[2]Stephan Leitner

1]Digital Age Research Centre, University of Klagenfurt, Universitätsstraße
65-67, Klagenfurt, 9020, Austria

2]Department of Management Control and Strategic Management, University of
Klagenfurt, Universitätsstraße 65-67, Klagenfurt, 9020, Austria

# Interactions between the individual and the group level in organizations

The case of learning and autonomous group adaptation

Dario Blanco-Fernandez<EMAIL_ADDRESS><EMAIL_ADDRESS>Alexandra
Rausch<EMAIL_ADDRESS>[  [

###### Abstract

Previous research on organizations often focuses on either the individual,
team, or organizational level. There is a lack of multidimensional research on
emergent phenomena and interactions between the mechanisms at different
levels. This paper takes a multifaceted perspective on individual learning and
autonomous group formation and adaptation. To analyze interactions between the
two levels, we introduce an agent-based model that captures an organization
with a population of heterogeneous agents who learn and are limited in their
rationality. To solve a task, agents form a group that can be adapted from
time to time. We explore organizations that promote learning and group
adaptation either simultaneously or sequentially and analyze the interactions
between the activities and the effects on performance. We observe
underproportional interactions when tasks are interdependent and show that
pushing learning and group adaptation too far might backfire and decrease
performance significantly.

###### keywords:

Agent-based modeling and simulation, Dynamic capabilities, Multi-level
research, Group composition, Interaction effect, NK framework

##  1 Introduction

Learning in organizational contexts and issues related to dynamic capabilities
are usually researched at different (and often isolated) levels. First, at the
level of the individual, research mainly addresses the enablers of learning
and adaptation, contingencies, and the effects of learning and adaptation for
the individual [60, 85]. Second, at the organizational level, research is
often concerned with developing and maintaining learning systems,
institutionalizing learning in terms of embedding it into the processes,
structures and strategies effective in the organization, promoting learning,
and issues related to how organizational capabilities emerge [42, 112]. Third,
at the intermediate level, the importance of groups to link the individual and
the organizational level is usually emphasized. It is addressed how
individually learned information can be integrated, shared and adjusted [85].
Research on learning at the intermediate level is, for example, concerned with
how to compose groups ideally, how to use individual knowledge optimally, and
how to take advantage of synergies [18, 75]. However, since the capabilities
at the individual level might be dynamic because of learning, the composition
of groups at the intermediate level may have to be dynamic too. Thus, the
layers are apparently interrelated. Previous studies often take an
unidimensional perspective and focus on one level only. This lack of
integration across levels reflects the micro-macro divide that is predominant,
for example, in the field of managerial science [2, 84]. The focus on only one
of the interrelated levels can, to some extent, be explained by disciplinary
borders [10]. Research in psychology, for example, tends to focus on issues
related to learning at the individual level. In contrast, research on
organizational design is likely to be more interested in matters concerning
the collective level. To overcome the gap between micro- and macro-level
research, [55] recommend (i) applying a multi-level design to existing models,
(ii) considering the consequences of micro-level activities for macro-level
performance, (iii) pushing disciplinary boundaries, and (iv) addressing
problems of practical relevance.

Our research follows the suggestions provided by Hitt et al [55]. We apply a
multi-level approach that connects the individual and group level. In
particular, we investigate whether individual learning and adaptive group
composition mutually reinforce or attenuate each other concerning their
effects on performance. We aim at answering the following research questions:
(i) How do individual learning and group adaptation interact and affect task
performance if they are promoted simultaneously? (ii) If individual learning
and group adaptation are promoted sequentially, what are the effects on
performance in the sequential stages? (iii) Are there any moderating effects
of task decomposability, i.e., the fact that the tasks assigned to different
agents are interdependent or not?

To answer these questions, we propose an agent-based model with a population
of heterogeneous agents who are limited in their rationality [104]. In our
model, the limitations imposed on agents are two-fold: First, the agents’
abilities to solve tasks are restricted to a particular area of expertise. In
an organizational context, we consider agents who are experts in fields such
as accounting, marketing, or production management. Second, agents have
limited information within their area of expertise and limited cognitive
capabilities. Consequently, they cannot oversee the entire room of actions
immediately but explore it only sequentially. This means they learn new ways
to perform their tasks over time [70]. The population of agents autonomously
forms a group following an auction-based mechanism, and agents can re-organize
their groups from time to time. We base our model on the well-known
$N\\!K$-framework that allows placing the agents in task environments of
different complexity [73, 117]. We deem our multi-level approach to analyze
interactions between individual learning and group adaptation a relevant
contribution to bridging the gap between the micro- and macro-level in
managerial science. Our results are also of relevance for decision making in
organizations since such multi-level approaches appear to be particularly
interesting for corporate practice [2].

The remainder of this paper is structured as follows: We discuss the related
literature on dynamic capabilities and multi-level considerations, individual
learning and bounded rationality, and group formation and adaptation in Sec.
2. The simulation model is introduced in Sec. 3. The results of the simulation
experiments are presented in Sec. 4 and discussed in Sec. 5. Finally, Sec. 6
concludes the paper.

##  2 Related research

###  2.1 Dynamic capabilities and multi-level considerations

An organization’s dynamic capabilities are usually regarded as some sort of
higher-order capability that affects how tasks are solved [120]. Dynamic
capabilities are manifold, making it particularly difficult to find a coherent
and tangible definition [107]. We follow the conceptualization provided in
[123], according to which dynamic capabilities are the stable and learned
patterns of activity within an organization, through which an organization
modifies its routines. In the dynamic-capabilities framework, the learning of
the routines needs to take place at both the individual and organizational
level and is driven by a steady demand for adaptation to the task environment.
Naturally, having learned the appropriate patterns of activity substantially
increases task performance [38, 112]. This means that individuals and
organizations need to reconfigure their capabilities (e.g., knowledge, skills,
abilities) to meet the requirements for solving the tasks they face
efficiently and successfully. Bendig et al [19] highlight that there are two
parallel developments in this research context: The first stream of research
focuses on the micro-foundations of how capabilities evolve and aims to
understand how learning by individual employees aggregates to an
organization’s capability [see, e.g., also 1]. The second stream of research
exclusively focuses on the macro-level and analyzes how managerial decisions
at the top level affect an organization’s performance [52]. While both streams
substantially advanced the knowledge on dynamic capabilities, the interaction
between the two levels is still under-researched [19, 40].

Research related to the emergence of dynamic capabilities is tightly connected
to the literature on learning in organizational contexts. The latter
distinguishes between three levels of learning, i.e., (i) individual learning,
(ii) team learning, and (iii) organizational learning [see, e.g., 37, 63, 85,
102]. By doing so, the literature on learning adds an intermediate layer to
the two levels already considered in the context of dynamic capabilities,
i.e., a group level between the individual and the organizational level.
Murray and Moses [85] emphasize the role of the intermediate layer and
particularly underline the role of learning at the level of teams as a link
between individual and organizational learning. Also, it is widely agreed in
previous research that individual learning is the foundation for learning at
higher levels and that the learning outcomes at the collective level are more
than the accumulation of individual learning [25, 33, 44, 51, 82, 92].111For
reviews of the literature on learning in organizational contexts, the reader
is referred to [9], [15], [35], and [88], amongst others. Reviews related to
the dynamic capabilities framework are, for example, provided by Barreto [11],
Schilke et al [101], and Wang and Ahmed [118]. Within this three-layer
framework, our focus follows an argument put forward by Simon [105]. He claims
that learning in organizations may occur in two ways: First, the individual
members of an organization may learn. Second, the organization may ingest new
members who have new knowledge that was not available to the organization
before. In line with this argumentation, we focus on understanding the effects
of a variation in two organizational design parameters, namely (i) individual
learning and (ii) group adaptation in the sense of changing a group’s
composition from time to time. For this purpose, we consider the probability
of learning at the individual level and groups of different lifetimes. Our
research analyzes how efficiently agents use the information they have
previously learned and how promoting learning and group adaptation affects the
performance of organizations.

###  2.2 Individual learning and bounded rationality

In the literature on learning in organizations, there are two main streams of
research. The first stream focuses on the enablers of learning, the second on
the results of learning in organizations [60]. Research focusing on the
enablers of learning explores and discusses ways and means to promote
individual learning. Such ways and means comprise, for example, the methods of
mentoring [67], learning by doing and/or exploration [7, 22], fostering a
creative working environment [6, 89], employee training [99, 113], and the
design of information flows [27]. In this paper, we are not concerned with the
efficiency of the different ways and means to promote learning but rather
consider them as given. We locate our research in the second stream. Hence, we
particularly emphasize the effects of promoting individual learning on
performance, its interplay with promoting group adaptation, and the results
for an organization. Thereby, we contribute to closing the gap between
(changes in) micro-level behavior, i.e., the behavior of individual agents,
and the organization’s macro-level outcome [2].

Following the concept of dynamic capabilities, individual learning allows for
adapting to the requirements posed on the organization by dynamic
environments, which, in turn, is key to organizational competitiveness and
survival [112]. As soon as we break down the problem of adaptation to smaller
entities within the organization – such as groups or individual decision
makers – and consider decentralized decision-making authority, a similar but
extended argumentation applies: From the perspective of the individual, it is
not just important to adapt to the organizational environment by learning, but
the individual agent is also well-advised to adapt to their individual
environment when making decisions. This individual environment captures, for
example, the decisions made by fellow agents within a group. This is
particularly relevant if the tasks assigned to different decision makers are
interdependent [96].

Learning at the individual level is affected by characteristics such as
cognitive capability, learning styles, and interpretative ability [30, 85,
86]. Research often considers limitations in these characteristics and
addresses, amongst others, decision makers who over-weight information in
favor of prior beliefs [32], slant information towards a preferred state [66],
and over- or under-react to information [72]. More generally speaking,
research often assumes that individual agents suffer from the limitations of
bounded rationality in the sense of Simon [104], to which Hendry [53] refers
as ‘incompetence’. In particular, he argues that humans might have limited
knowledge and foresight and face limitations of rational understanding and
communication that may arise from language, culture, and cognition [see also
79]. Of course, this directly translates into consequences for individual
learning. First, suppose humans suffer from limited foresight. In this case,
they might have problems in correctly predicting the outcomes of their future
actions, or they might not be able to form beliefs about the actions of others
[39]. Second, if the cognitive capabilities are limited, decision makers might
not be able to understand and oversee their entire room for actions. In
practical terms, one might not be aware of all feasible ways to carry out a
specific task at a time but rather sequentially explore this room [70]. Third,
the extent to which learning is successful might be affected, amongst others,
by the individuals’ cultural backgrounds [62] or their technical literacy
[94].

There is a long tradition of studying (individual) learning in organizational
contexts. [78], for example, relates decision making to a process by which
agents balance learning new solutions to a particular task (i.e., exploration)
and building on the solutions they already know (i.e., exploitation). By
choosing a specific mix of both strategies, agents adapt more or less
successfully to the organizational and individual environments. Previous
research suggests that an appropriate balance of exploration and exploitation
is the key to improving task performance [73, 78, 95]. In particular, it has
been found that exploration is important for increasing task performance in
the early stages of task-solving, but exploitation becomes more relevant in
later periods [71, 73, 121].

###  2.3 Group formation and adaptation

In Sec. 2.1 we addressed the intermediate level of the group that plays a
pivotal role in organizational learning since it links the individual and the
organizational level. One fundamental question in this context is how groups
should be ideally composed [54]. Mello and Ruckes [81] argue that
heterogeneous groups perform better than homogeneous groups since
heterogeneity assures access to various information. The argument that
heterogeneity might increase the performance is in line with the findings
presented in [75], who analyze the power of heterogeneous agents joining
forces in a group to solve tasks over a large solution space. They find, for
example, that larger groups can solve problems that individual agents cannot
solve alone. They also claim that positively correlated abilities of agents
require larger groups to solve tasks efficiently. However, they also show that
teaming up does not necessarily guarantee success in all cases. Further
empirical evidence corroborates this finding by relating the poor performance
of heterogeneous groups to differences in the agents’ abilities and
characteristics and to heterogeneous preferences, which cause different
behaviors and objectives [5, 21].

The characteristics that make a group heterogeneous are manifold. Previous
research argues that heterogeneity might result from differences in the social
background, age, gender, education, national culture, and professional
development, amongst others [17, 18, 56, 57, 81]. Krech et al [65] provide a
systematic analysis of the variables that might affect the performance of
groups and clusters them into four categories: (i) structural variables
(characters, talent, size, etc.), (ii) situative environmental variables
(e.g., functional position), (iii) task-related variables (type of task,
restrictions in, e.g., time, etc.), and (iv) intervening variables (personal
relations, level of interaction, etc.). Following the categorization provided
in [65], we focus on the role of the (i) structural variables in group
heterogeneity. Also in line with the arguments brought forward in [81], we
consider the professional background by explicitly modeling agents that are
experts in specific fields, such as accounting, marketing, or production
management. We do not take into account the (ii) situative and (iv)
intervening variables. However, our research actively controls for the (iii)
task-related variables by modeling tasks of different degrees of
decomposability and complexity.

We follow the argument raised by Simon [105] and use the intermediate group
level to endow the group with knowledge that was not available earlier. We do
so by ensuring that only those agents join forces in a group who are best
prepared for the task they face. Groups can be formed either by a top-down or
a bottom-up approach. The top-down approach corresponds to the idea of
classical organizational design and considers that managers conceptually
develop a group’s composition before implementation [98, 106]. The bottom-up
approach, on the contrary, follows an evolutionary perspective and regards a
group’s composition as an emergent property [114]. This corresponds to the
ideas of plastic control [91] and guided self-organization [93]. In our paper,
we follow the bottom-up approach to group formation and implement a
corresponding mechanism based on a second-price auction. The design of this
mechanism is inspired by previous research in the fields of robotics and
transportation research, which also employs auction-based mechanisms for
bottom-up task allocation and collaboration [31, 87, 97]. Since we consider
agents with dynamic capabilities, a group composition yielding the best
possible results at one particular point in time is not necessarily the
optimal composition at another point in time. This argumentation is in line
with research on the interface between dynamic capabilities and temporary
organizations. Spanuth et al [107], for example, claim that temporary
structures enhance an organization’s innovative capacity and strategic
flexibility, finally resulting in better performance. We account for this
relationship by controlling for a group’s lifetime and analyzing the effects
of different lifetimes on performance. Examples of autonomously formed
(temporal) groups can, amongst others, be found in the context of agricultural
cooperatives [50], consulting firms [28], and professional services
partnerships [45].

##  3 The Model

We set up an agent-based model of an organization formed by a population of
$P=30$ agents. This organization aims at solving a particular task. We endow
the agents with (heterogeneous) capabilities related to specific areas of
expertise.222We have implemented the agent-based model in Python 3.7.4. The
agents are limited in their cognitive capacities. For example, they might have
limited cognitive resources and they suffer from restrictions in information
processing. As a consequence, they cannot handle the task alone but have to
collaborate with other agents. That is why a subset of the agent population
autonomously forms a group of $M\in\mathbb{N}$ members who jointly solve the
task. The group-formation mechanism follows the idea of a second-price
auction. Depending on the studied scenario, we allow for individual learning
(see Sec. 3.2) and changing the group’s composition from time to time (see
Sec. 3.3). We run simulations and observe the agents’ behavior and the task
performance achieved by the group over $t=\\{1,\dots,T\\}\in\mathbb{N}$
periods. In the following subsections, we introduce the model’s three main
building blocks:  (i) the task environment in Sec. 3.1,  (ii) agents and
individual decision making in Sec. 3.2, and  (iii) the group-formation
mechanism in Sec. 3.3.  Section 3.4 discusses the key parameters and the
sequencing during simulation runs. Finally, in Sec. 3.5, we introduce the
performance measures.

###  3.1 Task environment

#### Task and performance contributions

We represent the task environment by a performance landscape based on the
$N\\!K$ framework with an $N$-dimensional decision-making task and $K$
interdependencies among decisions. We denote the task by the binary vector

$\mathbf{d}=\left(d_{1},\dots,d_{N}\right)~{},$ (1)

consisting of $N$ individual decisions $d_{n}$, where
$n=\\{1,\dots,N\\}\in\mathbb{N}$ and $d_{n}\in\left[0,1\right]$. For this
paper, we set $N=12$. Thus, there are $2^{12}$ feasible solutions to
$\mathbf{d}$ that all take the form of a 12-digit binary string.333For
readability, we suppress the subscript $t$ in Sec. 3.1.

There are $K\in\mathbb{N}_{0}$ interdependencies among the individual
decisions that determine task complexity and decomposability (i.e., the extent
to which tasks are decomposable). Higher values of $K$ stand for more
interdependencies and, thus, higher task complexity. In our framework, the
interdependencies indicate that the outcome of a single task is not only
affected by the decision associated with this task but also by $K$ other
decisions [73]. The decomposability is affected by the patterns of
interdependencies. We denote the performance contribution of an individual
decision $d_{n}$ by the pay-off function

$c_{n}=f(d_{n},d_{i_{1}},\dots,d_{i_{K}})~{},$ (2)

where $\\{i_{1},\dots,i_{K}\\}\subseteq\\{1,\dots,n-1,n+1,\dots,N\\}$.
Following the $N\\!K$ framework, we draw the performance contributions of
individual decisions from a uniform distribution so that $c_{n}\sim U(0,1)$
[119].

In line with the $N\\!K$ framework, we compute the overall task performance as
the mean of all individual performance contributions, such that

$C\left(\mathbf{d}\right)=\frac{1}{\|\mathbf{d}\|}\sum^{\|\mathbf{d}\|}_{n=1}c_{n}~{},$
(3)

where $\|\mathbf{d}\|$ indicates the length of vector $\mathbf{d}$.

#### Subtasks

Recall that the population of agents forms a group of $M$ members who jointly
perform the task. We symmetrically divide the $N$-dimensional task into $M$
subtasks of size $S=N/M$, and we sequentially assign agents a
subtask.444Please note that we allocate tasks to agents following a bijective
function. Thus, within the group, all agents $m=\\{1,\dots,M\\}\in\mathbb{N}$
are assigned their subtasks, such that

$\mathbf{d}_{m}=\left(d_{S\cdot(m-1)+1},\dots,d_{S\cdot m}\right)~{}.$ (4)

With a task consisting of $N=12$ decisions and a group of $M=3$ agents, agent
$1$ is responsible for individual decisions $1$ to $4$, agent $2$ is in charge
of decisions $5$ to $8$, and decisions $9$ to $12$ are assigned to agent $3$
(see Fig. 2). The subtasks can be related to specific areas of expertise that
reflect the need for different skills required to solve tasks [59]. During the
phase of model initialization, we randomly place every agent in the population
in one area of expertise. Agent $1$ might, for example, be an expert in
accounting, while agents $2$ and $3$ could be experts in, e.g., marketing and
production management, respectively. Once placed in an area of expertise, the
agents’ respective capabilities are limited to performing tasks that are
within this area [46, 59, 95].

#### Task decomposability

Since our aim is to understand the moderating role of task decomposability, we
explicitly control for the patterns of interdependencies. We consider the
following two cases:

  * •

In the case of a decomposable task, every decision affects the contributions
of three other decisions ($K=3$) of one subtask. Figure 1(A) (Decomposed
tasks) illustrates this case. The solid lines and ‘x’ indicate the subtasks
and interdependencies, respectively. There is complete interdependence within
subtasks but no cross-interdependencies with other agents’ subtasks. Thus, one
agent’s individual decisions do not affect the performance contributions
associated with decisions assigned to other agents.

  * •

In the case of an interdependent task, each decision affects the contributions
of five other decisions ($K=5$). Interdependent tasks are characterized by
cross-interdependencies between the agents’ respective subtasks. This
structure implies that an agent’s decisions also affect the performance
contributions associated with decisions assigned to other agents (see Fig.
1(B), Interdependent tasks).

Figure 1: Interdependence matrices.

#### Performance landscape

The number of decisions $N$, the task complexity $K$, and the performance
contributions jointly determine a particular performance landscape. We use the
pay-off function introduced in function Eq. 2 to map the $2^{N}$ feasible
solutions to a performance landscape. If there are no interdependencies
between individual decisions ($K=0$), the resulting performance landscape has
a single peak. Increasing $K$ to its maximum of $K=N-1$, results in maximally
complex performance landscapes since altering one single decision would affect
the outcomes associated with all tasks. Consequently, if $K$ increases, the
performance landscape gets more rugged with numerous local maxima in the
extreme case [4, 96]. By making decisions, the group moves in the performance
landscape and follows the objective to increase its performance step-wise. The
following section introduces the agents’ characteristics and their decision
rules.

###  3.2 Agents and individual decision making

#### Agents’ characteristics

Recall that we model a population of $P=30$ heterogeneous agents [104].
Bounded rationality imposes limits on them in two respects. First, agents are
limited in their cognitive capacity and can only perceive a subtask but not
the entire decision problem. In consequence, agents are experts in one area
such as accounting, marketing, or production management. Also, agents do not
monitor the history of solutions to the decision problem, which results in
myopic agents who only optimize their immediate utility and refrain from
making forecasts based on the history of solutions [8, 70]. Second, the
limitation in the cognitive capacity also affects the agents’ search behavior
and memory. Our model considers agents who, when making their decisions,
cannot oversee the entire solution space but only possess the cognitive
capabilities to evaluate some solutions at a time. We endow agents with the
ability to explore feasible solutions to their subtasks sequentially and
forget already explored solutions because of their limited memory.

#### Individual learning

To overcome the limitations at the level of the performance components, agents
explore the solution space over time, i.e., they learn.555Please note that
learning could take place at multiple levels [61]. We exclusively focus on the
individual agent. Recall, Eq. 4 denotes agent $m$’s subtasks by
$\mathbf{d}_{m}$. Since we model binary decisions, the set of feasible
solutions to subtask $\mathbf{d}_{m}$ includes $2^{\|\mathbf{d}_{m}\|}$
solutions, which we refer to as solution space. Above, we argue that agents
cannot oversee the entire solution space at a time. Let us refer to the
solutions that agent $m$ is aware of in period $t$ by

${\mathbf{S}}_{mt}=\left(\hat{\mathbf{d}}_{m1},\dots,\hat{\mathbf{d}}_{mI}\right)~{},$
(5)

where $\hat{\mathbf{d}}_{mi}$ are bitstrings that represent feasible solutions
to subtask $\mathbf{d}_{m}$, $i=\\{1,\dots,I\\}\in\mathbb{N}$ and $1\leq I\leq
2^{\|\mathbf{d}_{m}\|}$. If agent $m$ is, for example, an expert in production
management, ${\mathbf{S}}_{mt}$ represents all possible ways to organize the
production process she is aware of in period $t$. Consequently, the symbol $I$
is a proxy for agent $m$’s cognitive capacity. The higher (lower) $I$, the
more (fewer) solutions the agent knows to the subtask. The known solutions are
dynamic. At every period, agents might learn new solutions that differ in one
bit from any of their already known solutions, or they forget solutions that
are not utility-maximizing in the current period. Learning and forgetting are
independent events that occur with a fixed probability $\mathbb{P}$. We
consider the following three scenarios:

  * •

In the case of zero individual learning, agents neither learn nor forget any
solution they already know, so we set $\mathbb{P}=0$.

  * •

In the case of moderate individual learning, we set $\mathbb{P}=0.25$.

  * •

In the case of high individual learning, we set $\mathbb{P}=0.5$.

Initially, agents are aware of one solution to their subtask. Depending on the
value of $\mathbb{P}$, the agents’ known solution spaces are more or less
dynamic. Increases in the value of $\mathbb{P}$ could indicate that
organizations support employees in learning, e.g., they provide learning
resources, training, or incentives for the creation of new products,
procedures, and methods [28]. We refer to the actions taken by the
organization to increase the probability of learning (i.e., to increase
$\mathbb{P}$) as promoting individual learning.

#### Individual decision-making rule

In every period, every agent $m$ who is part of the group can decide which
solution in her solution space ${\mathbf{S}}_{mt}$ she wants to implement. We
denote the solution to agent $m$’s subtask implemented in period $t$ by
$\mathbf{d}_{mt}$. We formalize the corresponding decision rule in Eq. 8.

Agent $m$’s utility in period $t$ results from the performance contributions
generated by the implemented solution to her subtask (i.e., her own
performance) and the performance generated by the solutions implemented by the
remaining agents $r=\\{1,\dots,M\\}\in\mathbb{N}$, where $r\neq m$. We refer
to the latter as residual performance and denote the other agents’ solutions
by

$\mathbf{D}_{mt}=\left(\mathbf{d}_{1t},\dots,\mathbf{d}_{\\{m-1\\}t},\\\
\mathbf{d}_{\\{m+1\\}t},\dots,\mathbf{d}_{Mt}\right)~{}.$ (6)

Then, agent $m$’s utility follows the linear function

$U\left(\mathbf{d}_{mt},\mathbf{D}_{mt}\right)=\alpha\cdot
C(\mathbf{d}_{mt})+\beta\cdot\frac{1}{M-1}\sum_{\begin{subarray}{c}{r=1}\\\
{r\neq m}\end{subarray}}^{M}C(\mathbf{d}_{rt})~{}.$ (7)

We compute the performance in line with Eq. 3. The agent’s utility is affected
by a linear incentive scheme that is parameterized by $\alpha\in\mathbb{R}$
and $\beta\in\mathbb{R}$ to weight agent $m$’s own and residual performances,
respectively, and $\alpha+\beta=1$.

Every agent’s objective is to maximize their utility, which is only possible
by participating in the group.666Agents who are no group members receive zero
utility, and we omit outside options. We omit the coordination of decisions
between agents and allow them to act autonomously. Consequently, agent $m$ is
not aware of the solutions that the other $r\neq m$ group members intend to
implement in a period. However, agents can observe the solutions implemented
in the previous period, $\mathbf{D}_{m\\{t-1\\}}$. Agents use this information
and base their decisions in $t$ on the estimated utility, for which it is
assumed that the residual decisions do not change from the previous period.
Consequently, agent $m$’s decision rule takes the form of

$\mathbf{d}_{mt}:=\operatorname*{arg\,max}_{d^{\prime}\in\mathbf{S}_{mt}}~{}U\left(d^{\prime},\mathbf{D}_{m\\{t-1\\}}\right)~{}.$
(8)

The function ‘arg max’ returns the argument that maximizes the utility
function.

#### Group solution

Once all agents have made their decisions, we compute the entire solution to
the task in period $t$ by

$\mathbf{d}_{t}:=\mathbf{d}_{1t}{}^{\frown}\dots{}^{\frown}\mathbf{d}_{Mt}~{},$
(9)

where ⌢ indicates the concatenation of the solutions individually implemented
by the agents. Once the entire solution $\mathbf{d}_{t}$ is implemented, all
agents in the population can observe it.

###  3.3 Group-formation mechanism

The model considers that the population of agents autonomously forms one group
consisting of $M$ agents. In a group, there is one agent per area of
expertise. Thus, if the task at hand requires experts from accounting,
marketing, and production management, a group-formation mechanism makes sure
that one experts per area joins the group.

Recall, we consider an organization formed by $P=30$ agents, and we split the
overall task into $M$ subtasks. We symmetrically allocate the agent population
to areas of expertise. $P_{m}=P/M$ potential candidates could solve a
particular subtask and, consequently, be a group member.777Please note that
subscript $m=1,\dots,M$, thus, indicates (i) the group members assigned to a
subtask, (ii) the subtask, and (iii) the subsets of the population of agents
that are capable of solving the subtasks. Thus, with a population of $30$
agents and three subtasks, there would be ten experts that could solve a
particular subtask. Let us denote the agents who possess the capabilities to
solve the subtask $\mathbf{d}_{m}$ by $p_{m}^{j}$, where
$j=\\{1,\dots,P_{m}\\}\in\mathbb{N}$. The challenge is to identify those
agents in the areas of expertise who are best prepared to solve the task in a
group. To do so, agents employ a mechanism that follows the concept of a
second-price auction (see also Fig. 2). Every time an auction is held, the
agents use the following procedure to form a group [97]:

  1. 1.

Agents are informed about the auction, and they can place bids to join the
group. Since agents can only experience utility by joining the group, every
agent has the incentive to participate in the auction. Auctions are anonymous,
and agents have no information about the other agents’ bids. The bids are
independent, and the ‘price’ of joining the group is determined by the bids
only.

  2. 2.

Agents compute their bids by drawing on the information available to them in
the following way: In line with Eq. 5, let us denote agent $p_{m}^{j}$’s known
solution space in period $t$ by $\mathbf{S}_{mt}^{j}$. Then, agents compute
their bid, which is the maximum estimated utility they can attain given the
known solution space in the current period, according to

$b_{mt}^{j}=\max_{d^{\prime}\in\mathbf{S}_{{m}t}^{j}}U\left(d^{\prime},\mathbf{D}_{m\\{t-1\\}}\right)~{}.$
(10)

In line with Eq. 6, $\mathbf{D}_{m\\{t-1\\}}$ indicates the solutions
implemented outside of agent $p_{m}^{j}$’s subtask in the last period, which
agents can infer from $\mathbf{d}_{t-1}$.888Since agents are myopic, they do
not take future pay-offs into account and place their bids only on their
immediate utility.

  3. 3.

Since every agent has the incentive to participate in the auction, $P_{m}$
bids are submitted by the agents capable of solving the associated subtask for
each slot in the group.

  4. 4.

The agent who submitted the highest bid for task $\mathbf{d}_{m}$ wins the
auction, joins the group at slot $m$, and gets charged the second-highest
bid.999Auctions in which the top bidder pays the second-highest price are
optimal in revealing the bidders’ true preferences when the agents’
information about other agents’ bids is restricted or non-existent [116]. The
prices that agents get charged for joining the group are transferred to the
organization. Consequently, the group in that particular period is composed of
$M$ agents (one per subtask) who know the solutions that lead to the best
(estimated) performance.

Figure 2: Group formation process within the organization

Figure 2 summarizes the group-formation process. By recurrently holding such
auctions, the group adapts its composition so that it best fits the task.
Throughout $T$ periods, auctions occur every $\tau$ time steps. Lower (higher)
values of $\tau$ indicate a larger (smaller) interval between two auctions and
can be interpreted in terms of a long-term (short-term) group composition. An
auction always occurs in the first time step, irrespective of the value of
$\tau$. We consider three different scenarios:

  * •

In the case of long-term group composition, the group is formed once in the
first period. Only one auction is held. For practical purposes, we refer to
this case as $\tau=\emptyset$.

  * •

In the case of medium-term group composition, we set auctions taking place
every $\tau=10$ time steps.

  * •

To model a short-term group composition, we set $\tau=1$, so that auctions
occur at every time step.

From an organization’s perspective, $\tau$ is a design parameter since by
changing the value of $\tau$, organizations can control the lifetime of
groups. For example, if $\tau$ is very low (high), an organization gives a
group the opportunity of autonomously changing its composition in short (long)
intervals. We refer to the actions taken by the organization to reduce the
time between auctions (i.e., to decrease $\tau$) as promoting group
adaptation.

###  3.4 Scheduling and parameters for simulation experiments

We summarize the sequence of events during a simulation run in Fig. 3. In the
previous sections, we have introduced the following independent variables
considered in the model:  (i) The probability of individual learning
$\mathbb{P}$,  (ii) the number of auctions $\tau$ during the observation
period, and  (iii) the task complexity $K$.

Figure 3: Sequence of events during simulation runs.

Since organizations can control the learning probability $\mathbb{P}$ and the
number of auctions $\tau$ in a real-world setting, we regard those two
variables as design parameters. The task complexity $K$ and the structure of
interdependencies, on the contrary, are usually given and cannot be controlled
by an organization. All relevant parameters included in the model are
summarized in Tab. 1. The parameter settings result in a total number of
$3\cdot 2\cdot 3=18$ different scenarios. We set the observation period to
$T=200$ and fix the number of subtasks included in the task to $M=3$. We
perform $R=1,500$ simulations for every scenario and observe the performance
at the group level as the dependent variable.101010The number of simulations
was fixed after analyzing the results’ variance, as [76] suggested The
computation of the group’s solution and the corresponding performance is
formalized in Eqs. 9 and 3, respectively.

Table 1: Parameter setting Type | Variables | Notation | Values
---|---|---|---
Independent variables | Interval between auctions | $\tau$ |  {$\emptyset$, 1, 10}
Task complexity | $K$ | {3, 5}
Learning probability | $\mathbb{P}$ | {0, 0.25, 0.5}
Dependent variable | Group performance | $\bar{C}_{t}$ | $\left[0,1\right]$
Other parameters | Time steps | $t$ | $\\{1,...,200\\}$
Observation period | $T$ | 200
Number of decisions | $N$ | 12
Population of agents | $P$ | 30
Number of subtasks | $M$ | 3
Incentive parameters | $\alpha,\beta$ | 0.5
Number of simulations | $R$ | 1,500

###  3.5 Performance measures

#### Normalized performance

To assure that the results are comparable across all simulation runs, we
normalize the performance of the group solution by the maximum achievable
performance in every landscape. We compute the average normalized performance
in period $t$ according to

$\bar{C}_{t}=\frac{1}{R}\sum_{r=1}^{R}\frac{C\left(\mathbf{d}_{tr}\right)}{C^{\ast}_{r}}~{},$
(11)

where $C^{\ast}_{r}$ indicates the maximum performance in simulation run $r$,
$\mathbf{d}_{tr}$ stands for the group solution implemented in simulation run
$r$ and period $t$. The performance $C\left(\mathbf{d}_{tr}\right)$ is
computed according to Eq. 3.

#### Mean performance

To give a condensed performance measure, we also report the mean performance
over the entire observation period,

$\bar{\bar{C}}=\frac{1}{T}\sum_{t=1}^{T}\bar{C}_{t}~{}.$ (12)

Please note that the mean performance includes information about the
attainable performance and the speed at which this performance level is
reached in a specific scenario. Thus, the mean performance is also a proxy for
the convergence speed.

#### Interaction effect

In addition to the two measures of performance introduced above, we aim to
analyze the interaction between individual learning and group adaptation when
they are promoted simultaneously. To do so, we compute the interaction
coefficient based on the final and the mean performance. The analysis starts
at a baseline scenario in which neither learning nor group adaptation is
promoted. We refer to the parameter set for this case by
$\delta_{0}=\\{\mathbb{P}=0,\tau=\emptyset\\}$. To analyze the isolated
effects of promoting the design parameters, we start from the baseline
scenario and promote either learning or group adaptation, so that the
parameters are $\delta_{\mathbb{P}}=\\{\mathbb{P}>0,\tau=\emptyset\\}$ or
$\delta_{\tau}=\\{\mathbb{P}=0,\tau>0\\}$, respectively. Finally, we analyze
the case when learning and group adaptation are promoted simultaneously with
the corresponding parameters
$\delta_{\mathbb{P}\tau}=\\{\mathbb{P}>0,\tau>0\\}$.111111For $\mathbb{P}>0$
and $\tau>0$, we consider the values listed in Tab. 1. We denote the
normalized performance in the final period $t=200$ given a specific parameter
setting by $\bar{C}^{\delta}$, where
$\delta\in\\{\delta_{0},\delta_{\mathbb{P}},\delta_{\tau},\delta_{\mathbb{P}\tau}\\}$
(see also Eqs. 11 and 12). We suppress the subscript $t=200$ for readability.

Next, we compute the differences in the achieved final performances. We denote
the difference between the final performance achieved in the baseline scenario
and the final performance if either individual learning or group adaptation is
promoted by
$\bar{\Delta}_{0}^{{\mathbb{P}}}=\bar{C}^{\delta_{\mathbb{P}}}-\bar{C}^{\delta_{0}}$
and $\bar{\Delta}_{0}^{{\tau}}=\bar{C}^{\delta_{\tau}}-\bar{C}^{\delta_{0}}$,
respectively. The increase in the final performance when individual learning
and group adaptation are promoted simultaneously is denoted by
$\bar{\Delta}_{0}^{{\mathbb{P}\tau}}=\bar{C}^{\delta_{\mathbb{P}\tau}}-\bar{C}^{\delta_{0}}$.
Finally, we compute the interaction coefficient by

$I\\!E=\genfrac{}{}{}{}{\raisebox{3.0pt}{$\bar{\Delta}_{0}^{{\mathbb{P}\tau}}$}}{\raisebox{-3.0pt}{$\bar{\Delta}_{0}^{{\mathbb{P}}}+\bar{\Delta}_{0}^{{\tau}}$}}~{}.$
(13)

Consequently, an interaction coefficient $I\\!E>1$ and $I\\!E<1$ indicates
over- and underproportional interactions, respectively [68]. The effect of
simultaneously promoting individual learning and group adaptation on the mean
performance is computed correspondingly.

#### Offsetting effect

In addition to the interaction between individual learning and group
adaptation, we are interested in the potential offsetting effects when the
design parameters are changed sequentially. To do so, we analyze the relative
difference in the performance of promoting individual learning (group
adaptation) after group adaptation (individual learning) has been promoted.
When individual learning is promoted after group adaptation, we compute the
difference in the final performance by
$\bar{\Delta}_{\tau}^{{\mathbb{P}\tau}}=\bar{C}^{\delta_{\mathbb{P}\tau}}-\bar{C}^{\delta_{\tau}}$.
Consequently, the relative difference in the final performance follows

$O\\!E_{\mathbb{P}}=\genfrac{}{}{}{}{\raisebox{3.0pt}{$\bar{\Delta}_{\tau}^{{\mathbb{P}\tau}}$}}{\raisebox{-3.0pt}{$\bar{C}^{\delta_{\tau}}$}}~{}.$
(14)

For the case of promoting individual learning first and group adaptation
second, we compute the absolute differences in the final performance by
$\bar{\Delta}_{\mathbb{P}}^{{\mathbb{P}\tau}}=\bar{C}^{\delta_{\mathbb{P}\tau}}-\bar{C}^{\delta_{\mathbb{P}}}$
and the relative difference in the final performance by

$O\\!E_{\tau}=\genfrac{}{}{}{}{\raisebox{3.0pt}{$\bar{\Delta}_{\mathbb{P}}^{{\mathbb{P}\tau}}$}}{\raisebox{-3.0pt}{$\bar{C}^{\delta_{\mathbb{P}}}$}}$
(15)

The relative differences in the mean performances are computed
correspondingly.

##  4 Results

We analyze the effects of an adaptation at two levels - via individual
learning and changing group composition - on task performance and report the
final and mean performances for decomposed and interdependent tasks in Tab.
2.121212Please note that the mean performance is a condensed performance
measure that also includes information about the convergence speed. We
organize the results in three subsections: In Sec. 4.1, we analyze the
interaction effects between individual learning and group adaptation and study
how a simultaneous variation in both design parameters affects task
performance. In Secs. 4.2 and 4.3, we focus on a sequential promotion of
individual learning and group adaptation. In particular, Sec. 4.2 analyzes the
effects of promoting individual learning after group adaptation has been
promoted. Hence, a group’s lifetime is fixed before there is any promotion of
individual learning. We plot the effects of a subsequent variation in
individual learning on task performance in Figs. 5 and 6. Table 6 includes
information about whether or not subsequently promoting learning has
significant effects on the mean and final performances. In Sec. 4.3, we
explore the effects of promoting group adaptation after learning has been
promoted. Figures 7 and 8 and Tab. 7 show whether and how a variation in the
lifetime of a group after promoting learning affects task performance.

Our analysis also considers whether different configurations of the incentive
systems (i.e., $\alpha$ and $\beta$) and different structures of
interdependencies (i.e., Fig. 1) affect the results. In both cases, there are
slight differences in the overall performances achieved, but the findings
presented in Secs. 4.1 to 4.3 hold true for different configurations of the
incentive system. Also, when tasks are interdependent, we observe the same
patterns in the performances for different structures of interdependencies.
The details are provided in the Appendix.

Table 2: Mean and final performances

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero | Moderate | High | Zero | Moderate | High
Long-term | Mean | 0.8687 | 0.9750 | 0.9882 | 0.7524 | 0.9076 | 0.9190
Final | 0.8697 | 0.9994 | 1.0000 | 0.7529 | 0.9363 | 0.9341
Medium-term | Mean | 0.8723 | 0.9915 | 0.9934 | 0.7910 | 0.9227 | 0.9201
Final | 0.8734 | 1.0000 | 1.0000 | 0.7942 | 0.9411 | 0.9344
Short-term | Mean | 0.8692 | 0.9947 | 0.9963 | 0.7921 | 0.8864 | 0.8680
Final | 0.8702 | 1.0000 | 1.0000 | 0.7920 | 0.8933 | 0.8708

###  4.1 Interactions between promoting individual learning and group
adaptation

#### Interaction effects

First, this section analyzes the interaction effects if both individual
learning and group adaptation are simultaneously promoted. To do so, we
compute the interaction coefficient introduced in Eq. 13. This coefficient
gives information about whether promoting learning and group adaptation
reinforce or mitigate each other or interact linearly. A reinforcing
(mitigating) interaction effect means that the joint effect of simultaneously
promoting learning and group adaptation is larger (smaller) than the sum of
the isolated effects. Isolated effects refer to the mean and final performance
when either individual learning or group adaptation is promoted (see Tab. 2).
Reinforcing and mitigating effects indicate an over- and underproportional
interaction and are indicated by $I\\!E>1$ and $I\\!E<1$, respectively.

Following Eq. 13, we consider a baseline scenario in which neither individual
learning nor group adaptation is promoted (i.e., zero individual learning and
long-term group composition). Starting from there, we change both design
parameters simultaneously by promoting either learning (moderate or high) or
group composition (medium- or short-term). The logic of the interaction
coefficient is illustrated in Fig. 4, where we refer to one scenario included
in Tab. 2: In the baseline scenario for an interdependent task with no
learning and long-term group composition, a final performance of $0.7529$ can
be achieved. Promoting group adaptation towards a short-term composition
(individual learning towards a high probability) increases this performance by
$0.0391$ ($0.1812$). However, when both design parameters are changed at a
time, the performance increases by $0.1179$; see the first path indicated by
the solid line in Fig. 4. The resulting interaction coefficient is
$I\\!E=0.54$ and suggests that the joint effect is smaller than the sum of the
isolated effects. Thus, the effects of promoting individual learning and group
adaptation simultaneously interact underproportionally in this case.

Figure 4: Interaction effect and potential offsetting effects (numbers are for
the final performance and interdependent tasks).

The interaction coefficients for decomposed and interdependent tasks are
presented in Tab. 3. When tasks are decomposed, the effects of promoting
individual learning and group adaption interact more or less linearly. We
observe a slight overproportional effect on the mean performance only if the
probability of individual learning increases to $0.25$ with a simultaneous
promotion of group adaption to either a medium-term or short-term composition.
By contrast, when tasks are interdependent, the results indicate
underproportional interactions in all cases. This means that the joint effect
of simultaneously promoting individual learning and group adaptation is
smaller than the sum of the isolated effects.

Table 3: Interaction coefficients

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero to moderate | Zero to high | Zero to moderate | Zero to high
Long-term to medium-term | Mean | 1.12 | 1.01 | 0.88 | 0.82
Final | 0.98 | 0.97 | 0.84 | 0.82
Long-term to short-term | Mean | 1.18 | 1.06 | 0.69 | 0.56
Final | 1.00 | 1.00 | 0.63 | 0.54

#### Potential offsetting effects

Since there are underproportional interactions in all scenarios with
interdependent tasks, it would be interesting to know whether and, if so, to
what extent promoting individual learning offsets the increase in performance
produced by promoting group adaptation and vice versa.131313Recall, we observe
almost linear interactions for decomposed tasks which is why no offsetting
effects are expected in these cases. For this purpose, we consider two-stage
paths that promote individual learning and group adaptation sequentially in a
different order and compute the relative changes in performance in the second
stage (see Eqs. 14 and 15). The logic behind our analysis is illustrated in
Fig. 4. Again, we start at the baseline scenario with a performance of
$0.7529$. The first path, indicated by a solid line, captures the interaction
effect as described above. The second path, indicated by dashed lines,
considers promoting group adaptation first, resulting in a performance of
$0.7920$, and individual learning second. In the second stage, the performance
increases by $9.95\%$ to $0.8708$ (see Tabs. 2 and 4). Thus, promoting
individual learning after promoting group adaptation does not offset any
positive effects in this scenario. The third path, indicated by dotted lines
in Fig. 4, considers promoting individual learning first, resulting in a
performance of $0.9341$, and group adaptation second. In the second stage, the
performance decreases by $6.78\%$ to $0.8708$ (see Tabs. 2 and 5). Thus, in
this path, promoting group adaptation in the second stage offsets the
performance increase achieved by promoting individual learning at the first
stage.

Table 4: Relative performance changes caused by promoting individual learning (interdependent tasks). |  | Learning
---|---|---
Group composition | Performance | Zero to Moderate | Zero to High
Long-term | Mean | 20.63% | 22.15%
Final | 24.37% | 24.07%
Medium-term | Mean | 16.65% | 16.32%
Final | 18.49% | 17.64%
Short-term | Mean | 11.90% | 9.57%
Final | 12.79% | 9.95%

Table 4 shows the relative changes for promoting individual learning in the
second stage when tasks are interdependent. There are no offsetting effects
but, conversely, further increases in the final performance. This observation
is most pronounced if groups do not adapt at all (i.e., $24.37\%$ and
$24.07\%$) and least pronounced if groups change their composition in the
short-term (i.e., $12.79\%$ and $9.95\%$). However, these results also show
that it still pays off to promote individual learning in the second stage but,
at best, only up to a moderate level because any further increase in
individual learning would reduce the final performance: Promoting learning
from zero to moderate results in an increase of $12.79\%$, whereas promoting
from zero to high leads to an increase of only $9.95\%$.

Table 5: Relative performance changes caused by promoting group adaptation (interdependent tasks). |  | Group Composition
---|---|---
Learning | Performance | Long-term to medium-term | Long-term to short-term
Zero | Mean | 5.13% | 5.28%
Final | 5.49% | 5.20%
Moderate | Mean | 1.66% | -2.33%
Final | 0.51% | -4.60%
High | Mean | 0.12% | -5.55%
Final | 0.03% | -6.78%

Table 5 shows the relative changes for promoting group adaptation in the
second stage when tasks are interdependent. The performance only slightly
increases if there is no individual learning. At higher levels of individual
learning, promoting group adaptation offsets preceding positive effects so
that the performance achieved in the first stage even decreases. In
particular, in the case of no individual learning, changing a group’s
composition is somewhat beneficial to the performance in the medium-term
instead of not changing it at all (i.e., $5.49\%$). The marginal effects of
promoting a short-term instead of a medium-term group composition are
negligible since the final performances almost do not differ (i.e., $5.49\%$
and $5.20\%$). This means, promoting group adaptation towards a short-term
composition does not pay off even if individuals do not learn at all. Given a
moderate or high level of individual learning, there is no effect on
performance when changing from long-term to medium-term group composition in
the second stage (i.e., $0.51\%$ and $0.03\%$). Promoting group adaptation
towards a short-term composition even decreases the final performance (i.e.,
$-4.60\%$ and $-6.78\%$).

###  4.2 Promoting individual learning after promoting group adaptation

The results presented in Tab. 2 suggest that the performance always increases
as agents start to learn at the second stage, i.e., when the learning
probability increases from $\mathbb{P}=0$ to $\mathbb{P}=0.25$. This finding
follows intuition and can be observed for decomposed and interdependent tasks.
Since learning enables agents to find new and perhaps better-performing
solutions to their subtasks, performance may steadily increase. However, after
a certain number of periods, the performance cannot be further improved and
settles at a specific level, which we refer to as the limit performance (i.e.,
the maximum performance that agents can achieve on average, given the
conditions considered in the scenario). The limit performance increases
significantly with the learning probability. While both the final and mean
performances increase if agents only start to learn, further increases in the
learning probability lead to more differentiated effects that require taking
into account the lifetimes of groups.

#### Long-term group composition

In the case of promoting a long-term group composition at the first stage, the
convergence speed rises if the learning probability increases from moderate
($\mathbb{P}=0.25$) to high ($\mathbb{P}=0.5$). However, an increase in the
learning probability to values above $0.25$ has no significant effect on the
final performance. The results included in Tabs. 4 and 6 confirm this finding,
and Fig. 5 shows the same pattern for decomposed and interdependent tasks.

Figure 5: Effects of promoting individual learning in the case of a long-term
group composition. Table 6: Significance test for the effects of promoting
individual learning at the second stage

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero to moderate | Moderate to high | Zero to moderate | Moderate to high
Long-term | Mean | ** | ** | ** | **
Final | ** | n.s. | ** | n.s.
Medium-term | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Short-term | Mean | ** | n.s. | ** | **
Final | ** | n.s. | ** | **

** Indicates significance at the 99% level
n.s. Indicates not significant

#### Medium-term and short-term group composition

For the case of promoting a medium-term group composition at the first stage,
there are no moderating effects of task decomposability and no significant
effects of increasing the learning probability to $0.5$ on the final and the
mean performances (see Tab. 6). This corresponds to the results presented in
Sec. 4.1 and in particular in Tab. 4. There are marginal effects of an
increase in the learning probability on the speed of performance improvement
only in very early periods (see Fig. 6(A)).

For the case of promoting a short-term group composition at the first stage,
the results included in Fig. 6 and Tab. 6 indicate that the effects of
promoting learning are moderated by task decomposability. There is a
significant decrease in the final and mean performance when tasks are
interdependent. This result is in line with the decrease in the relative
performance changes presented in Tab. 4 (e.g., $12.79\%$ to $9.95\%$ for the
final performance in the case of a short-term group composition). Further,
there are no significant effects of promoting learning on performances when
tasks are decomposed, reflecting the linear interactions presented in Tab. 3.

Figure 6: Effects of promoting individual learning in the case of a medium-
term and short-term group composition.

###  4.3 Promoting group adaptation after promoting individual learning

While promoting individual learning at the second stage has significant
consequences for the performance in most cases, promoting group adaptation
after promoting individual learning has less pronounced effects.

#### No individual learning

In the case of not promoting individual learning at the first stage, the
results included in Fig. 7(B) and Tabs. 2 and 7 show that groups of a short-
term or medium-term composition achieve significantly higher performances than
groups that do not adapt at all. Still, this effect can only be observed when
tasks are interdependent. Thus, there is a moderating effect of task
decomposability. The observation that promoting group adaptation increases
performance follows intuition: When individuals do not learn and a group has
the opportunity to change its composition in the short- or medium term, new
group members may know better-performing solutions to the task at hand and,
consequently, performance may increase. Since groups that do not adapt at all
cannot acquire knowledge by attracting new members, the increase in task
performance comes to a standstill at a low level early in the observation
period. The effects on performance, however, are relatively small. Promoting a
short-term instead of a medium-term composition has no significant effect on
the final performance, if agents do not learn at the individual level. Yet,
the level of the final performance is achieved slightly faster (see Fig.7(B)).
This is also reflected in the results presented in Sec. 4.1 and in particular
in Tab. 5.

Table 7: Significance test for the effects of promoting group adaptation at
the second stage

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Group composition | Group composition
Learning | Performance |  Long-term to |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  Long-term to |  |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  |  |  |  |
Zero | Mean | n.s. | n.s. | ** | n.s.
Final | n.s. | n.s. | ** | n.s.
Moderate | Mean | ** | n.s. | ** | **
Final | n.s. | n.s. | n.s. | **
High | Mean | n.s. | n.s. | n.s. | **
Final | n.s. | n.s. | n.s. | **

** Indicates significance at the 99% level
n.s. Indicates not significant

#### Moderate individual learning

When moderate individual learning is promoted at the first stage, groups of a
medium-term composition improve their performance significantly faster than
groups that do not adapt at all. However, there are no significant differences
in the achieved final performances (see Tab. 7). While the same result can be
observed for both interdependent and decomposed tasks, task decomposability
appears to have a moderating effect on the convergence speed and the final
performance when group adaption is further promoted towards a short-term
composition. For interdependent tasks, the results presented in Fig. 7(B)
indicate that groups that are of a short-term composition achieve a
significantly lower performance than groups that adapt less frequently, i.e.,
are of a medium-term composition (see Tab. 2). That means, too much group
adaptation might unfold adverse effects on performance (see also the negative
values reported in Tab. 5 in Sec. 4.1).

Figure 7: Effects of increasing group adaptation in the case of no and
moderate individual learning.

#### High individual learning

If high individual learning is promoted at the first stage, the effects of
promoting group adaptation are similar to those identified for moderate
individual learning. However, if the probability of individual learning is
high as compared to lower learning probabilities the performance increases
faster when groups are of a long-term composition (see Fig. 8). For decomposed
tasks, the mean performances achieved by groups of a short-, medium- and long-
term composition become even more similar than in the case of moderate
learning (see Tab. 2). Thus, the benefits of promoting a relatively short-term
group composition decrease. If tasks are interdependent, groups of a short-
term composition perform significantly worse than groups of a longer lifetime
(see Tabs. 2 and 7). In contrast, the performances achieved by groups of a
long-term and a medium-term composition almost do not differ (see Fig. 8(B)).
Like for moderate individual learning, task decomposability has a moderating
effect when a group’s lifetime is short. Promoting group adaptation is not
necessarily advantageous if individual learning is already high but might even
unfold adverse effects, particularly when tasks are interdependent.

Figure 8: Effects of promoting group adaptation in the case of high individual
learning.

##  5 Discussion

Our research aims to gain insights into the interaction effects of variations
in two design parameters, namely  (i) individual learning and  (ii) group
adaptation.  In particular, we aim to understand how the simultaneous and
sequential promotion of individual learning and group adaptation affects task
performance and how task decomposability moderates any effects. To do so, we
have extended the NK-framework by a learning mechanism and a group formation
mechanism to account for adaptation at the level of the individual agent and
the group of agents, respectively.

###  5.1 Results related to interaction effects

The results presented in this paper can be related to the exploration-
exploitation dilemma, which is concerned with the trade-off between obtaining
new knowledge and using the available knowledge to improve performance [20].
Previous research points out that the key to improving performance is a proper
balance between exploration and exploitation [20, 73, 122]. Further factors,
such as the managerial initiative [90], feedback [46, 49], information about
the environment [70], task complexity [115], and organizational policies [108]
need to be taken into account.

The previous literature on the topic often employs a unidimensional
perspective, in which exploration and exploitation occur either at the
individual [46, 49, 70, 90] or group level [108, 115]. Please note that, from
the perspective of the organization, promoting individual learning and
promoting group adaptation are key design parameters to control whether groups
lean more towards exploration or exploitation. Naturally, promoting high
learning motivates agents to obtain new knowledge and, thereby, exploration is
fostered at the individual level. The group formation mechanism implemented in
our model makes sure that the agents who have the best knowledge to solve the
task join forces in a group. Thus, promoting group adaptation can be
interpreted as a mechanism to foster exploration from a group’s perspective.
The shorter (longer) the group’s lifetime, the more (less) often this
mechanism is carried out, and, consequently, the more exploration
(exploitation) is promoted. We contribute to this stream of literature by
analyzing the interactions between individual learning and group adaptation
and quantifying them in terms of the interaction coefficient.

In Sec. 4.1, we have shown that there are not just linear but also non-linear
interactions if individual learning and group adaptation are promoted
simultaneously. While these interactions are close to linear in all cases for
decomposed tasks, we find (highly) underproportional interactions in
interdependent tasks. Previous research has addressed the interaction between
the effects of promoting learning and group adaptation, too. Savelsbergh et al
[100], for example, found a positive relationship between a long-term group
composition and team learning. We show that a similar relation also holds true
for individual learning. Moreover, Bartsch et al [14], Edmondson [36], and
Sergeeva and Roehrich [103] claim that the promotion of a more frequent group
adaptation might offset the effects produced by an increase in individual
learning. This finding is in line with our observations, and we contribute to
this stream of research by showing that this offsetting indeed occurs, but
just if tasks are interdependent.

Moreover, previous research suggests that interactions across subtasks require
a broader exploration to develop well-performing solutions to tasks, mainly
when landscapes are characterized by a relatively high number of peaks [96].
This is in line with our findings since both promoting individual learning and
group adaptation particularly pay off in interdependent tasks. However, we
also show that overpromoting exploration might fire back via performance.
Thus, we regard it highly important, particularly for corporate practice, not
to overpromote learning and group adaptation. Billinger et al [23], for
example, find that human decision makers are indeed prone to overexploration.
We show that cheering this tendency might unfold unwanted behavioral dynamics
and might result in a decreasing performance.

###  5.2 Results related to promoting individual learning

In Sec. 4.2, we find that starting to learn at the second stage, i.e., after
group adaptation has been promoted before, increases performance in all cases.
In contrast, the effects of promoting high learning depend on a group’s
lifetime. Also, for higher levels of learning, the effects are moderated by
task decomposability. These results relate to previous research on the
relationship between means to promote individual learning and organizational
performance measures, such as productivity and financial and innovative
performance. These means include mentoring [67], allowing for ‘learning by
doing’ [7] and exploratory learning [22], determining the flow of information
that surrounds learning [27], and, more generally, creating work environments
that stimulate creativity and learning [6, 89, 110]. In addition, Stinchcombe
[109], Cohen [27], Salas et al [99], and Tharenou et al [113] highlight the
importance of employee training to promote individual learning since the
agents’ skills are the foundation of organizational capabilities and, hence,
employee training contributes substantially to organizational competitiveness.
In this regard, it is important to note that providing training to promote
individual learning is widely employed in corporate practice. This can – not
least – be seen in the enormous amounts of money spent on it [48].

Previous research on promoting learning through training mainly focuses on the
individual-level outcomes in terms of what was learned by the agent, while the
consequences of training for the organizational level is seldom in focus and,
consequently, still needs to be explored [47, 64]. The first large-scale
studies concerned with the link between training and performance are those
carried out by Holzer et al [58] and Bartel [13]. They found evidence for a
direct positive relationship between learning and productivity, which is in
line with our finding that performance increases if agents start to learn.

Becker [16] distinguishes between general and specific training. Specific
training exclusively increases the performance of a particular organization,
makes new employees familiar with the organization, and helps gain new
knowledge in monopolistic environments, where no other organizations exist for
which the knowledge would be useful. By contrast, in the case of general
training, the knowledge gained might be useful also for competitors [see also
12]. Becker [16] claims that most training is neither purely general nor
completely specific. We argue that the learning included in our model is of a
general nature since it is concerned with how to carry out a task that similar
organizations could face. In our model, specific knowledge could be the
agents’ knowledge about the functioning of the auction mechanism for group
adaptation. We assume specific knowledge to be given and, hence, do not focus
on it. The emphasis on general training allows us to connect our results to
the literature on gift-exchange: As soon as agents realize that the knowledge
gained from promoting learning might be useful in other employments as well,
they might regard it as a ‘gift’ [3, 12]. Following gift-exchange models [34],
employees would eventually repay the gift in one or the other form. In an
organizational setting, this repay could take the form of putting more effort
into solving the task at hand, which might result in higher performance [29,
41]. Thus, for the positive relationship between learning and performance
observed in our model, the theory of gift-exchange explains a similar pattern.
However, our agents act utility maximizing under bounded rationality, which
apparently leads to the same patterns at the macro-level. Moreover, it is well
known that increasing the ‘gift’ does not necessarily lead to agents making
more effort since the marginal effect decreases. For the context of monetary
incentives, this observation has been explained by a crowding-out effect of
rewards or individual earnings targets that pose an upper limit on effort [24,
43]. We observe that performance does not increase but perhaps even decreases
if learning is promoted too intensely. In these cases, the agents’ behavior
can, thus, be described by a decreasing marginal ‘gift’-effort relation. It
is, however, not a crowding-out effect or a compensation target that drives
our observation, but the maximum attainable performance. If agents learn with
a high probability, they can achieve this performance faster. However, any
further promotion of learning does not pay off because there is no more room
for further improvement.

Glaveli and Karassavidou [47] argue that, in particular, the factors that
mediate the outcomes of promoting learning by training have not yet been
substantively explored. We contribute in this respect by showing that
promoting learning for untrained individuals yields positive effects in all
cases. This finding contrasts that by Barrett and O’Connell [12], who found
that the positive effects of general training are robust against corporate
restructuring. In the context of our model, corporate restructuring can be
translated to the design parameter of group adaptation. We contribute to this
line of research by showing that task interdependence and promoting group
adaptation impair the positive effects of training on performance as soon as
agents learn with a higher probability; in the worst cases, learning can even
yield negative effects for performance.

###  5.3 Results related to promoting group adaptation

In Sec. 4.3, we have shown that promoting group adaptation at the second
stage, i.e., after individual learning has been promoted, particularly pays
off when tasks are interdependent. In this case, both the final and mean
performances increase. By contrast, for decomposed tasks, there are, at best,
increases in the convergence speed.

Previous research asks to take into account temporal aspects of a group’s
composition. Mathieu et al [80], for example, argue that including the aspect
of time into group composition research allows, amongst others, to model how
teams move through a lifecycle from birth to death, temporal norms, and the
future orientation of the organizational culture [83]. This is also
particularly relevant for organizational research, as Tannenbaum et al [111]
and Bell and Outland [18] claim, because organizations more frequently keep
relying on team-based structures. This means that groups are formed for a
predetermined time to solve a specific set of tasks [77]. In our research, we
account for the lifetime of groups by the decision about whether or not to
promote a long-term, medium-term, or short-term group composition. While
previous research argues that organizations might strategically use a limited
lifetime of groups because this allows them to redeploy their human capital
[18], our approach is different. We follow an (evolutionary) bottom-up
approach of group formation [114] that is driven by a second-price auction
[see, e.g., 69] to assure that the best-prepared agents join forces in a
group. Previous research, in contrast, sometimes appears to stick to the
concept of more classical top-down approaches to group composition [98].
Furthermore, it has already been argued that promoting group adaptation can
have different consequences. It either increases the performance because it
stimulates creativity within a group [26] or it decreases performance because
newly formed groups require some time to develop efficient modes of
collaboration [74]. We contribute to this line of research by shedding light
on the interactions between the effects of promoting individual learning and
group adaptation and by exploring the moderating role of task decomposability.
In addition, we show that only the convergence speed but not the final
performance might increase in the case of decomposed tasks, which supports the
argument brought forward in [26]. On the contrary, we indeed observe decreases
in the final performance if groups change their composition too frequently and
if tasks are interdependent. This insight supports the claim by Lewis et al
[74]. However, we add that it is not only the initial phase of coalescing that
might decrease the performance, but also over-promoted exploration (and the
interaction with promoting individual learning) might lead to significant
decreases in performance.

##  6 Conclusion

In this paper, we analyze and discuss how learning and adaptation at multiple
levels in an organization affect task performance. We contribute to previous
research by extending the traditionally unidimensional perspective on either
the individual or the group level and exploring the effects when individual
learning and group adaptation, in the sense of changing a group’s composition,
occur simultaneously and sequentially, respectively. Our results indicate
that, in general, organizations are well-advised to promote learning and
adaptation to increase task performance. However, we also show that individual
learning and group adaptation should not be pushed too far because there are
interaction effects between the two levels. Whether or not interactions are
close to linear is moderated by task decomposability. In particular, if the
group members are very much engaged in learning at the individual level,
changing the group composition may backfire and even decrease performance, at
least when tasks are interdependent. If individuals only learn to a rather
minor extent, changing the composition of a group from time to time is
beneficial to task performance. Still, very short intervals between group
adaptations do not pay off. However, if organizations allow groups to change
their composition in the short term, learning at the individual level is in
general beneficial. However, the marginal effects of pushing learning beyond a
moderate level are negligible. Our results shed new light on the consequences
of simultaneously or sequentially promoting individual learning and group
adaptation for performance. By revealing the effects of micro-level activities
on macro-level performance we contribute to closing the still predominant gap
between micro- and macro-level research in managerial science.

Our research, of course, is not without its limitations. First, we assume that
the agents are heterogeneous concerning their capabilities and limited in
their rationality. We do not consider the effects of any other individual
characteristics – such as social background, age, gender, education, or
national culture – on performance. Further research may want to elaborate on
this. Second, we omit communication and coordination between agents, and we
exclusively focus on one group within an organization. Further research could
extend our approach by adding communication channels between agents and
analyzing the co-evolution of multiple (potentially interdependent) groups.
Third, we do not take into account the potential costs of individual learning
and group adaptation. Finally, we fix the probability of individuals learning
and the intervals at which groups may change their composition exogenously.
Future research could consider some self-control and self-generated initiative
by individuals and groups and investigate how endogenous decisions on the
learning probability and the lifetime of groups affect the results.

## Availability of data

Simulation data is available https://gitlab.aau.at/dablancofern/nk-model-
multilevel-adaptation/-/blob/data/Results_-_simulations.xlsx

## Code availability

The code is available https://gitlab.aau.at/dablancofern/nk-model-multilevel-
adaptation/-/tree/master

##  Appendix A Alternative structure of interdependencies

Figure A.1 illustrates the interdependencies and subtasks for an alternative
structure of interdependencies, e.g., a roll structure. The interdependencies
are partly located outside the subtasks, irrespective of the level of task
complexity $K$.

Figure A.1: Interdependence matrices for a roll structure.

Table A.1 shows the mean and final performances for the roll structure. As
compared to Tab. 2, the mean performances are lower for the case of a roll
structure.

Table A.1: Mean and final performances for a roll structure

|  | Low task complexity | Medium task complexity
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero | Moderate | High | Zero | Moderate | High
Long-term | Mean | 0.8045 | 0.9251 | 0.9432 | 0.7310 | 0.8896 | 0.8992
Final | 0.8053 | 0.9542 | 0.9567 | 0.7314 | 0.9200 | 0.9199
Medium-term | Mean | 0.8287 | 0.9434 | 0.9475 | 0.7779 | 0.9070 | 0.8981
Final | 0.8313 | 0.9566 | 0.9568 | 0.7822 | 0.9301 | 0.9210
Short-term | Mean | 0.8278 | 0.9373 | 0.9353 | 0.7782 | 0.8506 | 0.8296
Final | 0.8292 | 0.9444 | 0.9387 | 0.7822 | 0.8577 | 0.8348

Table A.2 shows whether and how a variation in individual learning after
promoting group adaptation affects the mean and final performances under the
condition of a roll structure. Similarly, Tab. A.3 shows whether promoting
group adaptation after promoting individual learning has significant effects
on the mean and final performances. The general patterns described in Sec. 4
hold for true the roll structure.

Table A.2: Significance test for the effects of promoting individual learning
for a roll structure

|  | Low task complexity | Medium task complexity
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero to moderate | Moderate to high | Zero to moderate | Moderate to high
Long-term | Mean | ** | ** | ** | n.s.
Final | ** | n.s. | ** | n.s.
Medium-term | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Short-term | Mean | ** | n.s. | ** | **
Final | ** | n.s. | ** | **

** Indicates significance at the 99% level
n.s. Indicates not significant

Table A.3: Significance test for the effects of promoting group adaptation for
a roll structure

|  | Low task complexity | Medium task complexity
---|---|---|---
|  | Group composition | Group composition
Group composition | Performance |  Long-term to |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  Long-term to |  |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  |  |  |  |
Zero | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Moderate | Mean | ** | n.s. | ** | **
Final | ** | ** | n.s. | **
High | Mean | ** | ** | n.s. | **
Final | n.s. | ** | n.s. | **

** Indicates significance at the 99% level
n.s. Indicates not significant

##  Appendix B Alternative incentive schemes

Tables B.1 and B.2 show the mean and final performances for an incentive
scheme that favors individualism (i.e., $\alpha=0.75$ and $\beta=0.25$) and
collectivism (i.e., $\alpha=0.25$ and $\beta=0.75$), respectively. For
decomposed tasks, the performances are very similar to those reported in Tab.
2. For interdependent tasks, an incentive scheme that favors individualism
decreases group performance. By contrast, an incentive scheme that favors
collectivism is associated with higher performances.

Table B.1: Mean and final performances for an incentive scheme that favors
individualism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero | Moderate | High | Zero | Moderate | High
Long-term | Mean | 0.8699 | 0.9748 | 0.9883 | 0.7444 | 0.8901 | 0.9014
Final | 0.8709 | 0.9994 | 1.0000 | 0.7448 | 0.9182 | 0.9163
Medium-term | Mean | 0.8706 | 0.9916 | 0.9933 | 0.7892 | 0.9015 | 0.9029
Final | 0.8717 | 1.0000 | 1.0000 | 0.7945 | 0.9233 | 0.9166
Short-term | Mean | 0.8720 | 0.9946 | 0.9963 | 0.7878 | 0.8869 | 0.8736
Final | 0.8730 | 1.0000 | 1.0000 | 0.7880 | 0.8934 | 0.8805

Table B.2: Mean and final performances for an incentive scheme that favors
collectivism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero | Moderate | High | Zero | Moderate | High
Long-term | Mean | 0.8722 | 0.9742 | 0.9882 | 0.7502 | 0.9163 | 0.9290
Final | 0.8732 | 0.9992 | 1.0000 | 0.7507 | 0.9425 | 0.9429
Medium-term | Mean | 0.8719 | 0.9914 | 0.9933 | 0.7831 | 0.9286 | 0.9249
Final | 0.8729 | 1.0000 | 1.0000 | 0.7876 | 0.9435 | 0.9356
Short-term | Mean | 0.8703 | 0.9944 | 0.9963 | 0.7853 | 0.8644 | 0.8444
Final | 0.8714 | 1.0000 | 1.0000 | 0.7853 | 0.8700 | 0.8470

Tables B.3 and B.5 show whether promoting individual learning after promoting
group adaptation has significant effects on the mean and final performances,
if the incentive scheme favors individualism and collectivism, respectively.
Similarly, Tabs. B.4 and B.6 show whether promoting group adaptation after
promoting individual learning significantly affects task performance, if the
incentive scheme favors individualism and collectivism, respectively. The
results suggest that the general patterns described in Sec. 4.2 and 4.3 hold
true for alternative configurations of the incentive scheme.

Table B.3: Significance test for the effects of promoting individual learning
for an incentive scheme that favors individualism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero to moderate | Moderate to high | Zero to moderate | Moderate to high
Long-term | Mean | ** | ** | ** | **
Final | ** | n.s. | ** | n.s.
Medium-term | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Short-term | Mean | ** | n.s. | ** | **
Final | ** | n.s. | ** | **

** Indicates significance at the 99% level
n.s. Indicates not significant

Table B.4: Significance test for the effects of promoting group adaptation for
an incentive scheme that favors individualism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Group composition | Group composition
Group composition | Performance |  Long-term to |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  Long-term to |  |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  |  |  |  |
Zero | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Moderate | Mean | ** | n.s. | ** | **
Final | ** | ** | n.s. | **
High | Mean | ** | ** | n.s. | **
Final | n.s. | ** | n.s. | **

** Indicates significance at the 99% level
n.s. Indicates not significant

Table B.5: Significance test for the effects of promoting individual learning
for an incentive scheme that favors collectivism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Learning | Learning
Group composition | Performance | Zero to moderate | Moderate to high | Zero to moderate | Moderate to high
Long-term | Mean | ** | ** | ** | **
Final | ** | n.s. | ** | n.s.
Medium-term | Mean | ** | n.s. | ** | n.s.
Final | ** | n.s. | ** | n.s.
Short-term | Mean | ** | n.s. | ** | **
Final | ** | n.s. | ** | **

** Indicates significance at the 99% level
n.s. Indicates not significant

Table B.6: Significance test for the effects of promoting group adaptation for
an incentive scheme that favors collectivism

|  | Decomposed tasks | Interdependent tasks
---|---|---|---
|  | Group composition | Group composition
Group composition | Performance |  Long-term to |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  Long-term to |  |  |  |
medium-term |  Medium-term to |  |  |  |
short-term |  |  |  |  |
Zero | Mean | n.s. | n.s. | ** | n.s.
Final | n.s. | n.s. | ** | n.s.
Moderate | Mean | ** | n.s. | ** | **
Final | n.s. | n.s. | n.s. | **
High | Mean | n.s. | n.s. | n.s. | **
Final | n.s. | n.s. | n.s. | **

** Indicates significance at the 99% level
n.s. Indicates not significant

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