# Social cohesion V.S. task cohesion: An evolutionary game theory study

Xinglong Qu${}^{1,^{\star}}$ , Shun Kurokawa2, The Anh Han3

1 The Research Center of Information Technology & Social and Economic
Development, Hangzhou Dianzi University, Hangzhou, Zhejiang, China
2 Graduate School of Arts and Sciences, University of Tokyo, 3-8-1 Komaba
Meguro-ku, Tokyo 153-8902, Japan
3 School of Computing, Engineering and Digital Technologies, Teesside
University, Borough Road, Middlesbrough, UK TS1 3BA
⋆ Corresponding author: Xinglong Qu<EMAIL_ADDRESS>
## Abstract

Using methods from evolutionary game theory, this paper investigates the
difference between social cohesion and task cohesion in promoting the
evolution of cooperation in group interactions. Players engage in public goods
games and are allowed to leave their groups if too many defections occur. Both
social cohesion and task cohesion may prevent players from leaving. While a
higher level of social cohesion increases a player’s tolerance towards
defections, task cohesion is associated with her group performance in the
past. With a higher level of task cohesion, it is more likely that a
dissatisfied player will refer to the history and remains in her group if she
was satisfied in the past. Our results reveal that social cohesion is
detrimental to the evolution of cooperation while task cohesion facilitates
it. This is because social cohesion hinders the conditional dissociation
mechanism but task cohesion improves the robustness of cooperative groups
which are usually vulnerable to mistakes. We also discuss other potential
aspects of cohesion and how they can be investigated through our modelling.
Overall, our analysis provides novel insights into the relationship between
group cohesion and group performance through studying the group dynamics and
suggests further application of evolutionary game theory in this area.

Keywords: public goods game, cooperation, evolutionary game theory,
conditional dissociation, social cohesion, task cohesion

##  1 Introduction

Group cohesion or cohesiveness is one of the oldest and most widely studied
factors in the group dynamics literature (Casey-Campbell and Martens, 2009;
Mullen and Copper, 1994). Despite a large body of works showing that group
cohesion is positively related to the group performances, such as job
satisfaction, psychological well-being, and group efficiency (Beal et al.,
2003; Carless and De Paola, 2000; Mullen and Copper, 1994), there are also
some controversies (Friedkin, 2004; Dyaram and Kamalanabhan, 2005; Rovio et
al., 2009; Khoshsoroor et al., 2019). That includes groupthink (Janis and
Janis, 1982), a conjecture that the effort to reach an agreement may lead to a
dysfunctional decision-making outcome. This inconsistency highlights the fact
that our understanding of the mechanism of how group cohesion works on the
group performance is insufficient (Carron and Brawley, 2000; Casey-Campbell
and Martens, 2009; McLeod and Von Treuer, 2013). Notably, Drescher et al.
(2012) even call it a spectacular embarrassment in group theory and research.
While apparently the complex nature of cohesion is a major cause of such
conflicting results, there exist other crucial drawbacks in the current
literature.

The principal one rests in the inconsistency of the definition and measurement
of cohesion among researchers. To make it up, Friedkin (2004) and Casey-
Campbell and Martens (2009) suggest scientists to turn focus on what is
concrete and its interrelationship with other constructs. One milestone is the
distinction between social and task cohesion (Mikalachki and Administration,
1969; Dion and Evans, 1992; Casey-Campbell and Martens, 2009). The social
facet includes relationships within the group, while the task facet includes
collective performance, goals, and objectives (Carron et al., 1985; Carron and
Brawley, 2000). In general, it is found that task cohesion is more positively
related to group performance than social cohesion (Zaccaro, 1991; Bernthal and
Insko, 1993; Mullen and Copper, 1994). But unfortunately, the explanation for
it is in short either.

Another shortage in the conventional studies is the overlook of the dynamic
nature of groups (Mathieu et al., 2008; Casey-Campbell and Martens, 2009;
Drescher et al., 2012; Mathieu et al., 2015). In reality, individuals
continually join and leave different groups and in the meanwhile interact with
each other. However, most of the previous works fail to describe this
dynamical process, and more importantly, its influence on peoples’ decision-
making. Mathieu et al. (2015) contribute one of the most important works which
proves positive reciprocal relationships between cohesion and performance over
time. They firstly synthesise aforetime studies that are about the reciprocal
influence of group cohesion and performance. Then they apply the so-called
structural equation (SEM) method to analyse the results, and justify their
model with data collected from a series of business simulations that last over
10 weeks. While SEM is inspiring, scholars are cautious about drawing strong
causal inference from it (Mathieu et al., 2015), and serious scrutiny is
required before assuming the framework of the model111 Mathieu et al. (2015)
find only 17 papers that use SEM to study the reciprocal effect of cohesion
and performance, which is the most key question in the study of cohesion, not
to mention other aspect of cohesion..

To further reveal the mechanism regarding how cohesion impacts group
performance, we take players’ decision-making into consideration using an
evolutionary game theory model. In the later half of last century, game theory
was developed as a fundamental tool to analyze the conflicts and cooperation
among human society and other organisms (Aumann, 2019). However, there are
multiple deficiencies lie in the traditional theory of games that centres on
the calculation and analysis of equilibrium, such as complexity of the
calculation, equilibrium selection, hyperrational agents and so on. These
problems motivate social scientists in a variety of areas to turn their eyes
on evolutionary game theory (Alexander, 2019). It assumes the decision makers
are short sighted, their action patterns are heuristic, and very limited
information is available. Despite the simple rules and assumptions, its
effectiveness in explaining the world wasn’t overshadowed. Especially when
considering large population behavior, simple action rules could generate
complex social phenomena which coincide with the institution and social facts
(Newton, 2018).

As aforementioned, while the complex nature of cohesion makes it hard to draw
an overall conclusion, it is meaningful to focus on the concrete aspects of
cohesion and apply different theoretical tools to study it. An important
characteristic of cohesion could be drawn from the definition of Festinger
(1950) that it is “the resultant forces which are acting on the members to
stay in a group . That is, the more cohesive a group is, the more likely that
its members would stay within it. However, people in reality keep leaving and
joining new groups for different reasons. In game theory study, this
phenomenon is studied in the model of conditional dissociation, which allows
dissatisfied players to break off with their opponents. This mechanism is
shown to promote cooperation in the two-person prisoners’ dilemma game
(Aktipis, 2004; Izquierdo et al., 2010; Qu et al., 2016). Qu et al. (2019)
extend this model to the public goods game, which is a multi-person game, and
also incorporate cohesion into consideration. In the model, if cohesion
exists, group members who do not want to leave would be united together,
otherwise, all group members become single. Their main finding is that
cooperation is better promoted when cohesion exists as it results in a greater
chance for cooperators to play with each other, which is usually termed as
positive assortativity (Eshel and Cavalli-Sforza, 1982; Doebeli and Hauert,
2005; Fletcher and Doebeli, 2006, 2009).

In this paper, we examine other important aspects of cohesion in influencing
group cooperation. In particular, as motivated above and also emphasized by
Mathieu et al. (2015), we compare social and task cohesion. Our basic model is
in line with Qu et al. (2019) where a well mixed population of players
interacting with each other using the one-shot public goods game under
conditional dissociation mechanisms. Players’ leaving decisions depend on both
their own tolerance towards defections and the levels and types of cohesion.
Social and task cohesion are inspected separately in the first two settings
and synthetically afterwards. When some players are unsatisfied, both types of
cohesion may prevent the group from dismissing. More concretely, with social
cohesion, players tend to become more tolerant toward defections; while with
task cohesion, dissatisfied players become more patient only if their goal was
achieved in the past. To be more specific, while social cohesion is described
as forces irrelevant to the history of interactions, task cohesion consists in
the likelihood that a player might refer to the play history in the last
round.

Our model reveals that, social cohesion and task cohesion, defined as above,
have opposite effects on the evolution of cooperation where the former one
harms it but the latter benefits it. The main reason accounting for the
failure of social cohesion and success of task cohesion in our model is that
social cohesion counteracts while the latter enhances the positive
assortativity effect (Qu et al., 2019). With either type of cohesion, the
dissociation mechanism is hindered and players would stay together longer. So
the higher the levels of cohesion are, the less likely that dissociation
happens, which explains the negative effect of social cohesion. However, with
task cohesion, players would leave their groups for sure once they are
dissatisfied in two continuously rounds. Moreover, since the winning
cooperative strategy is intolerant towards defection, it is vulnerable to
mistakes. Task cohesion prevents groups from being dismissed by mistakenly
defecting, which keeps the cooperative groups play together longer.

This finding partially in line with the vast empirical literature that claim
task cohesion is more positively related to group performances than social
cohesion (Zaccaro and Lowe, 1988; Zaccaro, 1991; Mullen and Copper, 1994;
Chang and Bordia, 2001; Warner et al., 2012; Spink et al., 2014). Our finding
supports the common wisdom that organizations and practitioners should
encourage team members to share successful experience more often. Intuitively,
this would increase the attraction towards teams or group pride or morale,
which according to our analysis, also increases the task cohesion that bonds
group members based on shared commitment to achieving goals.

Despite the difference of our results from the classical conclusion about the
positive significance of social cohesion, we will highlight other potential
aspects of cohesion that may be associated with group performance. When
players are less likely to update their strategies, cooperation is higher. In
reality, people update their strategies to improve their utilities. High
levels of cohesion is usually associated with higher levels of job
satisfaction, which thus reduces the possibility they update strategies. As
the probability of making mistakes decreases, group cooperation increases. In
reality, it is reasonable to conjecture that groups with higher levels of
social cohesion would communicate more effectively and accurately and thus
make less mistakes, which improves group performances. Another more subtle one
may be that social cohesion is related with selection intensity. In
evolutionary game theory study, selection intensity is an important element of
the evolution dynamics. Rand et al. (2013) find that a moderate level of
selection intensity is optimal for the emergence of generosity behavior. In
their experiment, they come up with a sagacious measurements for selection
intensity by asking subjects that among those they interact with in daily
life, how clear is it which people are more or less successful?. The more
clear their answers are, the larger the selection intensity is implied. So
cohesion might be important in maintaining a relatively optimal level of
selection intensity.

The rest of this paper is organized as follows. We detail our model in Section
2 and present the simulation results in Section 3. We then briefly examine and
discuss other potential aspects of cohesion in Section 4. The conclusions are
described in Section 5, with some further discussion being followed in Section
6.

##  2 Models

###  2.1 Public goods game

Consider a finite population of individuals who are going to play the
following public goods games. $G$ persons are grouped together with everyone
being endowed with 1 unit of personal token. They simultaneously decide
whether or not to contribute their personal token to the public wealth which
would magnify by $r(1<r<G)$ times and be equally shared among all the group
members. As the reserved personal token would not increase, the payoff
function for a player is defined as:

$u_{i}=\frac{r}{G}\cdot\sum\limits_{j=1}^{G}a_{j}+(1-a_{i}),$ (1)

where $a_{j}=1$ if player $j$ cooperates by contributing and $a_{j}=0$ if she
reserves her personal wealth in which case we say she defects.

###  2.2 Conditional dissociation and strategies

Conditional dissociation mechanism (Izquierdo et al., 2010; Qu et al., 2016,
2019) has proven to be an efficient mechanism to promote the evolution of
cooperation. Initially, all players are single players and they are randomly
grouped to play the games. After each round of game, if a player feels
unsatisfied about her group, under conditional mechanism, she could decide
whether or not to stay with her current group opponents. By leaving, both she
and her group mates become single and enter the matching pool. All the single
players in the pool would be randomly regrouped to play the games in next
round of game.

So, for each player $i$, her strategy is denoted by a tuple $(a_{i},b_{i})$
where $a_{i}\in\\{0,1\\}$ indicates whether she is a cooperator $(a_{1}=1)$ or
defector $(a_{1}=0)$ when playing the game, and $b_{i}\in\\{0,1,\cdots,G-1\\}$
indicates her tolerance towards defections. Denote
$\Sigma=\\{0,1\\}\times\\{0,1,\cdots,G-1\\}$ as the strategy space for agents.
We assume mistakes may happen to each agent randomly with a fixed probability
of $\epsilon$, that is, a player who intended to cooperate may wrongly defect
and a defector may wrongly cooperate. Let $d_{i}$ denote the number of
defectors observed by player $i$. Players are satisfied with their groups and
choose to stay within current group only if

$d_{i}\leq b_{i}.$ (2)

A group dissolves if any of its members are dissatisfied and choose to leave.
After a group dissolves, all its members, whether satisfied or not in the last
round, become single players and enter the matching pool. Players in the
matching pool regroup with each other randomly and play in the next round.

In the above model, whenever some players are dissatisfied, their groups
dissolve immediately. In this paper, different types of cohesion that prevent
the dissolution of groups are investigated.

###  2.3 Social cohesion and task cohesion

We compare two constructs of cohesion, i.e. social cohesion and task cohesion.

Social cohesion is the attractiveness that players feel about their groups,
which is independent of the history of games. To be more specific, for each
player $i$, her perception of the social cohesion, denoted by
$\xi_{i}=B(G,p)$, is a random variable satisfying a Binomial distribution $p$
indicates the level of social cohesion. A player is satisfied if the number of
defections is no more than the sum of her tolerance and her perceived social
cohesiveness, that is,

$d_{i}\leq\xi_{i}+b_{i}.$ (3)

Compared with (2), social cohesion $\xi_{i}$ is added to the right side.
Apparently, the expected cohesion $E\xi_{i}=G\cdot p$ so the higher level of
cohesion the more likely that player stays within the group. Here we assume
$\xi_{i}$ to be random since it is player’s perception or feelings which
usually are affected by some other external factors and change over time.
After each round of play, if a player is dissatisfied in the sense that she
observes too many defections than her tolerance and social cohesion, she
chooses to leave and all her group members enter the matching pool.

Task cohesion is dependent on how well players achieve their goals while
playing together. If a group of players have played together in the last
rounds, players may refer to the history of the last round of game. And the
level of task cohesion is defined here as the probability that players refer
to the history. To be specific, with task cohesion being $q$, a dissatisfied
player (i.e. when the condition in Eq 2 is violated) would choose to leave the
current group with probability $1-q$ and would look back on the last round
with probability $q$. If she was satisfied in the last round, then she would
still choose to stay within current group. The higher the task cohesion is,
the more likely that player would refer to the history before they make their
final decisions about leave or stay. When $q=0$, players do not look back on
the history and leave their current group immediately if they are
dissatisfied. When $q=1$, players always refer to the history and stay within
the current group unless they are dissatisfied in two consecutive rounds.

In our work, we investigate the role of social cohesion and task cohesion
separately in the first two settings and then jointly in the third.

###  2.4 Evolution dynamics

After each round of game, every player updates her strategy with a probability
of $\delta$. If a player updates her strategy, she would learn the strategy
from another player in the population. The learning process is a Moran process
(Nowak et al., 2004; Taylor et al., 2004; Szabó and Fath, 2007), which means
the probability that one player’s strategy is learned by others is
$\frac{f_{i}}{\sum f_{i}}$ where $f_{i}$ is player’s fitness defined as:

$f_{i}=u_{i}^{s}.$

Here $u_{i}$ is the player’s payoff222Payoffs gained in each round are
consumed immediately and cannot be accumulated. and $s$ measures the selection
intensity: when $s=0$, the process is just the neutral drift and when
$s\rightarrow\infty$, it is the best-response dynamics. Meanwhile, mutations
may also happen with a fixed probability $\mu$ to the learner that she would
randomly choose a strategy from the strategy set $\Sigma$ regardless of its
performance. The strategy update in each round is simultaneous which means if
a player has just updated her strategy and is learned by others, it is her
former strategy rather than the new strategy could be learned.

##  3 Results

Our main results are derived from computer simulations. We run three series of
simulations to compare how social cohesion and task cohesion affect the group
cooperation.

(a) Evolution of cooperation (b) Distribution of strategies

Figure 1: The evolution of the percentages of cooperation after $10^{6}$
rounds of games for different levels of social cohesion. Mistake=0.01, Group
size =5, population $N=200$, mutation rate $\mu$=0.05, strategy updating rate
$\delta=0.001$, r=3, selection intensity $s=1$.

###  3.1 Only social cohesion exists

Firstly, we investigate how different levels of social cohesion influence the
emergence of cooperation. In this set-up, players’ decisions only depend on
the current status of play. They leave if their tolerance and the
attractiveness of their group are unable to make them satisfied.

As can be seen from Figure 1(a), the stronger social cohesion is, the less
cooperation is observed. When social cohesion is fully hypothesized, only
about 10 percent of players are cooperators. This verifies that conditional
dissociation is beneficial to the evolution of cooperation from another
direction but contradicts the general hypothesis that social cohesion is good
for group performances. As when social cohesion increases, conditional
dissociation mechanism is weakened and thus cooperation declines.

We can see from Figure 1(b) that when cooperation flourishes, the most
successful strategy is $(1,1)$ and the least tolerant strategy always performs
badly. However, as strategies become more tolerant, they perform worse. The
strategy distribution is quite different to those observed in (Qu et al.,
2019), where the only winning cooperative strategy is the most intolerant one
i.e.$(1,0)$. One major difference is that in their model, whenever some
players update their strategies, their groups dismiss. Compared with $(1,0)$
strategists, players using $(1,1)$ settle down more quickly as they could
tolerate one more defector. At this point, despite being exploited by the
single defector, their payoffs are higher than the average of the whole
population and they are more likely learned by other agents, which means they
are more successful in the evolution.

###  3.2 Only task cohesion exists

(a) Evolution of cooperation for different task cohesion (b) Evolution of
cooperation for different task cohesion

Figure 2: The evolution of the percentages of cooperation after $10^{6}$
rounds of games for different levels of task cohesion. Mistake=0.05, Group
size =5, population $N=200$, error rate=0.05, strategy updating rate
$\delta=0.001$, r=3, selection intensity $s=1$.

With task cohesion, individuals may make their leaving decisions based on both
the current and the previous rounds of play. If they are satisfied in either
round, they would stay.

As can be seen from Figure 2, the effect of task cohesion is indistinguishable
when cohesion is smaller than 0.5. However, as task cohesion increases,
cooperation increases substantially. The cooperation level is highest when
players always consider the play history in the last round.

The strategy distribution is similar to the situation where only social
cohesion exists. When task cohesion is 1, the winning cooperative strategy is
also $(1,1)$. A tiny difference is that, except for $(1,1)$ that performs
better and better as task cohesion increases, all the remaining cooperative
strategies perform almost the same in different settings.

Since positive assortativity is the explanation for how dissociation mechanism
promotes cooperation, it would be natural to investigate how task cohesion
enhances it. In the study of conditional dissociation mechanisms, players are
rather impulsive that whenever they feel dissatisfied, they would leave their
opponents immediately. Intuitively, when a group of cooperators meet together,
they would continue playing together until someone defects by mistake. In our
model, with higher levels of task cohesion, players get more discreet before
they decide whether to stay within the current group or not. And a pleasant
history of play would stop the dissolution of groups. So with higher levels of
task cohesion, cooperative groups are less likely to dissolve which means
enhanced positive assortativity.

Figure 3: The percentage of cooperation after $10^{6}$ rounds of simulations
with different levels social cohesion, task cohesion and mistake rate. The
colors and sizes represent the percentages of cooperation. The more
cooperation, the larger the circle filled with the hotter color. Group size
=5, population $N=200$, mutation rate $\mu$=0.05, strategy updating rate
$\delta=0.001$, r=3, selection intensity $s=1$.

In Figure 3, we present the results of evolution for different levels of
social cohesion, task cohesion and mistake rates. It can be easily seen that
cooperation rates always decrease as mistake rates increase. Both types of
cohesion reduce the differences of cooperation rates under low mistake rates
and high mistake rates but at different directions. When the mistake rate is
low, the effect of increasing task cohesion is rather limited while it is
obvious that increasing social cohesion is detrimental to cooperation. When
the mistake rate is higher, the advantage of higher level task cohesion
becomes more obvious, which suggests that task cohesion is effective in
preserving cooperation from mistakes.

###  3.3 Both social and task cohesion exist

We now combine both types of cohesion and examine whether there is any synergy
effect from the combination. In each round of the game, player $i$ is
satisfied if and only if the number of observed defections $d_{i}$, her
tolerance, and the perceived social cohesiveness satisfy equation (3). With
task cohesion, every player might also refer to the history to determine
whether or not to leave. If a dissatisfied player doesn’t refer to the
history, or she was also dissatisfied in the last round, she leaves.
Otherwise, a player would remain in her group if she is satisfied in current
round or she refers to the history and finds herself satisfied in the last
round.

Our results suggest the negative effect of social cohesion and positive effect
of task cohesion remain when both types of cohesion exist. In Figure 4, we
present the outcomes for different levels of social and task cohesion. For any
levels of task cohesion, the proportions of cooperation decrease with
increasing social cohesion, which indicates the negative effect of social
cohesion. On the other hand, if we increase the levels of task cohesion, we
observe higher levels of cooperation with higher levels of task cohesion. So
task cohesion promotes cooperation whatever levels of social cohesion are.

The different effects for the evolution of cooperation in group interactions
from the two types of cohesion further support the idea to differentiate
between them in real-world social and economic contexts.

Figure 4: The percentages of cooperation after $10^{6}$ rounds of games for
different levels of social and task cohesion. Mistake rate=0.01, Group size
=5, population $N=200$, mutation rate $\mu$=0.05, strategy updating rate
$\delta=0.001$, r=3, selection intensity $s=1$.

##  4 Potential aspects of cohesion

While contradicting the common viewpoint that suggests positive relationship
between social cohesion and group cooperation, our model sheds light on other
potential aspects of cohesion, which could improve group performances but are
not involved in the current model.

###  4.1 Strategy updating and mistake rate

Our model confirms with the classical conclusion that cooperation is more
abundant under lower strategy updating rate (Figure 3) and mistake rate
(Figure 5).

Cohesion may lead to lower levels of strategy updating rate since it increases
individuals’ sanctification. In human society, people change their strategies
continually for different reasons. The primary one would be to pursue better
outcomes so they learn it from more successful ones. With higher levels of
group cohesion, players are more easily to be satisfied, so their incentives
to pursue better outcomes decline, and update their strategies less.

Figure 5: The percentage of cooperation after $10^{6}$ rounds of simulations
with different levels social cohesion, task cohesion and strategy updating
rate. The colors and sizes represent the percentages of cooperation. The more
cooperation, the larger the circle filled with the hotter color. Group size
=5, population $N=200$, mutation rate $\mu$=0.05, selection intensity $s=1$,
r=3. The mistake rate is $0.01$ in figure a for social cohesion and it is 0.05
in figure b for task cohesion.

In reality, everyone makes mistakes but by being more careful, they are less
likely to do so. The sense of responsibility has been emphasized by some other
studies (Chan et al., 2006; Dickes et al., 2010; Schiefer and van der Noll,
2017) on cohesion. With higher cohesion, players feel more responsible to act
carefully, which thus could reduce the probability that individuals make
mistakes. As the winning cooperative strategy is also intolerant of
defections, less mistakes help them to play together longer and gain more
benefits.

###  4.2 Selection intensity

In the study of evolutionary game theory, selection intensity has determinant
influence on the outcomes. However, in reality, how it is related to human
society or even natural world is not so clear. Rand et al. (2013) measure
selection intensity by asking subjects how clear to tell that someone is more
successful. This measurement obviously is related to the social and cultural
aspects of our society, which have non-trivial influences on human behavior.
Inspired by Rand et al. (2013), our model also suggests that social cohesion
may forge a medium level of selection intensity is better for group
cooperation.

Figure 6: The percentage of cooperation after $10^{6}$ rounds of simulations
with different levels social cohesion, task cohesion and selection intensity.
The colors and sizes represent the percentages of cooperation. The more
cooperation, the larger the circle filled with the hotter color. Group size
=5, population $N=200$, mutation rate $\mu$=0.05, strategy updating rate
$\delta=0.001$, r=3. The mistake rate is $0.01$ in figure a for social
cohesion and it is 0.05 in figure b for task cohesion.

It is worthwhile to remind that, with tiny selection intensity (say it is
being 0.05), the distributions of strategies remain almost unchanged, so we
see similar levels of cooperation (near 50%) in all settings in Figure 6. When
only a low level of social cohesion exists, cooperation rates firstly increase
and then decrease as the selection intensities increase. When a high level of
social cohesion exists, cooperation always decreases as selection intensities
increase. With medium social cohesion (=0.5), if we increase the selection
intensities from 0.05 to 0.5, cooperation decreases because it is not favored
by evolution. However, we would see a larger proportion of cooperation if
selection intensity is 1 or 2. But again cooperation almost vanishes under too
high selection intensities.

Intuitively, when the selection intensity is too weak, then all the strategies
almost make no difference in the evolution so we could not expect high levels
of cooperation. When selection intensity is too strong, the lucky defectors
whose opponents are all cooperators would get the highest payoffs and gain
more success. So medium levels of selection intensity is best for cooperation.

##  5 Conclusion

Cohesion is one of the most widely studied concepts in small-group performance
an intra- and intergroup relations (Evans and Dion, 1991; Carron and Brawley,
2000; Chiocchio and Essiembre, 2009) as it is essential for teams where people
of different talents and background meet together to collaborate on common
tasks. However, due to the complicated and elusive nature, researches have to
face the difficulty in defining, measuring, operationalizing, and
experimentally manipulating cohesion (Carron and Brawley, 2000). And it always
gives people the feeling of inconsistency when reviewing the literature
relevant to cohesion(Rosh et al., 2012). Obviously, it is hard to unify all
the previous definitions and measurements without giving new ones, which in
turn only add to the incoherence within this field of study. Friedkin (2004)
and Dion (2000) call for emphasis on the solid aspects of cohesion rather than
trying to propose new ones, which is more practical and heuristic.

One of the most significant and generally admitted findings in the study of
cohesion is the distinction between social and task cohesion, which is
referred as a milestone by Evans and Dion (1991). In general various meta-
analyses of existing cohesion studies are in agreement that group performances
are more strongly related with task cohesion than social cohesion (Mullen and
Copper, 1994; Casey-Campbell and Martens, 2009; Chiocchio and Essiembre, 2009;
Castano et al., 2013; Grossman et al., 2015). In these studies, the
definitions and measurements of cohesion are usually different among the
literature, making it hard to compare which factors are more crucial in the
cohesion-performance relation. Salas et al. (2015) conduct a meta-analysis and
highlight that further study should give priority to social and task cohesion
and incorporate dynamical and temporal factors into study.

Inspired by these suggestions and empirical observations, we compare in this
work how social and task cohesion influence the emergence of cooperation by
resorting to evolutionary game theory, since in this field, there has been
significant success in understanding the dynamic processes of group and
agreement formation and interactive strategies among group members (Szabó and
Fath, 2007; Han et al., 2015, 2017; Newton, 2018) In particular, conditional
dissociation mechanisms allow players to leave their groups once they are
dissatisfied about their opponents and have been proven beneficial for
cooperation to evolve (Izquierdo et al., 2010; Qu et al., 2016, 2019; Aktipis,
2004). We introduce group cohesion into this process to study how group
cohesion influences the performance of groups and the evolutionary outcomes of
cooperation.

Unlike psychological study where the definitions of cohesion are largely
dependent on the measurement the researchers choose, our definitions of social
and task cohesion are provided according to the intrinsic nature of individual
choices and decision making. Social cohesion is defined as a strength that
prevents players from leaving their groups regardless of the history of the
play. With higher levels of social cohesion, players are more tolerant towards
defections and thus more refrain from leaving. Task cohesion is modeled as the
likelihood that unsatisfied players would look back into the history before
they choose to leave. If they were satisfied about the outcomes in the last
round of play, they would choose to stay.

Our primary finding is that social cohesion has a negative effect on the group
cooperation while task cohesion has a positive effect. The difference of the
effect of social and task cohesion could be illustrated from the perspective
of positive assortativity, which is termed as the common characters of almost
all mechanisms that promote cooperation, including conditional dissociation
(Eshel and Cavalli-Sforza, 1982; Doebeli and Hauert, 2005; Fletcher and
Doebeli, 2009; Izquierdo et al., 2010; Qu et al., 2019). With either type of
cohesion, dissatisfied cooperators are less likely to leave their groups,
which means cohesion hinders the conditional dissociation mechanism. However,
task cohesion enhances positive assortativity by enforcing cooperative groups.
As cooperation is vulnerable to defections or mistakes, task cohesion helps
players to distinguish if the defection is intentional. A higher level of task
cohesion enables players to be more patient and thus protects cooperators from
entering a more defective match pool. So for organizational practitioners,
when monitoring and improving cohesion, recalling the successful history
regularly would benefit the team for a more productive team fulfillment.

We also discuss other parameters of the evolutionary dynamics and their
influence on the cooperation, which is a standard approach in evolutionary
game study. Our discussion is central on how it is related to other aspects of
cohesion and its impact on group performances. Compared with the complicated
nature of cohesion, our model is rather simple yet very powerful in that it is
capable of revealing the underlying evolutionary dynamics related to cohesion,
shedding light on the mechanisms based on which cohesion influences group
cooperation.

##  6 Further discussion

Cohesion is such an important and complex concept that many aspects of it need
to be further examined in future.

The first one is to study other categories of games which enable us to explore
cohesion from different angles. Both our paper and Qu et al. (2019) apply the
public goods game, which is the most studied multi-person game. But,
apparently in reality, people engage in other types of interactions too.
Sometimes, people need to coordinate on a common action or idea, then the
coordination game could be a better choice for study. Other well-known game
models including stag-hunt game, battle of sex, ultimatum game and etc. (Szabó
and Fath, 2007) could also be applied to analyze different kinds of conflicts
and interests among group members.

In our model, while both social and task cohesion are random and change over
time, there is no direction of the changes. So our model doesn’t incorporate
the reciprocal effect between cohesion and performances. In previous works,
different or even contradictory patterns of cohesion are found in military
units (Bartone and Adler, 1999; Siebold, 2006). Grossman et al. (2015) suggest
that social cohesion emerges first then members shift attention to task
cohesion. After they achieve more resources by accomplishing group goals, they
in turn enhance social cohesion and finally both cohesion become stable over
time. So the second direction for further study would be to take the time
inconsistency into consideration and in particular, the reciprocal effect
between social and task cohesion.

Our models assume well-mixed population, and it is definitely necessary and
interesting to investigate the influence of cohesion in structured populations
or spatial networks where certain individuals are more likely to interact than
others. Aktipis (2011) and Ichinose et al. (2018) study conditional
dissociation mechanism on spatial networks and complex networks respectively.
There are multiple ways to introduce cohesion into their models. Cohesion may
be referred to as how often players want to leave, how far they can migrate to
other parts in the network, and when updating strategies, how far neighbors
can be imitated. Since different types of networks (e.g., homogeneous vs
heterogeneous) exhibit different tendencies in promoting cooperation, it would
even more exciting to investigate how these topological properties interact
with different aspects of cohesion.

Evolutionary game theory is a powerful tool in analyzing group dynamics and
behaviors. Cohesion is an important concept in understanding team dynamics and
performances. Our work suggests it is promising to apply evolutionary game
theory to study cohesion in more diverse directions, which may provide other
interesting results and help us better understand how cohesion facilitates
group performances.

## Acknowledgements

Xinglong Qu acknowledges support from the National Natural Science Foundation
of China (NO. 71701058). T.A.H. acknowledges support from the Leverhulme
Research Fellowship (RF-2020-603/9) and Future of Life Institute (grant
RFP2-154).

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