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2010 | Book

Agent-based models of economic interactions Read chapter Read first chapter

Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences

Editors: Giovanni Naldi, Lorenzo Pareschi, Giuseppe Toscani

Publisher: Birkhäuser Boston

Book Series : Modeling and Simulation in Science, Engineering and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Mathematical modeling using dynamical systems and partial differential equations is now playing an increasing role in the understanding of complex multi-scale phenomena. Behavior in seemingly different areas such as sociology, economics, and the life sciences can be described by closely related models. Using examples from financial markets and modern warfare to the flocking of birds and the swarming of bacteria, the collected research in this volume demonstrates the common methodological approaches and tools for modeling and simulating collective behavior. The topics presented point toward new and challenging frontiers of applied mathematics, making the volume a useful reference text for applied mathematicians, physicists, biologists, and economists involved in the modeling of socio-economic systems.

Table of Contents

Frontmatter

Economic modelling and financial markets

Frontmatter
Agent-based models of economic interactions
Summary
The interdisciplinary field of econophysics has enjoyed recently a surge of activities especially with numerous agent-based models, which have led to a substantial development of this field. We review three main application areas of agent-based models in econophysics: order books, distributions of wealth in conservative economies, and minority games.
Anirban Chakraborti, Guido Germano
On kinetic asset exchange models and beyond: microeconomic formulation,trade network, and all that
Summary
We review the kinetic models of asset exchange and report the latest microeconomic formulations. We also discuss the role of topology and disorder in such models. The role of such models from a trading network perspective are also discussed.
Arnab Chattejee
Microscopic and kinetic models in financial markets
Summary
We review different microscopic and kinetic models of financial markets which have been developed by economists, physicists, and mathematicians in the last years. We first give a summary of the microscopic models and then introduce the corresponding kinetic equations. Our selective review outlines the main ingredients of some influential models of multiagent dynamics in financial markets like Levy, Levy, and Solomon (Economics Letters, 45, 1994) and Lux and Marchesi (International Journal of Theoretical and Applied Finance, 3, 2000). The introduction of kinetic equations permits to study the asymptotic behavior of the wealth and the price distributions and to characterize the regimes of lognormal behavior and the ones with power-law tails.
Stephane Cordier, Dario Maldarella, Lorenzo Pareschi, Cyrille Piatecki
A mathematical theory for wealth distribution
Summary
We review a qualitative mathematical theory of kinetic models for wealth distribution in simple market economies. This theory is a unified approach that covers a wide class of such models which have been proposed in the recent literature on econophysics. Based on the analysis of the underlying homogeneous Boltzmann equation, a qualitative description of the evolution of wealth in the large-time regime is obtained. In particular, the most important features of the steady wealth distribution are classified, namely the fatness of the Pareto tail and the tails’ dynamical stability. Most of the applied methods are borrowed from the kinetic theory of rarefied gases. A concise description of the moment hierarchy and suitable metrics for probability measures are employed as key tools.
Bertram Düring, Daniel Matthes
Tolstoy’s dream and the quest for statistical equilibrium in economics and the social sciences
Summary
The meaning of the notion of statistical equilibrium in economics is discussed as well as its relevance for economic theory. A simple agent-based model of taxation and redistribution is presented. Its invariant equilibrium distribution is the generalized Pólya sampling distribution. It turns out that the expected wealth distribution is the dichotomous Pólya whose continuous limit is the Beta distribution and whose appropriate thermodynamic limit is the Gamma distribution, often found in describing empirical data. The shape parameter of the Gamma distribution is the inverse of the wealth preferential attachment α−1.
Ubaldo Garibaldi, Enrico Scalas

Social modelling and opinion formation

Frontmatter
New perspectives in the equilibrium statistical mechanics approach to social and economic sciences
Summary
In this chapter we review some recent development in the mathematical modeling of quantitative sociology by means of statistical mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for particle (agents) interactions on random graph.
Our approach is based on describing a social network as a graph whose nodes represent agents and links between two of them stand for a reciprocal interaction. Each agent has to choose among a dichotomic option (i.e., agree or disagree) with respect to a given matter and he is driven by external influences (as media) and peer to peer interactions. These mimic the imitative behavior of the collectivity and may possibly be zero if the two nodes are disconnected.
For this scenario we work out both the dynamics and, given the validity of the detailed balance, the corresponding equilibria (statics). Once the two-body theory is completely explored, we analyze, on the same random graph, a diffusive strategic dynamicswith pairwise interactions, where detailed balance constraint is relaxed. The dynamic encodes some relevant processes which are expected to play a crucial role in the approach to equilibrium in social systems, i.e., diffusion of information and strategic choices. We observe numerically that such a dynamics reaches a well defined steady state that fulfills a shiftproperty: the critical interaction strength for the canonical phase transition is higher with respect to the expected equilibrium one previously obtained with detailed balanced dynamical evolution.
Finally, we show how the stationary states of this kind of dynamics can be described by statistical mechanics equilibria of a diluted p-spin model, for a suitable noninteger real p>2. Several implications from a sociological perspective are discussed together with some general outlooks.
Elena Agliari, Adriano Barra, Raffaella Burioni, Pierluigi Contucci
Kinetic modelling of complex socio-economic systems
Summary
This chapter is devoted to the investigation of the complex mechanisms which rule the transition from the behavior of single individuals to the collective behavior of groups of people, in social phenomena. The dynamics of our societies are ruled by many complex socio-economic phenomena, which still lack a proper support of robust mathematical models. A deeper understanding can possibly lead to important improvements in the explanation of various events of our times.
By means of the kinetic theory of active particles (KTAP), a mathematical framework suitable to model complex socio-economic systems is derived. This framework is based on concepts already introduced in [1, 2, 3]. Once the mathematical framework has been established, the second part of the chapter focuses on one specific application: the spread of opinions in a multi-community population affected by media. Different individuals belonging to different groups dynamically interact according to rules that take into account their social state and condition. We show which distributions of social groups in the global population emerge, and how these distributions change according to some key parameters of the model.
Giulia Ajmone Marsan
Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion
Summary
In this chapter, we briefly review some opinion dynamics models starting from the classical Schelling model and other agent-based modelling examples. We consider both discrete and continuous models and we briefly describe different approaches: discrete dynamical systems and agent-based models, partial differential equations based models, kinetic framework. We also synthesized some comparisons between different methods with the main references in order to further analysis and remarks.
Giacomo Aletti, Ahmad K. Naimzada, Giovanni Naldi
Global dynamics in adaptive models of collective choice with social influence
Summary
In this chapter we present a unified approach for modelling the diffusion of alternative choices within a population of individuals in the presence of social externalities, starting from two particular discrete-time dynamic models – Galam’s model of rumors spreading [10] and a formalization of Schelling’s binary choices [7]. We describe some peculiar properties of discrete-time (or event-driven) dynamic processes and we show how some long-run (asymptotic) outcomes emerging from repeated short time decisions can be seen as emerging properties, sometimes unexpected, or difficult to be forecasted.
Gian-Italo Bischi, Ugo Merlone
Modelling opinion formation by means of kinetic equations
Summary
In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics.New opinions are always suspected, and usually opposed, without any other reason but because they are not already common.John Locke, An Essay Concerning Human Understanding
Laurent Boudin, Francesco Salvarani

Human behavior and swarming

Frontmatter
On the modelling of vehicular traffic and crowds by kinetic theory of active particles
Summary
This paper deals with developments and applications of the mathematical kinetic and stochastic games theory to the modelling of the dynamics of vehicular traffic and pedestrian crowds. The mathematical approach is focused on the derivation of the evolution equation for the probability distribution over the state, at the microscopic scale, of vehicles and pedestrians. Models take into account their heterogeneous behaviour.
Nicola Bellomo, Abdelghani Bellouquid
Particle, kinetic, and hydrodynamic models of swarming
Summary
We review the state-of-the-art in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individual-based models based on “particle”-like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.
José A. Carrillo, Massimo Fornasier, Giuseppe Toscani, Francesco Vecil
Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints
Summary
This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different self-organized patterns from nonlocality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration.
Emiliano Cristiani, Benedetto Piccoli, Andrea Tosin
Statistical physics and modern human warfare
Summary
Modern human conflicts, such as those ongoing in Iraq, Afghanistan and Colombia, typically involve a large conventional force (e.g., a state army) fighting a relatively small insurgency having a loose internal structure. In this chapter, we adopt this qualitative picture in order to study the dynamics – and in particular the duration – of modern wars involving a loose insurgent force. We generalize a coalescence-fragmentation model from the statistical physics community in order to describe the insurgent population, and find that the resulting behavior is qualitatively different from conventional mass-action approaches. One of our main results is a counterintuitive relationship between an insurgent war’s duration and the asymmetry between the two opposing forces, a prediction which is borne out by empirical observation.
Alex Dixon, Zhenyuan Zhao, Juan Camilo Bohorquez, Russell Denney, Neil Johnson
Diffusive and nondiffusive population models
Summary
This survey is concerned with the modeling and mathematical analysis of continuous population equations. These models describe the change of the number of species due to birth, death, spatial movements, or stage variations. Our main focus is on spatially inhomogeneous models, given by reaction-diffusion equations, but we review also age- and size-structured and time-delayed population models. Results on the existence and stability of solutions as well as their qualitative behavior are given.
Ansgar Jüngel
Backmatter
Metadata
Title
Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences
Editors
Giovanni Naldi
Lorenzo Pareschi
Giuseppe Toscani
Copyright Year
2010
Publisher
Birkhäuser Boston
Electronic ISBN
978-0-8176-4946-3
Print ISBN
978-0-8176-4945-6
DOI
https://doi.org/10.1007/978-0-8176-4946-3

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