defects. The normalization of
payoffs is accomplished by applying a monotonic linear transformation to the
original payoffs. See Stahl II (1991) and Table
LABEL:table:stagegametheorynormal for a formalization.

#### Repetition

An infinitely repeated prisoner’s dilemma with discount factor $\delta$ can be
seen as an interaction which continues with some probability. The probability
that an interaction will continue is given by the discount factor $\delta$:
$1>\delta>0$. Thus, the expected number of rounds equals $\frac{1}{1-\delta}$
(see Roth and Murnighan, 1978).

Following Dal Bó and Fréchette (2018) and the literature therein, we assume
that agents choose between two strategies at the inception of the game: grim
and always defect.666Always defect is a plan of action or strategy which
starts by defecting and continues to defect irrespective of the other agent’s
action. Grim starts by acting cooperatively and continues to act cooperatively
until there is defection, then, defect forever. The latter is the strongest
possible retaliation for defection by others.  There is ample experimental
evidence that a substantial fraction of subjects focus on grim and always
defect (see, e.g., Dal Bó and Fréchette, 2019).777The experimental evidence
shows that the majority of subjects choose three pure-strategies: grim, always
defect, and tit-for-tat. In games equal to and lower than $\delta=0.75$, a
majority of subjects choose grim and always defect (see, e.g., Dal Bó and
Fréchette, 2019). As we employ a continuation probability of $\delta=0.75$ in
the laboratory experiments, focusing on grim and always defect seems to be
plausible. Following grim yields a value of $\frac{1}{1-\delta}$ when the
other agent is following grim as well, and a value of $-l$ when the other
agent is following always defect. Whereas, following always defect yields a
value of $1+g$ when the other agent is following grim as well, and a value of
$0$ when the other agent is following always defect. Hence, following always
defect yields a value of $p\cdot(1+g)$ and following grim yields a value of
$p\cdot(\frac{1}{1-\delta})+(1-p)\cdot(-l)$. $p$ pexplenationis an agent’s
belief that the other agent is following grim: $1\geq p\geq 0$. Thus, $1-p$
pexplenation always defect. Accordingly, the condition for which grim will be
chosen is calculated as follows.

(1) $p\cdot\bigg{(}\frac{1}{1-\delta}\bigg{)}+(1-p)\cdot(-l)\geq p\cdot(1+g)$

This condition holds if $\delta$ is larger than or equal to the critical
discount factor $\delta^{*}$:

(2) $\delta^{*}(g,l,p)=\frac{p\cdot(g-l)+l}{p\cdot(1+g-l)+l}$

Thus, the critical discount factor determines the minimum discount factor for
which grim will be chosen.888Equation 2 shows the critical discount factor
based on the idea of the size of the basin of attraction of always defect
against grim by Dal Bó and Fréchette (2018). Theoretical studies built on
changes in the critical discount factor to predict cooperation (see, e.g.,
Athey et al., 2004, Gilo et al., 2006). In this literature, a common
interpretation of the critical discount factor is that cooperation is less
likely reached and sustained when the value increases, and more likely when it
decreases (see Blonski et al., 2011, and the literature therein).

For any infinitely repeated prisoner’s dilemma, we can calculate the minimum
critical discount factor required to support cooperation in a sub-game perfect
equilibrium in addition to defection by plugging $p=1$ into Equation 2. The
result is shown in Equation 3. Hence, given that $\delta\geq\delta^{pd}$,
there is a multiplicity of equilbria. This fact raises the question when
agents will select the cooperative one.

Following Blonski et al. (2011), this paper turns to investigating the
equilibrium selection criteria Pareto dominance and risk dominance. Pareto
dominance and risk dominance are equilibrium selection criteria to predict
when agents will choose grim.

#### Pareto dominance

Grim is Pareto dominant if an agent is sufficiently patient given that the
other agent follows grim: $p=1$. The critical discount factor $\delta^{pd}$,
which is required for cooperation being a Pareto dominant response, can be
calculated by plugging $p=1$ into Equation 2. The result is shown in Equation
3.

(3) $\delta^{pd}(g)=\frac{g}{1+g}$

The condition for cooperation to be Pareto dominant is given by
$\delta\geq\delta^{pd}$.

#### Risk dominance

The grim strategy is risk dominant if it is a best response to the other
agent, who is randomizing 50–50 between always defect and grim: $p=0.5$. The
critical discount factor $\delta^{rd}$ required to support cooperation as a
risk dominant response can be calculated by plugging $p=0.5$ into Equation 2.
The result is shown in Equation 4.

(4) $\delta^{rd}(g,l)=\frac{g+l}{1+g+l}$

For cooperation to be risk dominant, the condition $\delta\geq\delta^{rd}$
must be satisfied. Note, the risk dominant critical discount factor
$\delta^{rd}$ is strictly larger than the Pareto dominant critical discount
factor $\delta^{pd}$ (see Blonski et al., 2011).

From a comparison of Equation 3 and Equation 4, we can see that changes in the
sucker’s payoff $S$, i.e. the loss $l$, do affect risk dominance
$\delta^{rd}$, but do not affect Pareto dominance $\delta^{pd}$. If
$\overline{l}>\underline{l}$, then $\delta^{pd}$ in game
$\Gamma(g,\underline{l})$ is equal to $\delta^{pd}$ in game
$\Gamma(g,\overline{l})$. If $\overline{g}>\underline{g}$, then $\delta^{pd}$
in game $\Gamma(\underline{g},l)$ is lower than $\delta^{pd}$ in game
$\Gamma(\overline{g},l)$. Risk dominance $\delta^{rd}$ in game
$\Gamma(g,\underline{l})$ is lower than the risk dominance $\delta^{rd}$ in
game $\Gamma(g,\overline{l})$. Similarly, risk dominance $\delta^{rd}$ in game
$\Gamma(\underline{g},l)$ is lower than the risk dominance $\delta^{rd}$ in
game $\Gamma(\overline{g},l)$. Thus, Equation 3 and Equation 4 show that a
variation of $l$ affects risk dominance $\delta^{rd}$ and not Pareto dominance
$\delta^{pd}$, while a variation of $g$ affects both.

Note that both equilibrium selection criteria consider the same probability
$p$ for games where agents can and cannot communicate. However, there is
experimental evidence that communication results in an increase in the
probability $p$ (see Ellingsen et al., 2018, and the literature therein). One
may question how this increase in the probability $p$ relates to the effect of
changes in the sucker’s payoff $S$, i.e. the loss $l$, on the critical
discount factor.

#### Model

To answer this question, we formally incorporate this effect into a model.
First, we assume that risk dominance is useful in games without communication:
$p=0.5$. There is ample experimental evidence suggesting that this criterion
fares well in determining the level of cooperation in games without
communication (see Dal Bó and Fréchette, 2018, and the literature therein).
Second, we assume that communication results in an increased belief $p^{+}$
for the other agent to follow grim (determined by the agent him- or herself):
$1>p^{+}>0.5$. $p^{+}$ pexplenation grim in games with communication. Hence,
$1-p^{+}$ pexplenation always defect in games with communication. Thus, we
incorporate into our model that the belief in games with communication is
larger than in ones without communication: $p^{+}>0.5$. It is important to
note that the assumption entails that communication does not cause an agent to
be certain about the other agent following grim: $1>p^{+}$. Incorporating
$p^{+}$ into Equation 2 results in Equation 5.

(5) $\delta^{+}(g,l,p^{+})=\frac{p^{+}\cdot(g-l)+l}{p^{+}\cdot(1+g-l)+l}$

Before investigating the effect of an increase in beliefs on the critical
discount factor, we investigate whether a change in the sucker’s payoff $S$,
i.e. the loss $l$, affects the critical discount factor in games with
communication. We find that a change in the loss $l$
($\overline{l}>\underline{l}$) affects the critical discount factor in games
with communication

(6) $\delta^{+}_{\overline{l}}>\delta^{+}_{\underline{l}}$

if and only if $1>p^{+}$. See Appendix A.1 for a proof. $\delta^{+}$ in game
$\Gamma(g,\underline{l})$ is shown as $\delta^{+}_{\underline{l}}$ and
$\delta^{+}$ in $\Gamma(g,\overline{l})$ is shown as
$\delta^{+}_{\overline{l}}$.999Following Table
LABEL:table:stagegametheoryorginal, a higher sucker’s payoff $S$ results in a
lower $l$. Thus, $\overline{S}$ ($\underline{S}$) results in $\underline{l}$
($\overline{l}$) for $\overline{S}>\underline{S}$. Equation 6 shows that the
critical discount factor $\delta^{+}$ in game $\Gamma(g,\overline{l})$ is
higher than in game $\Gamma(g,\underline{l})$ if and only if $1>p^{+}$. In our
model, given the assumption above that $1>p^{+}$, changes in the sucker’s
payoff $S$ do affect the critical discount factor
($\delta^{+}_{\overline{l}}>\delta^{+}_{\underline{l}}$). Intuitively, if
agents are uncertain ($1>p^{+}$) that the other agent is following the
cooperative strategy, she or he must worry about the payoff from cooperating
when the other agent defects: $l$. If, however, the agent is certain ($p=1$)
about the cooperative strategy of the other agent, she or he must not consider
the loss $l$ (see Equation 3 and the Pareto dominance equilibrium selection
criterion). Thus, according to our model, the effect of a change in the
sucker’s payoff $S$, i.e. the loss $l$, on the critical discount factor,
depends on whether communication eliminates uncertainty or not.

Now, let us investigate the effect of an increase in beliefs on the critical
discount factor. We find that the increase in beliefs by communication
decreases the critical discount factor

(7) $\delta^{rd}>\delta^{+}$

if and only if $p^{+}>\frac{1}{2}$. For a proof of Equation 7, see Appendix
A.2. Equation 7 shows that the critical discount factor in games without
communication $\delta^{rd}$ is higher than the one in games with communication
$\delta^{+}$ if $p^{+}>\frac{1}{2}$. Given our assumptions above
($p^{+}>\frac{1}{2}$), this condition is clearly satisfied in our
model.101010Intuitively, $\delta^{+}>\delta^{pd}$ as long as $1>p$. Given our
assumptions above ($1>p^{+}$), this condition is clearly satisfied in our
model and, hence, the critical discount factor in games with communication
$\delta^{+}$ is larger than the Pareto dominant critical discount factor
$\delta^{pd}$. From Equation 7, it is apparent that the increase in beliefs by
allowing for communication causes the value from following grim to be higher
than the value from following always defect. This implies that the critical
discount factor in games with communication is lower than in games without
communication.

Overall, according to our model, changes in the sucker’s payoff $S$ do affect
the critical discount factor as long as communication reduces, but does not
eliminate, agents’ uncertainty about the other agent following the cooperative
strategy. If, however, communication does eliminate uncertainty, the model
suggests that changes in the sucker’s payoff $S$ do not affect the critical
discount factor. It is worth mentioning that the reduction of uncertainty
results in a lower critical discount factor. Hence, whether changes in the
sucker’s payoff $S$ do affect the critical discount factor depends on the
effect of communication on uncertainty.

##  3 Experimental design

In the laboratory experiments, we vary the sucker’s payoff $S$ and whether
subjects can communicate in a between-subject design. Across the treatments,
subjects participate in a number of infinitely repeated prisoner’s dilemma
games. Subjects interact in a number of supergames to grasp the payoff
structure. In each supergame, two subjects interact continuously (partner
matching protocol). Between each supergame, subjects are rematched using a
perfect stranger matching protocol. The perfect stranger matching protocol
eliminates the chance that a subject is recognizing another subject, which he
or she has met before, based on his or her style to communicate. Recognizing
others may set up a chance of reputation building and, hence, may affect the
choice to cooperate. The stage game and continuation probability is similar to
Blonski et al. (2011), who report stable results in their third
supergame.111111To adopt the stage game and continuation probability of
Blonski et al. (2011), has the distinct advantage that we know how many
supergames are needed for subject’s to grasp the payoff structure and, hence,
allows us to choose the number of supergames accordingly.  Accordingly, there
should be at least three supergames per treatment. We employ five supergames
per treatment. Ergo, a perfect (stranger) matching graph involves six subjects
(see Both et al., 2016, and the literature therein).

#### Communication

In each supergame, before subjects set their action $a_{i}$ in round one, a
free-form chat window opens for $60$ seconds. A pre-play free-form chat window
enables subjects to negotiate their strategy choice and, then, to choose a
strategy for a supergame.121212Communication in every round, for example,
would enable agents to say sorry for unilateral defection and, hence, enable
them to re-negotiate their strategy. To hinder them to re-negotiate, we choose
pre-play communication (see Farrell and Maskin, 1989).

#### Stage Game

After the free-form chat window closes, both subjects choose their action
$a_{i}$ simultaneously in the infinitely repeated prisoner’s dilemma
$\Gamma(T=100,R=90,P=80,S)$.131313Subjects choosing actions and not strategies
has the distinct advantage that we can compare our results to those of Blonski
et al. (2011) and other studies. In the literature, it is common that subjects
choose actions and not strategies. According to Dal Bó and Fréchette (2011)
and Dal Bó and Fréchette (2018), subjects focus on grim and always defect in
those studies.  To provide a neutral frame, $a_{i}=\\{C,D\\}$ is renamed into
$a_{i}=\\{A,B\\}$. Following each round, subjects receive feedback about their
own action, the action of the other agent and their own payoff in that round
and the supergame so far.

#### Repetition

After every round $r\geq 2$ the game continues with a probability $\delta$ and
ends with a probability $1-\delta$. A probability $\delta^{{}^{\prime}}$ is
drawn from a uniform distribution over $[0,1]$. If and only if
$\delta^{{}^{\prime}}\leq\delta$ another round starts for all pairs. The
continuation probability is $\delta=0.75$.

#### Treatments

We employ a between-subject design varying the sucker’s payoff $S$ and whether
agents can communicate.141414To eliminate the chances of an experimenter
demand effect, a between-subject design and not a within-subject design (as in
Blonski et al., 2011) is employed. Here, a within-subject design may result in
a non-constant experimenter demand effect between treatments where people can
and can not communicate. The reason is that communication can be used to
discuss what the experimenter wants and, hence, it may be more salient in
treatments where people can communicate than in treatments where they can not.
Thus, we consider four treatments: NoComm70, NoComm0, Comm70 and Comm0. In
NoComm70 no free-form chat window opens and the sucker’s payoff equals $70$
and in NoComm0, no free-form chat window opens and the sucker’s payoff equals
$0$. In Comm70 a free-form chat window opens and the sucker’s payoff equals
$70$ while in Comm0 a free-form chat window opens too, but the sucker’s payoff
equals $0$. Table 2 summarizes the experimental design.

|  | Treatment
---|---|---
|  | NoComm70 | NoComm0 | Comm70 | Comm0
Between-Subject Design | Communication | no | no | yes | yes
Sucker’s payoff $S$ | 70 | 0 | 70 | 0
Theoretical Prediction | $\delta^{pd}$ | 0.50 | 0.50 | 0.50 | 0.50
$\delta^{rd}$ | 0.67 | 0.90 | 0.67 | 0.90
Table 2: Between-subject design and theoretical predictions split up by
treatments.

#### Probability

Each subject $i$ is asked immediately after having chosen their action and
before they receive feedback to state their probability $\tilde{p}_{i}$ that
the other agent will cooperate, on an intuitive slider without a default. A
subject’s belief about the others action in round one is a proxy for her or
his assessment about the others strategy choice.151515For example, if a
subject is certain that the other subject cooperates in the first round, he or
she is certain that the other subject is following a cooperative strategy such
as grim. For the following reasons, we do not incentives the belief
elicitation procedure. First, due to the evidence that incentivized belief
elicitation affects subsequent actions choices and the chance of this
influencing the subsequent action choices differently for a change in the
sucker’s payoff $S$ and a communication possibility, respectively (see Gächter
and Renner, 2010). Second, due to the ample experimental evidence that
communication can increase an agents belief about the cooperation of the other
agent (see Ellingsen et al., 2018, and the literature therein). Third, to opt
for simplicity whenever possible (see Aoyagi et al., 2022). Following Gill and
Rosokha (2022), to make the belief elicitation procedure as minimally invasive
as possible, we only elicit beliefs in the first round of the first and the
last supergame. This procedure aims to prevent any contamination on subsequent
action choices by the probability elicitation stage.

#### Procedure

Assignment to different treatments is random in the sense that subjects
signing up for a session do not know which treatment is run. Before the
experiment starts, subjects are seated randomly at computer terminals.
Instructions are given in written form. The instructions are presented in
Appendix B. quizAfter the instructions are read, subjects are asked
comprehension questions on the screen to ensure and to make it common
knowledge they all understand the important parts of the experiment. The
comprehension questions are presented in Appendix C. Only after all subjects
passed the comprehension questions, the experiment starts. The experiment was
conducted in May $2022$ at the University of Potsdam and a total of $132$
students participated. The subject’s final earnings are the sum of their
payoffs in points, plus a show-up fee. They earned (participated), on average,
$14.83$ euro ($36$ minutes) with a minimum of $9.80$ euro ($29$ minutes) and a
maximum of $22.00$ euro ($45$ minutes). Across the subjects, $30$ participated
in NoComm70, Comm70 and Comm0, respectively, and $42$ in NoComm0. Similar to
Blonski et al. (2011), in total, we observe 1.401 stage game interactions.

##  4 Hypotheses

In the following, we set up hypotheses to state the effect of changes in the
sucker’s payoff $S$ and the effect of communication on the rate of
cooperation, respectively. Before turning to these hypotheses, we introduce a
set of hypotheses on the effect of communication on beliefs. In our model, we
argue that communication fosters certainty: the belief in games with
communication ($p^{+}$) is higher than in games without communication
($p=0.5$). Thus, we expect that beliefs in treatments with communication are,
on average, higher than in ones without communication.

######  Hypothesis 1a.

The mean belief in Comm70 is higher than in NoComm70.161616This hypothesis was
not preregistered.

######  Hypothesis 1b.

The mean belief in Comm0 is higher than in
NoComm0.${}^{\ref{footnote:hypothesis}}$

Next, this section introduces the hypotheses on the effect of changes in the
sucker’s payoff $S$, i.e., the loss $l$, on the rate of cooperation. It is
apparent from Table 2 that changes in the loss $l$, do not affect the Pareto
dominance critical discount factor $\delta^{pd}$. While according to our
model, changes in the loss $l$ do affect the critical discount factor in games
with communication ($\delta^{+}$) due to $1>p^{+}$ and in games without
communication ($\delta^{rd}$). Thus, according to our model, we expect the
rate of cooperation in treatments with a sucker’s payoff equal to $70$ to be
higher than in treatments with a sucker’s payoff equal to $0$ in games with
communication and in ones without communication, respectively.

######  Hypothesis 2a.

The rate of cooperation in Comm70 is higher than in Comm0.

######  Hypothesis 2b.

The rate of cooperation in NoComm70 is higher than in NoComm0.

Finally, this section turns to the hypotheses on the effect of communication
on the rate of cooperation. Table 2 shows that the Pareto dominance critical
discount factor $\delta^{pd}$ in games with communication is equal to the one
in games without communication. According to our model, however, the critical
discount factor in games without communication is higher than in ones with
communication ($\delta^{rd}>\delta^{+}$) in $\Gamma(g,\underline{l})$ and
$\Gamma(g,\overline{l})$, respectively, for $\overline{l}>\underline{l}$. This
aspect of the model indicates that the rate of cooperation in treatments with
communication is higher than in ones without communication.

######  Hypothesis 3a.

The rate of cooperation in Comm70 is higher than in
NoComm70.${}^{\ref{footnote:hypothesis}}$

######  Hypothesis 3b.

The rate of cooperation in Comm0 is higher than in
NoComm0.${}^{\ref{footnote:hypothesis}}$

Following Cooper and Kagel (2022) and Kartal and Müller (2022), we analyze the
communication content to better understand why subjects made certain choices.
This analyzes may provide suggestive evidence on whether subjects are indeed
uncertain about the cooperation of the other subject.

##  5 Results

In this section, we first study the effect of communication on beliefs.
Second, we study the effect of changes in the sucker’s payoff $S$ on
cooperation in games with and without communication, respectively. We then
continue to investigate the effect of communication on cooperation. Finally,
this section turns to the analyses of the communication content.

#### Beliefs

Figure 2 presents the beliefiguremean belief in the first and final supergame
split up by treatments. It is apparent from this figure that the mean belief
in treatments with communication is substantially higher than in ones without
communication. A one-sided Wilcoxon-Mann-Whitney test with continuity
correction171717Similar to Blonski et al. (2011), unless noted otherwise, all
$p$-values reported in this paper refer to a one-sided Wilcoxon-Mann-Whitney
test with continuity correction and graph-level clustering. The continuity
correction accounts for the discontinuity in small sample sizes and produces
more conservative $p$-values. shows that the mean belief in the first
supergame in Comm70 (Comm0) is significantly higher than in NoComm70
(NoComm0): $p<0.001$, $N=15$ ($p<0.001$, $N=15$ and $N=21$, respectively). The
result is very similar if we instead consider the final supergame: $p=0.010$
($p=0.005$). Thus, our data clearly supports Hypothesis 1a and Hypothesis 1b
that the mean belief in treatments with communication is higher than in ones
without communication.

(a) Communication

(b) No communication

Figure 2: The beliefigure Error bars indicate standard errors, clustered at
the graph level. Table 3 in Appendix D provides further information.

Another observation is worth mentioning. As can be seen from Figure 2(a), the
mean belief in treatments with communication is below one and above one half,
across supergames. This suggests that neither Pareto dominance, implies a
belief level of one, nor risk dominance, implies a belief level of one half,
is a useful equilibrium selection criterion in settings with communication.

Taken together, the fact that the mean belief is higher in treatments with
communication than in treatments without communication and the suggestive
evidence that the mean belief in treatments with communication is below the
certainty level of one and above the level of maximum uncertainty of one half,
indicates that communication reduces, not eliminates, uncertainty. Thus, our
data is in line with the proposed effect of communication on uncertainty
presented in Section 2.

#### Cooperation

Figure 3 shows the cooperationsupergamesrate of cooperation in round one over
supergames, split up by treatments. A subject’s cooperation choice in the
first round is a proxy for her or his strategy choice in that supergame. From
Figure 3(a), we can see that the rate of cooperation in Comm70 is higher than
in Comm0 in late supergames. A test shows that the rate of cooperation in
round one in the final supergame in Comm70 is significantly higher than in
Comm0: $p=0.035$. The result is similar over all supergames: $p=0.055$. The
rate of cooperation in round one in the first supergame, however, is not
significantly higher in Comm70 than in Comm0: $p=0.409$.181818Also if we
compare first round actions in supergame three (two), the rate of cooperation
is (marginally) significantly higher in NoComm70 than in NoComm0: $p=0.012$
($p=0.090$). While the result in the fourth supergame in Comm70 is higher than
in Comm0, it is not statistically significant ($p=0.144$). The result is very
much the same if we instead look at all rounds. See Table
LABEL:table:cooperationsupergamesallrounds in Appendix D for support. Thus, in
late supergames, our data supports Hypothesis 2a that changes in the sucker’s
payoff $S$ affect the rate of cooperation in games with communication.

(a) Communication

(b) No communication

Figure 3: The cooperationsupergames

As can be seen from Figure 3(b), in round one, the rate of cooperation in
NoComm70 is higher than in NoComm0 across supergames. We find that the rate of
cooperation in the first round in the final supergame in NoComm70 is
significantly higher than in NoComm0: $p=0.032$. Checking all supergames
($p=0.025$), we find comparable results. In the first supergame, however, the
rate of cooperation is not significantly ($p=0.375$) higher in NoComm70 than
in NoComm0.191919A test shows that the rate of cooperation in round one in
supergame three (four) in NoComm70 is significantly higher than in NoComm0:
$p=0.021$ ($p=0.035$). While this is not the case in the second supergame
($p=0.135$), the mean rate of cooperation points in the predicted direction.
The results are almost the same if we alternatively examine the rate of
cooperation in all rounds. To back this up, see Table
LABEL:table:cooperationsupergamesallrounds in Appendix D. Ergo, in late
supergames, our data supports Hypothesis 2b that changes in the sucker’s
payoff $S$ affect the rate of cooperation in games without communication.

It is apparent from Figure 3 that the rate of cooperation in treatments with
communication is higher than in ones without communication across supergames.
A test shows that the rate of cooperation in round one of the final supergame
(over supergames) in Comm70 is significantly higher than in NoComm70,
$p=0.012$ ($p=0.006$). The result is very much the same if we rather consider
the rate of cooperation in round one of the final supergame (over supergames)
between Comm0 and NoComm0: $p=0.015$ ($p=0.005$). Even if we look at round one
of the first supergame, the rate of cooperation in Comm70 (Comm0) is
significantly higher than in NoComm70 (NoComm0): $p=0.004$
($p=0.003$).202020It is apparent from this analyses, that the rate of
cooperation in round one in Comm70 (Comm0) is significantly higher than in
NoComm70 (NoComm0) in the second, third and fourth supergame: $p=0.003$,
$p=0.004$ and $p=0.022$ ($p=0.004$, $p=0.003$ and $p=0.006$), respectively.
The result is almost identical if we instead consider the rate of cooperation
in all rounds. See Table LABEL:table:cooperationsupergamesallrounds in
Appendix D for support. Thus, our data clearly supports Hypothesis 3a and
Hypothesis 3b that the rate of cooperation in treatments with communication is
higher than in ones without communication in games with a sucker’s payoff of
$70$ and a sucker’s payoff of $0$, respectively.

#### Communication

Before studying the communication content, let us first explain our approach
to analyze the text data. To investigate the communication content, we use an
unsupervised machine learning algorithm. This algorithm has the distinct
advantage that it does not rely on pre-defined clusters, which are typically
introduced through a process of human hand-coding. Human hand-coding is the
most common approach to cluster chat data in experimental economics (see
Andres et al., 2023). Such a process involves defining clusters based on an
analysis (of a subset) of the chat data, and then having raters assign the
data to these pre-defined clusters. Characterizing clusters manually, however,
may be subject to biases, for example, in the interpretation of the definition
of clusters (see, e.g., Kartal and Müller, 2022). For this reason, in addition
to human hand-coding, we use an unsupervised machine learning algorithm to
specify clusters.

Our approach starts by looking at the entire communication content across
supergames and treatments. Thus, we obtain $150$ chats or documents, which
together form the corpus. The corpus is subject to a systematic natural
language procedure, which includes the correction of spelling mistakes, the
reduction of words to their dictionary form, and the elimination of words that
are not meaningful. The processed corpus can be represented in a matrix
$\Lambda$, where the element $(\mu,\theta,\omega)$ shows the word embedding
value $\omega$ of the unique token $\theta$ that appeared in the document
$\mu$. A token $\theta$ can be a word or, for example, or a number. A word
embedding of a token $\omega_{\theta}$ is a real-valued vector, which encodes
the meaning of the token $\theta$ such that tokens that are closer in the
vector space are expected to be similar in meaning (see Joulin et al., 2016).
The matrix $\Lambda$ is subject to the sum operator such that the element
$(\mu,\omega)$ shows the sum of the word embedding $\omega$ for each token
$\theta$ in each document $\mu$. This representation $\overline{\Lambda}$ has
been proven particularly useful to cluster documents with similar meaning (see
Ash and Hansen, 2022, and the literature therein).

To cluster documents with similar meaning, we use $k$-means. This unsupervised
machine learning algorithm is especially suitable for separating a corpus into
$k$ clusters (see Steinbach et al., 2000). Documents within the same cluster
are as similar as possible, whereas documents from different clusters are as
dissimilar as possible. The basic idea behind this algorithm is to define
clusters such that the total within-cluster variation is minimized. To
identify the number of clusters $k$, we examine the total-within clusters sum
of square. The results of this method suggest two clusters: See Figure 5 in
Appendix D for support. Hence, we use two clusters to investigate the
communication content.

In Figure 4, we depict the $30$ most frequent tokens in the corpus and their
relative rank differential to show the key distinguishing features between
both clusters. It is apparent from this figure, that the key distinguishing
feature between the two clusters is whether subjects engage in a discussion
about the riskiness of the action choices. In the Talk about A and B cluster,
the tokens ‘B’, ‘risk’ and ’trust’ often appear together with the token ’A’,
indicating that subjects mainly discuss the riskiness of action choices. In
the Talk about A cluster, the token ‘A’ often appears together with tokens
related to agreeableness (‘perfect’, ‘super’ and ‘happy-smiley’), indicating
that subjects mainly talk about an agreement to choose the cooperative action.
The fact that words like ‘risk’, ‘trust’ and the defection action ‘B’ are
present in one cluster, but not in the other one, indicates that subjects are
uncertain about the action of the other subject before choosing an action – at
least in one cluster.

Figure 4: Frequency rankings of the 30 most used tokens in both clusters.
Figure 7 in Appendix E provides the tokens in German. Note: The two black
lines indicate the border, where the two relative rank differentials
$\frac{r_{2}-r_{1}}{r_{1}}$ and $\frac{r_{1}-r_{2}}{r_{2}}$, $r_{1}$ and
$r_{2}$ are the rank in cluster one and in cluster two, respectively, are
equal to 1. Tokens outside the shaded area have a relative rank differential
exceeding 1. Tokens are more frequent in one cluster than in the other if the
relative rank statistic is larger than or equal to 1 (see Fischer and Normann,
2019).

Compared to our machine learning approach presented above, we find
qualitatively similar results with human hand-coding. The hand-coding was
realized as follows. First, two student research assistants, independently
from each other, read the corpus to note clusters present in the data. Then,
they met and agreed on two relevant clusters: agreement and discussion. The
first cluster—agreement—captures chats where the subjects purely agree to
choose action ‘A’, i.e. to cooperate. In the second
cluster—discussion—subjects communicate on both actions, ‘A’ and ‘B’, and the
dilemma-aspect of the game. Second, the student assistants independently
clustered every chat to the respective clusters: Their work across all
clusters is consistent ($\kappa=0.71$) with each other. The Cohen’s $\kappa$
across all clusters between the human raters and our machine learning approach
is above $0.62$, indicating a substantial agreement. Thus, the data supports
our machine learning approach, indicating that subjects are uncertain about
the cooperation of the other subject before choosing an action.

To sum up, in line with the machine learning approach, a communication
analysis by human hand-coding indicates that subjects are uncertain about the
cooperation of the other subject before choosing an action. It is worth
mentioning that we find qualitatively similar results between the human hand-
coding and machine learning approach. This finding provides further evidence
in line with our assumption that subjects are uncertain about the cooperative
strategy of the other subject.

##  6 Conclusion

This paper studied, in laboratory experiments, whether incorporating the
assumption that communication increases the belief, but does not causes
certainty that the other agent cooperates, into the theory of equilibrium
selection in the infinitely repeated prisoner’s dilemma, yields a criterion
$\delta^{+}$, which fares much better as a tool for predicting cooperation
than traditional ones. Traditional ones are insensitive to communication
possibilities. Our results suggest that the equilibrium selection criterion
$\delta^{+}$ is a better tool for predicting the effects of communication
possibilities than traditional ones.

We expect this paper to be useful in at least two ways: First, there is an
influential body of literature on the predictability of cooperation to which
we add here, using equilibrium selection criteria to design policies and to
determine institutions that are more or less conducive to cooperation (see Dal
Bó and Fréchette, 2018). Extending the tools in this literature to games with
communication possibilities might improve the results in this literature.
Second, we use word embeddings and machine learning to evaluate the
communication content in the laboratory experiments. There is an emerging
literature, to which we add here, using this novel approach to evaluate text
corpora in economics (see Ash and Hansen, 2022). Our results provide empirical
evidence that this approach fares well in clustering content with similar
meaning, even in an experimental context.

In a next step, it would be interesting to identify focal beliefs for
different kinds of communication. This would allow us to set up equilibrium
selection criteria for games with communication possibilities that make point
predictions on whether cooperation prevails, such as the well known risk
dominance criterion. Our results point towards a focal belief of the fraction
of three fourths in games with communication.

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## Appendix

##  Appendix A Theoretical background

The following section presents the theoretical background of the model. Before
documenting that the model predicts the cooperation enhancing effect of
communication, this section shows that a change in the sucker’s payoff $S$
affects the rate of cooperation in games with communication.

###  A.1 The effect of a change in the sucker’s payoff S

Equation 6 ties into Equation 2 as follows.

(8)
$\frac{p^{+}\cdot(g-\overline{l})+\overline{l}}{p^{+}\cdot(1+g-\overline{l})+\overline{l}}>\frac{p^{+}\cdot(g-\underline{l})+\underline{l}}{p^{+}\cdot(1+g-\underline{l})+\underline{l}}$

Equation 8 results in Equation 9.

(9) $\begin{gathered}(p^{+})^{2}\cdot g+(p^{+})^{2}\cdot
g^{2}-(p^{+})^{2}\cdot\underline{l}\cdot g+p^{+}\cdot g\cdot\underline{l}\\\
-(p^{+})^{2}\cdot\overline{l}-(p^{+})^{2}\cdot
g\cdot\overline{l}+(p^{+})^{2}\cdot\overline{l}\cdot\underline{l}-p^{+}\cdot\overline{l}\cdot\underline{l}\\\
+p^{+}\cdot\overline{l}+p^{+}\cdot
g\cdot\overline{l}-p^{+}\cdot\overline{l}\cdot\underline{l}+\overline{l}\cdot\underline{l}\\\
>\\\ (p^{+})^{2}\cdot g+(p^{+})^{2}\cdot
g^{2}-(p^{+})^{2}\cdot\overline{l}\cdot g+p^{+}\cdot g\cdot\overline{l}\\\
-(p^{+})^{2}\cdot\underline{l}-(p^{+})^{2}\cdot
g\cdot\underline{l}+(p^{+})^{2}\cdot\overline{l}\cdot\underline{l}-p^{+}\cdot\overline{l}\cdot\underline{l}\\\
+p^{+}\cdot\underline{l}+p^{+}\cdot
g\cdot\underline{l}-p^{+}\cdot\overline{l}\cdot\underline{l}+\overline{l}\cdot\underline{l}\end{gathered}$

To simplify Equation 9, we subtract $(p^{+})^{2}\cdot g$, $(p^{+})^{2}\cdot
g^{2}$, $(p^{+})^{2}\cdot\overline{l}\cdot\underline{l}$,
$p^{+}\cdot\overline{l}\cdot g$, $\overline{l}\cdot\underline{l}$ and
$p^{+}\cdot g\cdot\underline{l}$. Further, we add
$(p^{+})^{2}\cdot\overline{l}\cdot g$, $(p^{+})^{2}\cdot g\cdot\underline{l}$
and $2\cdot p^{+}\cdot\overline{l}\cdot\underline{l}$. The result is shown in
Equation 10.

(10)
$-(p^{+})^{2}\cdot\overline{l}+p^{+}\cdot\overline{l}>-(p^{+})^{2}\cdot\underline{l}+p^{+}\cdot\underline{l}$

Next, to simplify Equation 10, we subtract $p^{+}\cdot\underline{l}$ and add
$(p^{+})^{2}\cdot\overline{l}$. The result is shown in Equation 11.

(11)
$p^{+}\cdot\overline{l}-p^{+}\cdot\underline{l}>(p^{+})^{2}\cdot\overline{l}-(p^{+})^{2}\cdot\underline{l}$

Equation 11 results in Equation 12.

(12)
$p^{+}\cdot(\overline{l}-\underline{l})>(p^{+})^{2}\cdot(\overline{l}-\underline{l})$

As $(\overline{l}-\underline{l})>0$, the division of Equation 12 by
$(\overline{l}-\underline{l})$ results in Equation 13.

(13) $p^{+}>(p^{+})^{2}$

In a final step, we divide Equation 13 by $p^{+}$. The result is shown in
Equation 14.

(14) $1>p^{+}$

Equation 14 documents the model predicting that a change in the loss $l$
affects the rate of cooperation
($\delta^{+}_{\overline{l}}>\delta^{+}_{\underline{l}}$) if and only if
$1>p^{+}$. This holds given our assumption.

###  A.2 The effect of communication

This section documents that the model implies the cooperation enhancing effect
of communication. Equation 7 ($\delta^{rd}>\delta^{+}$) written out, is shown
in Equation 15.

(15) $\frac{g+l}{1+g+l}>\frac{p^{+}\cdot(g-l)+l}{p^{+}\cdot(1+g-l)+l}$

Equation 15 results in Equation 16.

(16) $\begin{gathered}g\cdot p^{+}+g^{2}\cdot p^{+}+g\cdot l+l\cdot
p^{+}-l^{2}\cdot p^{+}+l^{2}\\\ >\\\ p^{+}\cdot g-p^{+}\cdot l+l+g^{2}\cdot
p^{+}+g\cdot l-p^{+}\cdot l^{2}+l^{2}\end{gathered}$

To simplify Equation 16, we subtract $l^{2}$, $g\cdot p^{+}$, $g^{2}\cdot
p^{+}$ and $l\cdot g$. Further, we add $p^{+}\cdot l^{2}$. The result is shown
in Equation 17.

(17) $2\cdot l\cdot p^{+}>l$

In a final step, we divide Equation 17 by $2\cdot l$. The result is shown in
Equation 18.

(18) $p^{+}>\frac{1}{2}$

Equation 18 documents the model predicting the cooperation enhancing effect of
communication ($\delta^{rd}>\delta^{+}$) if and only if $p^{+}>\frac{1}{2}$.
This holds given our assumption.

##  Appendix B Instructions

In the following, we present our instructions for participants in the Comm0
treatment. Parts that appear only in the instructions of a particular
treatment are clearly marked as such. Text in italics only appears in the
instructions if people can communicate. Numbers in square brackets appear in
the instructions inherent a sucker’s payoff $S$ of $0$ as $0$ and of $70$ as
$70$. The (original) instructions for the participants were in German.

Today you are taking part in a decision-making experiment. If you read the
following explanations carefully, you can earn money. The amount you receive
depends on your decisions and the decisions of other participants.

You are not allowed to communicate with other participants for the entire
duration of the experiment. We therefore ask you not to talk to each other.
Violation of this rule will result in exclusion from the experiment and
payment.

If there is anything you do not understand, please refer to these experiment
instructions again or give us a hand signal. We will then come to you and
answer your question personally.

During the experiment we do not talk about euros, but about points. The number
of points you score during the experiment will be converted into euros as
follows:

180 Points = 1 Euro

At the end of today’s experiment, you will receive the points you have
achieved from the experiment converted into euros plus 5 euros in cash as
basic equipment.

The instructions are the same for all participants. On the following pages, we
will explain the exact procedure of the experiment.

The Experiment

The experiment consists of 5 independent sub-experiments. All sub-experiments
have the same structure. Each sub-experiment consists of several rounds. All
rounds have the same structure. In each round you have the opportunity to
choose an action.

For each sub-experiment, the participants are randomly assigned into groups of
2 persons each. The grouping remains the same within a sub-experiment. Neither
you nor the other people learn anything about the identity of the participants
in the groups - neither before nor after the experiment.

The grouping changes after each sub-experiment. It is ensured that you will
meet the same person in a group at most once during the entire experiment. In
each new sub-experiment, you will meet a new person whom you have not met
before or will meet after the experiment.

At the beginning of each sub-experiment, you can communicate with the other
person in your group in writing via chat on the computer. The duration of the
chat is limited to 60 seconds. You can write whatever you like in the chat,
with the only restriction that you may not give any hint of your identity.

The Sub-Experiment

A sub-experiment consists of several rounds.

Exactly how many rounds there are depends on chance. Before each round, a
number between 0 and 100 is drawn. For technical reasons, the computer does
this. Each number has the same probability of being drawn. If the number is
less than 75, a new round starts, if the number is greater than or equal to
75, the sub-experiment is finished and if necessary a new sub-experiment with
a new person follows. The random numbers are drawn independently of each
other.

In each round you can earn points depending on your decision and the decision
of the other person you interact with. The points earned in each round are
added up and paid out in cash at the end of today’s experiment.

The Round

In each round, you and the other person are simultaneously asked to decide
between the two actions A and B. You make your decision for the action A or B
by clicking the corresponding red button on the screen with the mouse. After
clicking, the decision is irrevocably made. You should make your decision
within 30 seconds if possible. After that you will be warned by a flashing
display.

Your payoff in the round depends on your action and the action of the other
person you interact with. The payoff’s are as follows:

My decision | Payoff if the other person chooses
---|---
| A | B
A | For me 90 Points | For me [0] Points
For the other person 90 Points | For the other person 100 Points
B | For me 100 Points | For me 80 Points
For the other person [0] Points | For the other person 80 Points

At the end of each round, you will learn the decision of the other person you
are interacting with, your scored points in that round, and the sum of the
scored points in the current sub-experiment. The information remains visible
for 30 seconds. However, you can exit the information screen before that by
clicking the gray OK button. When all participants have left the screen by
clicking the gray OK button, but after 30 seconds at the latest, the next
round will begin, if applicable.

We will ask you to answer some comprehension questions on the computer in a
moment. This is to make sure that all participants have understood this
instructions well.

After the experiment, we will ask you to fill out a short questionnaire on the
computer. After that you will receive your payout.

##  Appendix C Quiz

In the following, we present our comprehension questions. quiz The
comprehension questions are:

(1) How many people (including you) are in a group?

(2) What payoff do you get in a round if you and the other person choose ”B”?

(3) What payoff do you get in a round if you and the other person choose ”A”?

(4) If the random number is less than what number, a new round starts?

(5) The random numbers are drawn independently?

(6) What is the total payoff you would get in a sub-experiment if the sub-
experiment lasted 4 rounds and you and the other person always chose ”A”?

(7) In each new sub-experiment, you will meet a new person whom you have not
met before or will meet after the experiment?

(8) What payoff do you get in a round if you choose ”A” and the other person
chooses ”B”?

##  Appendix D Data

In the following, we present the data discussed in this article.

###  D.1 The total-within clusters sum of square

A need for any $k$-means algorithm lies in choosing the number of cluster $k$.
To choose the number of cluster $k$, we rely on the total-within clusters sum
of square. Figure 5 shows the elbowtotal-within clusters sum of square per
number of cluster $k$. It is apparent from this figure that an increasing
number of cluster $k$ led to a lower total-within clusters sum of square. The
statistically optimal number of cluster lies at the point where choosing one
more cluster does not reduce the the total-within clusters sum of square much
further. In Figure 5, this is the number of cluster $k$ where the elbow lies.
It is apparent from this figure, that the number of cluster $k$ that we
consider for the $k$-means algorithm is $3$. Two of those cluster are
meaningful: they consider a conversation. In the third cluster, both subjects
only greet each other without further intention.212121The human-coders, as
described in the main text, find a similar third cluster that contains
greetings between the participants only. Thus, for the analysis of the
communication content, we just consider the two meaningful cluster.

Figure 5: The elbow

###  D.2 Belief split up by treatment

|  | Treatment
---|---|---
|  | NoComm70 | NoComm0 | Comm70 | Comm0
Belief | Mean |  54.59 (31.16) |  35.42 (29.37) |  81.51 (23.83) |  84.99 (17.45)
Median | 64.69 | 30.47 | 89.94 | 88.66
>Median | Cooperation | 14 | 14 | 15 | 15
Defection | 1 | 7 | 0 | 0
<Median | Cooperation | 2 | 1 | 12 | 11
Defection | 13 | 20 | 3 | 4
(a)

|  | Treatment
---|---|---
|  | NoComm70 | NoComm0 | Comm70 | Comm0
Belief | Mean |  40.56 (28.08) |  15.88 (25.51) |  81.80 (23.25) |  68.33 (37.76)
Median | 43.30 | 0.73 | 90.99 | 84.75
>Median | Cooperation | 13 | 8 | 15 | 13
Defection | 2 | 13 | 0 | 2
<Median | Cooperation | 3 | 0 | 15 | 8
Defection | 12 | 21 | 0 | 7
(b)

Table 3: Mean and median belief as well as above- and below-median cooperation
count split up by treatment in the first and final supergame. Numbers in
brackets are standard deviations.

###  D.3 Cooperation per supergame split up by treatment

Table 4: the $p$-value between NoComm70 and NoComm0 in all rounds is $0.012$,
$0.036$, $0.017$, $0.024$, $0.019$ and $0.007$ firstuswin the first, second,
third, fourth, fith and over all supergames, respectively. The $p$-value
between Comm70 and Comm0 in all rounds is $0.375$, $0.090$, $0.001$, $0.017$,
$0.193$ and $0.072$ firstusw The $p$-value between Comm70 and NoComm70 in all
rounds is $0.005$, $0.004$, $0.005$, $0.010$, $0.018$ and $0.006$ firstusw The
$p$-value between Comm0 and NoComm0 in all rounds is $0.003$, $0.003$,
$0.005$, $0.009$, $0.010$ and $0.005$ firstusw

Supergame | Treatment
---|---
| NoComm70 | NoComm0 | Comm70 | Comm0
1 |  0.53 (0.08) |  0.36 (0.24) |  0.90 (0.09) |  0.87 (0.14)
2 |  0.43 (0.15) |  0.21 (0.27) |  1.00 (0.00) |  0.83 (0.24)
3 |  0.50 (0.26) |  0.19 (0.20) |  1.00 (0.00) |  0.73 (0.19)
4 |  0.53 (0.27) |  0.17 (0.22) |  0.93 (0.15) |  0.77 (0.28)
5 |  0.53 (0.27) |  0.19 (0.26) |  1.00 (0.00) |  0.70 (0.34)
Mean |  0.51 (0.17) |  0.22 (0.23) |  0.97 (0.04) |  0.78 (0.22)
(a)

Supergame | Treatment
---|---
| NoComm70 | NoComm0 | Comm70 | Comm0
1 |  0.52 (0.04) |  0.25 (0.23) |  0.80 (0.60) |  0.77 (0.22)
2 |  0.39 (0.12) |  0.15 (0.18) |  1.00 (0.00) |  0.82 (0.27)
3 |  0.46 (0.20) |  0.12 (0.16) |  0.94 (0.09) |  0.67 (0.17)
4 |  0.43 (0.33) |  0.11 (0.16) |  0.95 (0.08) |  0.65 (0.35)
5 |  0.39 (0.30) |  0.10 (0.13) |  0.89 (0.13) |  0.67 (0.40)
Mean |  0.44 (0.18) |  0.15 (0.16) |  0.92 (0.04) |  0.72 (0.26)
(b)

Table 4: The rate of cooperation in the first round and all rounds over
supergames split up by treatments. Numbers in brackets are standard
deviations.

##  Appendix E Original German tokens in their corresponding Figure

Here, we show the original German tokens in their corresponding figure. We
translated the tokens only after the analyzes of the chats.

Figure 7: Frequency rankings of the 30 most used German tokens in both
clusters.