# Cooperation and Cognition in Social Networks111We thank Peter Bossaerts,
Matthew Elliott, and seminar participants at NYU Shanghai and the University
of Cambridge for their useful comments. We also thank Wang Song for excellent
research assistance. Financial supports from the Cambridge Endowment for
Research in Finance (CERF) grant awarded to Edoardo Gallo and the Academic
Research Fund (AcRF) Tier 1 Grant RG163/18 from the Ministry of Education of
Singapore awarded to Yohanes E. Riyanto are gratefully acknowledged.

Edoardo Gallo Faculty of Economics, University of Cambridge, Cambridge CB3
9DD, UK; and Magdalene College, CB3 0AG, UK. E-mail<EMAIL_ADDRESS>Joseph Lee††footnotemark:  Yohanes E. Riyanto Division of Economics, School of
Social Sciences, Nanyang Technological University, 48 Nanyang Avenue,
Singapore 639818. Y.E. Riyanto: e-mail<EMAIL_ADDRESS>Erwin Wong:
e-mail<EMAIL_ADDRESS>Erwin Wong††footnotemark:

(May 2023
)

###### Abstract

Social networks can sustain cooperation by amplifying the consequences of a
single defection through a cascade of relationship losses. Building on Jackson
et al., (2012), we introduce a novel robustness notion to characterize low
cognitive complexity (LCC) networks - a subset of equilibrium networks that
imposes a minimal cognitive burden to calculate and comprehend the
consequences of defection. We test our theory in a laboratory experiment and
find that cooperation is higher in equilibrium than in non-equilibrium
networks. Within equilibrium networks, LCC networks exhibit higher levels of
cooperation than non-LCC networks. Learning is essential for the emergence of
equilibrium play.
JEL: C91, C92, D85, Z13.
Keywords: cooperation, network, bounded rationality, experiment, learning.

##  1 Introduction

A hallmark of a well-functioning society is the ability of its members to
cooperate with each other in times of need. Social connections are a natural
source of help, and there is extensive empirical support for an association
between network structure and cooperative activity (e.g. Greif, (1993),
Putnam, (2000)). Ample experimental evidence highlights the causal link
between the ability to alter the network and cooperation. In particular,
social networks matter for cooperation because they enable people to punish
defectors by removing connections (e.g. Rand et al., (2011)), and therefore
cooperation may thrive in social network structures where there is a threat of
punishment through potential link removal. Despite these findings, the
majority of theoretical contributions take the network as exogenous and
unchangeable (Nava, (2016)). We therefore lack a tight link between
theoretical predictions and empirical evidence of how network structure, in
combination with the ability to alter the network, affects cooperative
activity.

Jackson et al., (2012) (JRT hereafter) develop a model to analyze the
emergence of cooperation in networks through the threat of punishment by link
removal. In their set-up, agents are embedded in a network and at each point
in time one individual is called upon to provide a favor to one of their
connections. Her choice is to either provide the favor at a cost to herself,
or not provide the favor and lose the connection. JRT characterize the set of
networks that are renegotiation-proof equilibria (RPE) – they are stable
because the consequences of removing a connection in terms of the chain of
punishment are too big for any agent not to provide a favor. A drawback of
this characterization is that the set of RPE networks is rather large. In
order to address this, JRT define a notion of robustness against social
contagion, and identify a subset of RPE networks that they dub _social
quilts_. In these networks, a failure to cooperate leads to a chain of
punishment that stays local, and therefore the overall network is robust to
such a mistake.

In this paper, we define a novel robustness notion in the context of the JRT
framework that is based on the cognitive complexity of figuring out the chain
of punishment that is a consequence of removing a connection. Understanding
this chain of defections requires iterative reasoning, and there is ample
evidence that humans are limited in their ability to do it (Crawford et al.,
(2013)). Moreover, empirical and experimental evidence show that individuals
struggle to cope with the complexity of network structure and its implications
for behavior (e.g. Krackhardt, (1990), Dessi et al., (2016)). In our
theoretical contribution, we introduce the concept of _cognitive complexity
number_ to classify RPE networks according to the cognitive burden required to
calculate RPE predictions. Networks with minimal cognitive complexity number
equal to one are what we define as _Low Cognitive Complexity_ (LCC) networks.
Such networks collapse to an empty network after a chain of defections
resulting from a single link removal. The core idea is that these networks are
more robust because it is less likely that any agent removes a link due to a
failure in iterative reasoning.

We conduct a lab experiment to test how different network structures (RPE,
social quilts, LCC networks) sustain cooperation. Participants are randomly
assigned to nodes of a network, and in each period of the game one
node/participant is picked at random to either provide a favor or delete one
of their links. After the first participant has made a decision, the computer
selects another node and the process is repeated until no links are left or
all participants have consecutively chosen not to delete any link. The
experiment has two phases. In Phase 1, participants play the above game on two
different networks in succession with the groups reshuffled every time they
get assigned to a new network. Between Phase 1 and 2, participants face
decision-making tasks against computer players. In Phase 2, participants play
the game three times on the two different networks used in Phase 1 with groups
again reshuffled every time they are assigned to a new network. Additionally,
we elicit social preferences using a Dictator Game.

The first result is that participants are less likely to delete links in RPE
networks compared to links in non-RPE networks. This shows that the RPE notion
in JRT has behavioral validity in predicting which network structures are more
conducive to cooperative activity. Restricting the attention to social quilts,
however, provides no additional benefit in terms of cooperation. In other
words, participants in a social quilt are as likely to delete links as
participants in an RPE network that is not a social quilt. This shows that
robustness against social contagion is not a primary determinant of
cooperative activity in the networks examined in the experiment.

The second result is that participants are less likely to delete links in LCC
networks compared to social quilts and/or other RPE networks that are not LCC.
This validates our hypothesis that bounded rationality considerations are a
primary driver of what types of network structures are conducive to
cooperation.

The third result is that learning matters. Participants need time to
understand the consequences of defection and are better able to do so in
networks with low cognitive complexity, further showing how cognitive
complexity is an important determinant of sustaining cooperation. In Phase 1,
social preferences are the primary determinant of link removal decisions,
while the type of network structure is not significant. In Phase 2, however,
after participants have had the chance to learn from playing against computer
players, network structure becomes a significant determinant of link removal
decisions and social preferences are insignificant. Moreover, participants are
less likely to remove links in LCC compared to any other type of network,
including social quilts.

The rest of this paper is organized as follows. Section 2 discusses the
related literature. Section 3 presents the theoretical results, and the
resulting hypotheses. Section 4 explains the set-up of the experiment. Section
5 presents the results of the study, and section 6 concludes. The main
Appendix contains the proofs, a more general version of the theoretical model,
and the instructions of the experiment.

##  2 Related literature

Our work is related to several strains of theoretical and experimental
research in repeated games, networks, and behavioral economics. After a brief
overview of related work outside of economics, we review each one of these in
turn and discuss how this paper draws novel connections across these areas.

The question of how social structure determines cooperative behavior is a
long-standing interest in many academic disciplines. In sociology, seminal
work by Simmel, (1950) and Coleman, (1988) argues that transitivity or closure
– whether someone’s friends are friends with each other – is crucial for
cooperation. Their contributions and subsequent work, however, lack firm
theoretical foundations (Sobel, (2002)). A vast empirical literature shows an
association between cooperation and social network characteristics in many
disciplines including sociology (e.g. Putnam, (2000)), management (e.g. Burt,
(2007)), anthropology (e.g. Apicella et al., (2012)), and political science
(e.g. Bond et al., (2012)). The endogeneity of the network coupled with the
difficulty of separating the impact of network structure from unobservable
individual characteristics (Manski, (2000)), however, means that it is
challenging for these contributions to identify how network structure causally
affects cooperative behavior.

Due to the difficulty of combining network formation with a prisoner’s dilemma
type of game to capture cooperation, the majority of theoretical contributions
in the economics literature take the network as exogenous. For instance, Haag
and Lagunoff, (2006) show that the optimal network to sustain cooperation
depends on the distribution of preferences, and completely connected cliques
are optimal when there is enough heterogeneity.222Wolitzky, (2013) defines a
novel centrality measure to show that more central players cooperate more in
the equilibrium with the maximum level of overall cooperation. Other
contributions include Bloch et al., (2008) and Lippert and Spagnolo, (2011).
See Nava, (2016) for a comprehensive review of this literature. Jackson et
al., (2012) is the first paper to incorporate a reduced model of network
formation with a cooperation game on a network. Section 3 describes their
framework in detail because the theoretical model in our paper is a reduced
form version of theirs. The main result in Jackson et al., (2012) is to
characterize the set of RPE networks, and identify a subset of social quilts
networks that are robust against social contagion. We build on their
theoretical work by introducing a novel notion to select among RPE networks
based on bounded rationality considerations, and characterize this equilibrium
subset. The main contribution of our paper is to test experimentally the
behavioral validity of these equilibria to sustain cooperation.

Early experimental work on cooperation in networks also focused on the case
where the network is exogenous. Cassar, (2007) finds no significant difference
in cooperation levels across three types of networks of 8 nodes. Subsequent
work confirmed this finding on networks with different structure and
sizes.333In particular, Kirchkamp and Nagel, (2007) find no substantial
differences across circle networks of different sizes, while Gracia-Lázaro et
al., (2012) find no difference between two large networks. An exception is
Rand et al., (2014) who find some effect of network structure if the benefits
from cooperation are large enough. A more recent experimental literature
endogenizes the network and shows that the ability to punish defectors by
removing connections and/or rejecting proposals to connect fosters cooperation
(Rand et al., (2011), Wang et al., (2012)).444The efficacy of this network
punishment mechanism depends on information about others’ reputation (Gallo
and Yan, (2015), Cuesta et al., (2015)), knowledge of the social network
structure (Gallo and Yan, (2015)), accuracy of the monitoring technology
(Gallo et al., (2022)), and the strength of the connections (Shirado et al.,
(2013), Gallo et al., (2019)). See Choi et al., (2016) for a comprehensive
review. All these prior works lack a theoretical framework, while our
experiment is explicitly designed to test precise theoretical predictions.

Experimental (Dessi et al., (2016)) and empirical (Breza et al., (2018))
evidence show that humans have a biased and limited perception of social
network structure. Insights from behavioral economics seem, therefore, to be
quite relevant to understanding how social network structure affects behavior.
Despite this clear link, however, there are very few papers at the
intersection of behavioral economics and networks.555Exceptions include
Ushchev and Zenou, (2020) who incorporate conformity in network games, and
Gallo and Langtry, (2020) who study the effect of confirmation bias on social
learning.

In particular, we examine how individuals’ bounded rationality helps in
identifying a subset of equilibria that predict how networks affect behavior.
Several empirical and experimental studies show that humans engage in
iterative thinking to solve strategic problems (Crawford et al., (2013)).
Structural models based on cognitive hierarchy (Camerer et al., (2004)) and/or
level-k reasoning (e.g. Nagel, (1995), Costa-Gomes et al., (2001)) have been
successfully applied to explain behavior in, amongst others, auctions
(Crawford and Iriberri, (2007)), market entry (Goldfarb and Xiao, (2011)), and
strategic communication (Crawford, (2003)). In this paper we apply the logic
of iterative thinking to identify a subset of RPE networks that require only
one step of iteration to figure out the full consequences of a defection, and
we show that cooperation is higher in these networks compared to other
equilibria.

We are the first paper, to our knowledge, to investigate the role of bounded
rationality in sustaining cooperation. Given that equilibrium multiplicity is
ubiquitous in network problems666Examples include network formation (Jackson
and Wolinsky, (1996)), public goods games (Bramoullé et al., (2014)), and
social learning (Bala and Goyal, (1998))., this insight may be applicable to
other contexts where networks matter.

##  3 Theory

###  3.1 Model

The theoretical framework examined in this paper is a reduced form version of
the model in Jackson et al., (2012) (JRT hereafter). This section introduces
relevant network concepts and the reduced form game. At the end we discuss the
main simplifications to the JRT framework.

Networks. A finite set $N=\\{1,...,n\\}$ of agents is connected by a social
network represented by an _unweighted, undirected graph_ $g$. $ij$ denotes a
link between $i$ and $j$, and the graph $g-ij$ represents the network obtained
from $g$ by deleting link $ij$.777Throughout this paper, we assume there is no
self-link, ie $ii=0$ for all $i$. The _neighborhood_ of agent $i$ in $g$ is
$N_{i}(g)=\\{j|ij\in g\\}$. $d_{i}(g)\equiv|N_{i}(g)|$ denotes the _degree_ of
$i$ in $g$. $D(g)$ denotes the profile of degrees associated with a network g,
i.e. $D(g)\equiv(d_{1}(g),...,d_{n}(g))$. We say that $D(g)>D(g^{\prime})$ if
$d_{i}(g)\geq d_{i}(g^{\prime})$ for all $i$ with strict inequality for some
$i$. The empty network is a network $g$ in which there are no links. A
_complete network_ is a network in which $ij\in g$ for all $i,j\in{N}$. A
_triangle network_ is a complete network of three nodes.888For the purpose of
this paper, a complete network of three nodes with other isolated nodes is
also called a “triangle network”. We say $g^{\prime}$ is a _subnetwork_ of $g$
if $ij\in g^{\prime}$ only if $ij\in g$. A _walk_ in a network $g$ refers to a
sequence of nodes $i_{1}$, $i_{2}$, … $i_{K}$ such that $i_{k}i_{k+1}\in g$
for each $k$ from $1$ to $K-1$. A _simple cycle_ in a network $g$ is a walk in
$g$, $i_{1}$, $i_{2}$, … $i_{K}$, such that $i_{1}=i_{K}$, and the only
repeated vertices are $i_{1}$ and $i_{K}$.

Game. Let $t=\\{0,1,...\\}$ denote discrete time periods. The game begins with
some initial network $g_{0}$ at time $t=0$. Consider time period $t$ and
$ij\in g_{t}$, there is a probability $p$ that $i$ will request a favour from
$j$ at $t$. Let $n(n-1)p\leq 1$ and the time between periods be small so that
we can assume that at most one favour is needed across all agents in any given
period. As JRT discuss in more detail, this is a proxy for a Poisson arrival
process.

When an agent $j$ is called to do a favour for agent $i$, agent $j$ can then
choose to either perform the favour or refuse the favour. Performing the
favour costs agent $j$ an amount of $c>0$, it provides agent $i$ a benefit
$v>c$, and involves no change in the network. Refusing the favour entails the
permanent loss of the $ij$ link in future periods so $g_{t+1}=g_{t}-ij$, but
there are no costs/benefits for either agent. Assuming agents discount future
payoffs according to a $\delta\in(0,1)$ factor, an agent $i$’s expected
utility from having a link $ij\in g$ in perpetuity is therefore $b\equiv
p(v-c)/(1-\delta)$. This represents the present discounted value of a
perpetual mutual favour-exchange relationship. We assume throughout this paper
that $2b>c>b$. Appendix A discusses the generalization of the analysis to all
$c$ and $b$. Note that this is a complete information game, i.e. agents are
fully aware of all moves in the game at every node.

Reduced Form. We have adopted a reduced form of the JRT model along the
following two dimensions in order to simplify exposition and match our
experimental set-up.

First, in the original JRT framework agents simultaneously announce the links
that they are willing to retain in each time period $t$ prior to favour
decisions. There is, however, no strategic benefit for agent $i$ to remove her
link with another agent $j$ because retaining the link is costless and allows
agent $i$ to potentially call on agent $j$ for a favour in the future. We
have, therefore, suppressed this stage both in the theory and the experiment
to focus on the favour exchange stage.

Second, we restrict our attention to the $2b>c>b$ range. The lower bound is
identical to JRT and allows the focus to be on the interesting case where
simple bilateral relations cannot sustain cooperation. The upper bound
simplifies the characterization of equilibrium networks, and it is consistent
with the calibration in the experiment.

###  3.2 Results

The first part of this section summarizes the characterization of equilibrium
predictions in JRT in the context of the reduced form model. Following their
approach, it characterizes the subset of _social quilts_ equilibria that are
robust against social contagion. In the second part we introduce the novel
notion of _cognitive complexity_ to provide an alternative method to select
equilibria robust against social contagion based on behavioural
considerations.

JRT analyze the game using the renegotiation-proof equilibrium (RPE hereafter)
concept, and we refer the reader to their paper for a detailed discussion on
their definition of RPE in this game. Here we briefly restate their
definitions and focus on results that are relevant to formulate our hypotheses
in section 3.3 within our reduced form model. In particular, the assumption
that $2b>c>b$ is crucial to simplify the ensuing notation, statements, and
exposition. First, the following is the adaptation of the definition of
Transitively-Critical (TC) networks to our reduced model.

######  Definition 1.

The class of TC networks are generated as follows: Define $k$ as the number of
links in the network. For $k=0$, define $TC_{0}=\\{g^{\emptyset}\\}$.
Inductively on $k$, $TC_{k}$ is such that $g\in TC_{k}$ if and only if
$\forall i,ij\subset g,\exists g^{\prime}\subset g-ij$ such that

  * •

$g^{\prime}\in TC_{k^{\prime}}$ for some $k^{\prime}\leq k-2$;

  * •

$d_{i}(g^{\prime})\leq d_{i}(g)-2$

  * •

$\nexists g^{\prime\prime}\in TC_{k^{\prime\prime}}$ s.t.
$g^{\prime\prime}\subset g-ij$ and $D(g^{\prime\prime})>D(g^{\prime})$.

Intuitively, the TC concept states that in all continuing sub-networks that
are also TC, individuals who lost links must have lost at least 2 links. A key
result in JRT is to show that there is a one-to-one correspondence between RPE
and the set of TC networks.

######  Proposition 1.

(JRT) A network is a RPE if and only if it is Transitively Critical.

The intuition is that in deciding whether to accept or refuse the favour,
agents consider the eventual resulting network from refusing the favour. If
the loss of favour-exchange relationships is more than the cost of the favour
(i.e. at least 2 links to the individual are deleted as a consequence of not
performing the favour given $2b>c>b$), then the agent will do the favour. If
it is optimal for all agents to do the favours for each of their links, then
the network is a RPE network.999See Appendix in JRT for a detailed proof.

A drawback of Proposition 1 is its limited predictive power because the class
of TC networks is quite large. Additionally, some TC networks are not robust
to the random deletion of a link due to an accident and/or failure by an agent
to calculate RPE predictions. JRT introduce a stronger equilibrium concept of
“robustness against social contagion” within the class of TC networks, where
the deletion of a link results in “local” rather than “global” damage. These
networks prevent the entire society from unravelling if an agent deletes links
contrary to RPE predictions.

######  Definition 2.

A network $g$ is robust against social contagion if it is RPE and sustained as
part of a pure strategy subgame perfect equilibrium with $g_{0}=g$ that
satisfies the following condition. In any subgame continuation from any RPE
$g^{\prime}\subset g$, and for any $i$ and $ij\in g^{\prime}$, if $i$ deletes
the link with $ij$, then the continuation leads to $g^{\prime\prime}$ such
that if $hl\not\in g^{\prime\prime}$ then $h\in N_{i}(g^{\prime})\cup\\{i\\}$
and $l\in N_{i}(g^{\prime})\cup\\{i\\}$.

For networks that are robust against social contagion, if an agent deletes
links contrary to RPE predictions then the only link deletions in steady-state
involve that agent and her neighbours. One of the key results in JRT is to
show that there is a one-to-one correspondence between networks that are
robust against social contagion and a class of networks that they dub “social
quilts”.

######  Definition 3.

A social quilt is a union of triangle networks such that there is no simple
cycle in the network involving more than 3 nodes.

######  Proposition 2.

(JRT) A network is robust against social contagion if and only if it is a
social quilt.

Intuitively, should an agent delete a link, only the triangle network that the
link is part of will be lost. The other triangle networks remain untouched as
each triangle network is RPE and is self-sustaining. Therefore, only the agent
deleting the link and her neighbours will be affected, and the damage remains
localised.

Social quilts provide a mechanic way to stem contagion by limiting it to a
small part of the network. This is an appealing solution if a random link
deletion is an event purely caused by chance, and any link is equally
susceptible to this. In reality, however, mistakes that lead to link deletions
may be the result of an agent’s bounded rationality that makes her unable to
calculate RPE predictions. If bounded rationality is the cause of deletion
mistakes then deletions will depend on an agent’s level of (bounded)
rationality as well as how difficult it is to compute RPE predictions in a
given network environment.

Here we propose an alternative notion to select among RPE equilibria based on
considerations of bounded rationality. In particular, we define the concept of
“cognitive complexity number” to classify RPE networks according to the
cognitive burden required to calculate RPE predictions, and focus on the class
of networks that have cognitive complexity number equal to 1.

######  Definition 4.

Denote by $cc(g)$ the cognitive complexity number of network $g$, and assume
$cc(g^{\emptyset})=0$. Suppose we have a TC network $g$ and the largest TC
subnetwork of $g$ is $g^{\prime}\subset g$, then
$cc(g)=cc(g^{{}^{\prime}})+1$. We define $g$ to be a Low Cognitive Complexity
(LCC) network if $cc(g)=1$.

Given that computing RPE predictions requires the calculation of TC networks,
the notion of cognitive complexity number is based upon the definition of TC
networks. In particular, an agent needs to ascertain the largest TC subnetwork
should she refuse a favour. If the subnetwork is not an empty network, the
agent must iteratively identify further subnetworks until she reaches the
empty network. The more iterations she has to go through, the more cognitively
complex the network is. We define LCC networks to be the ones that collapse to
the empty network after one link removal because from a cognitive point of
view these are the simplest environments to understand the consequences of a
favour refusal. The following proposition characterizes the class of LCC
networks.

######  Proposition 3.

A network is LCC if and only if it is a simple cycle.

The intuition is as follows. If an agent only has one link connected to her,
she will refuse any favour when called upon, since the cost of doing a favour
is more than any potential benefit from retaining the link. This means that if
any link is broken in a simple cycle, it will trigger a series of deletions
and the resultant network will be the empty network. Thus, a simple cycle is
LCC. Since agents with links have two or more links, all non-empty RPE
networks minimally contain a simple cycle. As such, only simple cycles are
LCC, since all other networks will have at least a simple cycle as the
resultant network instead of the empty network. An implication of this result
is that the simple cycle can be thought of as a “basic unit” to help us
generate RPE networks with higher cognitive complexity number. Appendix A
contains a detailed proof.

Within the $2b>c>b$ parameter range, there is one RPE network that satisfies
both equilibrium concepts of LCC and robustness against social contagion.

######  Corollary 1.

The triangle network is the only network which is LCC and SQ.

This follows from Definition 3 and Proposition 3, since triangle networks are
both social quilts and simple cycles. Other simple cycles involving more than
3 nodes would no longer be social quilts by Definition 3.

###  3.3 Hypotheses

This section lists and discusses the hypotheses we are investigating in our
experiment. The basic check of the predictive power of Proposition 1 is that
cooperation is higher in RPE networks compared to networks that are not RPE.
The top part of Figure 1 shows the six RPE networks we investigate in our
experiment, while the bottom part of Figure 1 displays the five non-RPE
networks that we compare them to. In the set-up of our experiment, cooperation
means not deleting links, so the following is our first hypothesis.

Figure 1: Networks in our experimental design

_H1: Participants are less likely to delete links in an RPE compared to a non-
RPE network._

As we discuss in section 3.1, the class of RPE networks is quite large so the
predictive power of this equilibrium concept is limited. For instance, it does
not discriminate among the six RPE networks in our experiment despite their
evident differences in structure. JRT propose social quilts as a refinement to
identify a set of networks where cooperation is more likely to be maintained
because they are robust to contagion. Given that contagion stays “local” in SQ
networks, participants ought to be less likely to delete links because there
is a lower probability that they will be involved in a contagion chain
compared to participants in other RPE networks that are not SQ. In our design,
we investigate three RPE networks that are social quilts. They are composed by
one, two or three rings of three nodes with a central node that belongs to all
the rings. These are labelled as $x$-Ring-3 with $x\in\\{1,2,3\\}$ as shown in
the top-left part of Figure 1. This leads to our second hypothesis.

_H2: Participants are less likely to delete links in an SQ network than in an
RPE network that is not SQ. In particular, participants are less likely to
delete links in x-Ring-3 networks compared to x-Ring-y networks, where
$x\in\\{1,2,3\\}$ and $y>3$._

In section 3.1 we propose the novel concept of LCC networks to identify a
subset of RPE networks where cooperation is more likely to be sustained. The
core idea is that participants are more likely to cooperate in networks whose
structure makes it cognitively easy to see the chain of contagion that is a
consequence of a failure to cooperate. This is a different approach to carve
out a set of robust RPE networks compared to the concept of social quilts. In
the context of our design, we investigate three RPE networks that are LCC
formed by one ring of three, four and five nodes. For this reason, we label
them 1-Ring-y with $y\in{3,4,5}$ as shown in the top-right part of Figure 1.
Note that, as shown in Figure 1, 1-Ring-3 is the only RPE network in our
experiment that is both a SQ network and a LCC network at the same time. This
leads to our third hypothesis.

_H3: Participants are less likely to delete links in an LCC network than in an
RPE network that is not LCC. In particular, participants are less likely to
delete links in 1-Ring-y networks compared to x-Ring-y networks, where $x>1$
and $y\in\\{3,4,5\\}$._

##  4 Experimental design and procedures

The experiments were conducted at Nanyang Technological University (NTU), and
programmed using oTree (Chen et al., (2016)). A total of 225 NTU students from
all majors participated in our experiments. At the beginning of each session,
we ensured anonymity by assigning each participant a random numerical ID as
their identifier throughout the experiment. Participants had a physical copy
of the instructions, and the instructions were also displayed on the computer
screen as they were read aloud by the experimenter. In the rest of this
section, we describe the game and explain the overall workflow of the
experiment.

In each network game, subjects in the session are split into groups according
to the size of the networks they play in.101010If the number of participants
is not divisible by the network size “excess” players are chosen at random to
sit out. Each subject in the group is then randomly assigned to a node in the
network. In each round, the computer randomly selects one of the nodes, and
the subject assigned to that node has to decide to delete one of their links
or to keep all of their links.111111If the participant does not make a choice
then the computer automatically selects no link deletion after 60 seconds.
After the first participant has made a decision, the computer selects another
node and the process is repeated for the next round. The sequence of node
selection is randomized, with the constraint that all other nodes have to be
selected before a node is selected a second time. Participants can only see
the network at the time they need to make a decision, and they do not know the
sequence of node selection nor the history of other participants’ decisions.

As it is not practical for the experiment to model time-discounting flows, the
design cannot exactly replicate favour exchanges as per the theoretical set up
in section 3.1. Instead of having a favour provide a benefit $v$ to the
recipient at the cost of $c$ to the giver, we proxy this by participants
receiving a one-time benefit of $b$ for each of the links connected to them
that survive at the end of the game. To maintain the dynamics and the RPE
predictions in section 3.1, we allow participants to gain $c$ for each link
they delete, which is the relative gain they would get from refusing to
perform a favour (since they would save on the cost $c$).

However, this new structure introduces a new strategic element not present in
the model in section 3.1. As an example, consider a triangle network with
participants $\\{i,j,k\\}$, and assume $i$ refuses a favour to $k$. In the JRT
framework, the $ik$ link breaks, the network is no longer RPE, and therefore
it immediately collapses to the empty network so $i$ immediately loses two
links as a result of her refusal. Consider, instead, our experimental set-up.
When $i$ removes the link and gains $c$, there are still opportunities for
participants to gain a total of $2c$ collectively from deleting the remaining
two links. This presents a possible corner case. Suppose the order of turns is
$i\rightarrow j\rightarrow k\rightarrow i$. Suppose that $i$ decides to cut
the link with $k$. After $i$’s decision, it is $j$’s turn to decide which link
to be deleted, and $j$ may choose between deleting against $i$ or $k$. If $j$
deletes the $jk$ link, then $k$ has no link to delete in her turn, and $i$ can
delete the $ij$ link in the next turn. While the probability $h$ of this
scenario occurring is low, we need to set $2b>c+hc>b$ to account for this
corner case. By setting $c$ to be close to $b$, we ensure that participants
who are risk-averse, risk-neutral or even slightly risk-loving will still find
it optimal to play according to equilibrium predictions.

In order to mitigate the impact of this strategic effect, we chose $b=100$
points for each of the participants’ links that survives deletion, and $c=110$
points for each link they delete. These parameters are within the $2b>c>b$
range in our main model and they account for the above corner case, given that
$c$ is close to $b$.

Another difference between this game and the theoretical set-up in section 3.1
is that nodes, rather than links, are selected for decisions. The reason is
that in the (relatively) small networks in the experiment selecting a link
would imply significant differences across participants in the likelihood of
being selected to make a decision (e.g. the center node in 3-Ring-3 would be
three times as likely to be picked than every other node). It would,
therefore, be challenging to set-up a randomized link selection process that
would not give a participant more opportunities to play the game in a way that
is excluded in the theory.

The workflow of the experiment is described in Table 1. In Phase 1,
participants play the above game on two different networks. After the first
network, the groups are reshuffled to minimise spillovers across games and
subsequently they play the game on the second network. Following Phase 1,
participants face an individual decision-making task against computer players
described in detail below. In Phase 2, participants play the game three times
on two different networks used in Phase 1 with the first and last plays on the
second network seen in Phase 1, and the second play on the first network seen
in Phase 1. To account for order effects, we run another experiment session
with the opposing order of networks seen in Phase 1. Similarly to Phase 1, the
groups are reshuffled after each network.

Table 1: Tasks, participants and rounds by treatment group.

|  |  | Treatment |  |  |  Totals
---|---|---|---|---|---|---
|  1 |  2 |  3 |  4 |  5 |  6 |
| Networks |
|  |  |  |  |  |  |
Phase 1 |  1R3, N1R3 |  1R4, N1R4 |  1R5, 2R3 |  2R4, 3R3 |  2R3, 1R3 |  2R3 |
|  |  |  |  |  |  |
Computer Task | All participants play each network in Figure 1 |
|  |  |  |  |  |  |
Phase 2 |  N1R3, 1R3, N1R3 |  N1R4, 1R4, N1R4 |  2R3, 1R5, 2R3 |  3R3, 2R4, 3R3 |  1R3, 2R3, 1R3 |  2R3 |
|  |  |  |  |  |  |
Participants |  48 |  40 |  50 |  42 |  30 |  15 |  225

  * •

Note: There are 2 sessions per treatment. Given the same treatment, the only
difference between the sessions is that the orders of networks seen in Phase 1
and Phase 2 are reversed. For each treatment, the specific networks, order of
networks played in each round, and number of participants are specified.

Between Phase 1 and 2 participants play the same game against computer
players, which we call the Computer Task. They are told that each of the other
nodes is a computer player, and that all computer players make optimal
decisions. Specifically, we programmed the computer player not to delete any
link if the network is RPE, and to delete one of its links if the network is
not RPE and this link should be deleted in equilibrium. This game is repeated
for the eleven different networks in Figure 1 so in each session we can
compare how a participant plays against other participants as opposed to
computer players for each network.121212In the Computer Task, the participant
is always assigned to (one of) the node(s) with the highest degree. When there
are multiple nodes with the highest degree, then the assignment is randomized.

There are two reasons to make participants play against computer players.
First, it generates useful data to disentangle social preferences from limited
cognitive ability as a driver of decisions. When a participant fails to remove
a link in a non-RPE network against other participants, it can be due to the
inability to understand the equilibrium or because of altruism – the latter
channel is clearly suppressed when playing against a computer. Second, it
allows us to disentangle whether a participant removes a link in a non-RPE
network because of limited cognitive ability or because it believes other
participants have limited cognitive ability (and therefore it is rational to
delete the link because it would not survive anyway). The latter driver of
decisions is clearly absent with computer players who always make rational
(optimal) decisions, and therefore this task allows us to partially identify
the cognitive sophistication of participants by exposing them to networks with
different cognitive complexity numbers.

At the very beginning of the experiment, we elicit social preferences using a
Dictator Game. Each participant has to decide how much of an endowment of 1
SGD to allocate to another randomly selected participant. We ensure that a
participant is never a recipient of her recipient’s allocation to exclude
reciprocity as a factor influencing their allocation decisions. The outcome of
the game is only communicated to participants at the end of the experiment to
ensure earnings do not affect decisions in the experiment.

Toward the end of the experiment, we elicit participants’ risk preferences
using the Holt-Laury multiple-price list risk elicitation (Holt and Laury,
(2002)). We also conduct a post-experimental questionnaire to collect data on
individual characteristics such as game theory experience, age, and gender. On
top of their earnings from the Dictator Game and the Holt-Laury risk
elicitation, one round from Phase 1 and two rounds from Phase 2 are randomly
selected to determine the total payment of each participant. The experimental
sessions lasted an average of 75 minutes, and the average earning for each
participant was 21 SGD.

##  5 Empirical Results

We begin by focusing our regression analysis at the decision-level. Every time
participants are assigned to a network, they have to make a decision on
whether to keep all links or delete one of them. For example, consider the
case where three participants have been assigned to play in the 1R3 network.
Clearly, there will be a minimum of 3 decisions made in this network game, and
therefore this network game will contribute to a minimum of three units of
observation to the decision-level data. We run logit regressions, where each
unit of observation is a decision made by a participant, with standard errors
clustered at the subject level in the following form.

$Y_{i,s}=\beta_{0}+\beta_{T}\textbf{T}_{i}+\beta_{C}\textbf{C}_{s}+\epsilon$

The dependent variable $Y_{i,s}$ is binary – it equals 1 if decision $i$ made
by subject $s$ is to delete a link, and equals 0 otherwise. $\textbf{T}_{i}$
denotes a vector of indicator variables capturing whether the network at which
decision $i$ is made is an RPE network, a SQ network, a LCC network, or a 1R3
network. $\textbf{C}_{s}$ denotes a vector of control variables unique to
subject $s$ making decision $i$. The full list of control variables used
consists of the subject’s gender, risk preferences elicited using the Holt-
Laury multiple price list risk-elicitation procedure, and an indicator
variable for (lack of) altruism which is equal to 1 if the individual
allocated all points to him(her)self in the dictator game and 0 otherwise.
$\beta_{0}$ and $\epsilon$ are the constant and residual terms respectively.

###  5.1 Equilibrium predictions

Table 2 presents our main regression analysis to test the hypotheses in
section 3.3. Hypothesis 1 (H1) states that participants are less likely to
delete links in RPE networks compared to non-RPE ones. Specification (1)
regresses the decision to delete links on the “Network is RPE”. The
coefficient is significantly negative ($p<0.001$) validating the hypothesis
that a subject is less likely to delete a link in an RPE network than in a
non-RPE network. Thus, RPE networks are more likely to be sustained than non-
RPE networks, as predicted by the theory.

Columns (2) and (3) replicate the analysis by splitting the RPE variable into
SQ and LCC networks respectively. Hypotheses 2 and 3, however, focus on a
comparison between different types of RPE networks. In order to address these
hypotheses directly, we also add specifications (4)-(7) to restrict the data
to decisions on RPE networks only. Moreover, as Figure 1 shows, a confounding
factor in the analysis is that the 1R3 network is both SQ and LCC. In order to
isolate its effect, columns (5) and (7) introduce a dummy variable to isolate
the effect of 1R3 compared to other SQ or LCC networks. We discuss each
hypothesis in turn.

For H2, column (2) in Table 2 splits RPE networks into “RPE & non-SQ” and
“SQ”. The coefficient is significantly negative for both these variables
($p<0.001$) indicating that subjects are less likely to delete links in both
types of RPE networks, which is again consistent with H1. Once we look at the
size of the coefficients, however, it is clear that the SQ one is not larger
than RPE & non-SQ so these results do not validate the statement in H2 that
subjects are less likely to delete links in an SQ network compared to an RPE
network that is not SQ.

Column (4) directly shows that subjects are not less likely to delete links in
SQ compared to other types of RPE networks. Moreover, once we isolate the
effect of 1R3 in column (5), subjects are actually significantly _more_ likely
to delete links in SQ networks compared to other types of RPE networks
($p<0.001$). The data, therefore, clearly does not support Hypothesis 2 that
social quilts is a refinement that identifies a set of networks that sustain
cooperation.

Table 2: Logit regressions of “Delete” at the decision-level

Sample: | All networks in Phase 1 and Phase 2 |  All RPE networks in Phase 1 and Phase 2
---|---|---
Baseline: | Non-RPE networks | {2R4, 1R4, 1R5} | {2R4, 2R3, 3R3}
| (1) | (2) | (3) | (4) | (5) | (6) | (7)
RPE | -0.904*** |  |  |  |  |  |
| (0.144) |  |  |  |  |  |
SQ |  | -0.787*** |  | 0.391 |  |  |
|  | (0.173) |  | (0.242) |  |  |
RPE & non-SQ |  | -1.120*** |  |  |  |  |
|  | (0.201) |  |  |  |  |
LCC |  |  | -1.206*** |  |  | -1.039*** |
|  |  | (0.159) |  |  | (0.259) |
RPE & non-LCC |  |  | -0.131 |  |  |  |
|  |  | (0.249) |  |  |  |
1R3 |  |  |  |  | 0.005 |  | -0.993***
|  |  |  |  | (0.257) |  | (0.269)
SQ & not 1R3 |  |  |  |  | 1.134*** |  |
|  |  |  |  | (0.330) |  |
LCC & not 1R3 |  |  |  |  |  |  | -1.103***
|  |  |  |  |  |  | (0.321)
$\beta_{0}$ | 0.578** | 0.554** | 0.605** | -1.145** | -1.045* | 0.032 | 0.007
| (0.243) | (0.245) | (0.246) | (0.556) | (0.558) | (0.514) | (0.520)
Observations | 1,501 | 1,501 | 1,501 | 367 | 367 | 367 | 367
Other Controls | Yes | Yes | Yes | Yes | Yes | Yes | Yes
Pseudo R2 | 0.0329 | 0.0341 | 0.0444 | 0.0228 | 0.0524 | 0.0528 | 0.0532
Likelihood | -872.1 | -871.0 | -861.8 | -245.9 | -238.4 | -238.3 | -238.2

  * •

Note: Logit regressions of “Delete” using decision-level data. Each
observation is a decision made by a subject, where “Delete” = 1 if the
subject’s decision is to delete a link. Robust standard errors are clustered
at the subject-level.

For H3, column (3) splits RPE networks into “LCC” and “RPE & non-LCC”.
Interestingly, the coefficient of “RPE & non-LCC” networks is now
insignificant, which shows that subjects are not less likely to delete links
in RPE compared to non-RPE networks once we exclude LCC networks from the RPE
set. In contrast, the LCC coefficient is significantly negative ($p<0.001$)
validating the statement in H3 that subjects are less likely to delete links
in LCC compared to other types of non-RPE networks.

Column (6) directly shows that subjects are less likely to delete links in LCC
compared to other types of RPE networks ($p<0.001$). Moreover, the effect
survives even if we separate the impact of 1R3 networks. Both the “1R3” and
“LCC & not 1R3” variables are significant in specification (7) ($p<0.001$)
indicating that cooperation is higher in LCC networks compared to other types
of RPE networks even after we isolate the impact of the only LCC network that
is also SQ in our experiment. The refinement of the set of RPE networks based
on cognitive complexity has behavioral validity – it identifies a set of
network where cooperation is higher because it is easier for participants to
understand the consequences of failing to cooperate.

###  5.2 Social preferences and learning

The complexity of understanding the consequences of failing to cooperate may
mean that subjects need time to learn about equilibrium play. A benefit of the
Computer Task between Phases 1 and 2 is to facilitate this learning process by
exposing participants to decisions on the same networks they play against
other participants, and knowing that the decisions made by the computer
players are optimal. In this section we analyse the participants’ decisions
for Phases 1 and 2 separately in order to investigate the importance of
learning in determining equilibrium play, and whether the role of learning
varies across types of equilibria and/or networks structures.

Table 3 conducts a similar analysis to Table 2 stratifying the data into
Phases 1 and 2. The columns labelled with odd numbers contain the analysis
using Phase 1 data only, while the columns labelled with even numbers contain
the analysis using Phase 2 data only.

Looking just at Phase 1 data, columns (1), (3), and (5) show that cooperation
is not more likely in either RPE or SQ networks compared to non-RPE networks.
Similarly, (7) shows that cooperation is only marginally more likely in LCC
compared to non-RPE networks ($p=0.09$), and (9) shows that the significance
is driven by marginally higher cooperation in 1R3 rather than other types of
LCC networks ($p=0.07$). In other words, network structure is poorly
correlated with cooperation in Phase 1. A significant predictor of cooperative
play is, instead, social preferences. In all odd columns with Phase 1 data
only, self-interested individuals who allocated all points of the Dictator
Game to themselves are more likely to delete links ($p<0.05$). When
participants are first exposed to the game, they tend to make decisions driven
by their social preferences rather than equilibrium considerations.

Sample: | All networks in either Phase 1 or Phase 2
---|---
Baseline: | Non-RPE networks
| Phase 1 | Phase 2 | Phase 1 | Phase 2 | Phase 1 | Phase 2 | Phase 1 | Phase 2 | Phase 1 | Phase 2
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
RPE | -0.135 | -1.285*** |  |  |  |  |  |  |  |
| (0.232) | (0.164) |  |  |  |  |  |  |  |
SQ |  |  | -0.194 | -1.103*** |  |  |  |  |  |
|  |  | (0.276) | (0.197) |  |  |  |  |  |
RPE & non-SQ |  |  | -0.003 | -1.608*** | -0.001 | -1.607*** |  |  |  |
|  |  | (0.376) | (0.233) | (0.376) | (0.233) |  |  |  |
LCC |  |  |  |  |  |  | -0.434* | -1.612*** |  |
|  |  |  |  |  |  | (0.256) | (0.182) |  |
RPE & non-LCC |  |  |  |  |  |  | 0.688 | -0.474* | 0.689 | -0.475*
|  |  |  |  |  |  | (0.469) | (0.281) | (0.469) | (0.281)
1R3 |  |  |  |  | -0.568* | -1.534*** |  |  | -0.568* | -1.535***
|  |  |  |  | (0.317) | (0.231) |  |  | (0.317) | (0.231)
SQ & not 1R3 |  |  |  |  | 0.507 | -0.234 |  |  |  |
|  |  |  |  | (0.480) | (0.314) |  |  |  |
LCC & not 1R3 |  |  |  |  |  |  |  |  | -0.221 | -1.721***
|  |  |  |  |  |  |  |  | (0.391) | (0.252)
Subject is purely selfish | 0.453** | -0.008 | 0.449** | 0.005 | 0.471** | -0.033 | 0.475** | -0.038 | 0.473** | -0.032
| (0.190) | (0.185) | (0.190) | (0.185) | (0.188) | (0.188) | (0.188) | (0.187) | (0.188) | (0.187)
$\beta_{0}$ | 0.524 | 0.661** | 0.535 | 0.622** | 0.541 | 0.683** | 0.527 | 0.710*** | 0.542 | 0.696***
| (0.350) | (0.305) | (0.352) | (0.307) | (0.354) | (0.314) | (0.352) | (0.308) | (0.353) | (0.310)
Observations | 599 | 902 | 599 | 902 | 599 | 902 | 599 | 902 | 599 | 902
Other Controls | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes
Pseudo R2 | 0.0098 | 0.0657 | 0.0101 | 0.0689 | 0.0162 | 0.0810 | 0.0179 | 0.0796 | 0.0187 | 0.0799
Likelihood | -344.2 | -516.7 | -344.1 | -514.9 | -342.0 | -508.2 | -341.4 | -509.0 | -341.2 | -508.8

  * •

Note: Logit regressions of “Delete” using decision-level data. Each
observation is a decision made by a subject, where “Delete” = 1 if the
subject’s decision is to delete a link. Robust standard errors are clustered
at the subject-level.

Table 3: Logit regressions of “Delete” at the decision-level

In Phase 2, after participants have had the opportunity to learn from the
Computer Task, however, equilibrium considerations become a significant
predictor of decisions to cooperate. Specification (2) shows that, as stated
in H1, participants are less likely to delete links in RPE compared to non-RPE
networks ($p<0.001$). If we limit our attention to SQ networks, column (4)
shows that participants are less likely to delete links in SQ compared to non-
RPE networks ($p<0.001$). However, column (6) shows that this effect is driven
by the 1R3 network that is both SQ and LCC – participants are not more likely
to cooperate when assigned to other types of SQ networks compared to non-RPE
networks ($p=0.44$), and therefore we find little evidence to support H2.

Given that the motivation to formulate the concept of LCC is based on the low
burden of computation to understand these equilibria, we would expect
equilibrium play to emerge in LCC networks after the Computer Task facilitates
learning. Column (8) confirms that in Phase 2 cooperation is higher in LCC
networks compared to non-RPE networks ($p<0.001$). Moreover, column (10) shows
that cooperation in LCC networks is not just driven by cooperation in the 1R3
network – the LCC coefficient is significant even after we exclude 1R3
networks ($p<0.001$).

Experimental data suggests that LCC is a better behavioral predictor of
cooperative play than SQ. The analysis in Table 4 delves deeper into the
comparison of cooperation in SQ and LCC networks. First, we exclude data from
the 2R4 network, which is the only RPE network that is neither SQ nor LCC.
Second, specifications (1) and (2) investigate cooperation levels in LCC and
SQ compared to non-RPE networks, and they separate the 1R3 network that is
both SQ and LCC from either category. As we have seen above, network structure
is not significant in Phase 1, but it becomes a predictor of cooperation in
Phase 2. In particular, column (2) shows cooperation is significantly higher
in both LCC ($p<0.001$) and 1R3 ($p<0.001$) compared to non-RPE networks, but
there is no difference between SQ and non-RPE networks ($p=0.45$). A Wald test
confirms that there is more cooperation in both LCC ($p<0.001$) and 1R3
($p<0.001$) compared to SQ networks.

Finally, specifications (3) and (4) limit the data to RPE networks and take
1R3 as the baseline comparison network. Column (4) with Phase 2 data only
shows cooperation in LCC networks other than 1R3 is indistinguishable from
cooperation in 1R3, while cooperation in 1R3 is higher than SQ networks other
than 1R3 ($p<0.001$). Again, a Wald test confirms that there is more
cooperation in LCC compared to SQ networks ($p<0.001$).

Table 4: Logit regressions of “Delete” at the decision-level

Sample: | All networks excl. 2R4 | RPE networks excl. 2R4
---|---|---
Baseline: | Non-RPE networks | 1R3 networks
| Phase 1 | Phase 2 | Phase 1 | Phase 2
| (1) | (2) | (3) | (4)
Logit regressions on “Delete” |  |  |  |
$\beta_{LCC}$: LCC & not 1R3 | -0.220 | -1.723*** | 0.410 | -0.319
| (0.391) | (0.252) | (0.486) | (0.321)
$\beta_{SQ}$: SQ & not 1R3 | 0.506 | -0.238 | 1.096** | 1.133***
| (0.484) | (0.314) | (0.528) | (0.348)
$\beta_{1R3}$: 1R3 | 0.569* | -1.538*** |  |
| (0.316) | (0.232) |  |
Observations | 593 | 885 | 114 | 230
Other Controls | Yes | Yes | Yes | Yes
Pesudo R2 | 0.0161 | 0.0820 | 0.0332 | 0.0790
Likelihood | -340.2 | -495.9 | -68.0 | -146.8
Wald tests |  |  |  |
$\beta_{LCC}-\beta_{SQ}$ | -0.726 | -1.485*** | -0.686 | -1.452***
| (0.605) | (0.393) | (0.616) | (0.402)
$\beta_{LCC}-\beta_{1R3}$ | 0.349 | -0.185 |  |
| (0.482) | (0.319) |  |
$\beta_{SQ}-\beta_{1R3}$ | 1.074** | 1.300*** |  |
| (0.531) | (0.356) |  |

  * •

Note: Logit regressions of “Delete” using decision-level data. Each
observation is a decision made by a subject, where “Delete” = 1 if the
subject’s decision is to delete a link. Robust standard errors are clustered
at the subject-level. 2R4 networks were removed from the regressions to
standardize the baselines. Wald tests for differences in coefficients are also
provided in the bottom panel.

The Computer Task also allows us to disentangle social preferences and/or
beliefs about other participants’ cognitive ability as a driver of decisions
because these two factors are not relevant when playing against computer
players. Table 5 compares participants’ play in Phases 1 and 2 with the
Computer Task, conditioning on the same networks seen by the same participant.
In particular, we compare the difference in the ratio of optimal decisions
made by participants in each pair of tasks (e.g. Phase 1 optimality ratio
versus Computer Task optimality ratio), stratifying the decisions by “All
networks”, “RPE networks only”, “LCC networks only”, and “SQ networks only”.
Here, we define an “optimal” decision as a decision that a computer player
would have made (i.e., delete a link if the network is not RPE, not delete a
link if the network is RPE). The “Delete Ratio” of a subject in a task
represents the ratio of number of delete decisions to total number of
decisions of that subject in that task.

Participants play less optimally in Phase 1 compared to the Computer Task. The
negative sign in column (1) shows that participants’ decisions in Phase 1 are
significantly less optimal on all networks, while the positive signs in
(2)-(4) show participants remove more links in RPE networks – a decision that
is not optimal – in Phase 1 compared to the Computer Task. Once we compare
play in Phase 2 and the Computer Task, however, none of the differences are
significant. While qualitatively it still holds that play tends to be more
optimal in the Computer Task, the lack of significance suggests that social
preferences and/or beliefs about other participants’ limited cognitive ability
are second-order considerations in determining cooperative play.

Sample: | All Networks | RPE networks | SQ Networks | LCC Networks
---|---|---|---|---
Ratio: | Optimality Ratio | Delete Ratio | Delete Ratio | Delete Ratio
| (1) | (2) | (3) | (4)
Wilcoxon Signed-rank tests |  |  |  |
Phase 1 vs Computer Task | -3.538*** | 3.704*** | 2.865*** | 3.715***
| [82] | [78] | [47] | [76]
Phase 2 vs Computer Task | -1.006 | 1.346 | 1.395 | 1.313
| [171] | [161] | [98] | [160]

  * •

Note: The square brackets denote the number of observations. Each observation
is a subject in a task (i.e., each subject contributes to some number
$x\in\\{0,1,2,3\\}$ of observations corresponding to Phase 1/Phase 2). We test
the hypothesis that the mean difference of the optimality ratios is less than
zero, and the hypothesis that the mean difference of the delete ratios is more
than zero. We report the z-score for the Wilcoxon signed-rank tests.

Table 5: Tests for differences in means between Phase 1 or 2 vs Computer Task
delete and optimality ratios at the subject-task-level.

##  6 Conclusion

This paper has shown both theoretically and experimentally the importance of
bounded rationality in mediating how social network structure determines
cooperative behavior. In the theory part, we defined a novel notion of _Low
Cognitive Complexity_ (LCC) to select among cooperative (RPE) equilibria in
the Jackson et al., (2012) (JRT) framework. Our two main contributions come
from the experiment. First, we validate the core prediction of the JRT model
that RPE equilibria are more cooperative that non-equilibrium network
structures. Second, we show that cooperation is higher in LCC compared to
other types of equilibrium (RPE) networks.

Our results point to the importance of further research at the intersection of
bounded rationality and networks. The intrinsic complexity of network
structure means that equilibrium predictions based on full rationality and
complete information about the network are unlikely to be descriptive of
actual behavior. Iterative thinking is an appealing way to deal with this
complexity in the network context because, loosely speaking, individuals tend
to think about the consequences of their actions up to some steps away from
their network position, and the “some” can be easily related to the steps of
iterative reasoning the individual is able to perform. This paper shows that
bounded rationality based on iterative reasoning helps to select among a
multiplicity of equilibria to predict how network structure affects
cooperative behavior. We believe this approach may help in resolving the
problem of equilibrium multiplicity that is present in many other contexts
where networks affect behavior.

Our results provide further evidence that the ability to remove links affects
behavior. Given that most theoretical contributions take the network as
exogenous and unchangeable, previous experimental research incorporating
network formation tended to be atheoretical with a lot of variation in
experimental protocols in terms of how links can be added/removed by
participants. In our paper, instead, we generate precise theoretical
predictions by adopting a reduced form of the network formation process where
only link removal is allowed in order to match the JRT theoretical framework.
A fruitful avenue for further research is applying this reduced form approach
to other games on networks to investigate theoretically and experimentally how
the inclusion of the network formation process matters in determining how
networks affect our behavior.

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## Appendix A

In this section, we prove the novel theoretical results in the paper and
discuss their validity beyond the $2b>c>b$ parametrization adopted in the
paper.

Let $m$ be a positive integer such that $mb>c>(m-1)b$. We focus on the case
where $m\geq 2$ since $m=1$ is a trivial case where all networks are RPE. Note
that when $m=2$ this is the parametrization in our paper. Definition 1 can
then be generalised as:

######  Definition 5.

The class of TC networks are generated as follows: For $k=0$, define
$TC_{0}(m)=\\{g^{\emptyset}\\}$. Inductively on $k$, $TC_{k}(m)$ is such that
$g\in TC_{k}(m)$ if and only if $\forall i,ij\subset g,\exists
g^{\prime}\subset g-ij$ such that

  * •

$g^{\prime}\in TC_{k^{\prime}}(m)$ for some $k^{\prime}\leq k-m$;

  * •

$d_{i}(g^{\prime})\leq d_{i}(g)-m$

  * •

$\nexists g^{\prime\prime}\in TC_{k^{\prime\prime}}$ s.t.
$g^{\prime\prime}\subset g-ij$ and $D(g^{\prime\prime})>D(g^{\prime})$.

Intuitively, this is the same as Definition 1 except that one would need to
lose $m$ links as a result of deleting a link, in order to discourage the
agent from deleting the link. JRT show that Proposition 1 and 2 extend to this
set-up with an appropriate generalization of the definition of social quilt.

Below we prove Proposition 3 for the $m=2$ case and discuss its extension to
the general case when $m>2$. We start by proving three lemmas that apply to
the general $m\geq 2$ case.

######  Lemma 1.

All nodes in RPE networks either have no links or have $m$ or more links.

###### Proof.

Suppose that a node in a RPE network only has $l$ links, such that
$l\leq(m-1)$. Given that $c>(m-1)b\geq lb$, the agent can get a higher payoff
by refusing a favour, since the maximum possible value of retaining all links
is $lb$. This network therefore cannot be RPE, and we have reached a
contradiction. ∎

######  Lemma 2.

The number of links in RPE networks are more than or equal to the number of
nodes with links.

###### Proof.

We limit the graph only to the nodes with links, $g_{e}\subseteq g$. Let the
number of nodes in $g_{e}$ be $N_{e}$ and the number links be $|E(g_{e})|$.

From Proposition 1, $d_{i}(g_{e})\geq m$ $\forall$ $i\in g_{e}$. This implies
that $\sum_{i\in g_{e}}d_{i}(g_{e})\geq m\times N_{e}$.

Given that, by definition, $2|E(g_{e})|=\sum_{i\in g_{e}}d_{i}(g_{e})$, this
implies that $2|E(g_{e})|\geq m\times N_{e}$ and therefore $|E(g_{e})|\geq
N_{e}$ since $m\geq 2$. ∎

######  Lemma 3.

All non-empty RPE networks contain a cycle.

###### Proof.

A graph with $n$ vertices is a tree if and only if it is connected and has
$n-1$ links. This means that a graph with $n$ vertices and at least $n$ links
contains a cycle.

From Lemma 2, this implies that all non-empty RPE networks, of which $g_{e}$
is a subset, contain a cycle. ∎

The above lemmas apply for the general case where $m\geq 2$. However, the
following lemma is only valid if $m=2$.

######  Lemma 4.

For $m=2$, a simple cycle is a RPE network.

###### Proof.

For $m=2$, in a simple cycle, any agent who deletes a link will cause the
agent with the deleted link to only have one link remaining. This agent will
thus delete the remaining link he has, and this process of deleting continues
until we reach an empty network. All agents would have lost two links in the
resulting equilibrium, and therefore no agent has an incentive to refuse a
favour, making this an RPE network. ∎

Note that Lemma 4 does not apply if $m>2$, since $d_{i}(g)=2$ where $i$ are
nodes in the simple cycle $g$, and this would contradict Lemma 1.

We now prove the following proposition stated in the paper for $m=2$.

Proposition 3. For m=2, a network is LCC if and only if it is a simple cycle.

###### Proof.

$(\Leftarrow)$ In Lemma 4, we have shown that any deletion of a link in a
simple cycle will cause a cascade of deletion that results in the empty
network. Simple cycles are therefore LCC.

$(\Rightarrow)$ We prove this by contradiction. Suppose that there exists a
LCC RPE network g that is not a simple cycle. The empty network is excluded.
From Lemma 3, there exists a simple cycle within the network. Let that simple
cycle be $g^{\prime}$. Since $g$ is not a simple cycle, the simple cycle
$g^{\prime}$ must be a strict-subset of $g$; i.e. $g^{\prime}\subset g$.

By Lemma 4, this simple cycle is RPE; i.e. $g^{\prime}\in RPE$. By definition
of LCC, the largest strict-subset RPE-network must be the empty network.
However, $g^{\prime}\subset g$, $g^{\prime}\in RPE$, and
$g^{\prime}\neq\\{g^{\emptyset}\\}$.

Thus, we have reached a contradiction since we have found a non-empty strict-
subset that is an RPE network. ∎

Since Lemma 4 does not hold for $m>2$, it follows that Proposition 3 does not
hold for $m>2$ since simple cycles are not RPE under $m>2$ in the first
instance. To our knowledge, there is no simple defining characteristic for LCC
networks under $m>2$, and we would have to use the Definition 5 to
characterise it, i.e. a RPE network where the largest TC subnetwork is the
empty network. Given that there are no simple defining characteristics for
$m>2$, it is likely that participants with bounded rationality will find it
difficult to predict equilibrium outcomes even in LCC networks for $m>2$.
Hence, while the theoretical results would still apply for $m>2$, caution
should be applied in assuming that LCC networks for $m>2$ would be
computationally easy for participants to understand the consequences of a
defection. Future work could explore experimenting with LCC networks for $m>2$
to test this hypothesis.

## Appendix B

This section provides screenshots of the specific instructions and tasks given
to participants in the experiment. In Task 1, participants’ levels of altruism
are measured via the Dictator Game, after which the instructions of the
network game are provided. Before they proceed, participants must pass a Quiz
which tests their knowledge on how they are being compensated in a random
example of decisions made in a given network game. This is to ensure that
participants have a basic understanding of the game.

Once participants pass the Quiz, they proceed to play the network game in
Tasks 2, 3, 4, and 5. Tasks 2, 4, and 5 are the tasks in which participants
play against each other, whereas participants play against computer players in
Task 3. In our paper, we refer to Task 2 as Phase 1, Task 3 as the Computer
Task, and Tasks 4 and 5 as Phase 2.

Towards the end of the experiment, we measure participants’ ability to perform
backward induction and risk aversion using Holt and Laury, (2002) in Tasks 6
and 7 respectively131313In Task 6, the computer and the subject take turns
adding 1 or 2 points to a pot with 0 points at the beginning. The computer
adds randomly every turn and makes the first move. The winner is the player
that reaches 21 points. Subjects with an inclination for backward induction
would realize that in order to achieve a guaranteed win, the pot needs to be
at 18, 15, 12, 9, 6 and 3 points during the computer’s turn. Unfortunately, we
failed to record the moves of the subjects –with the exception of the end
result – due to a code error. Although winning may be determined by the
ability to do backward induction, luck can play a significant role. To be
conservative, we did not consider subjects’ performance in Task 6 in our
analysis. Finally, the experiment concludes with a survey.