2.1 On the Derivation of Mathematical Models

Our goal is to build a bridge between mathematics and economics. Therefore, for the purposes of the tutorial, we provide some technical definitions that are useful for a general methodological approach to modelling dynamical systems. Indeed, it is important to follow a general rationale instead of applying heuristic methods that may be valid in certain case studies, but cannot be easily applied to other case studies.

We refer specifically to a large system of interacting, behavioural entities that constitute a system in economics. We consider systems with a constant number of entities. Therefore, the overall state of the system can be provided according to the specific scale chosen for representing and/or modelling the system. In particular, one can consider the following scales and related classes of equations that provide the mathematical structure for deriving models.

Microscopic when the overall state is defined by the state of each individual entity, it can be called active particle, or a-particle for short. The state of the a-particles is called activity. Microscopic models are usually described by systems of ordinary differential equations.

Mesoscopic when the overall state is defined by a probability distribution over the state of the microscale entities that make up the system, called active particles. Mesoscopic scale models are generally described by systems of transport equations with the structure of integro-differential equations.

Macroscopic when the overall state is defined by the local mean and higher-order moments of the activity. Microscopic models are usually described by systems of partial differential equations.

This brief introduction naturally leads to an understanding of the general concept of the mathematical model. We focus on the lower scales, while the macroscale quantities are obtained by local averaging of the microscale activity. The following definitions are proposed for this purpose.

Independent and Dependent Variables:

Independent variables are time and space, which are also defined independently of the system object of the modelling approach. Dependent variables are the quantities that depend on the time and space selected in the mathematical model to describe the overall state of the system. These variables are sometimes referred to as state variables. The state variable at the micro scale is defined by the set of all activities expressed by the a-particles, since the activity is heterogeneously distributed among the a-particles. At the mesoscale, the system is defined by a probability distribution over the microstate.

Definition of a Mathematical Model:

A mathematical model is an equation or computer description suitable for describing the dynamics of a system. Models are generally referred to as mathematical structures, which are related to the scale chosen to represent and model the system.

Functional Subsystems and Endogenous Networks:

A-particles can express different types of activities. Therefore, the whole system can be divided into different aggregations called functional subsystems, or FS for short. Endogenous networks correspond to a system of nodes, where each FS is localized in a node. The communication between the nodes, namely in the network, can take place through different communication devices, while spatial dynamics is not taken into account.

Exogenous Networks:

Space collocation is considered in exogenous networks, where nodes correspond to a physical collocation. Each node consists of an endogenous network. Therefore, the two networks are subject to a complex dynamics of interactions.

All the preceding definitions naturally lead to the definition of mathematical models.

2.2 On a Quest towards Model of Living Systems

Behavioural systems in economics can be considered as a special class of systems belonging to the broader class of living systems. Therefore, a mathematical approach has been developed that takes this special feature into account, as first reported in Reference BellomoBellomo (2008) and later in Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017). These books refer to the kinetic theory for active particles.

Subsequently, a pseudo-Newtonian approach was proposed in Reference Bellomo, Ha and OutadaBellomo, Ha, and Outada (2020) to pursue the same goal. This approach has been called the mathematical theory of behavioural swarms.

The next section presents an overview and critical analysis of the two aforementioned methods, while here we simply report on the philosophy that gave rise to these methods. Actually, we refer to the reasoning given in Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017) for the first method and show how it provides the pseudo-Newtonian methods as well. The key point is that the search for a mathematical theory of living systems requires a strategy suitable to replace the background theories, consisting of conservation and/or equilibrium equations, which refer to systems of inert matter and which may include source/sink terms related to competition and dissipation/decay somehow related to entropy functions.

In fact, living systems are different. This was noted by Immanuel Kant (1724–1804), who defined living systems as special structures organized and with the ability to pursue a purpose (Reference KantKant, 2000).

It may be noted that Lee Hartwell (born 1939), Nobel Laureate, expressed an analogous concept: ‘Although living systems obey the laws of physics and chemistry, the concept of function or purpose distinguishes biology from other sciences. In fact, cells are not molecules, but have living dynamics induced by the lower level of genes and are organized into organs’ (Reference Hartwell, Hopfield, Leibler and MurrayHartwell, Hopfield, Leibler, and Murray, 1999, p. 647).

This statement concerns the organizing capacity of cells to organize themselves into biological structures, which opens up the search for the complex connections between genes and cells, where the dynamics at the level of genes induce the dynamics at the level of cells. This multiscale vision applies not only to biology, but to all living systems in general, including the economy.

Furthermore, Herbert A. Simon (1916–2001) teaches us how behavioural features should be considered by economic systems, since several aspects of the dynamics of systems in economics are influenced by individual strategies generated by learning and the ability to express specific strategies up to decision making. These features, which are evolutionary in time, are heterogeneously distributed in the individual or aggregate active particles that play interactions (games) in economics; see Reference SimonSimon (1978), Reference Simon(2019). Simon, similar to Hartwell, states that formally structured theories are possible for economics and gives hints on how to pursue this goal. Such a theory should be developed within a multiscale vision in which the dynamics at all scales are constantly interacting.

It is not surprising that most of the search for a mathematical theory of living systems has been related to biology, as evidenced by some pioneering work; see Reference Bellomo and ForniBellomo and Forni (1994), Reference Bellouquid and DelitalaBellouquid and Delitala (2006), and Reference Jager and SegelJager and Segel (1992). These are examples where the preceding concepts have been used in different mathematical settings, but without the framework of a global strategy. More recently, however, research has focused on social systems and behavioural and evolutionary economics; see Reference Ajmone Marsan, Bellomo and GibelliAjmone Marsan et al. (2016) and Reference Bellomo, Dosi, Knopoff and VirgillitoBellomo, Dosi, Knopoff, and Virgillito (2020).

Given the preceding brief introduction, we can now present the modelling strategy, which consists of two steps. First, we consider the derivation of a mathematical structure suitable to capture the complexity features of living systems. Then, the second step consists in deriving models by inserting into such a structure models of interactions for the specific system object of the modelling approach.

Step 1 is performed through the following sequence of blocks shown in Fig. 1:

• Block 1 Phenomenological interpretation of living systems to extract the specific features of complexity, as well as the phenomenology of the dynamics of interactions at the micro-scale.

• Block 2 Identification, from the study in Step 1, of the activity expressed by the active particles, and subdivision of the overall systems into groups of the same interest, already defined functional subsystems.

• Block 3 Design of mathematical structures suitable for describing m-m and m-M interactions, where these abbreviations correspond, respectively, to interactions between microscopic entities and between microscopic and macroscopic entities.

• Block 4 Design of mathematical structures suitable for describing the external actions acting on active particles and on the system as a whole.

• Block 5 The derivation of a general mathematical structure capable of capturing the complexity features defined in Step 1, consistent with the chosen representation and modelling scale. This structure should transfer micro-scale interactions to the description of collective behaviour.

Figure 1 Strategy towards mathematical structures to model living systems.

The approach is performed through the following sequence of blocks, shown in Fig. 2:

• Step 1 Derivation of mathematical models (Block 8) by inserting into the mathematical structure (Block 5) the interactions between a-particles (Block 6) and the external actions acting on active particles and on the system as a whole (Block 7).

• Step 2 Validation of models (Block 11) requires simulations (Block 9) and qualitative analytical studies (Block 10).

• Step 3 Validated models can be applied to specific case studies using simulations and analytical results (Block 12).

Figure 2 From mathematical structures to models and applications.

A selection of the most important complexity features is necessary, since we cannot naively claim that all of them are effectively considered. Our proposal is based on the authors’ bias in some way consistent with Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017). Accordingly, the following five common features and sources of complexity are selected.

• Ability to express a strategy is modeled by the activity variable, say behavioural “soft” variable, then all components can have a mutual influence. If the strategy contains both behavioural and mechanical variables, then the latter are influenced by the former.

• Heterogeneity can be modeled by a distribution function over the activity to account for the heterogeneous behaviour of a-particles, which can be viewed as probability distributions.

• Nonlinearity of Interactions: Modelling interactions generally leads to nonlinear additive outputs, which may also depend on the distribution functions. For example, models of opinion formation include the sensitivity of a-particles not only to individual a-particles, but also to individuals as a whole.

• Learning Ability: A-particles learn from past experience, as shown in Reference Burini and De LilloBurini and De Lillo (2019) and Reference Burini, De Lillo and GibelliBurini, De Lillo, and Gibelli (2016). As a consequence, the dynamics of the interaction are modified by the level of learning that is heterogeneously acquired by each individual. A key issue to be considered in modelling learning are the physics of the human mind, in Reference Schoeller, Perlovsky and ArsenievSchoeller, Perlovsky, and Arseniev (2018), and the complexity of the cognitive process, in Reference Perlovsky and SchoellerPerlovsky and Schoeller (2019).

• Darwinian Mutation and Selection: All living systems are evolutionary, as birth processes can produce entities that are either less or more adapted to the environment. The less adapted may even die out, while the more adapted may produce new entities that are better adapted to the external environment.

These complexity features, and additional possible ones, imply that the modelling approach always needs multiscale methods, where the dynamics at the large scale should be properly related to the dynamics at lower scales. For example, the behaviour of individual entities depends on their individual ability to develop strategies based on the amount of information stored in their minds due to past experience.

Collective behaviours are observed at the macroscopic scale, with the understanding that the dynamics of a few entities do not directly lead to the collective dynamics as already observed in Reference AndersonAnderson (1972). In addition, the role of the external environment can have an important influence on the dynamics. Therefore, appropriate models and parameters are needed to account for this type of action. The interaction with the external environment can generate mutations and selections, which can lead to the strengthening or weakening of FSs.

2.3 On the Validation of Mathematical Models

Let us now consider the validation of models by examining their ability to quantitatively reproduce empirical data, if available. Furthermore, models are also required to represent, on a qualitative level, the observed emergent collective behaviour. This is not an easy task, as several technical difficulties have to be taken into account. These are related to the multiscale structure of the systems under consideration and to their properties as living systems. The validation process should take into account that different types of models can be derived according to the different types of predictive description they can offer. In particular, two main typologies of models can be identified:

• Predictive models, which aim to predict, qualitatively and quantitatively, the behaviour of systems in time, given appropriate initial conditions and, if necessary, constraints on the solutions. In addition, models should also reproduce qualitatively emerging behaviours that are often qualitatively repeated within appropriate variations of the parameters of any mathematical problem.

• Exploratory models, which aim at exploring the behaviour in time of systems under programmed variation of parameters and external actions. In fact, in the case of open systems, the role of external actions must be studied to verify whether or not these actions have the ability to address the system towards specific behaviours.

The validation of models generally encounters technical difficulties. For example:

1. 1. Models are often derived at a scale different from that used to observe and collect empirical data. For example, interaction models refer to the microscopic scale, but collective behaviour is observed at the macroscopic scale.

2. 2. Collective dynamics are subject to large deviations depending on the parameters that model the interactions as well as the initial conditions.

3. 3. The amount of empirical data needed to develop a detailed validation process generally corresponds to steady uniform states, in some cases equilibrium, while models should describe dynamics far from such states.

Models could hopefully demonstrate a very special case of emerging behaviour, which is the occurrence of unpredictable events with the characteristics of the so-called black swan, defined as follows according to Reference TalebTaleb (2007): “A Black Swan is a highly improbable event with three principal characteristics: It is unpredictable; it carries a massive impact; and, after the fact, we concoct an explanation that makes it appear less random, and more predictable, than it was.”

3.1 On the Kinetic Theory of Active Particles

Let us consider the dynamics of a large system of a-particles, which can be divided into $m$ FSs, denoted by the subscript $i$ . The number $N$ of a-particles is assumed to be constant in time, since we do not consider birth and death dynamics. However, a-particles can move across FSs. This overview of mathematical tools refers to Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017), reviewed also in Reference Bellomo, Esfahanian, Secchini and TernaBellomo, Esfahanian, Secchini, and Terna (2022).

The state of the system is one whose state is defined by the distribution functions:

$f_i=f_i(t,u):[0,T]\times D_u⟶R_{+},i=1,\dots ,m,$(3.1)

where $D_u$ is the domain of activity $u$ and $T$ is the maximal observation time.

If $f_i$ is known and integrable, the low order moments provide the macroscopic state of the system. Specifically, the fraction of a-particles in each FS, with respect to $N$ , is given by

$n_i=n_i\left(t\right)={\int }_{D_u}{f}_i(t,u)du,$(3.2)

where

$\sum _{i=1}^m{n}_i\left(t\right)=1,$(3.3)

since we have assumed that $N$ is constant.

Similar calculations lead to the mean:

$E_i=E_i\left(t\right)={\int }_{D_u}u{f}_i(t,u)du,$(3.4)

and variance

$V_i=E_i\left(t\right)={\int }_{D_u}(u-E_i{)}^2{f}_i(t,u)du.$(3.5)

Of course, analogous calculations can be applied to the whole system.

The KTAP approach considers the following types of statistically identified a-particle:

• Test particle, which is supposed to be representative of the whole system, of the $i$ th functional subsystem with microscopic state, at time $t$ , supplied by the variable $u$ , whose distribution function is $f_i=f_i(t,u)$ .

• Field particle of the $k$ th functional subsystem with microscopic state, at time $t$ , defined by the variable $u^{*}$ , whose distribution function is $f_k=f_k(t,u^{*})$ . By field particles we mean the complete set of particles.

• Candidate particles, of the $h$ th functional subsystem, with microscopic state, at time $t$ , defined by the variable $u_{*}$ , whose distribution function is $f_h=f_h(t,u_{*})$ . Candidate particles are the field particles that end up in the test particle state after interaction.

Interactions modify the activity of a-particles, which can also move across FS. Each a-particle has a sensitivity range $\Omega \subseteq D_u$ and is therefore only sensitive to activity within $\Omega$ . Interactions may be non-symmetric. In this case the notation ${\Omega }_u$ is used to denote an activity dependent sensitivity. There are two kinds of interactions:

• Micro-micro interactions, in short m-m, by which each a-particle interacts with other a-particles of all FSs whose state falls in $\Omega$ .

• Micro-macro interactions, in short m-M, through which each a-particle interacts with all FSs identified by the mean value $E$ whose state falls within  ${\Omega }_u$ .

The modelling of the interactions can be supplied by the following quantities:

• m-m interaction rate ${\eta }_{ik}[f_i,f_k](u_{*},u^{*})$ , which models the frequency of micro-micro interactions between a candidate $i$ -particle with state $u_{*}$ and a field $k$ -particle with state $u^{*}$ .

• m-M interaction rate ${\mu }_{ik}[f_i,f_k](u_{*},E_k)$ , which models the frequency of micro-micro interactions between a candidate $i$ particle with state $u_{*}$ and the $F{S}_k$ with macroscopic state given by the mean value of the activity $E_k$ .

• Transition probability density due to m-m interactions, which models the probability density that a candidate $i$ -particle with state $u_{*}$ , after a micro-micro interaction with a field $k$ particle of state $u^{*}$ , is denoted by $A_{ik}[f_h,f_k](u_{*}\to u|u_{*},u^{*})$ .

• Transition probability density due to m-m interactions, which models the probability density that a candidate $i$ -particle with state $u_{*}$ ends up in the state of the test particle of the $i$ th FS after a micro-macro interaction with the $k$ -FS, is denoted by $B_{ik}[f_h,f_k](u_{*}\to u|u_{*},E_k)$ .

• Loss terms due to m-m interactions, which models the loss of the number of particles corresponding to micro-micro interactions described by $A_{ik}$ , is denoted by $L_i[f_i,f_k](u,u^{*})$ .

• Loss terms due to m-M interactions, which models the loss of number of particles corresponding to micro-micro and micro-macro interactions due to conservative interactions described by $B_{ik}$ , is denoted by $K_{ik}[f_i,f_k](u,E_k)$ .

The derivation of the mathematical structures that provide the conceptual framework for the modelling is obtained by the number balance of a-particles within an elementary volume of the space of microscopic states of the active particles. More specifically, the rate of change of the number of active particles is equal to the input flux minus the output flux for both micro-micro and micro-macro interactions.

Balance within the space of microscopic states can be described as follows:

$\begin{array}{cc}& \text{Variation rate of the number of active particles}\\ & =\text{Inlet flux rate caused by conservative interactions}\\ & -\text{Outlet flux rate caused by conservative interactions}\\ & +\text{Inlet flux rate caused by proliferative interactions}\\ & -\text{Outlet flux rate caused by destructive interactions},\end{array}$(3.6)

where all the proceeding fluxes include both micro-micro and micro-macro interactions, as well as the dynamics of mutations.

If only conservative interactions are taken into account, the preceding relation corresponds to the following structure:

$\frac{\partial }{{\partial }_t}{f}_i(t,u)=(G_i-{ℒ}_i+{\mathscr{H}}_i-K_i)[f_h,f_k](t,u),$(3.7)

where

$\begin{array}{cc}{G}_i& =\sum _{k=1}^n{\int }_{{\Omega }_u\times {\Omega }_u}{\eta }_{ik}[f_i,f_k](u_{*},u^{*})A_{ik}[f_i,f_k]\left(u_{*}\to u|u_{*},u^{*}\right)\\ & \times f_i(t,u_{*})f_k(t,u^{*})d{u}_{*}d{u}^{*},\end{array}$(3.8)

$\begin{array}{cc}{L}_i& =f_i(t,u)\sum _{k=1}^n{\int }_{{\Omega }_u}{\eta }_{ik}[f_i,f_k](u,u^{*})f_k(t,u^{*})d{u}^{*},\end{array}$(3.9)

$\begin{array}{cc}{H}_i& =\sum _{k=1}^n{\int }_{{\Omega }_u}{\mu }_{hk}[f_i,f_k](u_{*},E_k(t\left)\right)B_{ik}[f_i,f_k]\left(u_{*}\to u|u_{*},u^{*}\right)\\ & \times f_i(t,u_{*})E_k\left(t\right)d{u}_{*},\end{array}$(3.10)

and

$K_i=f_i(t,u)\sum _{k=1}^n{\mu }_{ik}[f_i,f_k](u,E_k(t\left)\right).$(3.11)

Let us now consider the modelling of interactions, where different models of interactions are reported in the following examples:

Competitive (Dissenting):

One of the interacting a-particles increases its status by taking advantage of the other, which is forced to decrease its status. Competition brings advantage to only one of them.

Cooperative (Consensus):

The interacting a-particles exchange their states, namely the a-particles with higher states decrease their states, while the others with lower states increase their states. All a-particles show a tendency to share their microstate.

Learning:

One of the two a-particles changes its microstate independently of the other. It learns by reducing the distance between the activities of the interaction particles. The distance in measured by an appropriate metric technically related to the specific dynamics under consideration.

Hiding:

One of the two tries to increase the total distance from the other, which in turn tries to decrease it.

Mixed competitive-cooperative:

A-particles do not share the same strategy, but some of them act competitively while others act cooperatively.

The preceding mathematical tools have been reported for a dynamics with constant number of particles. This assumption is consistent with the specific models of economics studied according the KTAP approach. On the other hand, the theory is more general and includes a variable number of individuals due to proliferative and/or destructive events. Therefore, the preceding structures need further developments that are reported in the following with the aim of providing a more general framework consistent with Eq. (3.6). The assumption that the previously mentioned event occur in the state of $i$ -particles leads to the following balance of particles:

${\partial }_t{f}_i(t,u)=(C_i-{ℒ}_i+{ℳ}_i^M-{ℒ}_i^M+P_i-D_i+P_i^M-D_i^M)[f_i,f_k](t,u).$(3.12)

If we only consider proliferative and destructive terms that are generated by interactions at the micro-scale, the following additional terms are considered:

$\begin{array}{cc}{P}_i& =\sum _{k=1}^n{\int }_{\Omega \times \Omega }{\eta }_{hk}[f_i,f_k]({\text{u}}_{*},{\text{u}}^{*})P_{ik}[f_i,f_k](\text{u},{\text{u}}^{*})\\ & \times f_i(t,\text{u})f_k(t,{\text{u}}^{*})d{\text{u}}^{*},\end{array}$(3.13)

and

$D_i=f_i(t,\text{u})\sum _{k=1}^n{\int }_{\Omega }{\eta }_{ik}[f_i,f_k](\text{u},{\text{u}}^{*})D_{ik}(\text{u},f_k)f_k(t,{\text{u}}^{*})d{\text{u}}^{*},$(3.14)

with the meaning of the terms $P_{ik}$ and $D_{ik}$ analogous to that of the conservative terms.

• Transition probability density due to m-m interactions, which models the probability density that a candidate $i$ -particle with state $u_{*}$ , after a micro-micro interaction with a field $k$ particle of state $u^{*}$ , is denoted by $A_{ik}[f_h,f_k](u_{*}\to u|u_{*},u^{*})$ .

• Transition probability density due to m-m interactions, which models the probability density that a candidate $i$ -particle with state $u_{*}$ , after a micro-micro interaction with a field $k$ particle of state $u^{*}$ , is denoted by $A_{ik}[f_h,f_k](u_{*}\to u|u_{*},u^{*})$ .

The formal expressions of $P_i^M$ and $D_i^M$ is not reported here, as they can be obtained by analogous calculations.

The preceding mathematical structures do not consider transitions from one functional subsystem to the other. However, some specific models might require the modelling of this specific feature. Examples can be found in political dynamics, where individual entities might move from one political orientation to another. This topic is postponed to Section 5, where it is revisited considering both methods, that is, KTAP and behavioural swarms, including their synergetic applications in specific case studies.

3.2 Mathematical Tools of Behavioural Swarms

The so-called theory of behavioural swarms, or TBS for short, was introduced in Reference Bellomo, Ha and OutadaBellomo, Ha, and Outada (2020) and subsequently applied to modelling and simulation of problems in economics, which will be reviewed in what follows. As in the case of the kinetic theory of active particles, the TBS approach refers to a mechanical system. Specifically, this theory is based on the Cucker–Smale model; see  Reference Cucker and SmaleCucker and Smale (2007), which describes the dynamics of a fixed number of self-propelled particles. This model is therefore a pioneering description of the dynamics of animal swarms.

It is useful to write the model because it helped to clarify the conceptual developments that lead to a new class of models that can be of interest in economics as well as in other fields of science related to the study of living systems. Let us consider an ensemble of mechanical Cucker–Smale particles whose states are represented by position and velocity variables. More precisely, let ${\text{x}}_i$ and ${\text{v}}_i$ be the position and velocity of the $i$ th C-S particle, respectively, in the free Euclidean space $R^d$ . Then the time evolution of the mechanical variables $({\text{x}}_i,{\text{v}}_i)$ is governed by the Newton-like system:

$\left(\begin{array}{c}\frac{d{\text{x}}_i}{dt}={\text{v}}_i,t>0,1\le i\le N,\\ \frac{d{\text{v}}_i}{dt}=\frac{\kappa }{N}\sum _{j=1}^N\varphi (\parallel {\text{x}}_j-{\text{x}}_i\parallel )\left({\text{v}}_j-{\text{v}}_i\right),\end{array}\right)$(3.15)

where $\parallel \cdot \parallel$ denotes the standard ${\ell }^2$ -norm in $R^d$ , $\kappa$ represents the strength of coupling, and the communication weight $\varphi :[0,\infty )\to R_{+}$ is bounded, Lipschitz continuous, and monotonically decreasing.

Let us now review the mathematical tools necessary to apply the theory. First, we define the interacting entities, their aggregation, and the rules of interaction that lead to the mathematical framework of TBS theory.

General System:

We consider a system $N$ of interacting living entities, all with the structure of an a-particle whose state is activity. We consider systems where space and speed variables do not modify the dynamics which is homogeneous in space. The mathematical structures underlying the derivation of the models can be derived within a pseudo-Newtonian framework, where for each a-particle, the action of the other a-particles produces an acceleration for the activity variable.

Active Particles and Functional Subsystems

The individual entities, i.e., a-particles, are carriers of a behavioural state, called activity, which defines their microstate. Active particles can be grouped into $m$ functional subsystems, often referred to by the acronym $j$ -FS, with $j=1,\dots ,m$ . Each individual a-particle, that is, $ij$ -particle, is labeled by the subscripts $i$ , where $i=1,\dots ,n_j$ , and $j$ , identifying each $i$ -particle and each $j$ -FS, respectively.

Microstate of a-Particles

The state of each a-particle is a scalar denoted by the variable $u\in D_u$ , which corresponds to the activity of the $ij$ particles. In general, the range is $D_u=[-1,1]$ . More generally, the activity can be a vector.

Interactions, as in the case of the KTAP approach, can be both micro-micro and micro-macro. In general, interactions are nonlocal and nonlinearly additive. A key property of $ij$ particles is the sensitivity domain ${\Omega }_{ij}\subseteq D_u$ within the space of the activity variable. The $ij$ particle perceives the presence of all $hk$ particles whose microstate lies within ${\Omega }_{ij}$ . The functions that model the interactions are the following:

• ${\eta }_{ij}^{hk}={\eta }_{ij}^{hk}\left(u_{ij},u_{hk}|u_{hk}\in {\Omega }_{ij}\right)$ models the interaction rate of micro-scale interactions between the $ij$ -particle with all $hk$ -particles in ${\Omega }_{ij}$ .

• ${\mu }_{ij}^k$ models the micro-macro interaction of the individual-based interactions between the $ij$ -particle and the $k$ -FS as a whole, which can be the mean value of the activity $E_k$ . Then, ${\mu }_{ij}^k={\mu }_{ij}^k(u_{ij},E_k)$ .

• ${\phi }_{ij}^{hk}={\phi }_{ij}^{hk}\left(u_{ij},u_{hk}|u_{hk}\in {\Omega }_{ij}\right)$ models the micro-scale action occurring at the rate ${\eta }_{ij}^{hk}$ , applied by all $hk$ particles in ${\Omega }_{ij}$ to the activity variable of the $ij$ particle.

• ${\psi }_{ij}^k={\psi }_{ij}^k(u_{ij},E_k)$ models the micro-scale action that occurs at rate ${\mu }_{ij}^k$ on the activity variable over the $ij$ particle by each $j$ FS.

These definitions lead to the mathematical structures by connecting cause and effect, namely actions on the a-particles, with pseudo-accelerations. In detail, $u_{ij}$ is the activity of the $ij$ particle and $v_{ij}$ is its velocity, defined as the time derivative of $u_{ij}$ . Then the mathematical structure modelling the dynamics of these dependent variables is a second-order system consistent with a pseudo-Newtonian framework:

$\begin{array}{cc}\frac{d^2{u}_{ij}}{d{t}^2}& =\sum _{h=1}^n\sum _{k=1}^m{\eta }_{ij}^{hk}\left(u_{ij},u_{hk}|u_{hk}\in {\Omega }_{ij}\right){\phi }_{ij}(u_{ij},u_{hk})\\ & +\sum _{k=1}^m{\mu }_{ij}^k\left(u_{ij},E_k\right){\psi }_{ij}^k\left(u_{ij},E_k\right).\end{array}$(3.16)

The mathematical tools reviewed in this subsection refer to a pseudo-Newtonian framework. The system is nonlinear as interactions are, as mentioned, nonlinearly additive. The book by Wei-Bin Zhang presents a review and critical analysis, specifically focused on economics, based on Newton-type methods that go far beyond application of Lotka–Volterra methods economics; see Reference ZhangZhang (2023). This book shows also how a qualitative and computational study of systems in economics can provide useful information on the predictive ability of models.

3.3 Critical Analysis towards Unified Theories

The two mathematical methods discussed in this section have been derived according to the same principles, but presenting substantial differences. In addition to those we have already presented, we will propose some considerations that can lead to new structures closer to each other in the spirit of the unification of physical-mathematical theories proposed in the Hilbert’s sixth problem; see Reference HilbertHilbert (1902). The considerations in this subsection are presented on a qualitative level, since their formalization in a mathematical framework should, in our opinion, be treated for each specific case study.

The approach of the kinetic theory requires that the specific systems under consideration consist of a sufficiently large number of active particles, while the theory of behavioural swarms requires a sufficiently small number, not only to avoid excessive computation, but also to have the possibility of identifying each particle individually. Actually, this is not important when the state of all particles is the same, but it may become necessary when it is heterogeneously distributed. However, these considerations are too vague to be used as a quantitative criterion. In particular, if you say large and small in reference to the number of a-particles, these adjectives are not quantitatively defined.

Of course, computational approximation methods can deal with this difficulty. For example, agent methods or differential systems with random initial conditions can be used. However, it is worth exploring further development of the kinetic theory approach to make it somewhat consistent with the behavioural properties of each specific system under consideration.

One method worth considering is the use of discrete values to represent the probability distribution. In fact, this approach is not simply an approximation device, but can be viewed as an interpretation of the model based on the assumption that particles whose state is confined to a small domain behave according to the same rules. This approach was used in Reference Bellomo, Dosi, Knopoff and VirgillitoBellomo, Dosi, et al. (2020), resulting in a mathematical tool that preserves the descriptive power of kinetic theory, for example the ability to describe transitions across FSs, and captures the specificity of the mathematical theory of behavioural swarms.

All topics mentioned in this subsection, as mentioned earlier, will be considered and revisited in Section 5. In fact, applications demand a unified vision of the two approaches. Therefore some further developments are necessary in both methods also, as their contextual use is, in some specific case-studies, useful. In this framework, it is important that modelling of interactions follow the same rules. The following list indicates the most important features that should be shared in both approaches. Theoretical tools of game theory can contribute to model them.

• Stochastic game theory deals with entire population of players, where strategies with higher payoff might spread over the population.

• The strategy expressed by individuals, namely, active particles, is heterogeneously distributed over players.

• Players are modeled as random variables linked to a distribution function over the activity variable.

• The pay-off is heterogeneously distributed over players and it can be motivated by “rational” or even “irrational” strategies.

• The payoff depends on the actions of the coplayers as well as on the frequencies of interactions. Both quantities depend on the overall state of the system.

• In the virtual world the payoff is substituted by the so-calld Utility Function.

These interactions generate nonlinearities in the structure used for modelling. Let us consider the mathematical structures of the KTAP approach presented in Subsection 3.1. These are nonlinear due to the product of the dependent variables $f_i$ . This is a quadratic nonlinearity analogous to that of the nonlinear Boltzmann-type models. If the terms modelling interactions, namely $\eta ,\mu ,C,M,P,D,\dots$ , do not depend on $[f_h,f_k]$ , but only on the activity variable, interactions are defined as linear. On the other hand, when these terms depend on $[f_h,f_k]$ , the interactions are defined as nonlinear. In this case, the mathematical structure presents a double type of nonlinearities and occasionally the term multiple nonlinearity is used.

The same considerations can be addressed to the dynamics of swarms, but in this case the micro-state variables are the dependent variables. Therefore, nonlinearity is referred directly to these variables.

If interactions within the domain $\Omega$ are considered, it is useful to understand more about symmetries of interactions. Then, if a-particles are sensitive with respect to $u$ , uniformly in $\Omega$ , interactions are symmetric, and otherwise are non-symmetric. This considerations can be referred to both types of mathematical structures.

4.1 A Brief Review of Applications of Active Particles Methods

The kinetic theory of active particles, as reported in the recent survey Reference Burini, Chouhad and BellomoBurini et al. (2023) and book Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017), was first developed to model the competition between cancer cells and the immune system, then further developments focused on modelling social dynamics such as opinion formation; see Reference Bertotti and DelitalaBertotti and Delitala (2004), Reference Bertotti and Delitala(2010). These papers were soon followed by studies on the dynamics of economic interactions with social and political dynamics: Reference BertottiBertotti (2010), Reference Bertotti and ModaneseBertotti and Modanese (2011), Reference Dolfin, Knopoff, Leonida and PattiDolfin, Knopoff, Leonida, and Patti (2017), Reference Dolfin and LachowiczDolfin and Lachowicz (2014), Reference Dolfin and Lachowicz(2015), and Reference Dolfin, Leonida and OutadaDolfin, Leonida, and Outada (2017).

The idea of linking social dynamics and economics is somewhat consistent with the guiding philosophy of behavioural economics; see Reference ThalerThaler (2016) and Reference Thaler and SunsteinThaler and Sunstein (2009). Then the applications moved to evolutionary economics, Reference Dolfin, Knopoff, Leonida and PattiDolfin, Knopoff, et al. (2017); see Reference Dosi and FreemanDosi (1984), Reference Dosi, Fanti and VirgillitoDosi, Fanti, and Virgillito (2020), Reference Dosi, Pereira and VirgillitoDosi, Pereira, and Virgillito (2017), and Reference Dosi and VirgillitoDosi and Virgillito (2021). This literature has also been reviewed in Reference Bellomo, Esfahanian, Secchini and TernaBellomo et al. (2022), which is a useful reference for this Element. The recently published book provides an excellent overview of the foundation of evolving economics; see Reference DosiDosi (2023).

Turning now to the kinetic theory of active particles, we note that this quest for a mathematical theory of living systems has a long history. Indeed, the idea of developing kinetic theory models in which the microstate includes an internal behavioural state was first developed in seminal papers in which this concept was introduced. We refer specifically to Reference Jager and SegelJager and Segel (1992), which focused on modelling the social dynamics of certain populations of insects, where interactions led to the splitting of the populations into dominant and dominated; and to the modelling of evolutionary immune competition between tumor and immune cells from Reference Bellomo and ForniBellomo and Forni (1994) to Reference Bellouquid and DelitalaBellouquid and Delitala (2006).

The scientific literature on swarms has been arguably initiated by physicists, while the interest of mathematicians was stimulated by the visionary, so-called Cucker–Smale model, see Reference Cucker and SmaleCucker and Smale (2007). Specific models can be derived for particles carrying a social variable, such as a social or political opinion; an analogous structure has been used to study and control the dynamics of the collective behaviour of one or more populations; see Reference Furioli, Pulvirenti, Terraneo and ToscaniFurioli et al. (2017). A key feature of the dynamics is to understand how internal variables, which we have called activity, evolve over time due to interactions and modify the collective behaviour of the system; see Reference Bellomo, Ha and OutadaBellomo, Ha, and Outada (2020).

The idea of modifying the Cucker–Smale model to describe the dynamics of financial markets has been proposed in Reference Bellomo, De Nigris, Knopoff, Morini and TernaBellomo, De Nigris, et al. (2020), while a first step in applying the theory of behavioural swarms to economics is in Reference Bellomo, De Nigris, Knopoff, Morini and TernaBellomo, De Nigris, et al. (2020), where activity is modeled to reproduce a perfectly competitive market with decentralized prices and elementary processes of price adaptation through swarm dynamics. However, the mathematical theory of behavioural swarms is quite recent and needs further development.

Therefore, further applications are expected once dynamics such as mutation and selection are introduced into the theory. These are necessary ingredients to derive models of evolutionary biological systems. However, evolutionary systems present common features and we argue that applications of the theory in economics will also consider this specific feature.

4.2 Buyers and Sellers and Price Dynamics

An application, consistent with the guidelines of our Element, focused on the modelling of price dynamics is proposed in Reference Knopoff, Secchini and TernaKnopoff et al. (2020) by a further development and enrichment of the so-called cherry picking dynamics. In this approach, the price quality is an important factor in the collective dynamics. In particular, this paper develops an earlier approach to this problem in Reference Bellomo, De Nigris, Knopoff, Morini and TernaBellomo, De Nigris, et al. (2020).

We borrow the general description of the model from the open access paper Reference Knopoff, Secchini and TernaKnopoff et al. (2020). This allows us to provide a description of the assumptions that lead to the derivation of the model without repeating all the analytical details that lead to the differential systems in Eqs. (7) and (9) in Reference Knopoff, Secchini and TernaKnopoff et al. (2020), which define the mathematical models.

In detail, consider a market in which $N$ sellers and $M$ buyers trade a given good, where both numbers are fixed. Sellers and buyers can be considered as functional subsystems. Within each FS, particles (i.e., sellers or buyers) express a heterogeneous activity. The activity variables for sellers and buyers are as follows:

• $u_s$ ,   $s=1,\dots ,N$ corresponds to the first functional subsystem (sellers), where each s-firm expresses the price $u_s$ of the product (good) offered for sale. The activity variable is the price assigned by each seller.

• $w_b$ ,   $b=1,\dots ,M$ corresponds the second functional subsystem (buyers), where each $b$ -buyer expresses the price $w_b$ that he/she accepts to pay. The activity variable is the price that each buyer accepts to pay for the good.

• $v_s$ ,   $s=1,\dots ,N$ and $z_s$ ,   $s=1,\dots ,M$ denote the speed of the variables $u$ and $w$ .

The knowledge of micro-scale variables leads to macro-scale quantities such as the $m$ -order moments within each FS:

$E_s^m=\frac{1}{N}\sum _{s=1}^N{u}_s^{m}and{E}_b^m=\frac{1}{M}\sum _{b=1}^M{u}_b^m.$(4.1)

The assumptions underlying the derivation of the model are as follows:

• Micro-micro interactions take place only between FSs, not within the same FS. In these interactions, firms and customers adjust prices through direct contact. Macro-micro interactions take place within the same FS, but not between different FSs. Through these interactions, each seller adjusts its price according to the average flow of sellers, while customers adjust the price according to the average flow of buyers.

• At each interaction, each buyer interacts with the seller having the lowest $u_s$ among those offering at least the reservation quality of the buyer. The chosen seller has the lowest $u_s$ among those offering at least the reservation quality of the buyer. The exchange takes place if $w_b\ge u_s$ at the price of $u_s$ . If so, the buyer’s reservation price $w_b$ tends to decrease while the seller’s price $u_s$ tends to increase.

• The opposite happens when the exchange does not take place (when the buyer does not accept the seller’s offered price because $w_b<u_s$ ). The term “tends to” is used to account for the change after the interaction does not act directly on the price, but on its velocity, because the model is based on a second-order system through swarm dynamics.

• In micro-micro interactions (described earlier), each agent adjusts its price by learning from past experience if its price (or reservation price) is too high or too low to sell or buy in the market, and changes it accordingly.

• At each macro interaction, each seller interacts with the other sellers as a whole, comparing its price with the first-order moment of its FS’s prices (the average seller’s price) and trying to move it closer to it. Again, through the information extrapolated by the micro-macro interactions, the seller learns and adapts its a-variable.

The general structure which provides the conceptual framework towards the derivation of models is as follows:

$\left(\begin{array}{c}\frac{d{u}_s}{dt}=v_s,\\ \frac{d{w}_b}{dt}=z_b,\\ \frac{d{v}_s}{dt}=\frac{1}{M}\sum _{q=1}^M{\eta }_s^q(u_s,w_q){\phi }_s^q(u_s,w_q,v_s,z_q)\\ +{\mu }_s(u_s,E_s){\psi }_s(u_s,E_s),\\ \frac{d{z}_b}{dt}=\frac{1}{N}\sum _{q=1}^N{\eta }_b^q(w_b,u_q){\phi }_b^q(w_b,u_q,z_b,v_q)\\ +{\mu }_b(w_b,E_b){\psi }_b(u_b,E_b),\end{array}\right)$(4.2)

for $s=1,\dots N$ and $b=1,\dots ,M$ .

Models are obtained by inserting into Eq. (4.2) a detailed modelling of the interactions, which are described by the following quantities:

• – ${\eta }_s^b(u_s,w_b)$ models the rate at which a seller $s$ interacts with a buyer $b$ ;

• – ${\eta }_b^s(w_b,u_s)$ models the rate at which a buyer $b$ interacts with a seller $s$ ;

• – ${\mu }_s(u_s,E_s)$ models the micro-macro interaction rate between a seller $s$ and her/his own FS;

• – ${\mu }_b(w_b,E_b)$ models the micro-macro interaction rate between a buyer $b$ and her/his own FS;

• – ${\phi }_s^b(u_s,w_b,v_s,z_b)$ denotes the micro-micro action, which occurs with rate ${\eta }_s^b$ , of a buyer $b$ over a seller $s$ ;

• – ${\phi }_b^s(w_b,u_s,z_b,v_s)$ denotes the micro-micro action, which occurs with rate ${\eta }_b^s$ , of a seller $s$ over a buyer $b$ ;

• – ${\psi }_s(u_s,E_s)$ denotes the micro-macro action, which occurs with rate ${\mu }_s$ of the FS of sellers over a seller $s$ ;

• – ${\psi }_b(w_b,E_b)$ denotes the micro-macro action, which occurs with rate ${\mu }_b$ of the FS of buyers over a buyer $b$ .

As mentioned earlier, further details can be found in Reference Knopoff, Secchini and TernaKnopoff et al. (2020). Here we conclude the presentation of the models with a few remarks that can guide the development of simulations.

4.3 Some Key Simulations

The model includes several parameters, among which the most important are the quality of the seller’s product and the quality of the buyer’s reservation, which is the minimum quality the buyer is willing to accept for the offered product. Also to be explained is the Pareto market efficiency: this quantity gives us a measure of the buyer’s ‘gain’. More specifically, it measures the difference between the maximum price the buyer was willing to spend and what he actually spent during the exchange with the seller.

The simulations developed in Reference Knopoff, Secchini and TernaKnopoff et al. (2020) have shown the following characteristics of the dynamic description of the system:

1. 1. When we reduce the interaction among sellers (we reduce the adjustment of sellers’ prices based on other sellers’ prices), buyers’ prices begin to coordinate and divide into functional subsystems. The reduced interaction among sellers is necessary for buyers to express collective behaviour. Buyers divide themselves based on their reservation quality (buyers with the same reservation quality end up in the same clusters).

2. 2. The coordination of buyers can occur in different numbers of clusters (from one to five), depending on the random initial conditions. This means that the formation of clusters is an endogenous effect. From an economic point of view, if allowed to do so, buyers manage to coordinate and create their preferred markets or the ones that are most convenient for them.

3. 3. In the classical framework, goods with quality differences generate multiple markets. From this perspective, it is impossible to analyze buyers’ behaviour in the face of quality differences. By considering micro-transactions with prices disclosed by the sellers (so-called adhesion contracts), we can instead study the effects of consumer control over quality on the market.

4. 4. This buyer control of the market can also be seen in the study of Pareto market efficiency. When buyers manage to coordinate and be part of their most convenient market, we can observe a higher market efficiency compared to the case in which the collective behaviour is not achieved. This means that when sellers create some kind of “agreement” about their prices (in the model, when they have a high interaction among them), buyers suffer from a disadvantage.

The following simulations have been developed using the same parameters as the ones in the figures showing collective behaviour in Reference Knopoff, Secchini and TernaKnopoff et al. (2020). Specifically, in these simulations, the parameter determining the seller-sellers interaction is lowered by one order with respect to the case in which the clustering does not appear. We can observe the evolution of buyers’ prices and their division into functional subsystems depending on their reservation quality, identified by different colors, right before and during coordination, in the short and long run.

In detail, we observe the emergent capability of the agents to act in a coordinate way, showing the same reaction to price changes and so creating close groups (clusters). In Fig. 3 we can see the prices before they are fully divided and coordinated. In Fig. 4 we have a general and long-term view of the prices and we can see clusters divided by quality. A closer look at the prices once they are coordinated can be seen in Figs. 5 and 6 (in the latter we can observe a rare five-cluster coordination, in a closer look).

Figure 3 Buyers’ prices: before coordination, view on the short run.

Figure 4 Buyers’ prices: coordination and emergence of four clusters. Whole view of the long run.

Figure 5 Buyers’ prices: coordination and emergence of four clusters. Closer look on the long run.

Figure 6 Buyers’ prices: coordination and emergence of five clusters. Closer look on the long run.

5.1 Extensions of the Mathematical Theories

An important objective of the research activity in this field consists in exploring the common features of the two approaches and in designing new mathematical structures suitable to make technically possible the passage from one type of mathematical method to the other. In this specific case, the transition is from the microscopic to the mesoscopic scale.

We start from the consideration that the assumption of a continuous distribution function requires that the number of a-particles is sufficiently large to justify the continuity assumption of the distribution function. This difficulty can be overcome by mathematical structures with discrete states.

The first approach to this goal is to consider a structure for number-conserving interactions:

$f_{ij}=f_{ij}(t;u_j);i=1,\dots n;j=1,\dots m,\text{f}=\left\{f_{ij}\right\},$(5.1)

where the a-particle belonging to the $i$ th functional subsystem with state $u_j$ is denoted as ij-particle whose state is denoted by $f_{ij}$ , while the state of the a-particle is given by $\text{f}=\left\{f_{ij}\right\}$

A general mathematical structure suitable for dealing with systems that undergo only conservative interactions was proposed by Reference Bertotti and DelitalaBertotti and Delitala (2004) and applied to modeling the distribution of wealth within a population of individuals who compete to increase their wealth but are also forced, through tax systems, to transfer some wealth from high to low levels of wealth. Their mathematical structure considered only binary interaction. The structure is as follows:

$\frac{d}{dt}{f}_{ij}=\sum _{h=1}^n\sum _{q,k=1}^m{\eta }_{iq}^{hk}{A}_{iq}^{hk}(q\to j)f_{iq}{f}_{hk}-f_{ij}\sum _{h=1}^n\sum _{k=1}^m{\eta }_{ij}^{hk}{f}_{hk},$(5.2)

where:

• ${\eta }_{iq}^{hk}$ models the interaction rate for encounters between an iq-particle and a hk-particle.

• $A_{iq}^{hk}(q\to j)$ models the probability of transition of the iq-particle to the state of the ij-particle due to encounters with hk-particle.

Technical calculation can lead to a more general structure, where a-particles can move across functional subsystems. In addition the encounter rate and the transition probability can depend on $\text{f}$ . This structure is as follows:

$\begin{array}{cc}\frac{d}{dt}{f}_{ij}& =\sum _{p,h=1}^n\sum _{q,k=1}^m{\eta }_{pq}^{hk}\left[\text{f}\right]A_{pq}^{hk}(p\to i,q\to j)f_{pq}{f}_{hk}\\ & -f_{ij}\sum _{h=1}^n\sum _{k=1}^m{\eta }_{ij}^{hk}\left[\text{f}\right]f_{hk},\end{array}$(5.3)

where:

• ${\eta }_{pq}^{hk}$ models the interaction rate for encounters between an pq-particle and an hk-particle.

• $A_{oq}^{hk}(p\to i,q\to j)$ models the probability of transition of the pq-particle to the state of the ij-particle due to encounters with hk-particles.

5.2 Dynamics of the Collective Learning

As mentioned earlier, learning dynamics is almost always present in the various dynamics that can be studied by the mathematical tools presented in this Element. In fact, the strategy developed by a-particles starts from an interpretation of the dynamics of the system through a collective learning process that supports decision making, which depends on the amount and quality of information acquired by the a-particles.

It is a collective learning that involves people who correspond to the a-particles of a functional subsystem or, in some cases, leaders who are carriers of a high level of knowledge that they transfer to the people. In this case we have two functional subsystems. An example that becomes more and more frequent in our society is that of interaction of influencers and followers. Therefore, it is useful in this Element first to define the main characteristics of this specific dynamic and, subsequently, how recent studies have approached this topic consistent with the conceptual framework of this Element. We refer to Reference BanduraBandura (1989) and Reference Salomon and PerkinsSalomon and Perkins (1998) for the following definitions:

• Collective learning is generally defined as a social process of cumulative knowledge, based on a set of shared rules and procedures that allow individuals to coordinate their actions in search for problem solutions.

• It is a cumulative dynamical process: as it accumulates over time. In the learning dynamic both a term of growth and a term of loss come into play, but the former is greater than the latter.

• Interactive, as the knowledge is transferred between agents through an interactive process.

• Individual/cognitive, which occurs in the mind of the individual as a social and participatory process.

• Social learning, which occurs when the individual learns new behaviours and concepts from others. Social learning is often combined with other dynamics.

Collective learning dynamics are qualitatively different in each specific case study. However, it is important to develop a general modelling methodology that can be specialized for each case study. This type of study has been developed by Reference Burini, De Lillo and GibelliBurini et al. (2016), with further specialization by Reference Burini, Gibelli and OutadaBurini, Gibelli, and Outada (2017) for the study of modeling teaching in high school classrooms.

The study by the approach of the kinetic theory of active particles. The sequential steps of the approach are as follows:

1. 1. Perception: Each individual possesses a perception domain within which the presence of other individuals is perceived with different intensities that depend, for instance, on the ‘distance’.

2. 2. Interactions and Learning: Interactions induce a learning process in which individuals modify (increase) their level of knowledge.

3. 3. Micro-Macro Interactions: Each individual also learns from the collectivity viewed as a whole.

5.3 Multi-model Systems and Networks

As we have seen earlier, different types of dynamics generally interact in the systems under consideration. In particular, learning dynamics interact with security problems, wealth distributions, and others. Therefore, it is important that the methods reported in this Element are addressed to modeling the dialogue between functional subsystems with the aim of transferring the information appropriate for the specific dynamics object of the modeling approach. Then the dialog involves influencers and followers. As a perspective, the modelling of price dynamics proposed in Section 4 could be reconsidered in accordance with the preceding considerations.

Some of the previously mentioned considerations can be applied to exogenous networks with specific focus on the interaction among nodes. Social dynamics in exogenous networks have been treated by Reference KnopoffKnopoff (2014); see also Reference Aguiar, Dosi, Knopoff and VirgillitoAguiar et al. (2021). A development of these studies is that the output of interactions should also include the previously mentioned division into influencers and followers by a specialization of the nodes into dominant and dominated.

The literature reviewed in this section shows that the methods presented in our Element refer to an interdisciplinary environment, where economics interact with other scientific areas. The reader interested in these interactions can find out about them in the following website:

www.modelingnature.org/training

A program of seven open access lectures for PhD students and young researchers

• Title: Collective Dynamics in Science and Society;

• Lecturers: Nicola Bellomo, Diletta Burini, Giovanni Dosi, Livio Gibelli, Damian A. Knopoff, Pietro Terna, Nisrine Outada, Maria E. Virgillito.

6.1 Hard Sciences and Social-Economical Sciences

As we have seen in Sections 1 and 2, we first proposed a philosophy to build the previously mentioned bridge and then showed how the theoretical concepts could be transferred into a mathematical framework. Of course, the philosophy retains the authors’ point of view and reasoning, but in the context of the theory of the artificial world and bounded rationality. Now we will take the liberty of giving a limited space to our bias. Accordingly, three specific arguments will be selected from among several possible ones and brought to the attention of the reader. Then, the final remark refers to the reasoning of the authors from the perspective of a mathematical theory of economics.

1. 1. The main difference between classical and active kinetic theory: Classical kinetic theory and active particles methods share some common features, but they have significant differences. The main difference is that the interactions in the Boltzmann equation are binary, local, and reversible, whereas in the case of active particles, the interactions are nonreversible and local. In addition, the interactions can also be non-symmetric and generally involve more particles as we will see in the next subsection.

2. 2. What about the theory of behavioural swarms? The properties of active particles also apply to behavioural swarms. Therefore, although the formal aspects of the structure are analogous, the interactions are completely different, and it is necessary to use new concepts such as pseudo-velocity and pseudo-acceleration. Again, the action on a particle is neither binary nor local.

3. 3. On the connection between microscopic and macroscopic description: This issue has already been raised in Section 2 with respect to model validation. In fact, the two methods discussed in this Element provide a description at the microscopic scale, either through the statistical description of kinetic theory or through the individual-based picture provided by behavioural swarm methods. On the other hand, empirical data are generally observed at the macroscopic level. Then, the derivation of macroscopic models from the underlying description at the microscopic scale in the spirit of Hilbert’s sixth problem, see Reference HilbertHilbert (1902), is motivated by the need of interpretation of empirical data toward validation of models. Some recent results have been proposed in the case of multicellular systems; see Reference Burini and ChouhadBurini and Chouhad (2022), Reference Burini and Chouhad(2023). An interesting perspective consists in extending these results to the case of behavioural agents.

4. 4. What about parallel approaches and possible developments? The two methods discussed in this Element are not exhaustive of the mathematical tools that have been developed recently. Indeed, agent methods should also be considered; see Reference Mazzoli, Morini and TernaMazzoli et al. (2019) in economics, and Reference GalamGalam (2012) in the study of social systems. For an excellent excursus on the development of methods from statistical physics toward applications to the study of social systems, see Reference HelbingHelbing (2010). Closer to the kinetic theory of active particles is the Boltzmann–Fokker–Plank approach developed in Reference Pareschi and ToscaniPareschi and Toscani (2013). Rather than comparing all methods, we propose, as a challenging perspective, their development into a unified mathematical theory.

5. 5. Artifacts towards a new vision of mathematical economics: We have learned from Herbert Simon the theory of artifacts and the virtual world. Applying this theory to the mathematical tools discussed in this Element means providing a mathematical description of interactions with rules that evolve over time and are modified by the interacting entities; see Reference Bellomo and EgidiBellomo and Egidi (2024). This means going far beyond deterministic population dynamics toward a unified mathematical theory.

These considerations indicate that for both mathematical approaches, kinetic theory and swarm methods, the first step is the modelling of interactions, where the modelling should be guided by the interpretation of what living systems are. This is not an easy task, as we can learn from Celand (2019), since we should understand the physical essence of what living systems are. A possible interpretation is given in Section 2 concerning the search for the main complexity features that characterize them. The last point of the preceding list indicates that interaction rules evolve in time following the evolution of the artificial world in which the specific system is localized.

The modelling approaches are similar, but we have shown in the previous sections that the two methods transfer low-scale dynamics to collective motion in different ways. At present, we can say that the kinetic theory method is complete with respect to the swarm approach, since it takes into account the specific features of interactions mentioned in the first point. Interactions also take into account post-Darwinian mutations and selections related to competition through proliferative and destructive encounters. The mathematical theory of behavioural swarms can account for heterogeneity of interacting entities, multiple interactions, and various types of nonlinearities. However, further studies are needed to complete this theory.

6.2 On the Development of Mathematical Tools

The previous subsection has indicated that the mathematical tools of generalized kinetic theory are currently the most convincing approach to a complete mathematical theory of behavioural systems in general. Therefore, it may be interesting to present a brief historical reconstruction of how the method developed from some pioneering papers to very recent results.

1. 1. Ilia Prigogine, Nobel Laureate, proposed a mathematical theory of vehicular traffic on highways based on the development of Boltzmann-like methods; see Reference Prigogine and HermanPrigogine and Herman (1971). The state of the system is described by a one-particle distribution function over the individual states of the driver-vehicle subsystem.

   Reference Paveri-FontanaPaveri-Fontana (1975) shows how interactions should be modeled by heterogeneous rules and introduces a probability description of the desired velocity distributed over the system. The modelling of interactions takes this specific distribution into account.

2. 2. Reference Jager and SegelJager and Segel (1992) propose a generalized Fokker–Plank model to describe the social behaviour of certain populations. The dynamics is that interactions divide the population into dominant and dominated.

3. 3. Reference Bellomo and ForniBellomo and Forni (1994) developed a kinetic theory approach that includes progression and competition between tumor and immune cells. The ideas proposed in this paper have been developed by various authors. The book by Reference Bellouquid and DelitalaBellouquid and Delitala (2006) presents a unified vision of the in-host competition. The competition ends either with the prevalence of tumor cells or with the successful defense of the immune system.

4. 4. Some mathematicians rapidly understood that an analogy of biological dynamics with some behaviours of living systems could lead to the development of a mathematical theory of living systems. Then, several interesting scientific papers appeared in the literature, see Reference Bertotti and DelitalaBertotti and Delitala (2004), Reference Bertotti and Delitala(2010) and Reference Bertotti and ModaneseBertotti and Modanese (2011), as examples of a rich literature developed in this century.

5. 5. The validity of kinetic theory for the modelling of the behavioural dynamics in different real-world systems has suggested to provide a formal presentation of the mathematical tools. This activity is documented in Reference BellomoBellomo (2008) for a modelling approach of Boltzmann-like models, and in Reference Pareschi and ToscaniPareschi and Toscani (2013), mainly focused on specific features of Boltzmann–Fokker–Plank models.

While the mathematical theories based on binary interactions appeared, some applications in different fields, such as vehicular traffic, see Reference Coscia, Delitala and FrascaCoscia, Delitala, and Frasca (2007), showed the need to model interactions, where the output is described by nonlinear maps with multiple interactions, where the utility functions could even depend on the distribution functions, that is, on the dependent variable of the models. Other applications have shown how the mathematical approach could model post-Darwinian mutations and selections; see Reference Bellomo and CarbonaroBellomo and Carbonaro (2011). Both dynamics should be considered in various fields of economics as documented in Reference DosiDosi (2023).

Accordingly, applications in economics have been developed; see, for example, the study of the blockage between economic development and political competition, Reference Dolfin, Knopoff, Leonida and PattiDolfin, Knopoff, et al. (2017), and the study of the role of nonlinear interactions in altruism and selfishness in welfare dynamics, Reference Dolfin and LachowiczDolfin and Lachowicz (2014). A comprehensive review of the literature in this area is provided by the review papers Reference Ajmone Marsan, Bellomo and GibelliAjmone Marsan et al. (2016) and Reference Dolfin, Leonida and OutadaDolfin, Leonida, and Outada (2017). The role of networks is also considered; see Reference Dolfin and LachowiczDolfin and Lachowicz (2014) and Reference KnopoffKnopoff (2014). This research motivated the mathematical theory formalized in Reference Bellomo, Bellouquid, Gibelli and OutadaBellomo et al. (2017).

The final step on this path is the mathematical interpretation of the dynamics within the artificial world of Herbert A. Simon. The recent contribution in Reference Bellomo and EgidiBellomo and Egidi (2024) takes into account Simon’s philosophy, which teaches that not only do interactions exhibit nonlinearities, but that the rules of interaction evolve over time due to the dynamics of the artificial world in which the interactions take place. Some seminal considerations can already be found in Reference KantKant (2000). The translation of Simons’ theory into a mathematical framework has opened up a new and challenging line of research that is definitely worth developing. The main contribution in Reference Bellomo and EgidiBellomo and Egidi (2024) is a mathematical theory that describes how the interaction rules follow the utility function that evolves over time in the artificial world.

6.3 Towards a Mathematical Theory of Behavioural Economics

Let us now return to the remark made at the very beginning of this Element about the idea of moving beyond mathematical models to a mathematical theory. The first difficulty is that systems in economics are in several cases behavioural. Then, they are carriers of most of the complexity features of living systems. The difficulty of mathematical approaches to describing the dynamics of living systems has been demonstrated by eminent scientists. For example, Robert May notes that the study of living systems cannot rely on background theories such as those that produced the physical-mathematical theories of inert matter; see Reference MayMay (2004) and Reference ReedReed (2004). Moreover, living systems are evolutionary, as we learn from Reference MayrMayr (1981).

However, this Element has shown that the current state of the art has led to mathematical structures that can capture the key features of living systems. These structures provide a framework that merits the definition of a mathematical theory. It would then be useful to understand how this theory can be specialized into a mathematical theory in economics, similar to the methods that have been brought to the different branches of mathematical physics. We do not have an exhaustive answer to this challenging problem. Therefore, we simply offer the reader some preliminary considerations.

1. 1. Selection of a specific branch of economics specialized by specific features that support the theory. For instance, behavioural, evolutionary, and so on.

2. 2. Emphasis on modelling interactions, taking into account nonlinear additivity, evolution of interaction rules, micro-micro and micro-macro features of interaction rules.

3. 3. Implementation of models of interactions into the general mathematical structures and then proceeding to the validation of models.

This process can be interpreted as a rational approach to derive mathematical models that goes beyond the heuristic derivation of phenomenological approaches. In fact, it refers to the rationale reported in Fig. 2. The new concept is that the derivation is based on well-defined mathematical structures.

We do believe that this challenging perspective is worth being pursued according to the idea that new mathematical tools have to be invented in economics rather than using tools that are valid for inert matter. Indeed, the second step is the key passage to the derivation of the mathematical theory. We have seen that both mathematical approaches are based on the idea of a mathematical transfer of the dynamics at the microscopic scale to the collective dynamics of the whole system. Then, we can argue that the mathematical tools of the kinetic theory of active particles and those of behavioural swarms can effectively provide to the said tool. Therefore, the key problem is the modelling of the interactions.

New inventions are needed to provide significant progress in this topic. For instance, by understanding that the dynamics of interaction follow rules that are modified by the individuals playing the game and by the the external actions that have a strategy to modify them. As mentioned earlier, these rules evolve over time.

As we have seen, we have different tools suitable to transfer the dynamics at the micro-scale into collective dynamics. Hence, the key problem is the modelling of interactions. Bearing all the preceding in mind, let us define some important features of interactions whose rules evolve in time.

• Interactions can be modelled by theoretical tools of game theory, which should consider entire populations of players, where strategies with higher payoff might spread over the population.

• The strategy expressed by individuals, namely active particles, in the interactions is heterogeneously distributed over the players of the population.

• Players are modeled as random variables linked to a distribution function over the activity variable.

• The Utility Function guides the output of interactions. It is heterogeneously distributed over players and can be motivated by “rational” or even “irrational” strategies. In some cases, interactions are asymmetric; see Reference Lachowicz, Matusik and TopolskiLachowicz, Matusik, and Topolski (2024).

• In the virtual world the Utility Function depends also on the actions of the coplayers as well as on the frequencies of interactions. Both quantities depend on the overall state of the system.

The overview proposed in our Element, has shown that the path to a mathematical theory in economics has been traced. It is a challenging path; however, it is worth trying also as some achievements by the kinetic theory approach have been effectively obtained. But what about using swarm theories? Some studies have been recently developed by using mathematical tools borrowed by the physical-mathematical theory of swarms; see Reference Ha, Kim, Park and ZhangHa, Kim, Park, and Zhang (2019), Reference Bae, Cho, Lee, Yoo and YunBae, Cho, Lee, Yoo, and Yun (2019), and Reference Bae, Cho, Kim and YunBae, Cho, Kim, and Yun (2019).

What about developments? Let us start from the books by Reference ZhangZhang (1991), Reference Zhang(2023), who developed a mathematical theory of economics within the framework of the mathematical theory of ordinary differential equations and of the interpretation of the dynamics of real systems by causality principles. These models provide an immediate description of the dynamics and, at least in some cases, allow a qualitative analysis focused on asymptotic behaviours, stability, and bifurcation. An analogous approach is developed in Reference Bonacich and LuBonacich and Lu (2012), where a variety of specific applications are considered.

On the other hand, these methods fail in describing some specific features of real systems from the individual heterogeneity to all behavioural attitudes put in evidence in this Element. Indeed Reference Bonacich and LuBonacich and Lu (2012) indicate some open problems that have been tackled by the kinetic theory of active particles and that can be viewed as research perspectives for the mathematical theory of behavioural swarms. In particular, it is mentioned that models should go beyond the consensus dynamics and include heterogeneity of individual behaviors.

The recent literature, as shown in the preceding sections, has tackled these problems within the framework of mathematical tools motivated by statistical physics. The review by Reference Ajmone Marsan, Bellomo and GibelliAjmone Marsan et al. (2016) has reported about them. The mathematical theory of behavioural swarms has also taken into account these hints that go beyond the deterministic description. Further steps are discussed in the last chapter of this Element.

7.1 Parameters

The parameters shaping the dynamics are the following:

#Number of agents in the model
N = 10 #numbers of sellers
M = 50 #numbers of buyers
ratio = M/N #ratio sellers/buyers
#Parameters for the model
#micro-micro interaction
eta_0 = 1 #intensity of rate of micro-micro interaction
alpha = 1 #intensity of micro-micro interaction for sellers
rho = 2 #intensity of exponential argument in the rate of
micro-micro interaction for seller
beta = 0.1 #intensity of micro-micro interaction for buyers
#macro-micro interaction
gamma_s = 0.01 #intensity of macro-micro interaction for sellers
mu_0 = 1 #intensity of rate of macro-micro interaction for sellers
#Number of cycles
n = 50000

7.2 The Quantities in the Dynamical System

Here we initialize the arrays containing the quantities (variables in the differential system) defining the dynamics and define their initial conditions, which are necessary for the computational solution of the initial value problem.

#PRICES
b = np.full([n,M], 0.0) #buyers prices during time-steps
s = np.full([n,N], 0.0) #sellers prices during time-steps
#VELOCITIES of prices
x = np.full([n,M], 0.0) #buyers prices velocities during time-steps
y = np.full([n,N], 0.0) #sellers prices velocities during time-steps
#MEAN prices
MeanPrice_B = np.full([n,1], 0.0) #buyers mean prices during time-steps
MeanPrice_S = np.full([n,1], 0.0) #sellers mean prices during time-steps
#VARIANCE of prices
VariancePrice_S = np.full([n,1], 0.0) #buyers variance of prices during
time-steps
VariancePrice_B = np.full([n,1], 0.0) #sellers variance of prices during
time-steps
#PARETO MARKET EFFICIENCY
efficiency_S = np.full([n,1], 0.0) #total buyers efficiency during
time-steps
efficiency_B = np.full([n,1], 0.0) #total sellers efficiency during
time-steps
efficiency_Tot = np.full([n,1], 0.0) #total market efficiency during
time-steps
#INITIAL CONDITIONS renewed at every time-step (only for the modelling)
z_b0 = [0,0] #initial conditions of every time-step for buyers
z_s0 = [0,0] #initial conditions of every time-step for sellers
#QUALITIES
quality_s = np.full(N, 0.0) #sellers’ offered quality
quality_b = np.full(M, 0.0) #buyers’ reservation qualities
#SELLERS
#Sellers Initial Cost
s_cost = np.full(N, 0.0)
#BUYERS
#Buyers Chosen Seller at every time step
chosen_seller = np.full([n,M], 0)
#Buyer Chosen Price and Seller renewed at each timestep (only for
modelling)
b_choices = [0,0]
#SETTING INITIAL CONDITIONS FOR ALL QUANTITIES
#Buyers initial Prices, and Reservation Qualities
for i in range(0,M):
#Price
b[0,i] = random.random()*5 + 1000
#Quality
quality_b[i] = random.randint(999,1004)
print(’buyer ’+ str(i) + ’ quality ’ + str(quality_b[i]) )
#to make sure the lowest quality offering seller will have a buyer
quality_b[1] = 999
#Sellers initial Prices, Costs, Qualities
for j in range(0,N):
#Price
s[0,j] = random.random()*5 + 1000
#Cost
s_cost[j] = (s[0,j])*0.1
#Quality
quality_s[j] = random.randint(1000,1005)
print(’seller ’+ str(j) + ’ quality ’ + str(quality_s[j]) )
#to make sure every buyers chooses a seller
quality_s[N-1] = 1005
#Calculation of initial Mean Prices
MeanPrice_S[0] = st.mean(s[0]) #buyers initial mean prices
MeanPrice_B[0] = st.mean(b[0]) #sellers initial mean price
#Calculation of initial Variances of Prices
VariancePrice_S[0] = st.variance(s[0]) #buyers initial variance of prices
VariancePrice_B[0] = st.variance(b[0]) #sellers initial variance of prices

In addition, we consider the following functions:

Functions:

The functions of the dynamics both for buyers and for sellers. We will have two types of interactions for both agents:

• micro-micro interaction: between the single buyer and the single seller;

• macro-micro interaction: between the single agent and all the FS to which it belongs.

Note that, even if present in the general formulation of the dynamics, we will not consider the interaction among buyers.

Buyers:

Functions defining buyers’ prices adaptation (through interaction).

#BUYERS FUNCTIONS
#Micro-Micro
#Function defining rate of micro-micro interaction
def eta_b(bb,ss,ratio,eta_0):
if -0.01 < bb > 0.01:
argument = (np.abs((ss-bb)/0.01))
else:
argument = (np.abs((ss-bb)/bb))
xx = -(ratio)*(argument)
etab = eta_0 * (np.exp(xx))
return etab
#Function defining micro-micro interaction
def fi_b(bb,ss,beta):
fib = beta * (ss-bb)
return fib
#Function defining the seller with which the buyer will interact
def choice(s,quality_b,quality_s):
acc_s = [] #stands for acceptable sellers, the sellers with quality
high enough for the buyer
l = 0 #to count which are acceptable sellers
for m in range(0,N):
if quality_s[m] >= quality_b:
acc_s.append(m) #registering all sellers with the right
quality
l = l + 1
chosen_price = s[acc_s[0]]
b_choice = acc_s[0]
for j in range(0,l-1): #choosing the seller with the lowest price
among the acceptable ones
if (chosen_price > s[acc_s[j+1]]):
chosen_price = s[acc_s[j+1]]
b_choice = acc_s[j+1]
elif (chosen_price == s[acc_s[j+1]]):
chosen_price = s[acc_s[j]]
b_choice = random.choice([s[acc_s[j]], s[acc_s[j+1]]])
#returns price of the chosen seller and the chosen seller in this
order return chosen_price,b_choice
#Whole Dynamics
#Function defining the whole dynamics of buyers’ acceleration in ODE
def model_b(z_b,_,min_price,ratio,beta,N,eta_0):
b = z_b[0]
x = z_b[1]
sumx = (eta_b(b,min_price,ratio,eta_0)) * fi_b(b,min_price,beta)
dbdt = x
dxdt = (1/N) * sumx
return [dbdt,dxdt]

Sellers:

Functions defining sellers’ prices adaptation (through interaction).

#SELLERS FUNCTIONS
#Micro-Micro
#Function defining rate of micro-micro interaction
def eta_s(s,ratio,eta_0,rho):
xx = -(rho/ratio)*np.abs(s)
etas = eta_0 * (np.exp(xx))
return etas
#Function defining micro-micro interaction
def fi_s(b,s,alpha,chosen_s,s_counter):
if (chosen_s == s_counter): #here seller knows if it is the chosen
seller for buyer i
if (b<s):
sig = -1
else:
sig = 1
else:
sig = -1
fis = alpha*np.abs(s)*sig
return fis
#Macro-Micro
#Function defining macro-micro interaction
def psi_s(s,E1,gamma_s):
psis = gamma_s*(E1 - s)
return psis
#Whole Dynamics
#Function defining the whole dynamics of sellers’ acceleration in ODE
def model_s(z_s,_,b,E1,mu_0,eta_0,alpha,M,gamma_s,chosen_s,s_counter,rho):
s = z_s[0]
y = z_s[1]
sumy = 0.0 #sum on buyers (interaction of the seller j with all
buyers)
for i in range(0,M):
sumy = sumy + (eta_s(s,ratio,eta_0,rho)*fi_s(b[i],s,alpha,
chosen_s[i],s_counter))
dsdt = y
dydt = (1/M)*(sumy) + mu_0 * psi_s(s,E1,gamma_s)
return [dsdt,dydt]

7.3 Some Technical Considerations

As mentioned earlier, we have reported the Python code to obtain the simulation associated with the class of models proposed in Section 4. This type of programming is quite standard, but it considers the possibility of a sensitivity analysis of the parameters, so that the user can develop a study of the whole variety of predictions that can be delivered by the model; in particular, the different types of aggregations that the system seller-buyer exhibits. Indeed, the simulations in Section 4 have shown this specific feature of the dynamics.

Referring to the contents of Section 5, we can deduce that the same dynamics can be described by the kinetic theory of active particles with discrete states. Furthermore, one can develop models where behavioural swarms are used for sellers, while kinetic theory is used for buyers. The derivation of new codes to consider this evolution of models follows techniques close to those we have presented in this section. The interested reader should develop them autonomously thanks to the hints given in this section.