The Prisoner's Dilemma in the Workplace: How
Cooperative Behavior of Managers Influence
Organizational Performance and Stress
Josef Spurný, Ivan Kopeček, Radek Ošlejšek, Jaromír Plhák
Faculty of Informatics, Masaryk University, Brno, Czechia
Francesco Caputo
Department of Pharmacy, University of Salerno,
Via Giovanni Paolo Fisciano, Italy
Abstract
Purpose
The aim of the paper is to analyze the impact of cooperativeness of managers
who occupy central positions in interaction networks on the performance
and stress levels of a whole organization.
Design/ methodology/ approach
To explore this relationship, a multi-parameter agent-based model is proposed
which implements the Prisoner's Dilemma Game approach on a
scale-free network in the NetLogo environment. A description of the
socio-economic aspects and the key concepts implemented in the model
are provided. Stability and correctness have been tested through a series
of validation experiments, including sensitivity analysis. The source code
is available for further exploration and testing.
Findings
The simulations revealed that improving the stress resistance of all employees
moderately increases organizational performance. Analyzing managers'
roles showed that increasing only the stress resistance of managers
does not account for significantly higher overall performance. However, a
substantial increase in organizational performance and a decrease in stress
levels are achieved when managers are unconditionally cooperative. This
effect is stronger for the lowered stress resistance of employees. Therefore,
the willingness of managers to cooperate under all circumstances can be a
key factor in achieving better performance and building a more pleasant,
stress-free working environment.
1
© 2 0 2 2 Emerald Publishing Limited. This Auth Dr Accept ed Manuscript (AAM) is published under the C reative
Commons Attribution Non-commercial Internati onal Lice ice 4.0 (CC B Y - N C 4.0) Any reuse is allowed in accordance
with the terms outlined by the licence To reuse the A A M for commercia purposes, per -nission should
be sought by contacting permissions@emeraldin ight.com Cite this article as foil ws: Spurný, J , Kopec ek, I.,
Oslejsek, R., Plhak, J. and Caputo, F. (2022), 'The pris Dner's dilemma in the wo rkplace: how cooperat ve behavior
of managers influence organizational per formance and stress" , Kybernetes Vol. 51 No. 1, pp. 52-76.
doi: 10.110S/K-04-2020-0229.
Originality / value
This paper presents a model for analyzing cooperation, specifically in
the organizational context, extending the Prisoner's Dilemma with novel
concepts and mechanisms. While the results confirm the existing theories
about the importance of central nodes in complex networks, they also
provide further details on how the cooperative behavior of central nodes
(i.e., the managers) might benefit the organization.
Keywords: Prisoner's dilemma, cooperation, scale-free network, agentbased
modeling, organizational performance, stress
Paper type: Research Paper
1 Introduction
Organizational performance is the crucial parameter that companies aim to
achieve high results in, in order to succeed in the marketplace. According to
Afful-Dadzie et al. (2016), tracking organizational performance serves to highlight
progress being made by an organization to meet an agreed sustainability
target. It is vital to reach high levels of performance in a way that is sustainable
in the long-term, as temporal bursts of activity that are not accompanied by a
growth of underlying structure may eventually lead to collapse (Goerner, 1994).
Modern organizations usually adopt mature management systems to maintain
sustainable performance. However, to analyze how the internal structure of
an organization might be enhanced, it may be fruitful to adopt a systemic perspective.
Given that the hierarchical nature of companies is self-organized, they
can be perceived as complex systems (Banerjee, 2012, Schneider and Somers,
2006, Dominici and Levanti, 2011). Specifically, the theoretical framework used
in this paper draws from three areas related to the complexity research - complex
network analysis, agent-based modeling, and analysis of cooperation using
the Prisoner's Dilemma Game (henceforth "PDG").
1.1 Research Gap
The evolution and maintenance of cooperation represents a fundamental problem
in biology and social sciences. Over the past two decades, researchers have
been extending the original concept of P D G to discover novel micromechanisms
supporting the emergence of cooperation. Despite the effort invested into this
area, the origin of cooperation among individuals remains an open and unsolved
issue (Wu et al., 2014, Scata et al., 2016, Lyer and Killingback, 2016, Niu et al.,
2017, Bandyopadhyay and Kar, 2018). It is also unclear whether organizations
should encourage either a cooperative or competitive environment between coworkers
to promote performance (Livio and De Chiara, 2018). Based on a recent
systematic literature review (Perc et al., 2017), human cooperation research focused
specifically on organizational context is lacking.
2
Also, empirical research investigating the impact of employees' knowledgebased
interaction on organizational performance do not provide conclusive results
(Maurer et al., 2011). As a consequence, it remains unclear which mechanisms
support the cooperative behavior of employees to benefit an organization,
namely when their interaction topology is considered.
1.2 Research Question
To address this research gap, we focus on the investigation of the impact of central
nodes on organizational performance when stress is considered. Therefore,
we put emphasis on three aspects: Leadership, employees' diversity, and stress.
The research question being: How does the strategic behavior of hubs impact the
performance of organizations at various levels of stress?
Leadership: Existing research suggests that nodes with a central position
in the network (hubs) may have a significant impact on organizations, both
beneficial or detrimental. In this paper, an agent-based model is presented
that uses a framework of complex networks with prominent nodes (hubs) that
communicate more often. In the model, a node represents an employee, and a
link between two nodes (employees) represents their mutual interaction. The
interaction is thought to be a channel that allows resources to flow - be it information,
knowledge, decisions, tasks, or counseling (Henttonen, 2010, Caputo,
2017). The ability of a company to support the flow of resources is traditionally
perceived to be closely related to its performance (Arrow, 1974).
Diversity: Team diversity may foster the flow, as employees with different
sets of skills and abilities may efficiently produce creative and innovative solutions
through the mutual exchange of resources (von Hippel, 1988). The more
diverse the skill sets of team members are, the higher the potential is for resources
to flow through social links between them (Borgatti et al., 2013, Di Fatta
et al., 2016). Skill diversity is assumed to be implicitly present among employees
in our model, and it is applied when employees exchange a defined part of their
tasks to produce added value - see Section 3.3 for additional details.
Stress: High team diversity in specific settings may lead to increased levels of
stress that relate to miscommunication and conflicts (Guimera et al., 2015). According
to Csermely (2006), complex systems with an impaired ability to relax
stress may experience a disturbance of their functions, namely their capability
to distribute information or other resources. A n increased risk of cascading failures
or the stress overload of hubs is also attributed to be the consequence of
continuously increased stress levels. Thus, stress is implemented in the model
as an essential factor that affects the social interaction of employees. As a consequence,
the quality of employees' interaction impacts the overall performance
of the organization.
1.3 Contribution and Structure
This paper contributes to the tradition of analyzing cooperation through PDG,
and it provides a deeper understanding of the role of highly central employees
3
on organizational performance and stress. This paper presents a novel approach
for modeling cooperation in organizations which reflects systemic thinking. The
complexity of the model allows for the simulation of various organizational processes,
e.g., processes reflecting strategies of managing human resources and
organizational culture.
This paper is organized as follows: key concepts used to design and implement
the model are presented in Section 3. The model was used to perform
experiments addressing the research question, and the results are presented and
discussed in Section 5. Managerial and theoretical implications are summarized
in Section 6. Lastly, the conclusion to the paper and future direction of research
is to be found in Section 7.
2 Theoretical framework and related work
Today's markets and organizations are considered to be complex systems, producing
emerging properties through interdependence across multiple spatial and
temporal scales (Gomez-Cruz et al., 2017). Since the early 1990s, agent-based
modeling has been used in organizational studies as a platform and heuristic for
theorizing about organizational responses to complexity (Miller, 2015). Henrickson
and McKelvey (2002, p. 7295) argue that agent-based modeling should
emerge as preferred modeling method in social sciences, since it supports "a
model-centered social science that rests on strongly legitimated connectionist,
autonomous, and heterogeneous agent-based ontology and epistemology". Additionally,
they believe that, in the future, significant "social science contributions
will emerge more quickly if science-based beliefs are based the joint results of both
agent-modeling and subsequent empirical corroboration ".
Agent-based modeling deals with a certain level of simplification or reduction
of complexity, which can be justified when the analysis is focused on the system
behavior instead of individual status of agents (Rhodes et al., 2016). All models
simplify reality, which makes them wrong in an engineering sense. But they offer
formal abstraction, which can be used to get valuable and fruitful insights into
relationships that would remain hidden without such model (Epstein, 2008).
Any complex system may be represented and analyzed as a complex network,
as long as the basic components of the network are discernible (Guastello et al.,
2009, Bai and Wang, 2008). During the past twenty years, complex network
analysis has become an increasingly popular methodology for understanding
systems from a wide variety of scientific domains, from protein networks to
macroeconomics (Barabasi, 2016).
When representing an organization as a complex network, the employees
are represented by the nodes and their interaction is represented by the links.
According to Csermely (2006), many human interaction networks have a scalefree
topology. A characteristic feature of a scale-free network is the existence
of a low number of highly connected hubs with many sparsely connected peripheral
nodes. This topology was formally described by Barabasi and Albert
(1999), who discovered that the scale-free property is formed by a preferen-
4
tial attachment whose mechanism is commonly known as the "rich get richer"
phenomenon.
The lack of a scale refers to the fact that it is not possible to choose a representative
node that would characterize the network, as differences in connectivity
between hubs and peripheries might be orders of magnitude large (Barabäsi,
2016). As the scale-free network is a realistic approximation of a human social
network, it was used as an interaction topology in the model presented in this
paper.
Recent network-oriented management research aimed at the team performance
has reported a variety of beneficial effects that the individuals accrue
when they occupy a favorable (i.e., central) position in the network of social interaction
between employees. Some of these beneficial effects include improved
coordination of tasks (Tröster et al., 2014), more efficient usage of resources
(Baldwin et al., 1997), high knowledge capital (Goldenberg et al., 2009, L i
et al., 2009), increased knowledge flow (Tsai, 2001), more accurate decisionmaking
(Crawford and LePine, 2013), or higher overall performance of a team
when the central individual performs well (Vaz de Melo et al., 2008, Yang et al.,
2009). Being central in a social network means being more influential and visible
as a communication partner (Ahuja et al., 2003) or being perceived as a
leader (Mehra et al., 2009). Targeting central nodes is also an efficient way of
disseminating new information through the network to increase knowledge flow
(Dana, Caputo and Räcek, 2019, Fiorini, 2017). Interestingly, central nodes
might be temporally stable even in the cases where the whole network is highly
transitory, i.e., when numerous nodes are connecting and disconnecting from
the network (Howison et al., 2006).
Conversely, central nodes may be prone to information and stress overload.
When hubs are stress-flooded, it may lead to the disintegration of a network into
several disconnected or sparsely connected sub-networks, significantly impairing
the interaction capabilities of the network as a whole (Csermely, 2006). This
effect has been observed in a case study on the online open-source SW development
community (Zanetti et al., 2013), where the departure of a highly central
member resulted in a significant and persistent negative effect on community
performance. This observation is in accordance with a theoretical presumption
that scale-free networks are quite robust against random damage or the random
removal of nodes, but are highly vulnerable to attacks targeted on hubs (Albert
et al., 2000, Ichinose et al., 2012, Mazzochi, 2016).
Another important aspect related to organizational performance is the quality
of employee interaction in terms of cooperative behavior. The traditional
approach for studying cooperation is through game theory, using the Prisoner's
Dilemma Game (henceforth "PDG"). This widely-known framework illustrates
that for a game with only one round, the rational choice for each player is to
defect (i.e. betray) their partner. However, the situation changes when the
game is played repeatedly and with more than two players.
In the late 1970s, Robert Axelrod organized a series of tournaments where
scientists could propose a strategy that would overcome others when playing
an iterated Prisoner's Dilemma Game (Axelrod, 1980a,&). Axelrod's tourna-
•5
merits had an important implication - the actual outcome of the game and
profitability of individual strategies are highly dependent on the composition of
the population. In other words, the best way to play the game greatly depends
on how others play, which illustrates the logic of interdependence (NescolardeSelva
et al., 2019). Therefore, the outcome of the game cannot be predicted,
and the solution is to run simulations.
During the past few decades, many extensions to the original concept of the
PDG have been introduced to simulation processes to discover new mechanisms
on how cooperation can be promoted. The traditionally considered principles
explaining the origin of cooperation are kin relations, reciprocal altruism, and
group survival (Nowak and May, 1993). A n overview of selected mechanisms
that help to promote cooperation is provided in Table 1. The model presented
in this paper allows agents to withdraw from the game, similar to the voluntary
participation mechanism. Instead of a rational decision, here agents cease
interaction as a result of illness - see Section 3.8 for a detailed description.
Mechanism Source
Reputation-based recognition of
partners for safe cooperation
Kim (2010), Nowak and Sigmund
(1998)
Homophily - preference of agents to
interact with others of the same kind
Pacheco et al. (2006), Scata et al.
(2016)
Social punishment - agents may punish
defectors at own cost
Niu et al. (2017), Zimmerman and
Eguiluz (2005), Wang et al. (2013)
Donations between agents with similarity
in some arbitrary characteristic
Riolo et al. (2001)
Voluntary participation in the game
- agents may withdraw from interaction
with unfavourable outcomes
Chen et al. (2010), Jia et al. (2018),
Hauert et al. (2002), Szabo and
Hauert (2002)
Formation of coalitions - agents may
group with others to exclude defec-
tors
Peleteiro et al. (2011, 2012)
Historical payoff - agents remember
outcomes of previous games with the
same player
Ceng et al. (2017)
Table 1: Selected mechanisms extending the original concept of PDG.
Analyzing interaction networks has become a useful perspective for studying
cooperation, and network science has been successfully applied in the study
of the P D G . As reported by Pacheco and Santos (2005), both evolution and
maintenance of cooperation are highly sensitive to the topology of the players'
interaction network. This effect is called network reciprocity (Nowak, 2006).
Selected observations of network reciprocity are illustrated in Table 2, where the
scale-free topology and hub failure are closely related to the research question
6
being addressed in this paper.
Observation Source
Scale-free topology - the emergence
of cooperation is supported by the
topology of network with hubs, and
in other types of networks with high
levels of degree heterogeneity
Kun et al. (2010), Hirihara et al.
(2013), Ichinose et al. (2012), Poncela
et al. (2007), Pacheco and Santos
(2005), Santos and Pacheco (2005),
Santos et al. (2006), Yang et al.
(2009)
Hub failure - cooperation is robust
against random nodes defecting, but
targeting influential nodes results in
quick deterioration of cooperation
levels. A n attack targeted at the hub
may lead to the disintegration of a
scale-free network
Albert et al. (2000), K i m et al.
(2002), Perc (2009)
Community size - promoting cooperation
increases with the size of the
community. The effect of a large
community cannot be replaced with
a greater number of smaller communities.
Larger communities of cooperators
are less susceptible to invasion
by defectors
Liu and Lui (2016), Wang and Perc
(2008)
Cooperating supercommunity - cooperators
form single giant community
while several clusters of defectors
might coexist
Gomez-Gardehes et al. (2007)
Link density - in sparse networks, cooperating
communities may be easily
invaded by defectors as cooperating
individuals have fewer alternative interaction
partners
Kun et al. (2010)
Table 2: Selected observations related to network reciprocity.
The experiments presented in this paper focus on simulations of the Prisoner's
Dilemma in the workplace on static scale-free networks. Two relevant
reasons imply the use of a static network setup. First, it is assumed that people
do not usually change their typical behavior significantly, as personality traits
are relatively stable in time (Caspi et al., 2005). Second, organizations have their
structure defined, and their rate of change is slower than the rate of change observed
in traditional simulations of the PDG. Employees in the presented model
cannot choose with whom they interact, in addition, they cannot terminate for-
7
mal social links with members of their organizational unit. Hence, to simplify
the situation, the simulations are performed on static Barabási-Albert networks
where employees do not change their strategic behavior.
3 The Model Description
This chapter describes the fundamental properties and interactions of agents in
the presented model. The methodology used in this paper is based on agentbased
modeling, where the dynamics of the system emerge from attributes and
behavioral rules of individual agents. Therefore, a relatively simple model can
exhibit complex behavioral patterns and provide valuable information about the
real-worid system that it emulates (Bonabeau, 2008).
For the purposes of the model development, the NetLogo tool was used.
NetLogo is an environment for agent-based modeling of complex systems, and
it offers a library with existing models from a wide range of scientific disciplines.
NetLogo provides high-level programming language for users to develop their
own models (Tissue and Wilensky, 2004, Wilensky and Rand, 2015).
The model presented in this paper was designed and developed by the authors.
Partial functionality related to network formation processes has been
adopted from the existing NetLogo model library (Wilensky, 1999), yet the majority
of the source code is original. The main functionality of the model was
incrementally developed and tested from September 2017 to December 2018,
additional modules and minor updates are subject to ongoing development.
During the implementation, each functionality was evaluated through a parameter
sweep to analyze how implemented parameters impact global behavior and
the key outcomes of the model. Furthermore, the evaluation and testing allowed
us to determine default values for each parameter, ensuring that the model produces
reasonable outcomes, e.g., realistic employee sickness absence. The core
functionality of the model has been described in a previous study, as well as
results of validation experiments (Daňa, Kopeček, Ošlejšek and Plhák, 2019).
The model represents an abstract organization (e.g., a company or an academic
institution) where individuals interact to create value. The activity diagram
of the model is depicted in Figure 1, and it illustrates the sequence and
decision-making logic during the simulation. It consists of three nested parts.
The work activity of an employee part captures the actions of a single employee
in one workday in an organization. Diversity in individuals' skills creates
the potential for knowledge flow, which results in increased performance. Hence,
it is assumed that employees enter an agreement with one another to exchange
part of their work for additional productivity on a daily basis. This agreement
is subject to the Prisoner's Dilemma - the workers may cooperate or defect,
which has an impact on interpersonal relationships, and stress levels (see the
Collaborate with a neighbor in the topology activity in the diagram).
Employees may become ill and unable to work when they accumulate stress
faster because they are unable to relax. Other essential concepts of the model
8
simulation of an organization
V
Figure 1: The activity diagram of the model. Actions inside the workday of an
employee box represent the behavior of employees. Actions outside the box are
performed by the management of the organization.
include changes in employees' stress and motivation in reaction to salary change.
These mechanisms are shown in the workday of an employee section of the
diagram. The primary input for this step is the salary of an employee. It is
expected that each employee generates values comensurate with their salary.
The employee's personality, other states, and the results of collaboration define
to what extent this expectation is fulfilled.
The simulation of an organization section of the diagram shows management's
strategies for rewarding employees' performance and cooperativeness.
The structure of the organization affects the interaction of employees, as defined
at the beginning of the simulation. Scale-free networks are used, yet
the model allows simulations to be performed on other widely-known complex
network topologies - i.e., regular square lattice with periodic boundaries, the
Erdos-Renyi model for a randomly connected network, and the Watts-Strogatz
model of small worlds. In the context of this paper, hubs with a prominent
role in the scale-free network may represent a leader - i.e., an employee with
a high position in the company hierarchy who is responsible for the performed
tasks and who is frequently contacted by others for reporting, supervision or
mentoring.
The remainder of this Section describes the model in greater detail. The
numbering of actions in Figure 1 corresponds to subsections, where they are
discussed.
3.1 Employee Attributes
Employee attributes reflect characteristic patterns of behavior or responses to
external stimuli, as implemented in the presented model. For the sake of simplicity,
it is assumed that these individual properties do not change during
9
simulations. The values of these parameters are set as described in Table 5.
Productivity is the ability of an employee to create profit for the organization.
Productivity equal to one means that the employee is able to do the
work but produces no extra profit for the company. Productivity of less than
one creates a loss for the company. Only those employees whose productivity
value is greater than one, produce a corresponding profit.
Resistance to stress is modeled as the capacity of a virtual stress container.
If the stress level is below the given capacity threshold, the employee
is healthy and working. When the stress level exceeds the limit, the employee
becomes ill and unable to work. The higher the capacity of the stress container,
the longer it takes to exceed its threshold.
Cooperativeness is the typical behavior of an employee when interacting
with co-workers. Seven strategies commonly used in the P D G context have
been selected for the model, as listed in Table 3. A l l strategies, except for
defecting, are considered as cooperative, as they always initiate the interaction
with cooperation.
Strategy Label Description of behavior.
Defect D Always defects.
Cooperate
c Always cooperates.
Tit for Tat T Initiates with cooperation, then repeats
partner's last move.
Tit for two Tats T2 Initiates with cooperation. Defects only if
defected by the same partner in the last two
rounds, otherwise cooperates.
Tit for Tat - nT Repeats partner's last move. When defected
Naive Peacemaker in the last round, there is a probability
of cooperative response.
Pavlov P Initiates with cooperation. Repeats action
when won last round, switches action when
lost last round.
Unforgiving U Always defects once defected by a partner.
Otherwise cooperates.
Table 3: The description of employees' collaboration strategies.
Diligence represents employees' ability to transform salary incentives into
work motivation. Compared to the worker with low diligence, a highly diligent
worker will be more motivated after a salary increase, and less demotivated after
salary decrease. Diligence has normal statistical distribution among the population
of employees. See Section 3.7 for further details related to adjustments
of motivation.
10
3.2 Employee States
While employee attributes represent stable "personal traits" and are fixed during
simulations, this subsection describes the employee's state. They are continuously
changing, reflecting their actual "mood" - reactions to workplace environment
and interaction with others. Sensitivity to changes in mood is affected
by employee's attributes. In the model, three types of states are defined.
Effort is related to productivity and represents an employee's inner motivation
to work. While productivity describes the employee's ability or potential to
do work, effort describes to what extent the employee fulfills their potential at
a particular moment — 0% can be understood as "an employee does not work
at all" while 100% as "an employee does their best." The effort is adjusted as a
reaction to changes in salary (see Section 3.7).
Stress is induced by solving work tasks and reduced by regeneration. It
describes how much the current stress level approaches the maximum stress
capacity — 0% means no stress at all while 100% means depletion of stress
capacity and possible sickness. The less resistance to stress an employee has,
the sooner the stress capacity can be reached, as discussed in Sections 3.4, 3.8,
and 3.7.
Interpersonal relationships arise from interaction between co-workers.
Employees remember their interaction history with particular co-workers, which
is important if an employee uses a P D G strategy that reacts to past actions of
other players, e.g., the "Tit for Tat," see Table 3. The interaction history would
also be taken into account when selecting partners for cooperation. However,
for the sake of simplicity, interaction partners are chosen randomly, as discussed
in the following section.
3.3 Simulation of Work Activities
This part of the simulation algorithm can be considered as one working day during
which an employee produces a certain amount of profit to the organization.
The primary input for this step is a salary /3j £ N, which represents the
amount of money that management gives the i-th employee. Management expects
that the employee will produce the work at least commensurate with the
value of the salary.
The primary output is a generated value V{ € N which corresponds to the
input salary increased (or decreased) as a result of the work produced during
the day:
Vi = (3i((l-wv)Ve + wvVc + Vr), (1)
where wv represents the cooperation weight denoting what portion of the work
will be exchanged with a co-worker. Therefore, the computation is split into
three parts. A major portion of the work (Ve ) is done directly by an employee.
A minor portion (Vc) represent subtasks that might be exchanged with a coworker.
The mechanism of work exchange is based on a presumption that
specific subtasks might be done by particular employees more efficiently. The
ratio between own and exchanged work is fixed in the model during the whole
11
simulation. The third addend Vr represents the cooperation requested by other
co-workers.
Own work Ve reflects directly solved tasks. It is proportional to the productivity
9i and current effort Ki of individual employees:
If the effort is 0%, then the contribution is also zero. It means that no work
has been done and this part of the salary was only consumed. Another extreme
arises when an employee's effort is 100%. In this case, the contribution equals
to the productivity 9i.
Exchanged work Vc reflects the amount of work that an individual agreed
to exchange. This interaction is subject to PDG, and the outcome of the exchange
is affected by the following payoff function that satisfies standard requirements
put on P D G (Chong et al., 2007). From the perspective of an employee,
there are four possible outcomes of the interaction:
• Sucker's payoff: Vc = 0. I did the partner's work as was agreed, but my
partner did not do my work. I cannot earn anything from my exchanged
work because it was not done. M y cooperation part of the work is lost.
• Cheater's payoff: Vc = 2.25. I did not do the partner's work, but the
partner did my work as agreed. Because they did the work more efficiently
than me, I get the payoff 1.25. Moreover, I had time to do my own work,
and so the final payoff is 2.25.
• Punishment: Vc = 1. We did not agree on cooperation and did the work
as usual, with normal efficiency. The payoff value is 1.
• Reward: Vc = 1.25. Both of us met the agreement, and because we did
the work more efficiently, both of us get a payoff 1.25.
A partner for exchanging the work is selected randomly from employee's
neighbors, i.e., from co-workers that the employee is linked to with a tie on a
social network. Sick employees (see Section 3.8) are excluded from the interaction
as they are on sick leave.
During a single simulation round, an employee may initiate the exchange of
work only once. However, one employee may interact more than once, as the
interaction may be requested by other employees during that round. The more
social connections an employee has, the higher is the probability of multiple
exchanges of work. Results of these requested interactions are taken into the
computation as requested exchanged work Vr:
where j corresponds to the j-th requested cooperation, Wj is the cooperation
weight assigned to the co-worker (co-worker's wv in Equation 1), and <pj is the
employee's results of the iterated P D G played with the j-th co-worker.
Ve = 9iKi. (2)
(3)
12
3.4 Simulation of Stress
It is assumed that performing everyday work activities produces stress. The
presented model incorporates two types of stress factors.
Intra-individual stress is related to own work Ve and the effort K made
by an employee to solve tasks. The more effort an employee exerts, the more
stress is produced. The less resistant to stress an employee is, the faster the
stress increases. The impact of intra-individual stress on simulation results can
be controlled by adjusting a global variable in the model. Note that the intraindividual
stress is linked to the extent of an employee's effort, not to the actual
amount of work done. Consequently, two distinct employees may deliver the
same amount of work at different levels of effort and stress, depending on their
attributes and actual state.
Being defected by a co-worker is another source of stress. As it is induced
by social interaction (through the PDG), it is referred to as inter-individual
stress further in the text. The outcome of the P D G defines its amount. If both
employees cooperate, their stress does not increase. If both defect, stress is increased
proportionally to their resistance to stress and the amount of exchanged
work Ve. If one cooperates while the other defects, then the cooperating employee
is stressed twice as much as in the previous case, while the stress of the
defecting employee does not change.
Because intra- and inter-individual stress increase continuously during the
workday, the mechanism of regeneration has been introduced in the model.
Regeneration reflects the post-work relaxation, which counterbalances stress
production. The rate of continuous regeneration takes into account resistance
to stress - the more stress-resistant an employee is, the faster the stress is
regenerated. The regeneration rate may be adjusted by the model to simulate
various environments or organizational cultures.
3.5 Evaluation of Employees
After a certain number of simulation steps (workdays), management evaluates
the behavior of each employee. The goal of the evaluation is to achieve management
objectives such as increased profit, the sustainability of production,
or employees' satisfaction by evaluating employees' work results. However, in
the real world, the insight of management into the behavior of employees can
be restricted and inaccurate. Therefore, the model allows for incorporating the
desired level of randomness by adding noise.
Evaluation Ei £ [0,1] of i-th employee is calculated as a weighted sum of
real employee's contribution Fi £ [0,1] and a random number r £ [0,1]:
Ei = weFi + (1 - we)r. (4)
The weight we is fixed during the whole simulation.
Employee contribution Fi determines how management perceives the
benefit of an employee in terms of achieving the objectives of an organization.
The evaluation process considers two aspects: efficiency and cooperativeness of
13
an employee. The efficiency, expressed as the generated value in Equation 1,
already reflects the extent of the cooperative behavior of an employee (see Ve
and Vr values in the equation), whereas cheating co-workers reach better results
in comparison. However, cooperativeness is profitable for the organization in
the long-term perspective. Therefore, management may compensate cooperating
employees by virtually "returning them" the generated value that they lost
when interacting with defecting co-workers. If such compensation did not exist,
employees would be punished for being cooperative.
The calculation is performed in two steps. First, an added value Wi is
computed as the difference between the generated values that an employee accumulated
during the evaluation period (X)Vi) and the salary earned during
that period ft). AV; is the value that given employee lost when exchanging
work with defecting co-workers:
wi = j2v
i-J2/3i + AVi
- (5
)
The employee contribution is then calculated relative to other employees:
Fi = (Wi - Wmin)/(Wmax - Wmin), (6)
where Wmax and Wmin are the maximal and minimal Wi observed in the current
evaluation period.
3.6 Budget Redistribution
At the beginning of the simulation, the company earmarks a certain budget
a € N for all employees. This budget is evenly distributed among them as their
input salary Pi.
During the simulation, employees usually produce a profit ( ^ Vt > X^ft)
through which the overall value of the organization increases. Management
may decide to use part of the profit to increase the budget that employees
receive. If the company produces a loss, then the budget a remains the same
as in the previous evaluation period. It is assumed that this strategy reflects a
socially responsible organization that keeps the profit when productivity is high
to support employees to overcome the temporary period when productivity is
low. This strategy should also act as a prevention to a downward spiral of
performance through decreased motivation, poor evaluation leading to further
loss of motivation.
The redistribution of budget is driven by relative performance pi £
[pmin,pmax], which is continuously updated during the whole simulation. It
represents a relative long-term efficiency of individual employees as, potentially
imprecisely, perceived by the management. Initially, it is set to 1 for all employees
(100% performance), but then it is re-calculated after every evaluation:
E•
Pi = Pi + Pi{~ET- ~ l
)wp, (7)
^avg
14
where wp £ [0,1] defines the intensity of the impact of the evaluation process.
The result is replaced with predefined limits pmin or pmax whenever the value is
out of this range so that the relative performance remains in reasonable limits
around 100%.
The budget a is then redistributed taking pi into the account:
ft = a ^ - , (8)
where 2~2 Pi is the relative performances of all employees.
3.7 Impact of the Salary Change on Employees
Changes in salary have an impact on employees and their future behavior.
Impact on stress: A n increase in salary decreases the employee's stress and
vice versa. Sensitivity to salary change is affected by the relative performance
of the employee. Stress-wise, employees with lower salaries are more sensitive
to salary changes. Compared to high earners, employees with lower salaries
will be more stressed when given less money and more motivated when given
more money. This behavior reflects the so-called Easterlin paradox, which states
that increased income correlates with increased happiness, but the increments
of happiness diminish after reaching a certain threshold (Clark et al., 2008).
Impact on effort: Changes in salary may be motivating or demotivating,
depending on how difficult it was for the employee to make money. The subjective
difficulty Si is calculated as a ratio between a new salary Pi and the amount
of work done in the current evaluation period:
pt{l-we)M + piWeXi'
where pi is the relative performance, we represents the cooperation weight, M
is the length of the evaluation period, and \% is the total number of interactions
that employee participated in during given evaluation period.
To determine whether it was easier or harder for the employee to make
money, the coefficient 5i is compared with its value 5t from the previous evaluation
period. The difficulty of earning money versus the salary change defines
four situations captured in Table 4. Motivating conditions lead to increased
effort and vice versa. The shift in the effort is proportional to the extent of how
much added value an employee brought to the organization:
e = ft-ft, (io)
where ft is a new salary, while ft is the salary from the previous evaluation
period.
The motivational factor is also affected by employees' diligence, and the
effect is positive linear.
15
difficulty to earn money
easier harder
salary decrease
salary increase
motivating demotivating
demotivating motivating
Table 4: Impact of the salary change and the difficulty to make money on the
effort adjustment.
The effort change of sick employees is re-calculated in the same manner as
working employees. Nevertheless, the effect of effort change on their performance
is applied after they return to work and start to produce the generated
value again.
Impact on interpersonal relationships: In the current model, interpersonal
relationships of employees are affected only by the P D G strategy and the
history of interactions with a particular co-worker. Therefore, changes in salary
do not modify the collaborative behavior of employees.
3.8 Sickness
A metaphor of sickness is used in the model to simulate the impact of high
stress on the work activities of employees. If the income of stress is higher than
an employee's regeneration rate for a particular time such that the stress level
exceeds their stress capacity, then the employee becomes ill.
These employees are on sick leave; thus, they do not work. The organization
still supports them, so they receive normal salary pi, but they produce
zero generated value V{. Since they do not work, their stress and interpersonal
relationships remain unchanged.
The stress level is checked at the end of each simulation step. The length
of sick leave is fixed in the model to seven days (simulation steps). After that
period, the employee returns to work relieved, having stress level reset to zero. If
the current effort Ki is higher than the initial effort, then the value is decreased
slightly to reflect the fact that people usually do not work at their best after
an extended period of inactivity. In the same way and for the same reason,
also, the relative performance pi (see Section 3.6) is decreased towards 100%
whenever it is above this threshold.
4 Simulations Setup
The previous Section describes selected fundamental concepts and mechanisms
that define the behavior of the model. Interested readers may refer to online
repository (Dana et al., 2018) where description, documentation, and source
16
code with the model to be run in the NetLogo environment is available to
download.
Simulations presented in this study include 3,600 simulation steps where
each step may be perceived as one day. Therefore, a single simulation can be
seen as approximately ten years of organization lifespan. As discussed earlier,
in real-world institutions, individuals usually do not structurally change their
work habits and typically cannot voluntarily decide who their co-workers are.
For the sake of simplification, the network used in the experiments is assumed
to be static - the employees cannot create or terminate links, and they also
cannot change their strategy while playing PDG.
The size of the company is set to n = 100 employees. The number of hubs
representing managers or leaders depends on the network formation process,
which is probabilistic. Most commonly, there are 5 ± 2 hubs. Evaluation of
employees by management is performed once per 30 steps, i.e., monthly. The
values of key parameters used for the simulations are summarized in Table 5.
Fixed attributes of employees
cooperativeness 4:1
productivity
stress capacity
regeneration
diligence
Initial employees' states
effort 75%
stress 0
interpersonal relationships none
Other features of employees
cooperation weight 20%
Features of management
evaluation period 30 steps
Proportion of cooperative vs.
defecting employees
Mean value and SD
Mean value and SD
Implicit value of stress relaxation
Mean value and SD
Employees use 75% of their
productivity
Employees are relaxed
No cooperation history
Fixed part of work exchanged
with a co-worker when playing
P D G
Frequency of management
intervention
3.8 ± 1
300 ± 5
1.125
0.5 ±0.175
Table 5: Selected parameters of performed simulations (SD = standard devia-
tion.
The simulation process is stochastic, i.e., identical parameter settings yield
different global outcomes as a result of random processes in the model. In order
to obtain unbiased results, 500 runs were executed for each setup to eliminate
the effects of stochasticity. Validation experiments showed that 500 repetitions
are sufficient to accurately represent the results in terms of obtained mean value
17
and data dispersion.
During the model development, we intensively tested the stability of the obtained
complex system concerning various parameter settings. The validation
experiments included sensitivity analyses to set up default values of parameters
to produce reasonable outcomes. One of the analyses aimed at identifying implicit
value for the regeneration rate of employees, i.e., to what extent they will
be able to cope with stress. Based on this analysis (see Figure 2), the implicit
value for regeneration was set to 1.125.
Figure 2: A sensitivity analysis of employees' regeneration rate and its impact
on organizational performance for selected P D G strategies of hubs. A sigmoid
progression is observed, with sharp increases in organizational performance for
small increments of regeneration in region r « 1.05 and r « 1.2. For values
r > 1.2, the number of ill employees converges to zero, and there are no ill employees
for regeneration rate r > 1.6 and for any hub strategy setup. Observed
fluctuations in organizational performance are caused by other mechanisms implemented
in the model.
18
The aim of this study is to analyze the impact of a highly central node
behavior in a social network (i.e., a manager) on the overall organizational performance.
To generate a network with a few highly connected nodes (the hubs),
the Barabasi-Albert network model was used. For an illustrative network visualization,
see Figure 3 below. Strategy of the hubs was predefined to particular
strategy (e.g. Cooperate, Defect ...) for each set of 500 simulations per 3,600
days.
Figure 3: A n example of a scale-free network with n = 100 nodes, average degree
k m 2. Labels on the nodes denote employees P D G strategy (see Table 3). A
few highly connected hubs are highlighted.
5 Results
As described in Section 3 of this paper, organizational performance is not only
dependent on the productivity of employees, but also on the quality of their
interaction and management evaluation. These mechanisms impact motivation
and stress levels of individuals, which might significantly influence the overall
organizational performance. The results of performed simulations reveal that a
specific behavior of a manager (conceptualized as a hub in the social network)
might substantially impact the key parameters of the organization. Note that
the results presented in this section are normalized against the maximum value
obtained for a given set of simulations.
The organization achieves the highest levels of performance (Figure 4) and
19
100,0%
90,0%
80,0%
20,0»
10,0%
0,0%
• Cooperate "Defect "Tit for Tat • Tit for two Tats •TitforTat-npm "Pavlov "Unforgiving
Figure 4: The impact of hubs' strategy on organizational performance. The
highest organizational performance is achieved when hubs remain unconditionally
cooperative. Values inside boxplots represent average performance for the
given setting when compared to the highest result obtained from the experiment.
the lowest rates of illness (Figure 5) when the highly central employees in the
interaction network i.e., the managers, are unconditionally cooperative. The
sickness rate is calculated as a cumulative sum of employees who become ill as
a result of high levels of stress during the simulation.
It is widely accepted that the topology of the scale-free network, and the
degree heterogeneity in general, is a contributing factor that facilitates the
emergence of cooperation on dynamic complex networks. In such a setting,
the cooperation propagates through the network as individual agents copy the
strategy of successful cooperating hubs. However, the results presented in this
paper were obtained from static networks, where agents can neither change their
strategy, nor create or terminate social ties. When interpreting the data, it is
essential to focus on the effect of stress, as it may significantly influence the
total economic outcome of the organization. At high levels of stress, employees
frequently become ill and therefore produce no value.
When a manager who occupies a highly central position in the network defects,
their uncooperative behavior has an area effect of stress because of the
many social connections with other colleagues. Moreover, many employees at
the network periphery may be connected only with the nearest hub, as a consequence
of the scale-free network topology. These peripheral employees cannot
substitute the malfunctioning interaction with the manager because they do
not have any other social connections. Depending on their strategy, peripheral
20
100,0%
90,0%
80,0%
0,0%
• Cooperate "Defect "Tit for Tat • Tit for two Tats "Tit for Tat - npm "Pavlov "Unforgiving
Figure 5: The impact of hubs' strategy on the sickness rate of the employees for
implicit regeneration level r = 1.125. The highest level of stress is observed when
hubs do not cooperate. Values inside boxplots illustrate the average sickness rate
of all employees compared to the highest obtained result.
nodes either cooperate with the manager and receive stress, or they do not interact
at all. Consequently, their stress is not increased, but there is also no
generated value resulting from the part of the work exchanged between manager
and employee.
It is important to note that highly central nodes are susceptible to stress
overload, as discussed in Section 2. Figure 6 illustrates that hubs are up to
three times more likely to get sick when their resistance to stress is the same
as the resistance of peripheral nodes. With increasing stress resistance, the
likelihood of hubs becoming ill is substantially reduced. However, the results
suggest that adjusting the capacity of hubs to resist stress does not produce
significant differences in the overall performance of the organization.
To further analyze the effects of stress, additional experimental setups were
considered. Since increasing the stress resistance solely for hubs does not account
for significant improvement of organizational performance, increasing the
regeneration for all nodes was considered. As the global regeneration rate increases,
the beneficial effect of hubs cooperating unconditionally diminishes,
the difference between cooperative and defecting hubs becomes less significant,
and hubs with Tit for Tat strategy account for the highest organizational performance
(Figure 7). For the sake of illustrative visualization, only three hub
strategies are included - variants of Tit for Tat and Pavlov strategies scored similarly
to pure Tit for Tat, and Unforgiving strategy scored similarly to Defecting
21
100,0%
90,0%
80,0%
c 70,0%
60,0%
50,0%
E? 40,0%
1 30,0%
20,0%
10,0%
!!
53.4% 53 6% 54. 5%
35.5% 36.3%
41.0%
0,0%
i Cooperate • Defect • Tit for Tat
Figure 6: The impact of increased stress resistance of hubs on organizational
performance. For any strategy, darker hue of the same color illustrates increased
hubs' resistance to stress - 100% (left), 130% (middle), and 150% (right), while
stress resistance of peripheral nodes remains unchanged. Values of s refer to
an average number of hubs becoming sick during the simulation for given setup
(s = 60.9 for peripheral nodes). A n implicit regeneration level r = 1.125 was
used for all setups.
strategy.
An additional experiment was considered to compare the results of all implemented
hub strategies while eliminating stress. To control the effects of stress,
all parameters were set to their implicit values (see Table 5), and regeneration
r was set to r = 100 (note that the implicit value is r = 1.125). This setting
prevented any stress build-up among employees that could result in their
becoming ill. The results of this experiment illustrate that there are no significant
differences in organizational performance when changing the strategy of
central employees (Figure 8). Eliminating stress reduces the beneficial effect
of unconditionally cooperative hubs on organizational performance. Except for
defecting and unforgiving strategies, all strategies scored close to 60% of the
highest observed value.
22
90,0%
80,0%
£ 70,0%
o
"S 60,0%
•B 50,0%
40,0%
•-P
•S 30,0%
20,0%
10,0%
39.9%
51.4%
2 3 i-y,
35.3
30.9%
39.6%
0,0%
i Cooperate • Defect • Tit for Tat
Figure 7: The effect of increased regeneration level on organizational performance
for selected strategies of hubs. For each strategy, the darker hue of
boxplot color illustrates increased regeneration r with values r = 1.125 (left),
r = 1.135 (middle), r = 1.500 (right). As stress resistance increases, the beneficial
effect of being a cooperative hub diminishes, and Tit for Tat becomes an
optimal strategy for a hub to achieve the highest organizational performance.
6 Implications
The results of the performed experiments suggest the following managerial im-
plications.
A stressful work environment requires cooperative leaders. Organizations
experiencing increased levels of stress should carefully evaluate the
cooperative behavior of managers, team leaders and other employees in central
positions within the organizational hierarchy. The increased level of stress may
be observed indirectly through an increased employee turnover or increased rate
of employees' sick leave - in these situations, the cooperative behavior of hubs
has a significant beneficial effect for an organization.
Employees' stress resistance has a positive effect on performance.
The ability of employees to resist stress was shown to be an important mechanism
which contributes to organizational performance under a wide range of
215
100,0%
90,0%
80,0%
8
£ 70,0%
£
% 60,0%
a.
TO
D 50,0%
S 40,0%hp
O
| 30,0%
"flj
at
20,0%
I Cooperate • Defect • Tit for Tat • Tit for two Tats • Tit for Tat - npm • Pavlov • Unforgiving
Figure 8: The impact of hubs' strategy on organizational performance when
controlling the stress level to zero. Values inside boxplots illustrate an average
relative performance compared to the highest observed value. The significant
advantage of unconditionally cooperative hubs has been reduced.
conditions. As the employees' relaxation rate stems from organizational culture,
the results presented in this paper are in accordance with the traditionally
perceived link between organizational culture and performance.
Stress-resistant work teams perform steadily. Organizations whose
employees are highly resistant to stress can be also more tolerant to the behavior
of managers. In other words, managers may experiment with acting
cooperatively or competitively towards employees, as the potential detrimental
effect on organizational performance is low. Conversely, the cooperative behavior
of managers is extremely important when employees' resistance to stress is
low, as previously mentioned.
Aside from the specific managerial implications related to the findings, the
research has also brought important theoretical and technical implications to
light.
A contextual perspective is important for the evaluation of an
organization. From the theoretical point of view, the results underline the
general importance of contextual perspective when evaluating organizational
ecosystems. Following the logic offered by Henrickson and McKelvey (2002), the
behavior of individual employees at lower scales provides an important context
for the organization at a higher scale. In particular, our research shows that
such a higher-scale structural effect emerges through the mechanisms of stress
and its relaxation by employees. As parameters modifying stress change, the
24
importance of hubs in social networks of organization are affected significantly.
Our simulations reveal how specific stress-related conditions may cause the hubs
to have a less significant role than is traditionally perceived.
A simulation framework is now available to allow for further experimentation
and extension. The generic model that we are making publicly
available was developed with the aim of simulating the aforementioned contextual
perspectives. Agent-based modeling used in the model is inherently associated
with simplification and abstraction of complex reality. The presented
model simplifies human behavior to focus on the global, emergent, organizational
level. Moreover, the performed experiments aim to analyze how strategic
decision-making related to human resources may impact both organizational
performance (the macro level), and the quality of employees' lives (the micro
level). We are aware that the inherent simplification and abstraction related
to agent-based modeling complicate the validation of every included parameter.
However, the model was continuously tested and validated during the development
to be robust, reliable, and to produce reasonable results.
7 Conclusions and Future Research
The presented paper contributes to the tradition of research aimed at modeling
and simulations of the Prisoner's Dilemma Game. Over the decades, many
extensions of the original concept of the game have been introduced, illustrating
more detailed aspects of cooperation emergence. While some of the studies
focused on discovering new mechanisms that help to promote cooperation, others
researched the effects of interaction topology, i.e., how the structure of the social
network between agents influences the outcome of the game.
The model described in Section 3 of this paper addresses both of the approaches
mentioned above. It is a multi-parameter model developed in the NetLogo
environment for the modeling and simulation of complex systems. The
model extends the P D G by setting it in an organizational context where agents
represent employees who interact to create added value for the company. The
model also presents novel concepts that influence the outcome of the simulations.
Specifically, the key novel concept is stress, which is functionally connected
with other mechanisms implemented in the model - i.e. the interaction
between agents, motivation level, or illness. Through their joint effects, stress
has a significant impact on overall organizational performance, as presented and
discussed in Section 5.
The goal of this paper is to investigate whether the specific behavior of a
manager may have a significant influence on organizational performance. To
obtain a network with a few highly central nodes and many peripheral nodes,
a Barabasi-Albert model of the scale-free network was used. A set of simulations
was designed to allow manipulation with the cooperative behavior of hubs
while controlling for all other parameters. Simulations were performed on static
networks, where agents do not change their strategy or social ties.
The experiments aimed at the verification of the stability and the expected
25
behavior of our model in defined scenarios confirmed its correctness and usefulness
for complex simulations. The subsequent experiments evaluating the
impact of stress and cooperative behavior of the key employees on the organizational
performance provided results based on which several managerial implications
were formulated. We believe that further experiment with the presented
model might yield interesting outcomes in the future.
As for the future direction of research, several possible extensions to the
existing model are being considered. Currently, the model performs the simulations
with a fixed number of agents for the whole experiment. To bring the
model closer to reality, an algorithm that would adapt the number of employees
depending on the actual economic situation of the organization could be implemented.
When employees are 'hired', their P D G strategy could be either:
random, the most successful strategy, or the strategy of the nearest hub. Additionally,
the network formation rules may be updated such that the newcomers
would link to existing employees based on their similarity (e.g., to employees
with the same P D G strategy) - in existing literature, this process is called
assortativity.
When people are under pressure, they might inaccurately evaluate other
people's intentions or behavior. Therefore, another concept considered for implementation
is the introduction of noise whose level will be proportional to the
actual stress of an employee. The level of noise would represent the probability
that an act of co-worker's cooperative behavior will be perceived as defection.
Finally, the extension of an algorithm that rewards employees for their aboveaverage
performance is currently being implemented. The goal of this functionality
is to find a suitable approach that would balance employees' motivation
and righteousness of rewarding, beneficial for the organization as a whole.
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