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Mathematical Models and Methods in Applied Sciences
c
World Scientific Publishing Company
From Herbert A. Simon’s Legacy to the Evolutionary Artificial World
with Heterogeneous Collective Behaviors
Nicola Bellomo(1) and Massimo Egidi(2)
(1) University of Granada, Departamento de Matem´atica Aplicada, Granada, Spain.
Polytechnic University of Torino, Italy
IMATI, CNR, Pavia.
nicola.bellomo@polito.it
(2) Department of Political Science, LUISS, Roma, Italy
megidi@luiss.it
Received (27 July 2023)
Revised (11 September 2023)
Communicated by (Franco Brezzi)
This article focuses on Herbert A. Simon’s visionary theory of the Artificial World. The
artificial world evolves over time as a result of various actions, including interactions
with the external world as well as interactions among its internal components. This pa-
per proposes a mathematical theory of the conceptual framework of the artificial world.
This goal requires the development of new mathematical tools, inspired in some way
by statistical physics and stochastic game theory. The mathematical theory is applied
in particular to the study of the dynamics of organizational learning, where coopera-
tion and competition evolve through decomposition and recombination of organizational
structures; the effectiveness of the evolutionary changes depends on the dynamic preva-
lence of cooperative over selfish behaviors, showing features common to the evolution of
all living systems.
Keywords: Active particles, Artificial world, artifacts, complexity, functional subsystems,
learning, living systems, collective dynamics
AMS Subject Classification: 82D99, 91A22, 91B02
1. Aims and plan of the paper
The scientific interest towards the application of mathematics to the description of
living systems, generally by means of differential structures, is growing steadily in
this century, following some pioneering contributions proposed in the last century.
This interest is not only motivated by academic research, but it is also demanded
by the needs of our society, due to the awareness that we do live in a complex world
and that science can contribute to prevent and counteract crisis situations affecting
both our planet and our societies 8. The pursuit of this challenging objective should
hopefully lead to a new mathematical theory
1
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2N. Bellomo, M. Egidi
Recent research activity has developed tools somehow inspired by methods
of statistical physics 6. The interested reader may refer to Fokker-Plank and
Boltzmann-like approaches corresponding to the classical equations of the math-
ematical kinetic theory in space homogeneity, but with interactions modeled by a
phenomenological interpretation of physical reality 66, rather than by the dynamics
of classical particles 28. Indeed, the interactions of elastic particles are reversible
and mass, momentum and energy conservative, but this property is not respected
in the case of living systems. Applications have been developed by the study of
economics, opinion formation and, in general, of social systems 42 .
A different approach is that of the so-called kinetic theory of active particles,
KTAP for short, whose approach is first to derive a differential structure that is
believed to capture the complexity features of living systems, and then to derive
specific models by inserting into such structure models of interactions at the indi-
vidual based (microscopic) scale described by stochastic evolutionary games 2,12.
The key feature of the theory developed from 2to 13 is that the role of the overall
distribution over micro-states is taken into account in the modeling od interactions.
Recent developments and applications are reviewed in 24.
Applications to the study of real world systems naturally focus on the dynam-
ics of biological systems 18,36, social-economic systems 2,3,20, see the review 34 . The
mathematical theory of collective learning 25, based on the above approach, can
contribute to a deeper understanding of the complex interactions of living systems.
Applications have also been developed to model the dynamics in space, typically hu-
man crowd dynamics 17, see also the review 19, and over exogenous networks 1,33,57.
The two methods, e.g. 66 and 12, present similarities, but also substantial dif-
ferences. In both cases, the overall state of the system is given by the distribution
function of the microscopic state of the interacting entities, while the mathematical
structure underlying the derivation of models is obtained by a balance of the number
of the said entities within the elementary volume of the microscopic states. How-
ever, the main difference lies in the modeling of interactions which are nonlinearly
additive, nonlocal in space, and asymmetric 2.
Various successful results have been achieved by going beyond methods of de-
terministic population dynamics. Indeed, living systems are evolutionary, operate
far from equilibrium, and thus, present heterogeneous behaviors within each pop-
ulation or any sort of aggregation. However, it is not just an exclusive domain of
biology, but it pervades all sciences where individual behaviors affect the overall
dynamics 15.
Our work is dedicated to describing the dynamics of evolutionary systems
through mathematical structures, with particular reference to the visionary the-
ory of Herbert A. Simon 71,76 . This challenging goal leads us to go far beyond the
aforementioned approaches. Indeed, although we start from the kinetic theory of
active particles 12, the description of interactions must be rethought, since the rules
that guide them are not independent of the properties of the environment in which
the dynamics develop (learning, decision making, development of strategies, etc.),
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From Simon’s Legacy to the Heterogeneous Artificial World 3
but refer to the virtual world that hosts them. In fact, the virtual world follows its
own dynamic rules depending on external actions, for instance, due to actions by
politics, as well as on internal dynamics of the different components of such world.
The presentation focuses mainly on methodological aspects. In fact, the main
objective of this paper is to provide a conceptual framework for the applications.
Accordingly, the case study analyzed in the following sections is treated only to
show the application of the theory in the study of a real problem. Perspectives for
further applications, also at the computational level, are critically analyzed in the
last part of the paper.
Section 2 deals with the conceptual framework that leads to the study of the dynam-
ics of interactions focusing on Herbert Simon’s contribution to the understanding
and representation of human organizations as living systems, which can be traced
back to the idea that organizations are entities that possess the ability to adapt
to the environment and to reproduce themselves thanks to the collective abilities
of their components. Key considerations proposed therein relate to the role that
understanding living systems can play in interpreting of the dynamics of behavioral
systems. In particular, this section naturally leads to the interpretation of the dy-
namics of organizational learning where cooperation and competition characterize
the evolution of organizations and have internal dynamic properties. Cooperation
and competition evolve through decomposition and recombination of organizational
structures, see 53,75.
Section 3 first presents the philosophical framework from which a new mathematical
theory is supposed to be derived. Then, a strategy (rationale) for achieving the
aforementioned purpose of describing the dynamics of evolutionary systems through
mathematical structures is presented. In particular, the approach is carried out in
two steps. First, we consider a stationary virtual work where the interaction rules
do not evolve in time, and then we consider a dynamic interpretation.
Section 4 deals with the derivation of the mathematical theory considered as the
underlying basis to derive specific models to be applied selected as well-defined case
studies. The study is developed within a wide generality to be specialized in well-
defined applications. The dynamics of organizational learning is selected as a case
study to outline the application of the theory. Then, the application to the case
study proposed in Section 2 is developed consistently to the mathematical theory
proposed in his section 4.
Section 5 develops a critical analysis of the contents of our paper to understand
to what extent the theory is consistent with the effective behavior of real social-
political-economic systems and to what extent the theory can lead to the discovery
of new trends of behavioral dynamical systems in general. The focus is on the
development of a theoretical approach to the study of systems in which different
dynamics interact. The final challenging goal is still the search for a mathematical
theory of living systems.
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4N. Bellomo, M. Egidi
2. Reasonings on Organizations
This section reports about the concept of Organizations according to Herbert Si-
mon’s philosophy and develops some considerations on the interpretation of in-
teractions referred to the above structure. These considerations contribute to the
conceptual framework in which our theory is proposed. A specific application is
proposed in Section 2, according to the concept proposed in this section, which is
devoted to understand the complex interaction of general aspects of the dynamics of
living systems on the dynamics of social-economic systems. Therefore, this section
can be viewed as a bridge leading to the philosophical and mathematical contents
of the next two sections.
2.1. What are Organizations?
About thirty years after the first publication of “The Science of Artificial”, see 72, in
a paper dedicated to modularity in biology Leland Harwell (Nobel Prize Laureate
in Physiology in 2001), see 51, compared the properties of living organisms and the
ones of human artifacts in the following way:
Today’s organisms have an unbroken chain of ancestors stretching
back to the origin of life. This constraint has been successfully used
to understand protein functions, by comparing existing protein se-
quences from related species, finding conserved parts and inferring
their roles.
[. . .].
This evolutionary history is similar to that of man-made devices.
Particular solutions in computing, or for any engineered object, are
the result of an elaborate historical process of selection by techno-
logical, economical and sociological constraints. A familiar example
is the less than optimal QWERTY keyboard, originally invented to
prevent jammed keys on early manual typewriters. It can be viewed
as a living fossil.
Following the same line of reasoning, the internal organization of a modern
business company can be compared to a living system, as it represents the result of
a process of adaptation to changing market conditions based on innovations. Such
innovations are the product of a collective process of heuristic research, similar to
the one that characterizes biological evolution, see 9,38.
If the adaptation of both the business firm and biological species
to their respective environments are instances of heuristic search,
hence of local optimization or satisfying, we still have to account
for the mechanisms that bring the adaptation about. In biology the
mechanism is located in the genes and their success in reproducing
themselves. What is the gene’s counterpart in the business firm?
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From Simon’s Legacy to the Heterogeneous Artificial World 5
Nelson and Winter (1982) suggest that business firms accomplish
most of their work through standard operating procedures, i.e.,
algorithms for making daily decisions that become routinized and
are handed down from one generation of executives and employees
to the next, see 61,37. Evolution derives from all the processes that
produce innovation and change in these algorithms. This book has
opened a wide debate on the nature of routines for many decades,
see 9,38,40,41.
Hence, a feature recoverable in both human organizations and living organ-
isms is adaptation to external environmental conditions by modifying their internal
structure through a learning process.
This feature allows us to classify both human organizations and living organ-
isms as “artifacts”, in line with the definition proposed by Simon. This feature
allows us to classify both human organizations and living organisms as artifacts,
in line with the definition proposed by Simon, who offers an extremely broad no-
tion of what constitutes an artifact, which he defines as an interface between an
internal and an external environment. In the scope of such an ample definition fall
both objects that are commonly understood as artifacts (that is, manmade objects
and structures), as well as living entities that possess the ability to adapt to the
environment by self-reproducing.
Herbert Simon’s contribution to the understanding and representation of hu-
man organizations as living systems can be traced back to the idea that organi-
zations are entities that possess the ability to adapt to the environment and to
reproduce themselves thanks to the collective abilities of their components. The
main macro-characteristic of organizations (such as a company, but also a market
or a medieval village) is therefore the ability to decompose and reassemble its parts
in response to external changes or newly emerging internal goals.
In some cases this capability is the result of purposeful decisions at macro
level (as for example a government activity) while in others it remains the result of
unplanned collective action.
See Herbert Simon 72 , pp.33–34.
I retain vivid memories of the astonishment and disbelief expressed
by the architecture students to whom I taught urban land economics
many years ago when I pointed to medieval cities as marvelously
patterned systems that had mostly just ”grown” in response to myr-
iads of individual human decisions. To my students a pattern im-
plied a planner in whose mind it had been conceived and by whose
hand it had been implemented. The idea that a city could acquire
its pattern as naturally as a snowflake was foreign to them. They
reacted to it as many Christian fundamentalists responded to Dar-
win: no design without a Designer!
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6N. Bellomo, M. Egidi
This dual aspect of organizational change is clearly emphasized in the famous
book “Organizations” that Simon wrote with Jim March 58: At the time of the
publication of the book, the common attitude of scholars was to study organizations
as planned entities characterized by a top down hierarchical internal structure aimed
at the realization of the goals: Cyert, Simon, and Trow’s studies 30, too, initially
emphasized the decisions of top managers of the companies, made under conditions
of uncertainty and requiring innovation, as opposed to the routines, standardized
decisions of most employees and workers within the company. However, as their
studies progressed, it became clear that the activity of problem solving did not
exclusively characterize managerial decisions. As a result, it became clear that the
Taylorist view of organizations as rigidly planned entities, wonderfully lampooned
by Charlie Chaplin in Modern Times, was misleading: organizations cannot function
without the active and intelligent cooperation of their components.
March and Simon understood that any internal rules and hierarchical settings
could not survive external changes without the continuous cooperation of internal
groups and their ability to learn and discover at the local and micro level, see 58.
Then the framework of organizational studies shifted from planning to organiza-
tional learning. The nervous system of organizations was considered as composed
by procedures largely formalized as computer programs, that were evolutionary
changing under the forces of adaptation and innovation. Furthermore, some of the
procedures are not supposed to be formally representable, are related to ”tacit
knowledge” and could only be learned by imitation, see Polanyi 67.
Then, organizations learning is depicted as a complex process of adaptation
and innovation at the macro and micro levels, essentially involving the changes of
the organizational procedures or routines. The engine of organizational learning is
the human ability to solve problems.
March and Simon understood that any internal rules and hierarchical settings
could not survive external changes without the continuous cooperation of internal
groups and their ability to learn and discover at the local and micro level. Then the
framework of organizational studies shifted from planning to organizational learn-
ing. The nervous system of organizations was considered as composed by procedures
largely formalized as computer programs , that were evolutionary changing under
the forces of adaptation and innovation. Than organizations learning is depicted as
a complex process of adaptation and innovation at macro and micro level , involving
essentially the changes of the organizational procedures or routines. The engine of
organizational learning is the human ability to solve problems.
2.2. Organizations as distributed problem solving entities: macro
processes
Therefore, the process of problem solving plays a central role in the study of organi-
zations, both from the viewpoint of its psychological aspects at the micro level and
from the viewpoint of its formal mathematical properties at the macro level; the
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From Simon’s Legacy to the Heterogeneous Artificial World 7
former aspect was the origin of a stream of research on human cognitive processes,
the latter gave rise to the studies of the formal properties of decomposability and
quasi-decomposability of routines and networks of routines, and was in a way the
first important step on the way to understanding the dynamics of organizational
knowledge.
Note that one of the most relevant applications of decomposability is the study
of the division of labor within organizations, interpreted as the decomposition of
organizational procedures: Simon 72 claims that basically all complex systems, be
they physical, social, biological, or artificial, share the property of having a nearly
decomposable architecture. And the recursive decomposition of problems into sub-
problems is a property of both organizations and computer programs.
As every programmer knows, the fundamental element that characterizes large
programs is decomposability: the more the program is composed of independent
subprograms, the easier it is to find bugs, the easier it is to assign the construction
of different parts to independent individuals (then allowing parallel construction),
the easier it is to modify existing subprograms (evolution), and the more protected
the program is from external attacks.
The approach based on routines as programs opened the way to representing
organizations as artifacts, artificial structures endowed with internal intelligence.
Advances in one of the artificial sciences - the theory of computation - became
fundamental to advances in the other - the theory of organizational learning. A
brilliant example of this progress is the theory of genetic algorithms developed by
John Holland 53 and later, with Holyoak, Nisbett and Thagard 54.
2.3. The internal forces of the process of organizational learning:
micro processes
In this framework, evolution is interpreted as the result of a process of collective
learning that involves both macro and micro levels. Herbert Simon developed the
theory of problem solving at micro level strictly referring to experimental data on
human reasoning. For many decades, starting in the 1970s, he conducted studies
with Chase and later with Gobet on chess players, see 27 and 45. These studies
made it possible to discover the dualistic nature of the reasoning process and the
role of heuristics, which were then developed by Kahneman and Tversky and gave
an extraordinary impulse to the development of cognitive psychology, see 78.
Advances in theory in the macro context made clear the relevance of the pro-
cesses of cooperation and competition within organizations as essential elements of
organizational learning and evolution. This brought attention to the interactions
between the macro and micro levels: Indeed, cooperation and competition presup-
pose that the decisions of individuals cannot be based solely on personal, selfish
preferences, but are influenced by and affect the decisions of others. Moreover, as
the games of public goods have shown, see ?,?,44, cooperation is a complex process
involving reciprocity and altruistic behavior. Traditional assumptions of rationality
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8N. Bellomo, M. Egidi
were no longer tenable.
Then, once the Pandora’s vase of “rational behavior” was broken, new impor-
tant theoretical distinctions become necessary: while the process of human reasoning
become progressively clearer, Simon’s theory assumes two essential modifications of
the traditional approach to rationality: on the one hand, that rationality as a means-
end calculation is inherently limited, and on the other, that individuals within orga-
nizations, or more generally in social contexts, take into account, in addition to their
personal selfish interests, the interests of other individuals, that could be harmed by
their actions. Thus, in the paper dedicated to altruistic behavior, Simon dropped
the second requirement of the traditional definition of rationality: selfishness.
From there onwards a new picture of rationality emerges: for ”rational” individ-
uals, it should be necessary to make a strategic calculation that includes examining
the effects of their actions on others and vice versa; a calculation that, given the
limits of rationality, is made possible by the use of heuristics, that allow to dras-
tically simplify the burden of computation; heuristics can come from individual
experience, from social norms and rules, such as the aversion to injustice, or from
ethical norms, or from organizational procedures, see Gigerenzer 43 .
2.4. What are Organizations? The governance of cooperation and
competition
To summarize the discussion so far, there are two dimensions in which we must
consider organizational learning. Cooperation and competition characterize the evo-
lution of organizations and have internal dynamic properties that operate at two
levels. On the one hand, at the macro level, cooperation and competition evolve
through decomposition and recombination of organizational structures. On the
other hand, at the micro level, cooperation and competition are based on two charac-
teristics of individual problem-solving abilities: bounded rationality (which implies
dualism of reasoning and an important role of heuristics) and non-purely selfish ra-
tionality. Thus, the most obvious characteristic of all organizations, such as political
parties, business firms, charitable organizations or clubs, is that they are composed
by 314 individuals who interact in a network: they are partially aware of the net-
work of relationships in which they act and consequently exhibit a variety of micro
behaviors (selfishness, altruism, docility, revenge, envy, jealousy, opportunism) in
both cooperative and competitive conditions.
A good example of the relationship between cooperation and competition is
what happens in an organization when an internal group proposes some innovations:
the activation of the proposal creates competition because, as far as the action
of the group is successful, it attracts internal resources to the detriment of other
groups of the organization; then the internal evolution of an organization generates
internal conflicts that need to be managed and controlled. This point is beautifully
illustrated in the introduction to the second edition of “Organizations”, see March
and Simon 58:
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From Simon’s Legacy to the Heterogeneous Artificial World 9
This book is about the theory of formal organizations. Organizations
are systems of coordinated action among individuals and groups
whose preferences, information, interests, or knowledge differ. Or-
ganization theories describe the delicate conversion of conflict into
cooperation, the mobilization of resources, and the coordination of
effort that facilitate the joint survival of an organization and its
members.
[. . .]
Effective control over organizational processes is limited, however,
by the uncertainties and ambiguities of life, by the limited cogni-
tive and affective capabilities of human actors, by the complexities
of balancing trade-offs across time and space, and by threats of
competition. As organizational actors deal with each other, seeking
cooperative and competitive advantage, they cope with these limita-
tions by calculation, planning, and analysis, by learning from their
experience and the experience and knowledge of others, and by cre-
ating and using systems of rules, procedures, and interpretations
that store understandings in easily retrievable form. They weave
supportive cultures, agreements, structures, and beliefs around their
activities.
Then, the nature of business organizations rests on two fundamental pillars:
on the one hand, the organization is seen as a goal-oriented entity that pursues
its goals through distributed problem solving; on the other hand, it is seen as an
institution that manages internal conflicts. These two main issues are closely related
because conflicts are cyclically generated by the discrepancies in strategies and
interests that arise among groups involved in problem-solving activities: problem-
solving, innovation, and creativity generate internal conflicts that organizations
can successfully manage if they are able to govern the competition among different
internal groups.
2.5. The forces that limit instability
As empirical data show, there is a relevant mortality rate of the firm competing on
the markets, which depends on market selection processes, in the presence of imper-
fect information or uncertainty due to R&D efforts. See, Cefis and Marsili 26. Thus,
firm survival is correlated with the ability to adapt to changing external conditions
through internal innovation; of course, adaptation and innovation are risky activi-
ties, and firms may fail to achieve the expected benefits of their innovative actions,
which may lead to collapse. Problem solving is an essential process to ensure the
adaptation of the organization to the competitive environment, but as mentioned
above, the process of reorganization that follows an internal innovation gives rise
to potential conflicts and instability. Let us consider this point briefly.
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10 N. Bellomo, M. Egidi
Suppose a group introduces an organizational innovation that is expected to
increase the firm’s competitiveness and its survival probability. On the one hand,
it is likely that a proposed innovation will affect only part of the organization by
triggering a chain of adaptations or new innovations (and thus generating decom-
positions and recombination of organizational units). On the other hand, even if it
is likely that the innovation will benefit the entire organization in the long run, it
is possible that in the short run the innovation will benefit only some groups and
be detrimental to others. This condition may create an internal conflict between
groups that needs to be managed in order to achieve the expected benefits; this
warns that cooperation for the realization of an innovation is an unstable process
that may fail to achieve the expected goals if the conflict that arises is not managed
fairly.
Then, as noted above, the governance of cooperation and competition is a key
element in realizing internal change. It follows that the internal form of governance,
which can vary from a top-down, rigid hierarchy to a bottom-up form, plays an
important role in ensuring that competition among internal groups does not result
in mutual harm, but is kept within limits that allow cooperation to be fruitful.
Internal governance can be organized in a democratic form, where key decisions
are made by internally elected leaders, or (more commonly) in an autocratic form,
where leaders are externally appointed by a board of trustees representing the prop-
erty rights of the owners : but whatever the form of governance, it is essential that
every member of the organization, employees and managers, will comply with the
organizational strategies, otherwise the organization risks becoming ungovernable.
Compliance is likely to occur when employees and managers share the goals
of their group and, to some extent, the global goals of the firm: but this implies
that the members of the organization recognize as a common interest the good
health and successful development of the firm. This question is at the heart of
two papers written by Simon 73,74, and opens the way to explaining the evolution
of organizations through the theory of collective action that Elinor Ostrom was
extensively constructing in the same years, see 63.
In 74, page 159, Simon claims that one of the most effective element to make
the compliance effective is the loyalty to organization.
Among the most important loyalties in modern society is loyalty
to organizations: business corporations, universities, armies, vol-
untary associations, and their components. Since organizations are
principal players in modern economies, identification of employees
and executives with goals of companies and subunits plays a major
part in economic motivation (Simon, 1991). Group loyalties, have
not only a motivational but also a cognitive component. They define
the boundaries of the group over which “goods” are to be summed,
and they cause particular variables and simplified world models to
govern the thinking of group members.
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From Simon’s Legacy to the Heterogeneous Artificial World 11
Moreover, in 74, page 160,
The composite of goals and structured information that surrounds
the head and members of an organizational unit produces powerful
loyalties and definitions of the situation that are called organiza-
tional identifications. Most of the economy’s work goes on inside
organizations, influenced heavily by information that is internal to
them.
The picture of organizations that emerges from the works of March and Si-
mon 58 and Simon 73,74, identifies loyalty to organization as a crucial characteristic
to study the internal organizational evolution. For the member of a group the loy-
alty implies to accept to make actions that increase the group well being or the
group fitness, even if do not increase their own well being, i.e. are altruistic. Thus,
after the assumption of full rationality, the assumption of selfishness also falls out
of Simon’s definition of rationality.
2.6. Beyond bounded rationality: Rationality, altruism, and the
collective action
Simon attributes particular importance to the loyalty of the members to their orga-
nization and to their identification with organizational goals because is aware that
when we consider organizations from the viewpoint of the collective actions that
characterize them, a typical social dilemma arises. According to Ostrom 63.
Social dilemmas occur whenever individuals in interdependent sit-
uations face choices in which the maximization of short-term self-
interest yields outcomes leaving all participants worse off than fea-
sible alternatives. In a public-good dilemma, for example, all those
who would benefit from the provision of a public good-such as pollu-
tion control, radio broadcasts, or weather forecasting - find it costly
to contribute and would prefer others to pay for the good instead.
If everyone follows the equilibrium strategy, then the good is not
provided or is under-provided. Yet, everyone would be better off if
everyone were to contribute.
The most famous example of this dilemma is described in 48. If a number
of people have unlimited access to a finite, common resource such as a pasture,
they will tend to overuse it and may end up destroying its value altogether; this
behavior is rational and selfish. Voluntary restraint is not a rational choice for
individuals - if they did, other users would simply displace them - but the result of
the rational, selfish decision is the destruction of the common resource, i.e., a loss
for all. Ostrom 64 notes the following:
[....]
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12 N. Bellomo, M. Egidi
Much of our current public policy analysis - especially since Garrett
Hardin’s seminal paper, “The Tragedy of the Commons”, - is based
on the assumption that rational individuals are helplessly trapped in
social dilemmas from which they cannot extricate themselves with-
out external incentives or sanctions.
An assumption which was the “mantra” of neoclassical approach after the celebrated
“The Logic of Collective Behavior” by Mancur Olson 62.
She counters this assumption on the ground of a large and growing literature
of field and behavioral experiments showing that, under certain conditions, cooper-
ation can emerge and be maintained without coercion. The question is to identify
these conditions: according to Ostrom 64, for example, reciprocity, reputation, and
trust can lead to cooperative behavior by growing up in a multilateral relationship.
To some extent, this proposal has its ancestor in the ideas presented in 74, where
Simon claims that cooperation is maintained without coercion if the members of
the organization identify with the goals and perspectives of the organization.
But in a large number of real social dilemmas there is sustained cooperation
without sanctions or coercion. 64 thus counters the assumption of rational selfish-
ness because a large and growing literature of field and behavioral experiments
show that, under certain conditions, cooperation can emerge and be maintained
without coercion. The question is to identify these conditions : she suggest that
reciprocity, reputation, and trust can lead to cooperative behavior by growing up
in a multilateral relationship. To some extent, this proposal has its ancestors in
the ideas presented by in 74, where he claims that cooperation will be maintained
without coercion if the members of the organization self-identify with the goals and
perspectives of the organization.
To illustrate this point, consider again the organization’s efforts to maintain an
efficient position in the marketplace by adapting to external competitive conditions
through internal innovation. To this end members of the organization must actively
cooperate with other members in developing the internal changes to increase the
organization’s efficiency; but some of them can refrain from active cooperation and
routinely continue the previously agreed-upon behavior if they do not believe that
the changes will achieve the expected organizational improvement. Of course, if the
internal innovation is not successfully implemented, the organization risks failure.
Thus two types of behavior emerge: on the one hand, selfish rational individuals
who do not contribute to the prospective change and who, if the adaptation is
successful, will benefit from the outcome without having contributed to it. These
individuals abstain from participating and continue their previous activities for
many reasons: because they fear to be harmed by the novelties (for example, if
some of their competencies become obsolete) or because they do not believe in the
chances of success of the innovative process (risk aversion).
On the other hand, there are individuals who cooperate, whom I call “forward-
looking”. Since it is generally very difficult to measure one’s personal contribution
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From Simon’s Legacy to the Heterogeneous Artificial World 13
to a collective initiative, forward-looking individuals are aware that some agents
may be rewarded even if they abstain from participating in the collective effort.
Thus, they exhibit a form of altruistic behavior in that they are aware that their
contribution to the collective effort will also increase the well-being of the selfish
individuals who did not contribute.
Given these two alternative behaviors, the identification of employees and man-
agers with the goals of the organization and their loyalty, which Simon strongly
emphasizes, plays a fundamental role 74, Page 159. In fact, the greater the individ-
ual’s identification with its goals, the higher the likelihood that he will engage in
forward-looking, cooperative behavior.
One especially important form of altruism is loyalty to (identifica-
tion with) groups to which the individual belongs. The shape of this
loyalty, and the nature of the groups to which it attaches in a par-
ticular society go far toward determining the institutional structure
of that society and its social processes. In particular, identification
with organizations, especially by employees, allows organizational
effectiveness to reach far higher levels than could be attained by
rewards and punishments alone in the face of wholly selfish moti-
vation.
See 74, Page 159.
As noted above, if the number of forward-looking agents is large enough to
allow for the efficient development of the innovative changes, all agents, including
those who did not participate, will benefit in the long run (i.e., after the effects
of the innovation have taken place). Thus, agents may be ”rationally” tempted to
avoid participating in the collective effort, knowing that they will be rewarded either
way. Of course, if the number of forward-looking agents is reduced beyond a certain
threshold, the organization may fail. This means that the organizational change
we have described so far is a social dilemma that has the characteristics of the
”Tragedy of the Commons”; consequently, it can be modeled using the prisoner’s
dilemma schema 7and, more generally, the public goods schema, see Ostrom 64,65,
Archetti et al. 4,5, Xia et al. 79, which are the usual tools for the analysis of the
”Tragedy”.
The characteristic of most of the models already existing in the literature is to
give rise to evolutionary processes in which either a group of forward-looking indi-
viduals increases and ends up with a successful transformation of the organization,
or, conversely, selfish individuals take over and lead the organization to inefficiency
and disorder. This happens if the outcome of cooperators is linearly increasing with
the individuals’ contribution, while by introducing Simon’s idea of identification
and loyalty the relation becomes nonlinear and the dynamic of the process leads to
a stable equilibrium in which the proportion of forward looking and selfish individ-
uals remain constant. Various different trajectories can be obtained by modifying
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14 N. Bellomo, M. Egidi
the characteristics of the organization and of the members’ contribution. See also
Archetti et al. 4,5, Xia 79, Zhang 80.
The identification we have made so far between the Tragedy of the Commons
and the conditions for realizing organizational adaptation reveals deep similarities
between human organizations and the living system, as claimed by Hartwell 50
and Simon 74; the public goods model is widely applied in biology, ethology, and
natural sciences because cooperation abounds in nature even under circumstances
that constitute a social dilemma. Thus, from the ongoing advances in the study of
social dilemmas in nature, such as the relationship between altruism and Darwin’s
natural selection, see 73,74, we can expect suggestions for a better understanding of
human organizations; in this vein, in Subsection 4.3 we consider the evolution of
organizational structures under the contrast between innovation and inertia.
3. On a search towards a mathematical theory
In this section, we present a rationale that we believe can lead to the derivation
of mathematical tools suitable for providing an analytic (differential) description of
Simon’s theory. The presentation focuses on various concepts leading to the said
description, while the formalization is left to the next sections, where mathematical
tools somewhat inspired by statistical physics are developed.
We consider large systems of interacting entities and develop our reasoning first
through a brief philosophical excursus focusing on various concepts related to the
idea of describing collective motions from the dynamics at the scale of individuals.
Then, we present a concise summary of the theory in the case of a stationary
interaction rules and show how to move from such a world to the dynamical artificial
picture of Herbert Simon.
3.1. On a conceptual-philosophical framework
The main contribution of our paper is the study on how the collective dynamics of
a system of behavioral entities can emerge from to their individual and collective
interactions. The first key concept is that the collective motion depends on inter-
actions that are generally nonlinearly additive. Therefore, the collective dynamics
cannot simply be related to the dynamics of a few entities. Some of the concepts
proposed in this section integrates those in Section 2, but are somewhat related also
to methods inspired to statistical physics and to the classical kinetic theory.
The concept that the collective dynamics depends on interactions at the scale of
the interacting entities is well understood in the statistical physics of classical par-
ticles thanks to the studies of Maxwell and Boltzmann, but substantial differences
appear in the case of living entities. The following is a non-exhaustive selection of
seminal studies:
•Erwin Schr¨odinger (1887–1961) proposed some forward-looking ideas for a
systems approach to biology 69. He also argued that the dynamics of cells is driven
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From Simon’s Legacy to the Heterogeneous Artificial World 15
by the dynamics at the molecular level, motivated by the study of mutations (some
of which are also induced by external actions such as radiation). This vision is a key
reference for a recent mathematical literature devoted to a multiscale mathematical
methods in biology.
•Ilya Prigogine (1917–2003) contributed to the modeling of human behavior for
vehicular traffic on roads 68. His approach provides the first mathematical approach,
somewhat inspired by methods of statistical physics, by representing the overall
state of large populations as a probability distribution over the microscale state of
interacting units. The dynamics of this distribution function is referred to individual
based interactions.
•Lee Hartwell (born 1939) teaches that the mathematical approach to the
description of the dynamics of the inert matter cannot be straightforwardly applied
to living systems 50,51. Indeed, he observes that biological systems are very different
from the physical or chemical systems of the inanimate matter. Therefore, although
living systems obey the laws of physics and chemistry, the notion of function or
purpose differentiate biology from other natural sciences.
These concepts are critically examined with a focus on the dynamics of behav-
ioral systems of evolutionary economics 38, where the authors also refer to Simon’s
theory, which guides our paper. In particular, the theory suggests that, in the case
of living systems, the search for universal interaction rules governing interactions
should be conducted within a time-evolving environment, which he calls artificial
world. The events of the dynamics can be called artifacts.
We can find some perspective ideas on the complex links between the real and
sensitive world in the dissertation 55, which marks the transition to the “critical
period” leading to the dissertation by Immanuel Kant 56. This dissertation begins
with a definition of “mundus” and then defines the difference between “mundus
sensibilis” and “mundus intellegibilis”. An important feature of the mundus sensi-
bilis is the heterogeneity of the individual learning process. The concept of artificial
world is not explicitly defined as a dynamic determined by interactions subject to
external actions yet to be defined.
Herbert A. Simon goes far beyond the above concepts and links the whole path
from learning to process of decision to the gain and loss of rationality within the
artificial world.
The human being striving for rationality and restricted within the
limits of his knowledge has developed some working procedures that
partially overcome these difficulties. These procedures consist in
assuming that he can isolate from the rest of the world a closed
system containing a limited number of variables and a limited range
of consequences.
Our paper proposes a methodological approach to a mathematical theory of
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16 N. Bellomo, M. Egidi
economics. A brief description of the objective of our paper is to seek an answer to
the following key question:
To what extent can the mathematical theory in 12 be developed to
include the dynamics of the artificial world and how this leads to
the collective dynamics of living systems?
Therefore, we start our search from the mathematical theory proposed in 12,
which, however, does not explicitly include the dynamics of the artificial world and
does not explicitly refer to economics. The key concept proposed in 12 is that the
mathematical approach can first lead to the derivation of general structures suitable
to capture the main complexity features of living systems. Then, the insertion of
models of microscale interactions into these structures leads to the derivation of
mathematical models. On the other hand, the time evolution of the interaction rules
is not yet taken into account. This dynamic is due to the heterogeneous way in
which individuals learn and pursue the individual interpretation of well-being.
The answers are given in the following subsections. However, we anticipate that
we will try to go beyond the idea that the contribution of mathematics is limited
to technical support for economics. In fact, a mathematical-economic theory should
capture specific features of well-defined economic theories within an analytical and
computational framework. Indeed, a rational description given by mathematics can
contribute to the further development of theories in economics, while hints given
but the interactions can contribute to the further development of the mathematical
sciences.
The first step of the above strategy is the derivation of a general mathematical
structure capable of capturing the main features of living, hence complex, systems.
We refer this search to the theory proposed in 12, where this search was developed
to replace possible field theories, available in the case of inert matter, but generally
missing in the case of living systems, as observed by Robert May 59. The second
step shows how further developments of the above structure lead to a mathematical
description of Simon’s theory.
The whole path towards the derivation of the mathematical theory is shown
in the flow-chart of Fig. 1, where the first step refers to Blocks 1, . . . 5, while Step
2 initiates with Blocks 3,4 and 5 and subsequently moves to Blocks 6, . . . 9. In this
figure the concept of steady virtual world is introduced meaning that the active par-
ticles’ interaction rules constant in time, while dynamical virtual world correspond
to rules that evolve in time.
Step 1 corresponds to the above mentioned mathematical structure to cap-
ture the dynamics of living systems, while Step 2 corresponds to the mathematical
description within the artificial world. A detailed description follows in the next
two subsections, where the relevant terminology is fixed, while the last subsection
provides a critical analysis towards the mathematical approach.
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From Simon’s Legacy to the Heterogeneous Artificial World 17
B1: Phenomenology,
scaling, complexity,
representation
B2: Micro-scale
and macro-scale
variables
in a steady world
B5: Structures
of micro-scale
and macro-scale
interactions
B3: Learning
and
interactions
B4: Models
of external
actions
B6: Parametrization
of micro-scale
interactions
B7: Parametrization
of external
actions
B8: Modeling
the dynamics
of all parameters
B9: Dynamics
of the
artificial world
Fig. 1. Strategy towards a mathematical theory and derivation of models
3.2. A rationale towards a steady virtual world
We search for mathematical tools that can capture the essential features of living
systems. We refer to a mesoscopic approach that is somewhat inspired to methods
of statistical physics 12. The sequential steps are described in the following blocks
which leads to the the first step of the rationale.
Block 1: The interacting entities, that compose the whole system, are called active
particles, in short a-particles. These can aggregate into groups, sharing common
interests and strategies, called functional subsystems, i.e., FSs. The microscopic
scale corresponds to the dynamics of a-particles, while the collective dynamics is
described at the mesoscopic scale.
Block 2: All a-particles interact with all a-particles of each FS according to rules
that may depend on the specificity of each FS, but are constant in time. Each a-
particle expresses on the microscale a function modeled by a variable called activity,
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18 N. Bellomo, M. Egidi
by a level of expression that is heterogeneously distributed among the a-particles
of each FS. In each FS, the a-particles express the same activity and share the
same objectives. All the FSs express a collective activity called macro-activity, i.e.
M-activity, in general as moments of the m-activities.
Block 3: Two types of interactions are considered that correspond to different
scales, i.e. micro-scale interactions involving individual a-particles and micro-macro
interactions that refer to each FS as a whole and act on individual a-particles.
Block 4: The dynamics is analogous to that described in Block 3, but it refers
to the external actions on the a-particles and FSs. The implementation of both
descriptions contributes to the definition of the mathematical structure sought in
Block 4.
Block 5: Derivation of a mathematical structure suitable to include all interactions
related to Blocks 3 and 4. This structure is required to capture, as far as it is possible
the complexity features of living systems. The structure describes the dynamics of
the probability distribution over the microscopic state of a-particles referred to each
FS.
Remark 3.1. The contributions of Blocks 3 and 4 are an essential part of Block
5, as the collective dynamics is described by a structure that transfers interactions
involving a-particles at the microscale to the dynamics of the whole system. The
structure in Block 5 can be defined mathematical theory because it provides the
conceptual framework from which a wide variety of models can be derived. There-
fore, models can be defined mathematical models as they refer to a theory rather
than ad hoc assumptions for each specific case studies.
Remark 3.2. a-particles correspond to individual entities in a world of interacting
entities. For example, a-particles can be people in society, but also aggregates such
as corporations in an industrial world or political parties in a system in a democracy.
The micro scale corresponds to the a-particles independently of their size. Functional
subsystems correspond to aggregations based on common goals and modus operandi,
while the collection of FSs corresponds to the collective system placed in what
we can call small world. Macroscopic quantities correspond to average quantities
expressed by a-particles and FSs. Interactions are sensitive to both microscopic and
macroscopic quantities. External actions operate in the small world to act on the
dynamics of a-particles and FSs.
Remark 3.3. The key problem is the description of the output of said interactions
which depends on the specific properties of the artificial world as it is created by
the action of the external environment, see Block 5, and by interactions within the
internal world, see Block 4. Then, the interactions depend on the so-called artifacts,
see 76. Therefore, in the line with Simon’s philosophy, but not yet fully consistent
consistent with his theory. This objective is considered in Blocks 6–9, which refers
to the dynamics of the evolutive world.
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From Simon’s Legacy to the Heterogeneous Artificial World 19
I1: a-particle
with
high activity
I2: a-particle
with
low activity
I3: a-particle
with
decreased activity
I4: a-particle
with
increased activity
I1: a-particle
with
high activity
I2: a-particle
with
low activity
I3: a-particle
with
decreased activity
I4: a-particle
with
increased activity
Fig. 2. Up: High-to-low competition: The activity of one of the a-particle is lower than the activity
of the other a-particle. Then, activity of the first a-particle increases, while the activity of the
second a-particle decreases. Down: low-to-high competition: The activity of the first a-particle
decreases, while the activity of the second a-particle increases.
3.3. From a steady to a dynamical virtual world
Each a-particle as in mind a selfish-utility objective somewhat corresponding to a
payoff. Interactions can be quite complex and need a sharp selection of the objective.
Furthermore, interactions may have constraints imposed by the external action.
Let us first consider, as a particular example, a dynamics which shows how
interactions of a-particles with different level of activity might increase or decrease
their activity. The output depends on an appropriate activity function to be properly
defined as we shall see in Section 4.
These two dynamics are shown in Figure 2. Other examples can be considered.
For example, learning, which corresponds to a dynamics where the a-particle with
low activity increases its level by interacting with the one with high level, so that the
distance between the two states is progressively reduced. An additional example is
the dynamic of hiding-chasing which corresponds to an increase of the level of both
activities, since the one with low activity learns from the one with high activity to
reduce the distance, but the other one increases it to escape. In all cases, the physical
meaning of the activity depends on the specific dynamics under consideration.
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20 N. Bellomo, M. Egidi
Remark 3.4. The above considerations refer to individual-based interactions, such
as that between an a-particle and another a-particle. However, one can also consider
interactions of the individual entity with a set (or even the whole) of other particles,
as we will see in the next subsection. Specifically, interactions with the whole world
in addition to those with other a-particles. Accordingly, we will consider both type
of interactions, say micro-micro (among individuals) and micro-macro (individuals
with the whole).
Remark 3.5. An important feature to consider is heterogeneity, which may refer to
the individual expression of activity in individuals of a particular FS. In addition,
individuals in a FS may express a mixture of different strategies weighted by appro-
priate parameters. Heterogeneity leads to parametrization of the dynamics, which
plays an important role in the mathematical description of the dynamics of the vir-
tual world, as we will see in the following. As an example, we can see that the output
of interactions depends on parameters that combine two types of interactions. For
example, instead of assuming only high-to-low or low-to-high competition, a convex
combination can be used by a sum of αcorresponding to high-to-low and (1 −α)
corresponding to low-to-high, with α∈[0,1]. Additional parameters may be needed
to specify the general characteristics of the interaction.
We have now established enough concepts to describe the contents of the last
blocks of the flowchart in Figure 1. These blocks define the dynamics for systems
whose interactions evolve over time.
Blocks 6, 7: We consider two types of interactions corresponding to different scales,
i.e. micro-micro interactions involving individual a-particles and micro-macro in-
teractions referring to each FS as a whole and acting on individual a-particles, see
block 6. Block 7 studies the dynamics induced by external actions on the a-particles
of each functional subsystem.
Block 8: The above parameters are now considered to be variable with time. Mod-
eling their dynamics can lead to a general structure that can describe the collective
evolution of the system over time.
Block 9: The mathematical structure is obtained by implementing the dynamics
of the parameters into the mathematical structure of the steady virtual world.
Then specific models corresponding to well-defined case studies are obtained
by a detailed selection, for each specific system, of the functional subsystems that
play the game, the function expressed within each subsystem, and the modeling
of the dynamics of the virtual world and the micro- and macro-scale interactions
within such a world.
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From Simon’s Legacy to the Heterogeneous Artificial World 21
3.4. Critical analysis
The strategy towards the main goals of our work has been described in blocks 1-9.
The goal is to develop a mathematical theory to describe the dynamics of living
systems within the aforementioned time-evolving artificial world. The interpretation
and description of interactions is the key passage to the development of the new
theory. We discover this topic in the concluding parts of 21, where the authors
propose five key problems that are brought to the attention of researchers active in
the fields of social systems and economics. Let us quote one of them, as it is closely
related to the plan of our paper.
There are lots of models that show how groups arrive at consensus
but no generally accepted model of how groups become polarized or
how two groups can become more and more different and possibly
hostile.
Different types of interactions have been considered in 12, for example agree
and disagree,hide and chase, all supported by learn and forget. As mentioned above,
different types of dynamics coincide in the interaction. On the other hand, the study
in 12 does not describe how the interaction rules evolve over time. Hence, it can
be argued that one can look for general structures that can be searched to include
different types of interactions depending on specific parameters.
This topic cannot be limited to the technical definition of consensus and dissent,
while it should be referred to Simon’s philosophy 73, which considers the following:
In this paper, I shall consider how far the altruism in human be-
havior is reconcilable with neo-Darwinian and with neo-classic eco-
nomic theory, and explore some of the consequences for economic
theory of the presence of substantial altruism behavior.
Herbert Simon then provides a number of alternative definitions and analyzes
the consequences for social, political, and economic dynamics. The problem then
is to understand how Simon’s typologies of interactions can be translated into a
mathematical framework.
A.2. We generally trust behavior as altruistic if it sacrifices wealth
or power for the wealth or power of an other, and selfish if it seeks
to maximize wealth and/or power, disregarding the effects of its
decision on the others.
The real world dynamics of interaction are highly complex and cannot be
quickly confined to a simple interpretation universally valid and constant over time.
The very brief description provided in the above lines has shown that the rules
by which individuals interact evolve over time in a manner consistent with Simon’s
theory of artifacts. We do not naively believe that mathematics can capture all the
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22 N. Bellomo, M. Egidi
features of dynamics. On the other hand, if we focus on a particular topic, we can
define a detailed strategy by which mathematics can describe the above dynamics.
4. Dynamics within Simon’s artificial world
In this section, we transfer the philosophy presented in Sections 2 and 3 to the
derivation of a mathematical framework that refers to Simon’s theory of the ar-
tificial world 76. First, we consider the case of a virtual world, where the rules of
interactions, however artificial, do not evolve in time. We then show how interactions
and external actions can lead to a dynamic artificial world.
The approach is a substantial development of the kinetic theory of active par-
ticles, where the modeling of interactions is described by the theoretical tools re-
viewed in 12,13,24. This method deals with entire populations of players, where strate-
gies with high payoffs could spread across each population by learning in terms of
individual-based and collective interactions. However, the concept of payoff will be
revisited by what we can call bounded utility function which, as we will see, can
evolve over time.
The last subsection concerns the mathematical theory for the case study de-
scribed in Subsection 2.3. This specific case study will show, also for tutorial pur-
poses, how the mathematical theory can be applied and how far it can go in pre-
dicting future dynamics. For example, the following questions can be posed.
1. How can the whole system be divided into functional subsystems
and how can the activity variables be assigned to each FS?
2. How to model the dynamics with constant parameters?
3. How the parameters can be modeled as variables and how their
dynamics are determined by the interpretation of the utility func-
tion?
4.1. Dynamics in a steady virtual world
As mentioned above, the dynamics is described by mathematical tools of the kinetic
theory of active particles, which provides structures suitable for deriving models by
implementing models suitable for describing microscopic interactions and external
actions. We choose, from the most general framework reported in 13, a specialized
structure that seems appropriate to describe the dynamics under consideration.
With all of the above in mind, let us present the derivation of the mathematical
framework used in the following.
Subdivision into functional subsystems: The overall system consists of a large
number Nof a-particles. We do not consider birth and death events so that Nis
constant. The system is divided into functional subsystems, FSs, where a-particles
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From Simon’s Legacy to the Heterogeneous Artificial World 23
express the same activity which is heterogeneously distributed over the a-particles
of the same FS.
Transitions across FSs are not initially considered, but we will show how this
dynamics can be described by a further technical development of the mathematical
theory.
In the following, the notation ij-particle is used in the following to denote an
a-particle in the i-th FS with state j. Actually, the subscript ialso denotes the
specific activity of each FS, since the activity is specific to that FS.
Representation: Each FS is labeled by the subscript i∈[1,...n]. A discrete
variable is used for the activity uexpressed by a-particles. In detail, uis defined in
a bounded domain subdivided into tracts of the same size labeled by the subscript
j∈[1,...m]. Then, the overall state of the system is described by the probability:
fij =fij (t) = fi(t, uj),with i= 1, . . . n, j = 1, . . . m, (4.1)
where the probability distribution is divided by N, so that
n
X
i=1
m
X
j=1
fij (t) =
n
X
i=1
ni(t) = 1,for all t≥0,(4.2)
where niis the number, referred to N, of a-particles in the i-th FS. Then, the sum
of all nithat is constant in time and properly normalized. If the dynamics across
subsystems is not taken into account, then also each niis constant.
Interactions within each FS and across them, involving the test, field, and candi-
date a-particles, which are distinguished only on a statistical level. In detail:
Test particle is assumed to be representative of the whole system by the probability
distribution fij (t);
Field particle denotes the particle of the h-FS with state k, so that the components
of the probability distribution is fhk(t), with h= 1, . . . n, k = 1,...m.
Candidate particle denotes the field particle of the i-th FS, with state fip(t), with
p= 1,...m, which, after interacting with the field a-particles fhk(t), ends up in the
state of the test particle.
The modeling of interactions requires a set of parameters which are denoted by w0
(the application of the case study can show how it can be modeled in practice). The
following quantities describe the dynamics of micro-scale interactions:
•The interaction rate ηhk
ip [f;w0] models the rate of interactions between a candidate
(test) ip-particle (ij-particle) and a field hk-particle, where fdenotes set of all {fij }
and w0denotes the set of all constant parameters needed to model the interactions
at the individual based scale.
•The dynamic of micro-micro interactions is described by Ahk
ip [f;w0](p→j) that
models the transition probability that a ip-candidate particle ends up into the state
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24 N. Bellomo, M. Egidi
of the test ij-particle after an interaction with a field hk-particle. Test ij-particle
loose their state after interaction with a field hk-particle. Also in this case a set of
parameters can be defined and inserted in w0.
•Micro-macro interactions model the dynamics of candidate and test particles that
are subject to the action of the other FSs viewed as a whole represented by their
mean value of the activity Ek. Interaction occur with rate µk
ip[fi,Ek;w0], while the
output of interactions is described by transition probability Bk
ip[fip ,Ek;w0].
•Square brackets are used to denote functional dependence in η,A,µand Bon f
and w0.
The derivation of the mathematical structures is obtained by a balance of the
number of a-particles within an elementary volume of the space of microscopic states
of the active particles. Technical calculations yield:
d
dt fij =Gm
ij [f;w0]− Lm
ij [f;w0] + GmM
ij [f;w0]− LM
ij [f;w0]
=
n
X
p=1
n
X
h=1
m
X
k=1
ηhk
ip [f;w0]Ahk
ip [f;w0](p→j)fipfhk
−fij
n
X
h=1
m
X
k=1
ηhk
ij [f;w0]fhk
+
n
X
p=1
m
X
k=1
µk
ip[fip ,Ek;w0]Bhk
ip [f;w0](p→j)fip Ek
−fij
m
X
k=1
µk
ij [fip,Ek;w0]Ek,(4.3)
where the operators Gij and Lij denote the gain and loss of the number of a-particles
in the elementary volume of the space of the micro-states. The superscript refers to
the type of interactions considered in the model, i.e. mand mM for micro-micro
and micro-Macro, respectively,
A technical generalization of the structure can take into account transition
across functional subsystems. In this case the structure of the transition probability
should be modified as follows:
Ahk
ip [f;w0](p→j)(i→r) = Bhk
ip [f;w0](p→j)× Chk
ip [f;w0](i→r),(4.4)
while summations should also include those with respect to i. Then, the model can
also represent aggregation-fragmentation dynamics.
The derivation of models can be obtained by the a mathematical description
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From Simon’s Legacy to the Heterogeneous Artificial World 25
of the map from the pre-interaction states to the post-interaction ones according
to by a phenomenological description of interactions in the real world. Various
applications, see 2, have shown that each type of model needs specific interactions
and related parameters which include the utility functions.
Some applications using different approaches based on the kinetic theory are
known in the literature. These are reviewed in 2, by models based on the assumption
of constant interaction rules. A few models, see 32, have also considered how the
dynamics can bring advantages/disadvantages to the whole population by taking
into account both the increase/decrease of wealth and the distribution of wealth.
In particular, this distribution provides an indication on the size of the middle
class. This is an important indication as a population in which the middle class is
excessively reduced may split into antagonistic groups, i.e., very poor or wealthy.
In some cases, this radical division may be difficult to manage and unproductive in
terms of initial consumption of products.
4.2. Towards modeling a dynamical world
Let us now consider how the modeling of steady world dynamics can be further
developed to describe dynamics in an artificial world, where both micro-micro and
micro-macro interactions can be modified by internal dynamics and external actions.
The transfer of the approach reported in Subsection 4.1 to evolutionary dynamics,
consistent with the vision of Herbert Simon, requires a dynamical interpretation of
the interactions that appear in the formal framework defined in Eqs. (4.3). This
provides a formalized interpretation of the blocks B6–B9 in Fig. 1.
Modeling the transition probability can be developed using two types of pa-
rameters:
w=w(t;f) refers to modeling the pre-interaction to the post interaction map,
where wcorresponds to the dynamical interpretation of w0. In general wcan be a
vector.
α=α(t;f,w) refers to modeling the relative weight of two types of interactions, as
an example: consensus and contrast interactions.
The dynamic interpretation of αconsiders that the selection of one of the
two dynamics is not equally shared in the whole population. Let Γ1and Γ2define
the map from the state of the interacting entities corresponding to consensus and
contrast. Then, the resulting interaction is weighted as follows:
I=α(t;f,w)(t) Γ1[f] + (1 −α(t;f,w)(t))Γ2[f],(4.5)
with 0 ≤α=α(t;f,w(t)) ≤1, ∀t≥0, while the dynamics both of αand w
correspond to the bounded utility functions.
In practice, the outcome of the interactions between a-particles with states
uip and uhk depends on to individual strategies involving candidate and field (test
and field) a-particles and on the search for individual utility, and can be formally
September 9, 2023 17:22 WSPC/INSTRUCTION FILE NB-ME-Sub-10-
09
26 N. Bellomo, M. Egidi
expressed as follows:
uij = Γ[f](uip, uhk ;α(t;f,w(t)),w(t))
=α(t;f,w)Γ1[f](uip, uhk ;w(t))
+(1 −α(t;f,w))Γ2[f](uip, uhk ;w(t))).(4.6)
The qualitative representation of the dynamics of interactions is shown, for
continuous activity, in Figs. 2 and 3, which refer to interactions corresponding to
consensus and dissent. The actual structure of the map Γ and of the parameters
αand wshould be referred to specific applications as we will see in the next
subsection. A quantitative description requires additional heuristic assumptions on
the structure of the map.
As a simple example, one might suppose a map of the type:
Ahk
ip [f](p→j) = δu−Γ[f](uip, uhk ;α(t;f,w),w)(4.7)
where δdenotes the delta-Dirac function and where the modeling of Γ[f] takes into
account the utility function. The same reasonings can be focused on micro-macro
interactions.
Let us now consider the key problem of the passage from the steady to the
dynamical world. The approach we propose consists in modeling the parameters
αand was variables whose dynamics might be modeled as a system of ordinary
differential equations:
d
dt fij =
n
X
p=1
n
X
h=1
m
X
k=1
ηhk
ip [f, α, w]Ahk
ip [f, α, w](p→j)fip fhk
−fij
n
X
h=1
m
X
k=1
ηhk
ij [f, α, w]fhk,
+
n
X
p=1
m
X
k=1
µk
ip[fip ,Ek, α, w]Bk
ip[f, α, w](p→j)fip Ek
−fij
m
X
k=1
m
X
k=1
µk
ij [fip,Ek, α, w]Ek,
dw
dt =K(f, α, w),
dα
dt =a(f, α, w).
(4.8)
The dynamical system (4.8) can be particularized for each specific system under
consideration. However, before going to applications, it is useful making precise the
key features of the general approach. In detail:
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From Simon’s Legacy to the Heterogeneous Artificial World 27
•The mathematical structures in Eqs.(4.3) and (4.8) can capture, as far as
it is possible, the main features of living systems. Mathematical models are
derived by implementing this structure with a mathematical description
of the interactions for each specific case study. The theory accounts for
the heterogeneous behavior of the interacting entities. These undergo both
individual based interaction and those with the whole system corresponding
to low order moment, for instance the mean value of each FS.
•The steady world studied in Subsection 4.2 already captures, by means of a
differential system, see Eq.(4.3), some of the key philosophical statements
of Simon’s theory. The dynamical description delivered by the augmented
system in Eq.(4.8) provides a further development to take into account how
interaction rules evolve in time.
•Interacting entities have in mind a bounded utility function that affects the
set of variables wand αand, consequently, the outcome of interactions. This
function may evolve over time under the overall influence of the dynamics
of the system. The term bounded is used to denote that the design of this
function should take into account the not fully rational behavior of the
interacting entities.
•The mathematical structures can be further developed by taking into ac-
count the role of external actions by introducing an additional transition
probability corresponding to the interaction between the internal and the
external system. These actions can modify the rules of interaction. We refer
to the so-called invisible hand, see 37,38, one could think of governments in
each country or across them, or other agents such as the action of financial
markets.
•The difficulty in measuring the activity variable faces the the lack of phys-
ical units. A pragmatic way is to find the highest observed value uM, take
the minimum value um= 0 and normalize the variable with respect to
uM, so that u∈[0,1]. Special cases are when the activity coexists with
the expression of its opposite. Then the assumption u∈[−1,1] denotes
the expression by positive values and the opposite by negative values. This
reasoning holds for discrete states defined as follows:
j=−m, . . . , j = 0,and j= 1, . . . , m = 1,
with u−m=−1, um= 1 and u0= 0.
•A technically simplified structure can be obtained from (4.8) by assuming
that an individual entity with state Pcan move, after interactions, only to
p−1 or p+ 1.
•In specific cases αmay be a variable obtained by a dynamic that takes into
account the variation in time of w.
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09
28 N. Bellomo, M. Egidi
In the case of the last two items, that can be considered in the application
dealt with in the next subsection, the mathematical structure is as follows:
d
dt fij =
p=j+1
X
p=j−1
n
X
h=1
m
X
k=1
ηhk
ip [f,w]Ahk
ip [f,w](p→j)fip fhk
−fij
n
X
h=1
m
X
k=1
ηhk
ij [f,w]fhk,
+
p=j+1
X
p=j−1
m
X
k=1
µk
ip[fip ,Ek,w]Bk
ip[fip ,Ek,w](p→j)fipEk
−fij
m
X
k=1
µk
ij [fij ,Ek,w]Ek,
dw
dt =K(f,w).
(4.9)
4.3. On the modeling of structures and organization
The application of the theory is shown with reference to the case study described in
Section 2. In particular, we consider Organizational Structures, where the internal
dynamics include support and contrast to innovation programs. These dynamics are
considered in a general framework without specializing in the type of innovation. We
do refer to the flowcharts in Fig1. 1, integrated by Fig. 3. Therefore, we consider
the sequence of blocks and, for each (or aggregation) of them, we show how the
formalization into specific models can be derived.
The following abbreviations are occasionally used:
OSs for Organizational Structures;
m-m for micro-micro interactions;
m-M for micro-macro interactions;
U-F for the utility function.
This application is presented with tutorial aims by a qualitative descriptio that
is preliminary to the quantitative study to be developed by inserting real input data.
We first consider models that include both m-m and m-M interactions referring to
the structure in Eq. (4.9). We then show how interactions with the external world
can be considered and, in general, how the model can be further refined.
The sequential steps of the derivation of the models are reported below, refer-
ring where useful, to the blocks of Figures 1 and 3, where Block 5 corresponds to the
already derived mathematical structure reported in Eq. (4.3) and to the transfer of
this equation to Eq. (4.9) through Eq. (4.8) .
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From Simon’s Legacy to the Heterogeneous Artificial World 29
D1: Selection
of interactions
micro-micro
and micro-macro
D2: Learning
and design of
a bounded
utility function
D5: Dynamics of the
interaction rules
w0→w(t),
α0→α(t).
D3: Learning
by
interactions
D4: Learning
by
external actions
D6:Mathematical
models of the
artificial world
D7:Parameter
sensitivity study
towards validation
Fig. 3. From learning, to the utility function, and to the dynamics in the artificial world
Step 1. This step refers to blocks B1 and B2 and considers Organizational Struc-
tures that promote an innovation structure through internal actions and interac-
tions. We first consider two FSs, where the first one is interested in supporting
innovation, while the second one opposes it. The activities, they express are con-
sensus and opposition (contrast), respectively.
This description considers a dynamic, in which one of the two FSs becomes
dominant, whereas the other progressively weakens, while the dynamic of the vari-
able αis determined by the description of the system delivered by the initial value
problem for Eq. (4.9).
Step 2. We refer to block D2 of Fig. 3, see also block B5 of Fig. 1. This step
corresponds to the design of the UFs, which is different for each FS:
The trend in FS1 corresponds to the additional economic profit. In general, the UF
induces an increase in consensus.
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30 N. Bellomo, M. Egidi
The trend in FS2 is simply motivated by maintaining the status quo, the UF does
not modify the trend, but the increase of the UF weakens the trend to contrast.
Step 3. The focus is on the modeling of the interactions depending on each type
of interaction both in terms of the types of m-m and m-M interactions and the FSs
involved in the dynamics. These considerations refer to the interaction rates and
transition probabilities, see blocks B3 and B6. The modeling can be developed ac-
cording to the phenomenological assumptions proposed in what follows by a simple
model characterized by a small number of parameters. This strategy can contribute
to the parameter sensitivity analysis.
Interaction Rates: A simple modeling approach is to assume a constant value for
all types of interactions, i.e. ηhk
ip =η0and µh
ip =µ0for all i, p, h, k.
Transition Probability: A simple assumption is that single entities, i.e. a-particles,
with state pcan only go, in probability, to p−1 or p+ 1, or keep the state p. The
model should consider that the sum of the probability of the three event is equal to
one.
Interactions within FS1: The m−minteractions produce interactions that, in prob-
ability, show a tendency to improve consensus. This trend decreases with increasing
state as A1k
1p=w1(1 −p). The same trend is expressed by M−minteractions, but
only when Ek> p:B1k
1p=w2(1 −p).
Interactions within FS2: Individual entities of FS2 try to maintain the status quo.
Therefore, m-m interactions are insensitive to UF and maintain their state even
if UF increases. Interactions could generate aggregation of low contrast entities to
high contrast state by interaction of particles with higher state. On the other hand,
a-particles of FS2 are sensitive to the overall mean of the whole system and show
a tendency to decrease their contrast when the mean shows an increasing trend
towards consensus.
Interactions between FS1 and FS2: This type of interactions do not give significant
modifications. However, the macroscopic quantity Efrom the whole system act on
both FSs.
Step 4. The focus is on the dynamics of the parameters w, specifically on modeling
the dynamics of these parameters, see blocks D5, D6.
Step 5. This step refers to simulations developed to solve the initial value problem
for the system in Eq. (4.9), given initial conditions for the probabilities fip(t= 0) for
all iand pand for α0and w0. Numerical methods to solve the problem are standard.
Increasing numbers of FSs and j-states may require algorithms specifically designed
for swarms 46.
The above steps can be further developed to improve and generalize the mod-
eling approach. In addition, modeling and simulation can be used to understand
how external policies can contribute to improving consensus and, consequently, the
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From Simon’s Legacy to the Heterogeneous Artificial World 31
overall well-being of populations. The following points suggest how the model can
be further developed to improve the contribution of mathematics to economics.
(1) Subdivision into FSs: Specific applications may require a further subdivision
of FSs by specializing the role of each individual in the organization, but the
activity they express is the same. For example, in the case of subdivision into
levels of responsibility, these may influence the level of expression.
(2) Transition across FSs: The Activity variable can be modeled with negative
and positive values, i.e. u∈[−1,1], where negative values for FS1 correspond
to contrast-opposition, while for FS2 they correspond to consensus. Transitions
occur whenever activity reaches the value u= 0.
(3) Utility Function: The output of the interaction depends on the utility func-
tion. This quantity evolves over time as it depends on both the internal dynam-
ics and the interaction with the external world, which can be considered as an
open market.
(4) External actions: These actions can be considered by adding one of several
FSs, for example corresponding to government incentives, can be modeled by
FSs acting on the internal system through models that depend on the specific
feature of the action.
(5) Simulations should be developed not only to show the dynamic response of
the system, but also to develop a sensitivity analysis for the parameters, utility
function, and external actions. This general study is necessary study to the
mathematical model.
5. A forward look at perspectives
This paper has shown how a new vision of the mathematical theory of active par-
ticles can be developed to describe the dynamics of large systems of interacting
behavioral entities, consistent with the Simon’s theory of the artificial world. The
key contribution is a sharp study of the rules that describe the output of interactions
as they evolve in time.
According to the achievement of this paper, we can state that the theory pro-
posed in this paper represents a further contribution to the science of living systems,
specifically, to the method reviewed in 13. Indeed, the mathematical formalization
of Simon’s theory of the artificial world and artifacts has suggested a revisiting of
the mathematical study of interactions, i.e., the key topic of the KTAP theory.
Once a mathematical model has been derived, simulations can show the time
evolution predicted by the model and its sensitivity to parameters. This study can
lead to emergent behaviors. Validation of models can also be based on the study
of emergent behaviors, including those that are not easily predicted. Specifically,
those defined by Taleb 77 as black swan.
In some cases, simulations show an emergent behavior that has not been ob-
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32 N. Bellomo, M. Egidi
served in empirical data. Then it is worthwhile to develop further studies to discover
the existence of the trend to the aforementioned behavior.
In parallel to the numerical study, analytical problems can be studied. In gen-
eral, these are challenging because the above models present different types of non-
linearity. Indeed, the mathematical structure involves a product of the dependent
variables. Furthermore, the interactions depend not only on the state at the mi-
croscopic scale, but also on the micro- and macro-state. This multiscale character
of the mathematical tools motivates the challenging analytical problem of deriving
macroscopic equations from the underlying microscopic description. This problem,
somewhat inspired by Hilbert’s Sixth Problem 52, has been treated in 11,22,23 for
models where interactions follow time-constant rules. The study of this problem for
the class of equations treated in this article is still open.
Without repeating the reasoning already discussed in the previous section, we
believe that the main contribution of our paper is to propose a general framework
to which the qualitative and computational study of complex behavioral systems
can refer.
•Economics: An important field of application is that of the sciences of eco-
nomics. The literature on the application of mathematical tools inspired by statis-
tical physics (in particular to the mathematical kinetic theory) has been reviewed
in 34 and, more recently, in 16, where a variety of applications have been reported
and critically analyzed. Uncertainty is a common feature of behavioral systems 35.
The mathematical tools proposed in our paper can provide a quantitative estimate
of the probability distributions that describe the state of these systems.
A first perspective is to revisit the above reviewed articles within the framework
proposed in our paper. The aim is to understand to what extent the interaction by
time-dependent rules modifies the dynamics corresponding to time-invariant rules.
An important objective of simulations is the study of emergent behaviors that
appear for specific values of the model parameters. In fact, the validation of mod-
els should also consider their ability to reproduce emergent behaviors observed in
reality.
•Multi-dynamics: The dynamics of living systems often, actually always, appear
within the framework of a multi-dynamics, where this term is used here to denote the
interaction of different types of dynamics. For example, social, dynamic, economic,
political, biological, etc. These studies have recently been motivated by the SARS-
CoV-2 pandemic, where a multi-dynamic study soon appeared necessary, see 14.
Then, interesting interdisciplinary studies have been generated, see 1,49,31. The
mathematical-economic theory proposed in our paper can be further developed
towards a multi-dynamics theory. This challenging goal requires to overcome addi-
tional conceptual difficulties related to the interpretation of interactions, where the
utility functions should take into account all variables of the interacting functional
subsystems.
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From Simon’s Legacy to the Heterogeneous Artificial World 33
•What is life? This key question belongs to the celebrated Schroedinger’s essay 69.
It was also used in the title of 13. This question opens to the mathematics of living
systems. The answer cannot, definitely, be simple and concise. Indeed, it is not even
available in the literature. The key difficulty in dealing with the modeling of living
systems has been clearly stated by Robert May 59 who identifies this difficulty with
the lack of field theories somewhat analogous to those of physics. Another key point
is given by Ernts Mayr 60 concerning the need to include evolution and selection in
the description of living systems.
As we have seen, analogous hints are given by Leeland Harwell 51, further de-
veloped by Herbert Simon’s philosophy of the artificial world. May’s and Mayr’s
hints are tackled in 12 according to the idea of searching for mathematical struc-
tures suitable to capture the main complexity features of living systems. Additional
interesting philosophical reflections on living systems are proposed by Carol Cle-
land 29, whereas a collection of contributions to this topic under consideration are
given in 10. These two contributions show that the topic is meeting great attention
in the scientific world and that the path that leads to a mathematical theory needs
the design of a philosophical foundation as we have done with specific to Simon’s
legacy.
Indeed, we have interpreted Simon’s legacy as a hint to develop his theory
towards the mathematical description of interactions by rules that evolve in time
also under the action of the individual entities that play the game. Indeed, our paper
has specifically tackled the above problem. In addition to the above reasonings, we
focus on the following statement:
Life is a multi-dynamic and multi-scale complex system.
Therefore, the development of the aforementioned multi-dynamic and multiscale
theory can, of course not yet exhaustively, contribute to the science of living systems.
Acknowledgments
Nicola Bellomo acknowledges the partial support by the State Research Agency
(SRA) of the Spanish Ministry of Science and Innovation and European Regional
Development Fund (ERDF), project PID2022-137228OB-I00, and by Modeling Na-
ture Research Unit, project QUAL21-011.
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