New paradigms and mathematical methods for complex systems in behavioral economics
Creators: name: Ajmone Marsan, Giulia
Publisher: name: IMT Alti Studi Lucca
Year: 2009
Subjects: HB Economic Theory

description: This dissertation is devoted to the mathematical investigation of properties of complex socio-economic systems, where individual behaviors, and their interactions,  exert a crucial influence on the overall dynamics of the whole system. In order to understand the importance of such an investigation, it is necessary to briefly analyze some conceptual aspects relating to the interaction between applied mathematics and socio-economic sciences. The main issue in this field consists in coupling the usual qualitative interpretation of socio-economic phenomena with an innovative quantitative description by means of mathematical equations. This dialogue, however difficult, is necessary to reach a deeper understanding of socio-economic phenomena, where deterministic rules may be stochastically perturbed by individual behaviors. The difficulty mostly stems from the fact that the  behavior of socio-economic systems, where the collective dynamics differ from the sum of the individual behaviors, is a paradigmatic example of a complex system. These aspects are discussed in the introductory section that follows. The mathematical framework presented in this dissertation is built by suitable developments of the so-called mathematical kinetic theory for active particles,  which proved to be a useful reference for applications in many fields of life sciences. The description of a system by the methods of the mathematical kinetic theory  essentially implies the definition of the microscopic state space of the interacting entities and of the distribution function over this state space. In the case of living systems, the identification of the microscopic state space  requires the definition of an additional variable, called activity, which captures the specific dynamical aspects of the system under consideration. Entities of living systems, called active particles, may be organized into several interacting populations. This dissertation presents in a unified context the results of the doctoral work, mostly described in four peer-reviewed research papers that are included as appendices of this dissertation. The essential ideas of each of the papers are introduced and summarized next. The first three papers exploit tools and developments of the kinetic theory for active particles, while the fourth paper is based on a different tool, namely on agents’ methods. The first paper,[3], develops a mathematical framework based on the kinetic theory for active particles, which describes the evolution of large systems of interacting entities. These entities are carriers of specific functions, in this case economic activities. The mathematical framework is constructed by means of a suitable decomposition into functional subsystems, namely aggregations of entities, which have the ability of expressing socio-economic purposes and  unctions. The paper shows how this framework can be implemented to describe some specific complex economic applications. Specifically, the applications are focused on  opinion dynamics and job mobility phenomena. These two examples offer a first insight into multi-scale issues: starting from the application, a preliminary mathematical  framework which takes into account both microscopic and macroscopic interactions is developed. This framework may be adapted to a great variety of complex phenomena. The second paper,[4], contains the initial elements of the development of the mathematical theory for complex socio-economic systems, already introduced in the first paper. The approach is based on the methods of the mathematical  kinetic theory for active particles. The key concept of functional subsystem is analyzed in detail, developing suitable mathematical models, which involve the decomposition of the overall system into functional subsystems. Different examples of socio-economic phenomena are taken into account, in order to provide an application background. The theoretical framework is then adapted to a specific application, dealing with opinion formation dynamics, which leads to numerical simulations, that show some preliminary interesting results. The third paper,[5], further develops the theory introduced in the first and second papers, with the setting of a mathematical model, where external actions play a key role. The aim of [5] consists in showing the emergence of collective behaviors or macroscopic trends from interactions at the microscopic scale, where agents are grouped into functional subsystems. The approach is, again, based on the methods of the mathematical kinetic theory for active particles: in this application the specific functions expressed by the interacting entities are socio-political activities subjected to the influence of media. Also in this case, numerical simulations show a direct application of the theoretical model, by means of specific settings of key parameters of the model. Finally, the fourth paper,[6], derives an agent-based model, which allows the investigation of the socio-economic phenomenon of fashion. The model introduces two classes of agents, common agents and trend-setters, which play the dynamics that rule the emerging behaviors investigated by means of numerical  simulations. Both the numerical and analytical tools used in this last paper differentiate it from the previous three works. Nevertheless, except for some conceptual differences, the approach is still focused on extracting emerging behaviors from individual-based interactions. The goal is to illustrate and implement different methodologies within the same research environment.