# Cooperation and Punishment Mechanisms in Uncertain and Dynamic Social
Networks††thanks: We are grateful to Marco Battaglini (editor), an anonymous
associate editor, and two anonymous referees whose comments substantially
improved the paper. We acknowledge the financial support from the Singapore
Ministry of Education Academic Research Fund (AcRF) Tier 2 grant
(MOE2014-T2-1-105), City University of Hong Kong (CityU 11500017), and the
Cambridge Endowment for Research in Finance (CERF). The paper benefited from
discussions during presentations at the 2019 North American ESA Meeting in Los
Angeles (USA) and the SEA 89th Annual Meetings in Fort Lauderdale (USA).
Declarations of interest: none.

Edoardo Gallo Faculty of Economics, University of Cambridge, Sidgwick Avenue,
Cambridge CB3 9DD, UK and Magdalene College, CB3 0AG, UK, phone:
+44-1223-335286, e-mail<EMAIL_ADDRESS>Yohanes E. Riyanto Division of
Economics, School of Social Sciences, Nanyang Technological University, 48
Nanyang Avenue, HSS 06-15A, Singapore 639818, phone: +65-6592-1578, e-mail:
<EMAIL_ADDRESS>Nilanjan Roy Department of Economics and Finance,
College of Business, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon
Tong, Hong Kong, phone: +852-3442-2659, e-mail<EMAIL_ADDRESS>Tat-How
Teh School of Management and Economics and Shenzhen Finance Institute, The
Chinese University of Hong Kong, Shenzhen, 2001 Longxiang Road, Longgang
District, Shenzhen, China, e-mail<EMAIL_ADDRESS>
###### Abstract

This paper examines experimentally how reputational uncertainty and the rate
of change of the social environment determine cooperation. Reputational
uncertainty significantly decreases cooperation, while a fast-changing social
environment only causes a second-order qualitative increase in cooperation. At
the individual level, reputational uncertainty induces more leniency and
forgiveness in imposing network punishment through the link proposal and
removal processes, inhibiting the formation of cooperative clusters. However,
this effect is significant only in the fast-changing environment and not in
the slow-changing environment. A substitution pattern between network
punishment and action punishment (retaliatory defection) explains this
discrepancy across the two social environments.

JEL Classification Numbers: C72, C73, C92, D81, D85

Keywords: cooperation, experiments, prisoner’s dilemma, uncertainty, repeated
games, networks.

##  1 Introduction

Understanding when and how repeated interactions lead to cooperation is
fundamental to comprehend many economic phenomena. Seminal theoretical
contributions have demonstrated that cooperative play is possible when
individuals are sufficiently patient (e.g., Friedman (1971), Fudenberg and
Maskin (1986)), and several experimental papers have shown that people
cooperate if the gains to cooperation are sufficiently large (e.g., Dal Bó
(2005), Dreber et al. (2008)). Many of these studies, however, focus on a
setup with two players interacting with each other with perfect monitoring of
actions, thus abstracting away from some of the features present in real
contexts where cooperation operates, e.g., among several individuals connected
through social networks.

A central feature of cooperative activities is that they often occur in a
network of relationships that change over time and consequently influence the
diffusion of behavior. Taking the network into consideration affects the
effectiveness of the known mechanisms to sustain cooperation in non-network
(bilateral) relationships and introduces novel ones. For instance, action
punishment by retaliating against a defector is a well-documented channel to
enforce cooperation in a bilateral setting. However, the presence of a network
may render action punishment less effective because a defection affects not
only the original culprit but also other well-behaving (cooperative) partners.
In the network context, instead, the removal of a link is a directly targeted
and potentially more effective way to punish a defector.

The purpose of this paper is to investigate the emergence and evolution of
cooperation in a dynamic social environment. In our experiment, each subject
is assigned to a group of 12 players who play a repeated Prisoner’s Dilemma
(PD) game. Subjects are connected by a network in which a link (or a
connection) denotes a pair of subjects who play a PD game against each other
in that round, and each subject is constrained to choose either to cooperate
or defect with all their neighbors. At the beginning of a round, some subjects
have the opportunity to remove some of their existing links and/or propose new
links to others. When making their decisions, subjects have access to
information about the reputation of each of their connections in the form of
their last five actions.

We adopt a $2\times 2$ treatment design. The first dimension we investigate is
the rate of change of the social environment, defined as the frequency with
which opportunities to update one’s links or form new links become available
(“ _Slow_ ” and “ _Fast_ ” environments). The second dimension we investigate
is the presence of uncertainty on whether each player’s observed action
corresponds to her intended action (“ _No uncertainty_ ” and “ _Reputational
uncertainty_ ”). These two aspects have been investigated previously in
isolation (e.g., Fudenberg et al. (2012), Rand et al. (2011)). We extend the
analysis of Fudenberg et al. (2012) in a bilateral interaction setting to a
more realistic social network setting.

The prototypical example of criminal activity to explain PD games is an ideal
setting to illustrate the relevance of our design. Social networks are
essential to conduct a large range of criminal endeavors (see, e.g., Varese
(2011)). In recent decades, the development of the internet has arguably made
it easier for criminals to form and remove connections with partners in crime
with the rise of various cybercrime platforms (Lusthaus (2018)). Our setup
where links among individuals are evolving captures such a setting. Next,
uncertainty also plays a prominent role in criminal activity in both offline
and online settings, and the difficulty of monitoring makes it challenging to
always learn the true intended actions from observed outcomes. In cybercrime
markets, for instance, it is common for delivered services to be defective.
For instance, stolen credit card numbers may not work because the seller has
intentionally fabricated them or because of external factors such as the
intervention of a law enforcement agency (e.g., Sebagh et al. (2021)). The
reputational uncertainty element in our setting is meant to capture an
environment with imperfect monitoring and uncertainty described in the example
above.

Our first result is that reputational uncertainty has a first-order effect on
cooperation rate and welfare, while the rate of change of the social
environment has, at best, a second-order effect. Specifically, cooperation
rate and welfare significantly decrease with reputational uncertainty in both
slow and fast environments. Meanwhile, they only qualitatively increase in a
fast-changing social environment, holding the presence of uncertainty
constant.

Even though the first result bears a resemblance to previous studies with two-
player PD (see the survey at the end of this section), the reason why
reputational uncertainty reduces cooperation in our setting is novel and
distinct from these studies. In our dynamic social network setting, action
choices of players co-evolve with the network structure, so that there are two
levers for individuals to punish defectors and enhance cooperation:

  * •

_Network punishment_ deters defectors with the threat of having their links
with cooperators removed; and/or cooperators avoiding to form new connections
with defectors.

  * •

_Action punishment_ is the standard threat of responding to a defector by
engaging in retaliatory defection (which is also present in two-player PD
games).

The use of network punishment has been documented in previous studies
involving endogenous updates to the interaction neighborhood without
uncertainty in various types of network games (e.g., Rand et al. (2011), Riedl
et al. (2016)). Compared to network punishment, action punishment is much less
targeted in network settings because any retaliatory action/defection affects
not only the defector but also the cooperative neighbors. This feature is
absent in a two-player PD game because the consequence of an action punishment
is felt solely by the target in such a setting. By exploring how the use of
network and action punishments vary with our treatment variables, we can shed
light on the impact of reputational uncertainty on cooperative behavior.

Our second set of results show that reputational uncertainty makes network
punishment less prevalent – in the sense that participants become more
_lenient_ and _forgiving_ towards defectors. However, the magnitude of this
effect differs across fast and slow environments. Our concepts of leniency and
forgiveness extend the notions formulated by Fudenberg et al. (2012) (for
bilateral PD games) to a dynamic network setting involving more than two
players. We show that, in the fast-changing social environment, reputational
uncertainty makes players: (i) employ lenient strategies – players are
significantly less likely to terminate an existing relationship with someone
having a history of low cooperative play in an environment with uncertainty
than without it; and (ii) employ forgiving strategies – players are
significantly more likely to propose a link to someone with a history of low
cooperative play. While these effects of reputational uncertainty similarly
occur in the slow-changing environment, they are marginally significant at
best.

Why is it that reputational uncertainty affects the usage of network
punishment differently across fast and slow environments? By design,
reputational uncertainty means that a player’s history of low cooperative play
does not necessarily reflect the true intention of the player. As such, all
else being equal, other players who are making network punishment decisions
would be more willing to give the benefit of the doubt and to extend a second
chance to this player with low cooperative history. In a fast environment,
opportunities to use network punishment arrive frequently, so it is relatively
inexpensive for players to be lenient and forgiving. In contrast, in a slow
environment, being lenient and forgiving entails huge opportunity costs —
chances to use network punishment only arrive occasionally, so there is a risk
of being forced to stay connected with a low-cooperativeness player for a long
time. In broader contexts, our observation highlights that it is essential to
consider the frequency with which punishment opportunities are available when
researchers analyze the notions of leniency and forgiveness in punishment
decisions.

A novel consequence of leniency and forgiveness in network punishment (as a
result of reputational uncertainty) is that cooperative clusters fail to
emerge, a feature absent in two-player games. Previous studies (e.g., Rand et
al. (2011), Wang et al. (2012), Gallo and Yan (2015), Gallo et al. (2019))
have shown that reputational information and a fast-changing social
environment promote cooperation by enabling the formation of clusters of
cooperators who associate with each other and shun away defectors. We show
that reputational uncertainty undermines this process because defectors are
more likely to remain embedded in cooperators’ neighborhoods, thereby creating
an obstacle to the formation of cooperative clusters that would have driven up
the cooperation level of the whole network.

Our final set of results shows that reputational uncertainty makes action
punishment less prevalent, regardless of the updating rate of the social
environment. Specifically, players’ likelihood of switching from cooperation
to defection becomes significantly less sensitive towards their neighbors’
past defections when there is uncertainty. This weakens the extent of action
punishment, which implies fewer repercussions from defecting, thus making
defection more prevalent. Moreover, reputational uncertainty also lowers the
cooperation rate through promoting opportunistic play – players become more
likely to defect after a history of highly cooperative play than in a setting
without uncertainty. This is because players can hide behind the veil of
uncertainty, which makes it difficult for others to infer whether the
defection was intentional or not.

Combining our results on both network and action punishments highlights a
novel interplay between these two dimensions in dynamic networks. In a slow-
changing environment, players primarily rely on action punishment, given that
chances to use network punishment are, by design, infrequent. As such,
reputational uncertainty lowers cooperation primarily by dampening action
punishment usage. In a fast-changing environment, players substitute action
punishment with the more targeted network punishment so that reputational
uncertainty reduces cooperation through its effect over both dimensions. To
the best of our knowledge, such a substitution pattern between different
punishment methods and its interaction with reputational uncertainty is new to
the literature.

This paper contributes to a growing literature that sits at the intersection
of, first, the vast body of work investigating the emergence and sustenance of
cooperation using variations of the PD game, and, second, research on how
networks affect behavior and economic activity. These literature span
different disciplines, and it is not feasible to have an exhaustive review of
all related contributions here. We aim, therefore, to position our paper in
the context of the most relevant contributions with a bias toward the economic
literature.

In experiments with an infinitely repeated two-player PD game under imperfect
public monitoring, Aoyagi and Fréchette (2009) show that subjects cooperate in
an environment with a continuous public signal and that their payoff decreases
with the level of noise in the public signal. Fudenberg et al. (2012) study a
model of a repeated PD with reputational uncertainty in players’ actions and
observe that, compared to the setting with no uncertainty, individuals are
slow to anger and fast to forgive. Specifically, with uncertainty, subjects
use strategies that, firstly, do not prescribe reverting to punishment after a
single bad signal, and secondly, entail a return to cooperation after
punishing the opponent. Aoyagi et al. (2019) find similar behavior with
private monitoring.111Early experimental studies have explored the effect of
manipulating players’ information in repeated games in less standard ways. For
example, Cason and Khan (1999) investigate a public goods game where
information on others’ contributions are presented with some delay. Bolton et
al. (2005) study cooperation in an environment with limited information about
reputation. Bereby-Meyer and Roth (2006) consider a repeated PD game with
noisy payoffs, even though in their experiment players could monitor others’
past actions perfectly.,222Related studies of repeated PD games under
uncertainty include Rojas (2012), Rand et al. (2015), Embrey et al. (2013),
and Arechar et al. (2017).

Beyond two-person games with uncertainty, Ambrus and Greiner (2012) study the
effects of a costly punishment option on cooperation and social welfare in
long, finitely repeated three-person public good contribution games. They show
that the strong positive effects of increasing the severity of punishment as
identified in Nikiforakis and Normann (2008) and Egas and Riedl (2008) do not
hold when there is uncertainty. With five-person repeated public goods game
with imperfect monitoring, Ambrus and Greiner (2019) further investigate the
impact of democratic punishment, when members of a group decide by majority
voting whether to inflict punishment on another member, relative to individual
peer-to-peer punishment.333See also DeAngelo and Gee (2020) for a recent
contribution on how endogenous monitoring structure interacts with punishment
in the context of public good provision game. They also explore whether a
subject’s actions are automatically monitored and broadcasted to the whole
group or whether the probability of monitoring is below 100%. While we only
consider the case where monitoring is automatic, the possibility of non-
automatic detection offers an interesting avenue for future research. Our
contribution to this strand of work on cooperation in groups in an uncertain
reputational environment is to introduce the possibility for individuals to
choose whom to interact with given their (possibly imprecise) information on
others’ reputations.

When there is perfect monitoring of others’ actions, recent research has
demonstrated that the freedom to choose whom to interact with promotes
cooperative behavior. Rand et al. (2011) and Wang et al. (2012) find that when
subjects can update their links frequently, cooperation is maintained at a
high level with participants preferentially removing links with defectors and
forming new links with cooperators. Shirado et al. (2013) report that high
cooperation is achieved at an intermediate frequency of link updating.444When
the frequency is too low, subjects choose to have many links, even if they
attach to defectors. When it is too high, cooperators cannot detach from
defectors as much as defectors re-attach and, hence, subjects resort to
behavioral reciprocity and switch their behavior to defection. Gallo and Yan
(2015) further show that in this context, knowledge about everybody’s
reputation is crucial to achieve a high level of cooperation.555A substantial
related literature studies how endogenous group formation affects play in PD,
public good and weak link games. Contributions include Ehrhart et al. (1999),
Hauk and Nagel (2001), Riedl and Ule (2002), Coricelli et al. (2004),
Cinyabuguma et al. (2005), Page et al. (2005), Güth et al. (2007), Ahn et al.
(2009), Brandts et al. (2009), Charness and Yang (2014), Riedl et al. (2016),
Yang et al. (2017), Riyanto and Teh (2020). See Choi et al. (2016) for a
comprehensive review. Our study contributes to this growing literature by
introducing reputational uncertainty to investigate how it affects the
effectiveness of the network formation mechanism in sustaining cooperation.

It is possible to view the link formation and termination decisions in our
networked setting as analogous to the ability of individuals to employ
targeted punishment in public goods games that allow for costly punishment.
With regards to the role of punishment in inducing cooperation, starting with
Fehr and Gächter (2000), vast literature has demonstrated that the possibility
of costly punishment facilitates cooperation in public goods games. However,
in the presence of noise, Ambrus and Greiner (2012) show that costly
punishment is less effective in establishing cooperation. We investigate
whether the punishment mechanism through link formation/termination is less
effective once we introduce reputational uncertainty.

An alternative form of punishment that has been shown to promote cooperation
is ostracism (Cinyabuguma et al. (2005), Maier-Rigaud et al. (2010)). Under
ostracism, individuals can expel each other from the group using various
voting schemes. The expelled players are then excluded from the benefits of
any cooperation undertaken by those still remaining in the group. We can think
of our set-up as allowing decentralized ostracism through the removal of links
by single individuals in an environment with reputational uncertainty.
Interestingly, we find that ostracism of defectors can be effective even with
a decentralized system, but only when there is no reputational uncertainty.

The remainder of the paper is organized as follows. Section 2 describes the
design and procedures of the experiment. The results are presented in section
3, and the last section concludes.

##  2 Experimental Design and Procedures

In our experiment, each participant is assigned to a group of 12 who play a
game that is repeated for 25 rounds with random termination thereafter.666We
opt for 25 rounds to ensure we have a long enough sequence of rounds to
observe stationary play. The random termination protocol implemented after
round 25 is to eliminate end-game effects. An alternative design would be to
either implement a _random termination protocol_ in every round proposed by
Roth and Murnighan (1978) or _a block random termination protocol_ proposed by
Fréchette and Yuksel (2017). The former is done by making a random draw with a
pre-specified termination probability in every round to decide whether the
round will be the terminal round. There are drawbacks of this method. First,
we could potentially end up with a varying length of the replication rounds.
Second, some of these replication rounds might be short while others might be
long, making it difficult to observe stable patterns of network evolution. The
block random termination protocol is a variant of the random termination
protocol, in which participants interact in blocks of a pre-announced fixed
number of rounds, for example, three blocks of 10 rounds. In each block, a
terminal round is randomly drawn, but participants were not told which round
it is. So they would treat all rounds within a block as payoff-relevant. Once
the block is completed, participants are told whether there is a terminal
round in the block and, if there is, which round it was. In hindsight, we
could have adopted this protocol, which would overcome the drawbacks of the
standard random termination protocol described earlier and ensure stationarity
across all rounds of the experiment. However, we were not aware of this
protocol when we ran our experiments. At the beginning of a round, some
participants have the chance to propose and/or remove links with some of the
other group members. After the links have been updated, each participant plays
a PD game with the other participants who are directly connected to her. We
now describe the protocol of the game in detail, which consists of three
stages in each round.

Figure 1 illustrates the three stages that constitute a round. In the initial
round, all participants are always linked with each other, but in any round
thereafter (e.g., round T in Figure 1) they may not be.777In principle, any
initial network structure that is not complete or not empty is arbitrary.
Between these two contrasting configurations, a complete network has the
advantage of allowing interaction between subjects to emerge more quickly,
thus reducing the required length of experimental sessions. In stage 1 of
round T, $x$ random pairs of participants, i.e., those drawn with dashed
circles, may receive opportunities to propose and/or remove links with each
other. Suppose $i$ and $j$ is a random pair picked at time $T$, each of them
faces one of these choices:

Figure 1: Schema of the experimental design.

  * •

Remove the link if $i$ and $j$ were connected at round $T-1$. Deletion of a
link is unilateral and does not require mutual consent.888This is a natural
assumption in the context of a relationship that is not legally binding and is
common in the economics literature (e.g., Jackson and Wolinsky (1996)).

  * •

Propose a link if $i$ and $j$ were not connected at round $T-1$. The formation
of a link is bilateral and requires the consent of the recipient of the
proposal in Stage 2.

Stage 2 occurs if one and only one between $i$ and $j$ has proposed a link.
Given that link formation requires mutual consent, in stage 2, the recipient
of a link proposal has to decide whether to accept or reject it, as shown in
the third row of Figure 1. Notice that if two participants send a link
proposal to one another in Stage 1, the link is automatically formed, and
there is no prompt in Stage 2.

In stage 3, participants play the PD game shown at the bottom right of Figure
1 with each one of their neighbors, subject to the restriction that they have
to choose either action $C$ (cooperation) or action $D$ (defection) for all
neighbors.999We used neutral labels $A$ and $B$ in our experiment to avoid the
experimenter-demand effect. The selection of a payoff structure symmetric
around zero provides the correct incentives to allow the formation of
meaningful and realistic network structures because the symmetry of payoffs in
the gains/losses domains means that both the absence of a connection and
connections between a defector and a cooperator lead to no change in social
surplus, i.e., the sum of payoffs of all the participants. The only way to
produce a social surplus is a connection between two cooperators, and,
conversely, the only way to reduce the social surplus by an equal amount is a
connection between two defectors. This is in contrast to studies that have
non-negative or small negative payoffs, which lead to the emergence of over-
connected networks because the losses from being connected to a defector are
non-existent or negligible. Choosing a single action simultaneously toward all
neighbors is a standard restriction employed in the literature to allow for
network effects, otherwise the network game becomes a collection of
independent PD games between pairs of players in the group.101010See for
example Morris (2000), Charness et al. (2014), Riedl et al. (2016), and Yang
et al. (2017). If a participant has no neighbors, then she has no action to
choose. Once all individuals make their choices, they see the points they
receive from the interaction with each of the other participants. They receive
zero points in this stage from participants with whom they are not connected
to. Subjects are also reminded of their chosen action and the action chosen by
their neighbors.

Appendix C shows screenshots for each of the three stages. For each
participant, the left area of the interface displays a visualization of their
links to their neighbors in which the nodes are colored in either green or
blue depending on whether their action in the last round was to cooperate or
defect respectively.111111Note that participants have information about the
local network structure (i.e., their neighbors), but they do not know the
overall structure of the network. This is consistent with previous
contributions in the literature, and it prevents information overload. Using a
similar set-up, Gallo and Yan (2015) show that providing information about the
whole network structure does not affect the aggregate level of cooperation,
but it changes the distribution of cooperative activity. The central part of
the interface shows the list of current neighbors with their past five actions
and, at the end of stage 3, the points obtained in the PD game with each of
the neighbors.121212Cuesta et al. (2015) show that cooperation is robust to
changing the number of past instances of play that participants see as long as
they can observe at least the last instance of play. The right area of the
interface has the list of other participants who are not currently neighbors.
The bottom part prompts the participant to make the relevant decision in each
stage, e.g., the participants with whom she can propose/remove a link in stage
1 and the choice of action in stage 3.

End of experiment. Each group of 12 participants plays at least 25 identical
rounds of this game. After round 25, there is a $50\%$ chance of termination
to reduce any endgame effects. Participants’ earnings are based on six
randomly selected rounds. For each participant, two other participants are
randomly picked for each of the selected rounds, and the sum of the points
obtained in the interaction with these 12 randomly picked participants
constitutes the total sum of points earned by the participant in this part.
Notice that it is possible to pick an unconnected pair of participants and, in
that case, the earnings from that interaction are equal to zero.131313This
random selection of pairs for payment, independent of whether a connection
exists or not, ensures that there are uniform incentives throughout the
experiment in forming connections, so the payment system does not introduce
biases in the emerging network structure. For instance, if we had excluded
unconnected pairs from the random selection for payment, participants would
have incentives to form just one link with a cooperator to ensure that a
specific pairing is picked. Some prior studies (e.g., Wang et al. (2012),
Shirado et al. (2013)) pay the cumulative number of points participants have
earned, which may lead to satisficing in the later rounds and therefore lower
incentives to change the network. Following the completion of the first part
of the experiment, participants take a Holt and Laury (2002) multiple price
list risk-elicitation test and fill in a socio-demographic questionnaire,
which includes the standard interpersonal trust question taken from the World
Values Survey (WVS).141414The interpersonal trust question is: “Generally
speaking, would you say that most people can be trusted or that you need to be
very careful in dealing with people?”.

Treatment variations. The boxes with round edges in Figure 1 illustrate the
two treatment dimensions in the experiment, namely the _updating rate_ (_slow_
and _fast_) of the social environment and _whether uncertainty is present_. We
conduct four treatments and use a 2$\times$2 between-subject design to test
the impact of uncertainty under two different link formation/removal rates.
Under no reputational uncertainty treatments, there is no uncertainty, so
participants receive payoffs according to their own action and neighbors’
actions. Under reputational uncertainty treatments, there is a commonly known
15% probability that an intended action is changed to the opposite
action.151515Previous experiments by Fudenberg et al. (2012) and Cason and Mui
(2019) select a slightly lower 12.5% error rate. Subjects are informed of
their intended action as well as their implemented one. However, they are only
notified of the neighbors’ actual actions, not their intended ones.

The way we model reputational uncertainty in our setup follows the paradigm
proposed by Fudenberg et al. (2012). In terms of the updating rate dimension,
we consider a slow $x=6$ condition in which only 9% of pairings potentially
change in each round, and a fast $x=33$ condition in which 50% of pairings
potentially change in each round.

The data was gathered in 16 experimental sessions conducted at the Nanyang
Technological University (NTU) in Singapore, and the experiment involved 384
undergraduate students from various majors. No subject participated in more
than one session, and we used a between-subject design so that each subject
participated in one and only one treatment. There are four sessions for each
of the four treatments. Each session consists of 2 groups (networks) of size
12. This gives rise to a total of 8 independent observations at the network
level per treatment. The choice of a group size of 12 ensures that the network
is large enough to allow the emergence of interesting structural features
while at the same time it is small enough to make it feasible for us to
collect a sufficient number of independent observations per treatment in a
laboratory setting. The experiment was programmed with the software z-Tree
(Fischbacher, 2007). Upon arrival, subjects were seated at visually isolated
computer workstations. Instructions were read aloud, and subjects also
received a copy of the instructions.161616A sample copy of the instructions
for the fast updating treatment with uncertainty is provided in Appendix C.
Participants, who were each given a random user ID, remained completely
anonymous to each other throughout the experiment.

The sessions lasted approximately two hours, and participants earned on
average S$13.73 in addition to the S$2 show-up fee. Out of the 384
participants, 53% are female, 62.4% are Singaporeans, and the average age is
21 years. About 23% of the participants have had prior exposure to game
theory, and exactly half had participated in at least one laboratory
experiment previously. From the WVS interpersonal trust question, 22% believe
that others can be trusted. In the post-experiment questionnaire, almost all
participants stated that they had no problem understanding the
experiment.171717The descriptive statistics of the participants in each of the
four treatments are provided in Appendix A.

##  3 Results

###  3.1 Overall cooperativeness

We start by analyzing how cooperation outcomes vary at the network level
across treatments, where there are eight independent observations for each
treatment. We adopt the following two network-level measures:

  * •

Cooperation rate. The number of pairs with mutual (intended) cooperation at
the end of a round, divided by 66, which is the total number of possible pairs
in a complete network with 12 nodes.181818Results remain the same if we
measure cooperation based on the fraction of intended cooperators, defined as
the ratio of participants in a group who choose $C$ as the intended action.

  * •

Welfare. The total number of points obtained by all participants from all
interactions at the end of the round.

Figure 2 shows the evolution of cooperation rate and welfare over time. After
some initial volatility, cooperation rate and welfare in different treatments
tend to stabilize over time, aside from the end-game effects in the rounds
toward the end of a session where we observe steep drops in both measures. To
identify the treatment effects formally, we utilize data from rounds 2-21 and
exclude the last four rounds to avoid end-game effects, which are common in
games of cooperation (Gächter et al. (2008), Ambrus and Greiner (2012), Rand
et al. (2011), Wang et al. (2012)).191919We focus on rounds 2-21 throughout
the paper. The only exception is analysis conditioned on reputation history,
where we focus on rounds 6-21 because the first five rounds are needed to
build a complete history of past plays. Nonetheless, our results remain
unchanged if we include the last four rounds as well by using data from rounds
2-25.

Figure 2: Cooperation rate and welfare over time.

Table 1 provides summary statistics of the average cooperation rate and
welfare for each treatment, using the eight network-level observations per
treatment. We apply the non-parametric Kruskal-Wallis test to detect treatment
effects, followed by the Dunn’s (_DT_) test to examine the statistical
significance of the differences between any two treatments.202020This
methodology has been used in prior work involving network experiments in which
the sample size at the network level is limited due to the number of
participants required per network (e.g., Gallo and Yan (2015)). It is typical
for network experiments to have 6-10 independent network-level observations in
each treatment (Rand et al. (2011), Riedl et al. (2016), Yang et al. (2017),
Gallo et al. (2019)).

Table 1: Cooperation rate and welfare across treatments.

| Slow |  | Fast
---|---|---|---
| No uncertainty | Uncertainty |  | No uncertainty | Uncertainty
Cooperation rate | 0.255 (0.063) | 0.144 (0.052) |  | 0.285 (0.080) | 0.198 (0.073)
Welfare | 56.29 (74.93) | -20.29 (63.02) |  | 96.11 (66.11) | 20.96 (49.64)

  * •

Note: we report mean values over rounds 2-21 with standard errors in
parentheses.

Holding the rate of network updating constant, the presence of reputational
uncertainty leads to significantly lower cooperation rate and welfare. In the
slow condition, reputational uncertainty reduces cooperation rate from 0.255
to 0.144 ($p=0.003$, $DT$) and welfare from 56.29 to -20.29 ($p=0.017$, $DT$).
In the fast condition, reputational uncertainty reduces cooperation rate from
0.285 to 0.198 ($p=0.024$, $DT$) and welfare from 96.11 to 20.96 ($p=0.028$,
$DT$).

Meanwhile, holding the presence of reputational uncertainty constant, the
introduction of a fast-changing social environment increases cooperation rate
and welfare, but the effects are at best marginally significant. Without
reputational uncertainty, the p-values ($DT$) for the difference between the
_slow_ and _fast_ conditions are $p=0.345$ for cooperation rate and $p=0.182$
for welfare. With reputational uncertainty, the p-values for the difference
between the _slow_ and _fast_ conditions are $p=0.071$ for cooperation rate
and $p=0.132$ for welfare. We are, therefore, unable to replicate the strong
finding from previous studies on cooperation in dynamic network settings
showing that a fast-changing social environment increases cooperation (e.g.,
Rand et al. (2011), Wang et al. (2012)). This is consistent with the recent
evidence showing that the role of network formation is not robust to changes
in the experimental environment (Rand et al. (2014)) suggesting that its role
is second order with respect to reputation.

Finally, note that the effects we find are not driven by initial conditions.
There is no significant difference in the cooperation rate between any two
treatments in round 1 ($p>0.1$ for all comparisons, $DT$). Because in all
treatments the groups start, on average, from the same level of cooperation
rate, it must be that reputational uncertainty and the dynamics of network
evolution are the factors that drive the differences in the cooperation rate
that emerge over time.

Our first result is, therefore, the following.

Result 1: (i) Reputational uncertainty significantly reduces cooperation rate
and welfare in both slow- and fast-changing conditions. (ii) The fast-changing
social environment increases cooperation rate and welfare, but only
qualitatively, regardless of the presence of reputational uncertainty.

The significant finding in the first part of Result 1 is consistent with the
insights from the existing studies on two-person cooperation games and public
good games without any network structure. These studies have shown that the
presence of reputational uncertainty (defined in a broad sense) results in
strictly lower cooperation (Aoyagi and Fréchette (2009), Aoyagi et al. (2019))
and lower contributions (Ambrus and Greiner (2012), Ambrus and Greiner
(2019)). However, in our dynamic social network setting, the reason why
reputational uncertainty reduces cooperation is different from these studies.
In the subsequent sections, we explore two individual-level mechanisms driving
Result 1, which we briefly outline below.

In Section 3.2, we show that reputational uncertainty makes subjects more
_lenient_ and _forgiving_ (to be made precise below) in their link
removal/formation decisions, i.e., it lowers the use of network punishment
(Results 2 and 3 below). Cooperators are unable to precisely break links with
defectors and form links with other cooperators (Result 4 below).
Consequently, reputational uncertainty disrupts the so-called “assortative
matching effect” of network dynamics, whereby cooperators cluster among
themselves and shun away defectors to sustain cooperation (e.g., Rand et al.
(2011), Wang et al. (2012), Gallo and Yan (2015), Gallo et al.
(2019)).212121See also Riedl et al. (2016), Yang et al. (2017), and Riyanto
and Teh (2020) for the same mechanism in the context of dynamic network
coordination games. While this mechanism of reputational uncertainty is
present in both _slow_ and _fast_ conditions, it is statistically significant
only in the fast condition. We explain this difference by pointing out a
substitution pattern between network and action punishments across _slow_ and
_fast_ conditions (Result 6 below).

In Section 3.3, we show that reputational uncertainty makes subjects’ choice
of action (to cooperate or to defect) less sensitive toward past defections by
other players. In other words, action punishment is less prevalent in the
presence of reputational uncertainty, in both _slow_ and _fast_ conditions
(Result 5 below). This reduction in punishment, in turn, promotes defections
and opportunistic plays with reputational uncertainty (Result 7 below).

###  3.2 Network punishment

_Network punishment_ is a primary tool to maintain cooperation when players
can choose their interaction neighborhood. It allows cooperators to remove
links with defectors in order to shun them away while at the same time
selectively form connections with other cooperators. We distinguish between
link removal and link formation decisions and define the following two notions
that describe subjects’ strategies in these decisions.

  * •

Leniency. Strategies are lenient if, conditional on receiving an opportunity
to remove a link, the deciding player does not remove the existing link with a
neighbor who has a history of low cooperative play (i.e., whose actual action
was $C$ at most once in the last five rounds).

  * •

Forgiveness. Strategies are forgiving if, conditional on receiving an
opportunity to propose a link, the deciding player proposes a link to someone
with a history of low cooperative play (i.e., whose actual action was $C$ at
most once in the last five rounds).

These notions of leniency and forgiveness are novel as they are designed to
capture variations in link formation/removal rather than action punishment
which is the focus in two-person PD games (Fudenberg et al. (2012), Embrey et
al. (2013), Aoyagi et al. (2019)).222222In these studies, strategies are
defined to be lenient if they do not prescribe sure defection following a
single defection by the opponent, and forgiving if they return to cooperation
after having defected. The main finding is that in the presence of
reputational uncertainty subjects use strategies that are both lenient and
forgiving. We condition our definitions on play histories of length-5 because
in our experiment the computer interface shows each subject the list of her
current neighbors with their past five actions.

It should be emphasized that the exact definitions of leniency and forgiveness
above are empirically motivated for our analysis. Our results remain robust to
other definitions. For example, we can alternatively define “a history of low
cooperative play” as actual action being $C$ at most two or three times in the
last five rounds. Furthermore, we can broaden the definition of forgiveness by
including link acceptance decisions as well. See Appendices B.1 and B.2 for
further details.

In what follows, we focus on identifying the effect of reputational
uncertainty on leniency and forgiveness in network punishment.232323For
completeness, in Appendix B.3, we report the effect of the network updating
rate on leniency and forgiveness. Given the definitions of leniency and
forgiveness, throughout this section, we focus on observations from rounds
6-21, unless otherwise stated. Before proceeding further, we first describe
the several standard network metrics prominently used in the
analysis.242424See Jackson (2008) for more details.

  * •

Degree. The degree $n_{i}$ measures the number of links or neighbors that a
player $i$ has.

  * •

Clustering coefficient. The local clustering coefficient measures the
connectedness of a player’s neighbors. A player $i$ with $n_{i}$ neighbor has
$\binom{n_{i}}{2}$ possible pairwise links between her neighbors. The
clustering coefficient measures the ratio of the number of links that player
$i$’s neighbors have to the total number of possible links that can
potentially be formed among player $i$’s neighbors.

  * •

Betweenness centrality.252525Our main insights remain robust if we use
closeness centrality instead. The betweenness centrality $c_{B}\left(i\right)$
measures the importance of a player $i$ in a network in terms of the flow in
the network. Formally,

$c_{B}\left(i\right)=\sum_{j\neq k\neq i}\frac{\sigma_{jk}(i)}{\sigma_{jk}},$

where $\sigma_{jk}$ is the number of shortest paths between players $j$ and
$k$, and $\sigma_{jk}(i)$ is the number of those shortest paths that go
through player $i$.

####  3.2.1 Reputational uncertainty promotes leniency and forgiveness

Analyzing Leniency. We begin by analyzing link removal decisions. Figure 3
displays the average likelihood of removing an existing link, conditional on
receiving an opportunity to do so (recall that such an opportunity arrives at
an exogenous rate). We consider three different histories of a neighbor’s
cooperative play: 0 or 1 $C$ (low), 2 or 3 $C$ (medium) and 4 or 5 $C$ (high).
From Figure 3, we can see that subjects seldom remove existing links that are
highly cooperative (around 2% of the total observations in each treatment),
with or without reputational uncertainty. In contrast, in instances of low or
medium cooperative play of the existing link, a subject is less likely to
remove a link when there is reputational uncertainty, suggesting that the
subjects become more lenient.

Figure 3: Average probability to remove an existing link, conditional on
receiving an opportunity. Notes: Separated by the last five rounds’ history,
where Low: the actual action was $C$ at most once in the last five rounds;
Medium: $C$ was the actual action two or three times; High: $C$ was the actual
action more than three times. We report the average values over rounds 6-21,
with the number of observations in parentheses.

To test the effect of reputational uncertainty on leniency, we report in Table
2 the results from a random-effects logit regression with the dependent
variable being the link removal decision ($=1$ if the link is unilaterally
removed). We conduct the regressions separately for the _slow_ and _fast_
conditions, and for neighbors with histories of _low_ and _medium_ cooperative
plays. We exclude link removal decisions with neighbors with histories of
_high_ cooperative play to simplify the presentation, especially given that
subjects virtually never remove links with such neighbors, as observed from
Figure 3. We cluster the standard errors at the group level and use
observations from rounds 6-21.

The independent variables include the following: a treatment dummy variable
for _Reputational uncertainty_ ($=1$ if there is uncertainty) and network
metrics of the deciding player (_Degree_ , _Local_ _clustering_ ,
_Centrality_). To control for the recent cooperativeness of the deciding
player, we also include the last intended action chosen by the deciding player
(_Own C Action t-1_), which equals $1$ if it is $C$ and $0$ if it is
$D$.262626Results 2 and 3 below remain valid if we replace this control
variable with the deciding player’s “innate cooperativeness” defined in
Section 3.2.2. In addition to these variables, we also include seven other
control variables: round number, age, gender, trust obtained from the World
Values Survey question, risk profile elicited from the Holt-Laury procedure,
and prior exposure to game theory concepts and incentivized economic
experiments.272727All regression results reported in the main text remain
robust if we exclude network metrics and these seven control variables. See
Appendix B.1 for further details.

Table 2: Random-effects logit regressions - effect of reputational uncertainty
on leniency.

| Slow |  | Fast
---|---|---|---
History of the target neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link removal (conditional on receiving an opportunity)
Reputational uncertainty | -0.397 | -0.372 |  | -0.949** | -0.820***
| (0.394) | (0.323) |  | (0.375) | (0.207)
Own C Actiont-1 | 1.380*** | 0.703*** |  | 0.360** | 0.404
(recent cooperativeness) | (0.381) | (0.153) |  | (0.181) | (0.253)
Degree | 0.193 | -0.219** |  | 0.014 | -0.068
| (0.155) | (0.091) |  | (0.051) | (0.044)
Local clustering | 1.223 | 1.971** |  | -0.401 | 0.589
| (1.211) | (0.978) |  | (0.346) | (0.468)
Betweenness centrality | -4.723 | 5.967 |  | -3.842** | -0.938
| (6.785) | (4.905) |  | (1.842) | (1.538)
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 508 | 754 |  | 1009 | 3735

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) are in parentheses, * $p<0.10$, ** $p<0.05$,
*** $p<0.01$. Control variables include round number, age, gender, trust and
risk aversion profiles, and prior exposure to game theory and economic
experiments.

The main insight from Table 2 comes from the coefficient of _Reputational
uncertainty_. It shows that reputational uncertainty results in subjects using
strategies that are more lenient in the _fast_ condition, in the sense that
they become significantly less likely to remove existing links with neighbors
of low and medium cooperative play ($p<0.05$ and $p<0.01$, respectively). In
contrast, in the _slow_ condition, reputational uncertainty does not
significantly affect leniency, given that the presence of uncertainty does not
significantly affect the probability of link removal for any history
($p>0.1$). In addition, such removal decisions are primarily triggered by
deciding players whose action is $C$ in the previous round (coefficient of
_Own C Action ${}_{t-1},$ _$p<0.05$), confirming the intuition that players
who are “recently” cooperative (prior to the link removal decision) punish
defectors (Low $C$) by removing links with them.

Result 2: Reputational uncertainty increases leniency in link removal
decisions, but the effect is statistically significant only in the fast
condition.

In Appendix B.4, we show that Result 2 remains robust even if we take into
account that subjects who are making link removal decisions may treat links
that were formed from their own proposal differently compared to links they
accepted but did not initially propose. Interestingly, we find that subjects
are indeed significantly less lenient toward the latter type of links (but
only in the fast condition).282828We thank an anonymous referee for suggesting
this exercise. Thus, leniency depends on how a relationship was initially
formed, a novel feature that is absent in non-network settings.

Analyzing Forgiveness. Next, we analyze forgiveness in link formation
decisions. Figure 4 displays the average probability of sending a proposal
(conditional on receiving an opportunity to do so) for the three different
histories of the potential neighbor’s cooperative play defined earlier. We
observe that the likelihood of sending a proposal is consistently high (close
to 0.90) for recipients with a history of high cooperative play, with or
without reputational uncertainty. In contrast, this likelihood is generally
less than 0.58 when a potential neighbor has a history of low or medium
cooperative play. However, reputational uncertainty results in subjects being
more willing to send a proposal to such potential neighbors, suggesting that
the subjects become more forgiving.

Figure 4: Average probability to send a proposal, conditional on receiving an
opportunity. Notes: Separated by the last five rounds’ history, where Low: the
actual action was $C$ at most once in the last five rounds; Medium: $C$ was
the actual action two or three times; High : $C$ was the actual action more
than three times. We report the average values over rounds 6-21, with the
number of observations in parentheses.

To test the effect of reputational uncertainty on forgiveness, we report in
Table 3 the results from a random-effects logit regression with the dependent
variable being the link proposal decision ($=1$ if the participant sends a
proposal). Our regression specifications and control variables used in the
regressions shown in Table 3 are the same as those in Table 2. Again, the main
insight from Table 3 comes from the coefficient of _Reputational uncertainty_.
In the fast condition, reputational uncertainty makes the subjects more
forgiving when forming new links. They become significantly more likely to
propose new links with target potential neighbors of low and medium
cooperative play ($p<0.05$ and $p<0.01$, respectively).292929This result
remains robust even if we additionally control for the number of links of the
target potential neighbor. In contrast, in the _slow_ condition, reputational
uncertainty does not significantly affect forgiveness given that the
coefficient of _Reputational uncertainty_ is insignificant for any history
($p>0.1$). Finally, consistent with the intuition discussed at the beginning
of Section 3.2, recent cooperative players are less likely to form new links
with neighbors of low and medium cooperative play (coefficient of _Own C
Action ${}_{t-1},$ _$p<0.05$).

Table 3: Random-effects logit regressions - effect of reputational uncertainty
on forgiveness.

| Slow |  | Fast
---|---|---|---
History of the potential neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link proposal (conditional on receiving an opportunity)
Reputational uncertainty | 1.684 | 0.145 |  | 0.789** | 0.763***
| (1.760) | (0.341) |  | (0.393) | (0.205)
Own C Actiont-1 | -1.043 | -1.077*** |  | -0.424** | -0.402***
(recent cooperativeness) | (0.679) | (0.251) |  | (0.178) | (0.154)
Degree | -0.294 | -0.174 |  | -0.189** | -0.141***
| (0.272) | (0.124) |  | (0.074) | (0.046)
Local clustering | -3.809* | -0.609 |  | -1.044*** | -0.244
| (2.160) | (1.116) |  | (0.317) | (0.310)
Betweenness centrality | 14.463* | 2.401 |  | 2.855 | 4.984**
| (8.452) | (8.932) |  | (2.051) | (2.023)
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 381 | 436 |  | 1979 | 2623

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) are in parentheses, * $p<0.10$, ** $p<0.05$,
*** $p<0.01$. Control variables include round number, age, gender, trust and
risk aversion profiles, and prior exposure to game theory and economic
experiments.

Result 3: Reputational uncertainty increases forgiveness in link proposal
decisions, but the effect is statistically significant only in the fast
condition.

From Results 2 and 3, we note that reputational uncertainty affects the usage
of network punishment differently across the _slow_ and _fast_ conditions.
Intuitively, when there is reputational uncertainty, a deciding player finds
it difficult to ascertain whether a specific _target player_ with a history of
low/medium cooperative is a “true defector”. The observed history of
low/medium cooperative play could be a result of unfortunate random draws that
reverse the target player’s intention to cooperate. The standard intuition
from two-person PD games (Fudenberg et al. (2012), Embrey et al. (2013),
Aoyagi et al. (2019)) suggests that, facing such uncertainty, the deciding
player would give the target player a “benefit of doubt”, and adopt strategies
that are more lenient and forgiving. This is precisely the intuition in the
_fast_ condition in Results 2 and 3.

In contrast, in the _slow_ condition, there is a high potential burden for
maintaining/starting a relationship with a target player having a history of
low cooperative play. If the target player turns out to be a genuine defector,
the next opportunity for the deciding player to remove this link would only
arise in the distant future. In other words, in the _slow_ condition, giving a
“benefit of doubt” is costly, thus explaining why the effects of reputational
uncertainty are not statistically significant.303030To complement this
argument, we show in Appendix B.3 that subjects indeed become significantly
more lenient in the fast condition than in the slow condition, holding the
presence of reputational uncertainty constant.

####  3.2.2 Reputational uncertainty disrupts clustering of cooperators

In this section, we examine how the increase in subjects’ leniency and
forgiveness in network punishment (as a consequence of the reputational
uncertainty) affects the network-level metrics and the tendency for
cooperators to connect to each other.

We classify a subject as an _innate cooperator_ (Type-$C$) if she chose $C$ as
her first-round action, and an _innate defector_ (Type-$D$) if she chose $D$.
This categorization is suitable for our homophily analysis below because it is
exogenous to the evolution of cooperation in the group so that it is fixed
before participants receive any feedback about game play. The average number
of innate cooperators ranges from 6.63 to 6.88 across treatments, and there
are no significant differences across treatments ($p>0.1$, $DT$). Appendix B.5
provides further details on the comparison between these two categories of
players and the statistical analyses. Most importantly, as one would expect if
our categorization into these types is meaningful, we observe that Type-$C$
players have significantly more intended cooperative actions compared to
Type-$D$ players ($p<0.05$).

Following Currarini et al. (2009), we measure the tendency for players of the
same type $i$ to connect to each other as follows. Consider a population with
two categories of players $i\in\left\\{C,D\right\\}$. Let ${w_{i}}$ be the
relative fraction of $i$ players in the population, where ${w_{C}+w}_{D}=1$.
Let ${s_{i}}$ denote the average number of links that individuals of category
$i$ have with players who are of the same category and let ${o_{i}}$ be the
average number of links that category $i$ players have with players of the
opposite category. The homophily index is

${H_{i}=}\frac{{s_{i}}}{{s_{i}+o_{i}}},$

and it captures the fraction of the links of players of category $i$ that are
with that same category. A key drawback of the homophily index is that it is
sensitive to the relative size of the two categories in the population, so
that it is _not a suitable measure_ to evaluate the extent of homophily
between categories of different sizes (e.g., in our setting). For this reason,
we instead focus on the following measure in our analysis:

  * •

Inbreeding Homophily index (IH). Inbreeding homophily index of category $i$ is
defined as

${IH_{i}=}\frac{{H_{i}-w_{i}}}{{1-w_{i}}}\text{{.}}$

Intuitively, it corrects the homophily index by the relative sizes of each
category. We say that there is inbreeding homophily between individuals of
category $i$ if ${IH_{i}>0}$, and there is baseline homophily if ${IH_{i}=0}$
.

Table 4 provides summary statistics of the standard network-level metrics,
$IH$ of type-$C$ players, and the average betweenness centrality of each
type-$C$ player. We focus on rounds 2-21 (as in Table 1) because we do not
need to condition the analysis on play histories of length-5 unlike in the
previous subsection. We then apply the Mann-Whitney rank-sum test (_MW_) to
evaluate the effect of reputational uncertainty, holding constant the rate of
network updating.

Table 4: Network-level metrics across treatments.

| Slow |  | Fast
---|---|---|---
| No uncertainty | Uncertainty |  | No uncertainty | Uncertainty
Avg. degree | 7.485 (1.537) | 7.293 (1.621) |  | 6.184 (1.316) | 6.729 (1.150)
Avg. local clustering | 0.710 (0.149) | 0.681 (0.156) |  | 0.675 (0.113) | 0.712 (0.107)
IH of type-$C$ players | 0.061 (0.064) | -0.024 (0.126) |  | 0.122 (0.137) | -0.006 (0.104)
Avg. btw. centrality of type-$C$ players | 0.038 (0.009) | 0.038 (0.009) |  | 0.051 (0.010) | 0.041 (0.011)

  * •

Note: we report mean values over rounds 2-21 with standard errors in
parentheses.

The main finding from Table 4 is that, in the fast condition, reputational
uncertainty significantly (i) decreases the inbreeding homophily index ($IH$)
of type-$C$ players from 0.122 to -0.006 ($MW$, $p=0.046$); and (iii)
decreases the average betweenness centrality of each type-$C$ player from
0.051 to 0.041 ($MW$, $p=0.059$). However, the differences in these measures
are statistically insignificant in the slow condition ($MW$, $p>0.1$).
Finally, reputational uncertainty has no significant effect on average degree
and local clustering in both fast and slow conditions.313131We also perform
Kolmogorov-Smirnov equality test to check the effect of reputational
uncertainty on the overall degree distributions. We find no significant effect
($p>0.1$), conditioned on either fast or slow treatment.

Result 4: Reputational uncertainty decreases (i) inbreeding homophily among
innate cooperators and (ii) average centrality of each innate cooperator, but
only in the fast condition.

Result 4 reflects the intuition that innate cooperators may find it more
difficult to recognize each other and connect when there is reputational
uncertainty. This results in a lower level of cooperativeness-based homophily
and a lower centrality of each innate cooperator, especially when the network
updating happens quickly.

###  3.3 Action punishment and opportunistic play

Action punishment is the primary channel to ensure cooperative behavior in
two-player PD games. It entails defecting as a response to defection by the
other player. The following definition extends this notion to our set-up with
more than two players in a linked network.

  * •

Action Punishment. The likelihood of a player choosing $D$ as the intended
action as an increasing function of the fraction of neighbors who choose $D$.
A steep reaction function indicates a strong extent of action punishment and
vice versa.

####  3.3.1 Reputational uncertainty erodes action punishment

Intuitively, the presence of reputational uncertainty could affect a player’s
choice of action in two ways. First, a player finds it difficult to ascertain
whether defection choices by her neighbors reflect their true intentions or
are merely random noise (Fudenberg et al. (2012), Embrey et al. (2013), Aoyagi
et al. (2019)). Thus, with uncertainty, the player’s likelihood of choosing
$D$ would be less sensitive towards her neighbors’ past defections, weakening
the extent of action punishment. Second, the weaker action punishment implies
less repercussions from defecting, making defection more tempting.
Consequently, holding constant the neighbors’ behaviors, we expect a player’s
likelihood of choosing $D$ to be higher when there is uncertainty. In sum, if
we plot a subject’s likelihood of choosing $D$ as a function of the fraction
of neighbors who choose $D$, reputational uncertainty should raise the
intercept while flattening the gradient of the plot.

We report in Table 5 the results from a random-effects logit regression with
the dependent variable being each subject’s choice of intended defecting
action ($=1$ if it is $D$) in each round $t$. We focus on the switching
behaviors by limiting the analysis to the subsample of instances when a
subject has selected $C$ as the intended action in round $t-1$ and focus on
rounds 2-21.323232The results remain the same if we do not limit the analysis
to this subsample. Note we use rounds 2-21 here instead of 6-21 because the
analysis does not condition on the established reputation (information on the
history of plays of length-5) of the players, unlike in the previous
subsection Again, we conduct the regressions separately for the _slow_ and
_fast_ conditions. Specifications (1)-(3) focus on the _slow_ conditions,
while (4)-(6) investigate the _fast_ conditions.

The key independent variable is _Fraction of_ $D$_neighbors_ t-1, which
denotes the fraction of a player’s neighbors whose actual action was $D$ in
the last round. In the baseline specifications (1) and (4), we focus on
_Fraction of_ $D$_neighbors_ t-1, without including any treatment dummy
variables. In specifications (2) and (5), we include an intercept dummy
variable for treatments with _Reputational uncertainty_ , and interact the
dummy variable with _Fraction of_ $D$ _neighbors_ t-1. Finally, in
specifications (3) and (6), we check for robustness by including network
metrics of the deciding player. The remaining control variables are the same
as in Tables 2 and 3.

Table 5: Random-effects logit regressions - effect of rep. uncertainty on
action punishment.

| Slow |  | Fast
---|---|---|---
| (1) | (2) | (3) |  | (4) | (5) | (6)
Dependent variable: Intended defecting action (=1 if $D$)
Fraction of $D$ neighborst-1 | 3.872*** | 4.602*** | 4.651*** |  | 2.656*** | 3.375*** | 3.652***
| (0.397) | (0.567) | (0.584) |  | (0.254) | (0.426) | (0.470)
Fraction of $D$ neighborst-1 |  | -1.478* | -1.527** |  |  | -1.504** | -1.690**
$\times$ Reputational uncertainty |  | (0.778) | (0.762) |  |  | (0.645) | (0.687)
Reputational uncertainty |  | 0.878 | 1.003 |  |  | 0.786*** | 0.813**
|  | (0.674) | (0.649) |  |  | (0.290) | (0.320)
Degree |  |  | 0.253** |  |  |  | 0.130**
|  |  | (0.113) |  |  |  | (0.062)
Local clustering |  |  | -1.018 |  |  |  | 0.384
|  |  | (0.874) |  |  |  | (0.439)
Betweenness centrality |  |  | -8.576** |  |  |  | 1.198
|  |  | (3.417) |  |  |  | (1.485)
Control and intercept | Yes | Yes | Yes |  | Yes | Yes | Yes
Observations | 1985 |  | 2335

  * •

Notes: Standard errors (clustered at the group level) are in parentheses, *
$p<0.10$, ** $p<0.05$, *** $p<0.01$. Control variables include round number,
age, gender, trust and risk aversion profiles, and prior exposure to game
theory and economic experiments.

The main finding from Table 5 is that the probability of choosing $D$ as the
intended action in a round is significantly increasing ($p<0.01$ for all
specifications) in the fraction of defecting neighbors, suggesting that
subjects engage in action punishment. Then, specifications (3) and (6) show
that the presence of reputational uncertainty alters this dependency, thus
dampening action punishment in both _slow_ and _fast_ conditions.
Specifically, with reputational uncertainty, players become less sensitive
toward past actions of their neighbors (the coefficient of the interaction
term between _Reputational uncertainty_ and _Fraction of_ $D$_neighbors_ t-1,
$p<0.05$ in both conditions) and become more likely to defect (the coefficient
of _Reputational uncertainty_ , $p=0.122$ and $p<0.05$ in _slow_ and _fast_
conditions respectively). Figure 5 illustrates that these effects of
reputational uncertainty are consistent with the intuition discussed at the
beginning of this subsection.

Result 5: In both fast and slow conditions: (i) individual subject’s
probability of choosing $D$ is significantly increasing in the fraction of
defecting neighbors; (ii) reputational uncertainty makes this probability
significantly less sensitive towards the fraction of defecting neighbors
(i.e., less action punishment).

Figure 5: Estimated likelihood of participants continuing to choose $D$ as
their intended action in a round, conditioned on the fraction of defecting
neighbors in the last round (based on (3) and (6) in Table 5). Vertical lines
indicate 95% confidence intervals.

####  3.3.2 Substituting action punishment with network punishment

At the end of Section 3.2, we pointed out that network punishment is more
effective in a fast environment than in a slow environment. This suggests that
subjects in a slow environment may primarily rely on action punishment to
punish defectors. In contrast, subjects in a fast environment may substitute
action punishment with the more targeted network punishment.

To examine this conjecture, we replicate the analysis in Table 5 but focus on
the effect of the fast condition. We conduct the regression separately for
treatments without reputational uncertainty and with reputational uncertainty.
The independent variables of interest are the treatment dummy variable Fast
and its interaction with Fraction of $D$ neighborst-1. All other
specifications are the same as in Table 5.

Table 6: Random-effects logit regressions - effect of fast condition on action
punishment.

| No uncertainty |  | Reputational uncertainty
---|---|---|---
| (1) | (2) | (3) |  | (4) | (5) | (6)
Dependent variable: Intended action (=1 if $D$)
Fraction of $D$ neighborst-1 | 3.900*** | 4.667*** | 4.629*** |  | 2.363*** | 3.073*** | 3.176***
| (0.273) | (0.311) | (0.329) |  | (0.322) | (0.295) | (0.262)
Fraction of $D$ neighborst-1 $\times$ Fast |  | -1.262*** | -0.958* |  |  | -1.216** | -1.210**
|  | (0.487) | (0.580) |  |  | (0.475) | (0.497)
Fast |  | 0.514 | 0.531 |  |  | 0.504 | 0.563
|  | (0.323) | (0.333) |  |  | (0.298) | (0.331)
Network metrics | No | No | Yes |  | No | No | Yes
Control and intercept | Yes | Yes | Yes |  | Yes | Yes | Yes
Observations | 1985 |  | 2335

  * •

Notes: Standard errors (clustered at the group level) are in parentheses, *
$p<0.10$, ** $p<0.05$, *** $p<0.01$. Network metrics include degree, local
clustering coefficient, and centrality. Control variables include round
number, age, gender, trust and risk aversion profiles, and prior exposure to
game theory and experiments.

The main finding from Table 6 comes from specifications (3) and (6). Holding
the presence of reputational uncertainty constant, introducing the fast
condition makes players less sensitive towards past actions of their neighbors
(the coefficient of the interaction term between _Fast_ and _Fraction of_
$D$_neighbors_ t-1, $p<0.10$ without uncertainty and $p<0.05$ with
uncertainty). This is consistent with the intuition that a faster-updating
environment induces players to substitute away from action punishment.

Result 6: Holding constant the presence of reputational uncertainty: (i)
individual subject’s probability of choosing $D$ is significantly increasing
in the fraction of defecting neighbors; (ii) fast-changing social environment
makes this probability significantly less sensitive towards the fraction of
defecting neighbors (i.e., substituting away from action punishment).

####  3.3.3 Reputational uncertainty promotes opportunistic play

Apart from its effect on action punishment, reputational uncertainty could
undermine overall cooperation by promoting “opportunistic play”. Specifically,
in the presence of reputational uncertainty, cooperators could occasionally
defect, knowing that others would be unable to ascertain whether or not the
defection is intentional, i.e., they can hide their occasional defections
“behind the veil of noise”. For the purpose of analysis, we define this
behavior as follows.

  * •

Opportunistic Play. The act of a player choosing $D$ as the intended action
after establishing a history of high cooperative play (i.e., actual action was
$C$ at least four times in the last five rounds).333333 Note that this
definition of high cooperative play is consistent with the categorization in
Section 3.2.1. Our results remain robust if we define this based on exactly 4
$C$ instead.

Figure 6 reports the average probability of opportunistic play in each
treatment over rounds 6-21. We then apply the Mann-Whitney rank-sum test
(_MW_) to evaluate the effect of reputational uncertainty, holding constant
the rate of network updating. We find that reputational uncertainty
significantly increases the likelihood of opportunistic play from 0.16 to 0.30
in the _fast_ condition ($p=0.021$, $MW$), and from 0.14 to 0.33 in the _slow_
condition ($p=0.009$, $MW$).

Figure 6: Probability of intended action being $D$ given specific history of
own actual actions. Notes: we focus on players who had at least 4 occurrence
of the actual action being $C$ out of the last 5 rounds. We report the average
values over rounds 6-21. The number of observations are in parentheses.

Result 7: In both fast and slow conditions, reputational uncertainty
significantly promotes opportunistic play.

The rise in opportunistic play in Result 7 provides a complementary
explanation for the drop in the overall cooperation level when reputational
uncertainty is present. Previous experimental research on repeated games has
found that the presence of reputational uncertainty can result in a higher
share of players adopting non-cooperative strategies like “always defecting”
or a “defection-inclined version of tit-for-tat” (Fudenberg et al. (2012),
Aoyagi et al. (2019)). While our set-up differs from these studies, we have
obtained a similar result: opportunistic play, which can be interpreted as a
type of non-cooperative strategy, becomes more common when there is
reputational uncertainty.

##  4 Conclusion

We provide empirical evidence regarding cooperation in an uncertain world with
individuals receiving opportunities to update their interaction partners. Our
experiment demonstrates that the presence of noise, defined as uncertainty on
whether other people’s observed actions correspond to their intended actions,
affects individual behavior along the network as well as the action dimension.
We further show that the extent of the impact of noise depends fundamentally
on the frequency at which individuals receive opportunities to propose and/or
remove links with others. In doing so, we extend our knowledge on the
characterization of behavior under noise with endogenous network formation.

Our study raises several questions that suggest exciting possibilities for
follow-up studies as future research. First, we do not impose an explicit cost
on forming, maintaining, or removing links in our experiment.343434 Real-world
relationships take time to form, and maintenance of such links involves time
and effort, both of which are costly. Exploring the effect of such cost might
provide us with additional insights than mere opportunity costs of not
interacting as in our setup.

Second, there are few studies that investigate the effect of allowing
communication on cooperation in a two-person repeated PD game (Arechar et al.,
2017) and a partnership game (Embrey et al., 2013). It would be fruitful to
study the effect of communication in our framework.

Third, viewing cooperation as a risky choice, the strategic decisions made by
individuals affect the distribution of income and wealth in society. In a
static, non-strategic, and non-networked context, Cappelen et al. (2013)
demonstrate that, even though many subjects focus exclusively on ex-ante
opportunities, most favor some redistribution ex-post, and several individuals
make a distinction between ex-post inequalities that reflect differences in
luck and ex-post inequalities that reflect differences in choices. It would be
interesting to explore the fairness and inequality consequences of the action
and link proposal/removal decisions in our setup.

Finally, noisy information is publicly observable in our setting, so our game
has an imperfect public monitoring structure. However, it is possible that
players do not even know what signals other players have observed about their
own play, thereby making the monitoring structure private. Aoyagi et al.
(2019) and Kayaba et al. (2020) study cooperation in two-person experimental
games with such a monitoring structure. Our setup could be extended to include
this type of challenging monitoring technology or, more generally, alternative
formulations of reputational uncertainties.

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Appendix: Cooperation and Punishment Mechanisms in Uncertain and Dynamic
Social Networks

##  Appendix A Demographics

Table A1 summarizes the main socio-demographic characteristics of our
participants. We verified that there are no significant differences ($p>0.10$)
across the four treatments in terms of these socio-demographic
characteristics.

Table A1: Average values of demographic variables.

Treatment (No Uncertainty, (Uncertainty, (No Uncertainty, (Uncertainty, Slow)
Slow) Fast) Fast) Age 20.91 21.31 20.84 21.01 Gender (Female) 57% 45% 59% 49%
Risk Aversion 5.42 5.62 5.91 6.03 Game Theory 25% 24% 23% 22% Prior
Participation 63% 46% 46% 47% Trust 25% 17% 28% 20%

  * •

_Risk Aversion_ shows the score from Holt and Laury (2002) risk-elicitation
test, in which a score of 0 indicates minimal risk aversion; _Trust_ shows the
percentage of participants indicating “yes” in the standard interpersonal
trust question taken from the World Values Survey (WVS).

##  Appendix B Additional results

###  B.1 Alternative definitions for leniency and forgiveness

Our results remain robust if we redefine Leniency and Forgiveness in Section
3.2 by specifying “history of low cooperative play” as actual action (of the
target neighbor/potential neighbor) being $C$ at most two times in the last 5
rounds.

Tables A2 and A3 below replicate Results 2 and 3. Given the new definitions,
in what follows, we focus on observations where the target neighbor/potential
neighbor has actual action being $C$ at most two times in the last five
rounds. All other specifications and controls used are the same as Tables 2
and 3. We find that the coefficient of Reputational uncertainty in the Fast
condition remains significant ($p<0.05$) and that the results remain robust if
we exclude the demographic variables and the network metrics from the
regressions.

Table A2: Random-effects logit regressions - effect of rep. uncertainty on
leniency (new definition).

| Slow |  | Fast
---|---|---|---
History of the target neighbor: Actual action being $C$ at most twice
Dependent variable: Link removal (conditional on receiving an opportunity)
Reputational uncertainty | -0.330 | -0.247 | -0.240 |  | -0.662** | -0.733*** | -0.720**
| (0.325) | (0.295) | (0.304) |  | (0.261) | (0.257) | (0.286)
Own C Actiont-1 | 0.957*** | 0.913*** | 0.904*** |  | 0.393** | 0.376** | 0.369**
| (0.199) | (0.204) | (0.197) |  | (0.183) | (0.179) | (0.186)
Degree |  |  | 0.035 |  |  |  | 0.005
|  |  | (0.117) |  |  |  | (0.043)
Local clustering |  |  | 0.384 |  |  |  | -0.784**
|  |  | (0.686) |  |  |  | (0.326)
Betweenness centrality |  |  | -1.628 |  |  |  | -3.113**
|  |  | (4.367) |  |  |  | (1.370)
Control | No | Yes | Yes |  | No | Yes | Yes
Intercept | Yes | Yes | Yes |  | Yes | Yes | Yes
Observations | 899 |  | 2606

  * •

Notes: Standard errors (clustered at the group level) are in parentheses, *
$p<0.10$, ** $p<0.05$, *** $p<0.01$. Control variables include round number,
age, gender, trust and risk aversion profiles, and prior exposure to game
theory and economic experiments.

Table A3: Random-effects logit regressions - effect of rep. uncertainty on
forgiveness (new definition).

| Slow |  | Fast
---|---|---|---
History of the target neighbor: Actual action being $C$ at most twice
Dependent variable: Link proposal (conditional on receiving an opportunity)
Reputational uncertainty | 0.364 | 0.363 | 0.298 |  | 0.752** | 0.865*** | 0.966***
| (0.329) | (0.337) | (0.418) |  | (0.297) | (0.294) | (0.274)
Own C Actiont-1 | -0.806** | -0.790** | -0.668** |  | -0.418*** | -0.374*** | -0.375***
| (0.373) | (0.361) | (0.335) |  | (0.111) | (0.130) | (0.123)
Degree |  |  | -0.313*** |  |  |  | -0.188***
|  |  | (0.112) |  |  |  | (0.043)
Local clustering |  |  | -0.712 |  |  |  | -0.438*
|  |  | (0.473) |  |  |  | (0.236)
Betweenness centrality |  |  | 5.669 |  |  |  | 3.079*
|  |  | (6.886) |  |  |  | (1.625)
Control | No | Yes | Yes |  | No | Yes | Yes
Intercept | Yes | Yes | Yes |  | Yes | Yes | Yes
Observations | 628 |  | 3340

  * •

Notes: Standard errors (clustered at the group level) are in parentheses, *
$p<0.10$, ** $p<0.05$, *** $p<0.01$. Control variables include round number,
age, gender, trust and risk aversion profiles, and prior exposure to game
theory and economic experiments.

###  B.2 Forgiveness in both proposal and acceptance

In Section 3.2, our definition of _Forgiveness_ is based on link proposal
decisions. We can alternatively broaden the definition by considering both
link proposal and link acceptance decisions. That is, strategies are forgiving
if, conditional on receiving an opportunity to propose a link, the deciding
player proposes a link to or accepts a link proposal (after initially deciding
not to propose) from someone with a history of low cooperative play (i.e.,
whose actual action was $C$ at most once in the last five rounds).

Table A4 is analogous to Table 3, except that we now additionally include
observations that are link acceptance decisions. In other words, we treat
observations of link proposal decisions and link acceptance decisions
(whenever it arises) pertaining to the potential neighbor as two separate
observations. The dependent variable is the link forming decision ($=1$ if the
participant sends a proposal to or accepts a link proposal from the potential
neighbor). We also included a dummy variable _Is acceptance decision_ (=1 if
the observation is a link acceptance decision).

Table A4: Random-effects logit regressions - effect of uncertainty on
forgiveness (incl. acceptance decisions).

| Slow |  | Fast
---|---|---|---
History of the target neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link proposal or acceptance (conditional on receiving an
opportunity)
Reputational uncertainty | 1.784 | 0.404 |  | 1.376*** | 0.815***
| (3.276) | (0.399) |  | (0.510) | (0.313)
Own C Actiont-1 | -1.222 | -1.311*** |  | -0.064 | -0.227
| (0.823) | (0.339) |  | (0.159) | (0.169)
Is acceptance decision | -1.615*** | -1.845*** |  | -0.937*** | -1.219***
| (0.453) | (0.259) |  | (0.144) | (0.145)
Network metrics | Yes | Yes |  | Yes | Yes
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 591 | 569 |  | 3227 | 3545

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) are in parentheses, * $p<0.10$, ** $p<0.05$,
*** $p<0.01$. Network metrics include degree, local clustering coefficient,
and betweenness centrality. Control variables include round number, age,
gender, trust and risk aversion profiles, and prior exposure to game theory
and economic experiments.

There are two main findings from Table A4. First, Result 3 remains robust as
indicated by the coefficient of Reputational uncertainty in the fast condition
($p<0.01$ for both Low $C$ and Medium $C$, respectively). Second, all else
being equal, players are more likely to say “yes” when faced with a decision
to send a link proposal initially than to say “yes” in an acceptance decision
(after initially deciding not to propose), as indicated by the coefficient of
Is acceptance decision ($p<0.01$).

To illustrate the second finding, Figure A1 is analogous to Figure 4 but
focuses on acceptance decisions. Notice that the number of observations is low
compared to the decisions of sending a proposal in Figure 4. Also, the
likelihood of players saying “yes” to an incoming proposal (after not sending
a proposal initially) is generally below 0.50 across all play histories of the
potential neighbor.

Figure A1: Average likelihood to accept an incoming proposal (after not
proposing initially). Notes: Separated by the last five rounds’ history, where
Low: the actual action was $C$ at most once in the last five rounds; Medium:
$C$ was the actual action two or three times; High: $C$ was the actual action
more than three times. We report the average values over rounds 6-21, with the
number of observations in parentheses.

###  B.3 Effect of network updating rate on leniency and forgiveness

Tables A5 and A6 are similar to the analysis in Section 3.2.1, except that
here we focus on the effect of Fast condition. All other specifications and
controls used are the same as Tables 2 and 3. Introducing fast condition
significantly raises leniency ($p<0.05$), regardless of the level of
reputational uncertainty. Meanwhile, it has no significant impact on
forgiveness ($p>0.1$).

Table A5: Random-effects logit regressions - effect of network updating rate
on leniency.

| No uncertainty |  | Reputational uncertainty
---|---|---|---
History of the target neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link removal (conditional on receiving an opportunity)
Fast | -1.032*** | 0.023 |  | -1.377*** | -0.604**
| (0.319) | (1.822) |  | (0.328) | (0.294)
Own C Actiont-1 | 0.694* | 0.494 |  | 0.583*** | 0.421
| (0.421) | (0.348) |  | (0.175) | (0.262)
Degree | -0.009 | 0.024 |  | 0.071 | -0.142***
| (0.083) | (0.319) |  | (0.048) | (0.047)
Local clustering | 1.791** | 0.422 |  | -1.374** | 0.628
| (0.805) | (1.601) |  | (0.603) | (0.602)
Betweenness centrality | -0.806 | -0.148 |  | -6.866*** | -2.999
| (2.474) | (4.996) |  | (2.440) | (2.577)
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 565 | 1490 |  | 952 | 2999

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) are in parentheses, * $p<0.10$, ** $p<0.05$,
*** $p<0.01$. Control variables include round number, age, gender, trust and
risk aversion profiles, and prior exposure to game theory and economic
experiments.

Table A6: Random-effects logit regressions - effect of network updating rate
on forgiveness.

| No uncertainty |  | Reputational uncertainty
---|---|---|---
History of the potential neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link proposal (conditional on receiving an opportunity)
Fast | 0.130 | -0.526 |  | -0.330 | -0.054
| (0.477) | (0.393) |  | (0.547) | (0.251)
Own C Actiont-1 | -0.595** | -0.736** |  | -0.296 | -0.357***
| (0.233) | (0.339) |  | (0.213) | (0.137)
Degree | -0.161 | -0.211*** |  | -0.221** | -0.126**
| (0.101) | (0.047) |  | (0.101) | (0.060)
Local clustering | -1.857*** | -0.541 |  | -0.405 | -0.198
| (0.599) | (0.356) |  | (0.315) | (0.364)
Betweenness centrality | 0.190 | 5.122* |  | 7.900** | 5.580*
| (3.020) | (2.645) |  | (3.713) | (2.888)
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 1211 | 1228 |  | 1149 | 1831

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) are in parentheses, * $p<0.10$, ** $p<0.05$,
*** $p<0.01$. Control variables include round number, age, gender, trust and
risk aversion profiles, and prior exposure to game theory and economic
experiments.

###  B.4 Leniency and type of link formation

In our experimental environment, broadly speaking, there are three ways
through which a link between two players can be formed: (i) imposed by default
starting from round 1; (ii) formed through mutual proposal; (iii) formed
through a mixture of proposal and acceptance decisions of the two players. We
summarize the properties of each type of link below (those that lasted for at
least 1 round):

| Default links | Mutually-proposed links | Proposal-acceptance links
---|---|---|---
Link duration | 11.56 (8.23) | 6.75 (5.40) | 5.64 (4.76)
Observations | 1972 | 1070 | 524

  * •

Note: we report mean values in all treatments throughout all rounds with
standard errors in parentheses.

Observe that the second type of links tends to last longer than the third
type. This suggests that players making link removal decisions may treat links
formed from their own proposal differently than links they accepted but did
not initially propose. To explore this question, Table A7 below replicates the
leniency analysis in Table 2 but includes the following additional dummy
variables: (i) Was a default link (=1 if the link was imposed from round 1),
and (ii) Was an own accepted link (=1 if the link was formed from the deciding
player’s acceptance after not proposing a link initially). This specification
means that the baseline case (when both dummy variables are equal to zero)
refers to links formed from the deciding player’s proposal.

Table A7: Random-effects logit regressions - effects of rep. uncertainty and
types of links on leniency.

| Slow |  | Fast
---|---|---|---
History of the target neighbor: |  Low $C$ |  Medium $C$ |  |  Low $C$ |  Medium $C$
Dependent variable: Link removal (conditional on receiving an opportunity)
Reputational uncertainty | -0.404 | -0.374 |  | -0.907** | -0.784***
| (0.416) | (0.348) |  | (0.373) | (0.191)
Own C Actiont-1 | 1.449*** | 0.737*** |  | 0.347* | 0.403
| (0.398) | (0.156) |  | (0.184) | (0.255)
Was an own accepted link | 1.304 | 1.092 |  | 0.373* | 0.427***
| (1.964) | (1.234) |  | (0.220) | (0.146)
Was a default link | 1.279** | 0.712 |  | -0.059 | -0.541***
| (0.553) | (0.562) |  | (0.101) | (0.124)
Network metrics | Yes | Yes |  | Yes | Yes
Control and intercept | Yes | Yes |  | Yes | Yes
Observations | 508 | 754 |  | 1009 | 3735

  * •

Notes: Low $C$\- $C$ was the actual action at most one time in last 5 rounds;
Medium $C$\- $C$ was the actual action two or three times. Standard errors
(clustered at the group level) in parentheses, * $p<0.10$, ** $p<0.05$, ***
$p<0.01$. Network metrics include degree, local clustering coefficient, and
centrality. Control variables include round number, age, gender, trust and
risk aversion profiles, and prior exposure to game theory and experiments.

There are two main findings from Table A7. First, Result 2 remains robust as
indicated by the coefficient of Reputational uncertainty in the fast condition
($p<0.05$ and $p<0.01$ for Low $C$ and Medium $C$, respectively). Second, all
else being equal, in fast condition, players are more likely to remove links
they accepted but did not initially propose (relative to those that they
proposed), as indicated by the coefficient of Was an own accepted link
($p<0.1$ and $p<0.01$ for Low $C$ and Medium $C$, respectively).

###  B.5 Classification of Type-C and Type-D players

In Section 3.2.2, we classified players as either type-C or type-D according
to their intended action in the first round. Table A8 summarizes the key
properties of type-C and type-D players in each treatment. To check the
statistical significance of the observed differences between these two types
of players, we adopt the Wilcoxon paired sign-rank test for each treatment,
with eight observations per treatment.

For each treatment, we find that the difference is statistically significant
($p<0.05$) for Intended action, Average degree, IH, Homophily index, and
Centrality. The only exceptions are the two treatments with reputational
uncertainties, whereby the difference between type-C and type-D players in
homophily index and centrality are statistically insignificant ($p>0.1$).

Table A8: Descriptions of Type-C and Type-D players.

|  | Slow | Fast
---|---|---|---
|  | No uncertainty | Uncertainty | No uncertainty | Uncertainty
Number of players | C | 6.625 | 6.750 | 6.875 | 6.750
| D | 5.375 | 5.250 | 5.125 | 5.250
Intended action (=1 if C) | C | 0.691 | 0.538 | 0.770 | 0.666
| D | 0.482 | 0.304 | 0.559 | 0.424
Average degree | C | 7.872 | 7.639 | 6.821 | 7.000
| D | 7.021 | 6.782 | 5.324 | 6.327
IH | C | 0.061 | -0.024 | 0.122 | -0.006
| D | -0.059 | -0.153 | -0.132 | -0.116
Homophily index | C | 0.581 | 0.541 | 0.625 | 0.556
| D | 0.415 | 0.359 | 0.352 | 0.373
Centrality | C | 0.038 | 0.038 | 0.051 | 0.041
| D | 0.028 | 0.031 | 0.035 | 0.039

  * •

Note: we report mean values of these variables over rounds 2-21.

##  Appendix C Experimental Instructions for (Uncertainty, Fast) treatment

Welcome to all of you! You are now taking part in an interactive study on
decision making. Please pay attention to the information provided here and
make your decisions carefully. If at any time you have questions to ask,
please raise your hand and we will attend to you in private.

Please note that unauthorized communication is prohibited. Failure to adhere
to this rule would force us to stop this study and you may be held liable for
the cost incurred in this experiment.

Your participation in this study is voluntary. You will receive SGD 2 show-up
fee for participating in this study. You may decide to leave the study at any
time. Unfortunately, if you withdraw before you complete the study, we can
only pay you for the decisions that you have made up to the time of
withdrawal, which could be substantially less than you will earn if you
complete the entire study.

The amount of your earnings from this study depends on the decisions you and
others make and also on chance. At the end of this session, your earnings will
be paid to you privately and in cash. They will be contained in an envelope
(indicated with your unique user ID).

General Information

Each of you will be given a unique user ID and it will be clearly stated on
your computer screen. At the end of the study, you will be asked to fill in
your user ID and other information pertaining to your earnings from this study
in the payment receipt. Please fill in the correct user ID to make sure that
you will get the correct amount of payment.

Rest assured that your anonymity will be preserved throughout the study. You
will never be aware of the personal identities of other participants during or
after the study. Similarly, other participants will also never be aware of
your personal identities during or after the study. You will only be
identified by your user ID in our data collection. All information collected
will strictly be kept confidential for the sole purpose of this study.

Specific Information

The total duration of this study is approximately 2 hours.

Your total earnings = earnings from Part I + earnings from Part II + show up
fee (S$\$2$)

All incentives will be denominated in experimental dollars (expressed as ECU).
The exact conversion rate will be detailed later.

Part I

You and 11 other participants will engage in several rounds of interactions.
Each round is the same and consists of 3 Stages.

In Stages 1 and 2, you can form and delete links with any of the other
participants, which we will explain shortly. If you are linked with another
participant then this participant is a “neighbor”.

In Stage 3, you choose either action A or action B, and the choice of action A
or B applies to all your neighbors. Action A is color-coded in green and
action B is color-coded in blue. The colors are only a visual aid to
distinguish the actions and have no meaning. You get points for the action you
choose and the action each of the other participants chooses, in the following
way:

You get 0 points if you are not linked with another participant regardless of
your choice of action.

The number of points you get depends on the actions you and your neighbor
choose, according to the table below:

Neighbor
---
You |  | A | B
A | 3 | -5
B | 5 | -3

This is what the table says:

  * •

If you choose A and the neighbor chooses A, you get 3 points

  * •

If you choose A and the neighbor chooses B, you get -5 points

  * •

If you choose B and the neighbor chooses A, you get 5 points

  * •

If you choose B and the neighbor chooses B, you get -3 points

At the end of each round you will see a summary of the number of points you
get with each of the other participants.

You will play 25 rounds of this game for sure. After round 25, there is a
50$\%$ chance that the game will terminate in the following round. In other
words, in every round after round 25 the experimenter will “flip a coin” and
if the outcome is “Heads” then the game will terminate.

Stage 1 - Link Decisions

In Stage 1, you will have the opportunity to (i) form a link with another
player who is not linked to you or (ii) to delete an existing link with one of
your partner(s). However, you will not have the complete freedom to form or to
delete links. In particular, 33 random pairs of players (out of the 66
possible pairs of players in this network) will be selected to be updated such
that:

(i) If you and another player that is selected are not linked currently, you
will need to decide whether to propose a new link between you two.

(ii) If you and another player that is selected are linked currently, you will
need to decide if you would like to delete the link.

For illustration purposes, the screenshots from Round 4 are shown (see also
the attachment (stage 1).

Top-left of the screen

You will see the following figure that visualizes your current neighbors.

The neighbors you have at the start of each round are the same as the
neighbors you had at the end of the previous round. A link means that if you
are linked to participant Z then participant Z is linked to you.

Each circle denotes a participant with an ID. The “You” circle is always
positioned at the center. For example, above you are linked with Z, Y, Q, and
N. The color of the circles represents the last actual action (A or B) that
has been chosen by each participant.

Note that in Round 1 all participants are connected to each other.

Center of the screen

This table allows you to make your decisions to delete links in Stage 1. The
first row of the table (“Your Last 5 Actions”) reminds you of the actions you
have chosen in the last 5 rounds. For example, the sequence below means that
you chose action A in the last round (the leftmost), action A two rounds ago
(the second-leftmost slot), action A three rounds ago (the middle), and action
B four rounds ago (the second rightmost), and action B five rounds ago (the
rightmost).

For the pairs of players selected to be updated, a box will be shown next to
the ID of the participant. You can delete a link to the neighbor by ticking
the corresponding box under the “Unlink” column. There is no limit to how many
boxes among the shown boxes you can tick, and you can also choose not to
delete any link by leaving the shown boxes unticked.

The table has 3 columns, from left to right:

  * •

“Neighbor”: lists the IDs of the participants who are currently linked with
you

  * •

“Past actions”: lists the last 5 actual actions by each of your neighbors

  * •

“Unlink”: allows you to delete your link to the neighbors that have been
randomly selected by the system. For example, ticking the box in the last row
will delete your link to participant Z

The color-codings of the other participants’ actions are the same as before.
The green letter A denotes the choice of action A. The blue letter B denotes
the choice of action B. The “-” denotes that the participant did not have any
neighbor in that round, and therefore the participant did not choose an
action.

Right of the screen

This table allows you to make your decisions to form links in Stage 1.

For the pairs of players selected to be updated, a box will be shown next to
the ID of the participant. You can propose a link to the participant by
ticking the corresponding box under the “Link” column. There is no limit to
how many boxes among the shown boxes you can tick, and you can also choose not
to propose any link by leaving all the boxes unticked.

The table has 3 columns, from left to right:

  * •

“Others”: lists the IDs of the participants who are not linked with you

  * •

“Past actions”: lists the last 5 actual actions by each of these participants

  * •

“Link”: allows you to propose a link to the participants randomly selected by
the system you are not linked with. For example, ticking the box in the last
row will propose a link to participant M

Bottom-right of the screen

Note that the participants whom you can propose or delete links with are
decided randomly by the computer. The text reminds you about the decisions you
need to make and lists the participants whom you can propose or delete links
with. You can propose new links and/or delete existing links, or you can
choose not to propose and/or delete any links. In either case, you need to
click the “Submit” button to confirm your decisions.

Stage 2 - Yes/No Decision

In case you decided not to link to another participant in Stage 1 but this
other participant has opted to send you a link proposal, then in Stage 2 we
ask you to confirm your earlier decision not to link to this other
participant. You will be shown the following table where you need to respond
to link proposals in Stage 2.

Please refer to the attachment (Stage 2) for illustration purpose.

As in Stage 1, the table has 3 columns, from left to right:

  * •

“Others”: lists the IDs of the participants who are not linked with you

  * •

“Past actions”: lists the last 5 actual actions by each of these participants

  * •

“Respond to link proposal”: allows you to accept or reject a link proposal
from a participant that is not currently linked with you

For example, look at participants P, S and T: under the “Respond to link
proposals” column a “No” and a “Yes” buttons appear. This means that P, S and
T have sent you a link proposal. For each proposal, you need to click on “Yes”
to accept it or “No” to reject it. In the example shown, you have chosen to
reject the proposals from S and T, and accept the proposal from P.

Note that if you have sent a proposal to connect to a participant in Stage 1
and that participant has done the same then the link is automatically formed
without the need of further approval in Stage 2, and hence no button will be
shown.

Stage 3 - Action Choice

In Stage 3, you choose either action A or action B. Please refer to the
attachment (Stage 3) for illustration purpose.

Bottom-right of the screen

This is where you choose your action in Stage 3 by clicking either the A or B
button. You must pick an action in order to proceed.

Note that it is possible that you are not linked with any participant (i.e.
you have no neighbor) as the result of yours and others’ earlier decisions. In
such case, you will instead see the message reminding you that you do not have
any neighbor and so you do not need to choose an action.

Action Outcome

After Stage 3, you will see a summary of the points you got with each of the
other participants in the round.

The action you have chosen (intended action) during Stage 3 may or may not be
implemented correctly. Specifically, there is a 15% chance that your intended
action A is implemented as actual action B and a 15% chance that your intended
action B is implemented as actual action A. Note that other players only
observe your implemented (actual) action and not your intended action.

Please refer to the attachment (Action Outcome) for illustration purpose.

The table shows your neighbors and your neighbors’ actual actions in this
round. The first row of this table reminds you of your intended action and the
actual action implemented. For example, in the table below your intended
action was A and your actual action was A. Note that the table is updated with
yours and the others’ decisions in the previous Stages.

The table has 3 columns, from left to right:

  * •

“Neighbor”: lists the IDs of the participants who are currently linked with
you

  * •

“Action”: lists the actual action by each of your neighbors in this round

  * •

“Points”: lists the points you gained from your and each neighbor’s actual
action

If your intended action is implemented correctly, you will see the following
box.

If your intended action is not implemented correctly, you will see the
following box. You need to click the “Continue” button to move to Stage 1 of
the next round.

Earnings in Part I

At the end of the Experiment, we will randomly select 6 rounds for payment. In
each of these 6 rounds, we will randomly pick 2 of your potential link pairs
(There are 11 potential link pairs in total). Note that a participant who was
not linked with you can be picked, and in that case you will get 0 points for
the interaction with that participant.

To determine your earnings, we sum the number of points you got with each of
the picked participants in each of the 6 rounds. The exchange rate from ECU to
SGD is:

1.5 ECU = 1 SGD

Part II

In this part of the study you will be asked to make a series of choices. How
much you receive will depend partly on chance and partly on the choices you
make. The decision problems are not designed to test you. What we want to know
is what choices you would make in them.

For each line in the table you will see in this stage, please indicate whether
you prefer option A or option B. There will be a total of 10 lines in the
table but just one line will be randomly selected for payment. You do not know
which line will be paid when you make your choices. Hence you should pay
attention to the choice you make in every line.

After you have completed all your choices, the computer will randomly generate
a number, which determines which line is going to be paid out.

Your earnings for the selected line depend on which option you chose: If you
chose option A in that line, you will receive 20 ECU. If you chose option B in
that line, you will receive either 60 ECU or 0. To determine your earnings in
the case you chose option B, there will be a second random draw. The computer
will randomly determine if your payoff is 0 or 60, with the chances set by the
computer as they are stated in Option B.

Earning in Part II

In Part II, the conversion rate from experimental dollars (ECU) to Singaporean
dollars is:

20 ECU = 1 SGD

Attachment (Stage 1) Attachment (Stage 2)

Attachment (Stage 3) Attachment (Action Outcome)