Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-22T15:54:25.023Z Has data issue: false hasContentIssue false
Series:   Elements in Complexity and Agent-based Economics

Active Particles Methods in Economics

New Perspectives in the Interaction between Mathematics and Economics

Published online by Cambridge University Press:  22 November 2024

Nicola Bellomo ,
Diletta Burini ,
Valeria Secchini  and
Pietro Terna
Nicola Bellomo
Affiliation:
Universidad de Granada
Diletta Burini
Affiliation:
Liceo statale classico e musicale ‘A. Mariotti’
Valeria Secchini
Affiliation:
Charles University, Prague
Pietro Terna
Affiliation:
Università di Torino, Italy

Summary

The aim of this Element is to understand how far mathematical theories based on active particle methods have been applied to describe the dynamics of complex systems in economics, and to look forward to further research perspectives in the interaction between mathematics and economics. The mathematical theory of active particles and the theory of behavioural swarms are selected for the above interaction. The mathematical approach considered in this work takes into account the complexity of living systems, which is a key feature of behavioural economics. The modelling and simulation of the dynamics of prices within a heterogeneous population is reviewed to show how mathematical tools can be used in real applications.
Get access

Keywords

economic modelinginteracting agentsactive particleprice emergencemathematical modelling
Type
Element
Information
Series: Elements in Complexity and Agent-based Economics
Online ISBN: 9781009548755
Publisher: Cambridge University Press
Print publication: 02 January 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aguiar, M., Dosi, G., Knopoff, D. A., & Virgillito, M. E. (2021). A multiscale network-based model of contagion dynamics: Heterogeneity, spatial distancing and vaccination. Mathematical Models and Methods in Applied Sciences, 31, 2425–2454.CrossRefGoogle Scholar
Ajmone Marsan, G., Bellomo, N., & Gibelli, L. (2016). Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics. Mathematical Models and Methods in Applied Sciences, 26(06), 1051–1093.CrossRefGoogle Scholar
Anderson, P. W. (1972). More is different. Science, 177(4047), 393–396.CrossRefGoogle ScholarPubMed
Aristov, V. (2019). Biological systems as nonequilibrium structures described by kinetic methods. Results in Physics, 13, 102232.CrossRefGoogle Scholar
Bae, H.-O., Cho, S.-Y., Kim, J., & Yun, S.-B. (2019). A kinetic description for the herding behavior in financial market. Journal of Statistical Physics, 176, 398–424.CrossRefGoogle Scholar
Bae, H.-O., Cho, S.-Y., Lee, S.-H., Yoo, J., & Yun, S.-B. (2019). A particle model for the herding phenomena induced by dynamic market signals. Journal of Statistical Physics, 177, 365–398.CrossRefGoogle Scholar
Ball, P. (2012). Why society is a complex matter. Springer Science & Business Media.CrossRefGoogle Scholar
Bandura, A. (1989). Human agency in social cognitive theory. American Psychologist, 44(9), 1175–1184.CrossRefGoogle ScholarPubMed
Bellomo, N. (2008). Living systems. Birkhäuser-Springer.Google Scholar
Bellomo, N., Bellouquid, A., Gibelli, L., & Outada, N. (2017). A quest towards a mathematical theory of living systems. Birkhäuser-Springer.CrossRefGoogle Scholar
Bellomo, N., Burini, D., Dosi, G., Gibelli, L., Knopoff, D., Outada, N., Terna, P., and Virgillito, M. E. (2021). What is life? A perspective of the mathematical kinetic theory of active particles. Mathematical Models and Methods in Applied Sciences, 31(09), 1821–1866.CrossRefGoogle Scholar
Bellomo, N., & Carbonaro, B. (2011). Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives. Physics of Life Reviews, 8(1), 1–18.CrossRefGoogle Scholar
Bellomo, N., Colasuonno, F., Knopoff, D., & Soler, J. (2015). From a systems theory of sociology to modeling the onset and evolution of criminality. Networks and Heterogeneous Media, 10, 421–441.CrossRefGoogle Scholar
Bellomo, N., De Nigris, S., Knopoff, D., Morini, M., & Terna, P. (2020). Swarms dynamics towards a systems approach to social sciences and behavioral economy. Networks and Heterogeneous Media, 15, 353–368.CrossRefGoogle Scholar
Bellomo, N., Dosi, G., Knopoff, D. A., & Virgillito, M. E. (2020). From particles to firms: On the kinetic theory of climbing up evolutionary landscapes. Mathematical Models and Methods in Applied Sciences, 30(07), 1441–1460.CrossRefGoogle Scholar
Bellomo, N., & Egidi, M. (2024). From Herbert A. Simon’s legacy to the evolutionary artificial world with heterogeneous collective behaviorss. Mathematical Models and Methods in Applied Sciences, 34(1), 145–180. https://doi.org/10.1142/S0218202524400049.CrossRefGoogle Scholar
Bellomo, N., Esfahanian, M., Secchini, V., & Terna, P. (2022). What is life? Active particles tools towards behavioral dynamics in social-biology and economics. Physics of Life Reviews, 43, 189–207.CrossRefGoogle ScholarPubMed
Bellomo, N., & Forni, G. (1994). Dynamics of tumor interaction with the host immune system. Mathematical and Computer Modelling, 20(1), 107–122.CrossRefGoogle Scholar
Bellomo, N., Ha, S.-Y., & Outada, N. (2020). Towards a mathematical theory of behavioral swarms. ESAIM: Control, Optimisation and Calculus of Variations, 26, 125.Google Scholar
Bellomo, N., Herrero, M. A., & Tosin, A. (2013). On the dynamics of social conflicts looking for the Black Swan. Kinetic Related Models, 6(3), 459–479.CrossRefGoogle Scholar
Bellouquid, A., & Delitala, M. (2006). Modelling complex biological systems: A kinetic theory approach. Birkhäuser-Springer.Google Scholar
Bertotti, M. L. (2010). Modelling taxation and redistribution: A discrete active particle kinetic approach. Applied Mathematics and Computation, 217(2), 752–762.CrossRefGoogle Scholar
Bertotti, M. L., & Delitala, M. (2004). From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences. Mathematical Models and Methods in Applied Sciences, 14(07), 1061–1084.CrossRefGoogle Scholar
Bertotti, M. L., & Delitala, M. (2010). Cluster formation in opinion dynamics: A qualitative analysis. Zeitschrift für angewandte Mathematik und Physik, 61, 583–602.CrossRefGoogle Scholar
Bertotti, M. L., & Modanese, G. (2011). From microscopic taxation and redistribution models to macroscopic income distributions. Physica A: Statistical Mechanics and Its Applications, 390(21–22), 3782–3793.CrossRefGoogle Scholar
Black, F., & Scholes, M. (1972). The valuation of option contracts and a test of market efficiency. The Journal of Finance, 27(2), 399–417.CrossRefGoogle Scholar
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.CrossRefGoogle Scholar
Bonacich, P., & Lu, P. (2012). Introduction to mathematical sociology. Princeton University Press.Google Scholar
Burini, D., & Chouhad, N. (2022). Virus models in complex frameworks: Towards modeling space patterns of SARS-CoV-2 epidemics. Mathematical Models and Methods in Applied Sciences, 32(10), 2017–2036.CrossRefGoogle Scholar
Burini, D., & Chouhad, N. (2023). Cross diffusion models in complex frameworks from microscopic to macroscopic. Mathematical Models and Methods in Applied Sciences, 33(9), 158–162.CrossRefGoogle Scholar
Burini, D., Chouhad, N., & Bellomo, N. (2023). Waiting for a mathematical theory of living systems from a critical review to research perspectives. Symmetry, 15(2), 351.CrossRefGoogle Scholar
Burini, D., & De Lillo, S. (2019). On the complex interaction between collective learning and social dynamics. Symmetry, 11(8), 967.CrossRefGoogle Scholar
Burini, D., De Lillo, S., & Gibelli, L. (2016). Collective learning modeling based on the kinetic theory of active particles. Physics of Life Reviews, 16, 123–139.CrossRefGoogle ScholarPubMed
Burini, D., Gibelli, L., & Outada, N. (2017). A kinetic theory approach to the modeling of complex living systems. Active Particles, 1, 229–258.CrossRefGoogle Scholar
Burini, D., & Knopoff, D. A. (2024). Epidemics and society: A multiscale vision from the Small World to the Globally Interconnected World. Mathematical Models and Methods in Applied Sciences, 34(08), 1567–1596.CrossRefGoogle Scholar
Compte, A. (2012, 1839). Cours de philosophie positive. Hermann.Google Scholar
Coscia, V., Delitala, M., & Frasca, P. (2007). On the mathematical theory of vehicular traffic flow models II. Discrete velocity kinetic models. International Journal of Non-Linear Mechanics, 42(3), 411–421.CrossRefGoogle Scholar
Cucker, F., & Smale, S. (2007). Emergent behavior in flocks. IEEE Transactions on Automatic Control, 52(5), 852–862.CrossRefGoogle Scholar
Diamond, J. (1987). Soft sciences are often harder than hard sciences. Discover, 8(8), 34–39.Google Scholar
Dolfin, M., Knopoff, D., Leonida, L., & Patti, D. M. A. (2017). Escaping the trap of “blocking”: A kinetic model linking economic development and political competition. Kinetic and Related Models, 24, 2361–2381.Google Scholar
Dolfin, M., & Lachowicz, M. (2014). Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions. Mathematical Models and Methods in Applied Sciences, 24(12), 2361–2381.CrossRefGoogle Scholar
Dolfin, M., & Lachowicz, M. (2015). Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 10(4), 877–896.CrossRefGoogle Scholar
Dolfin, M., Leonida, L., & Outada, N. (2017). Modeling human behavior in economics and social science. Physics of Life Reviews, 22, 1–21.CrossRefGoogle ScholarPubMed
Dosi, G. (1984). Technology and conditions of macroeconomic development. In Freeman, C. (Ed.), Design, innovation and long cycles in economic development (pp. 99–125). Design Research Publications.Google Scholar
Dosi, G. (2023). The foundations of complex evolving economies: Part one: Innovation, organization, and industrial dynamics. Oxford University Press.CrossRefGoogle Scholar
Dosi, G., Fanti, L., & Virgillito, M. E. (2020). Unequal societies in usual times, unjust societies in pandemic ones. Journal of Industrial and Business Economics, 47, 371–389.CrossRefGoogle Scholar
Dosi, G., Pereira, M. C., & Virgillito, M. E. (2017). The footprint of evolutionary processes of learning and selection upon the statistical properties of industrial dynamics. Industrial and Corporate Change, 26(2), 187–210.Google Scholar
Dosi, G., & Virgillito, M. E. (2021). In order to stand up you must keep cycling: Change and coordination in complex evolving economies. Structural Change and Economic Dynamics, 56, 353–364.CrossRefGoogle Scholar
Furioli, G., Pulvirenti, A., Terraneo, E., & Toscani, G. (2017). Fokker–Planck equations in the modeling of socio-economic phenomena. Mathematical Models and Methods in Applied Sciences, 27(01), 115–158.CrossRefGoogle Scholar
Galam, S. (2012). Sociophysics: A physicist’s modeling of psycho-political phenomena (understanding complex systems). Berlin: Springer.CrossRefGoogle Scholar
Ha, S.-Y., Kim, J., Park, J., & Zhang, X. (2019). Complete cluster predictability of the Cucker–Smale flocking model on the real line. Archive for Rational Mechanics and Analysis, 231, 319–365.CrossRefGoogle Scholar
Hartwell, L. H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999). From molecular to modular cell biology. Nature, 402(Suppl 6761), C47–C52.CrossRefGoogle ScholarPubMed
Helbing, D. (2010). Quantitative sociodynamics: Stochastic methods and models of social interaction processes. Springer Science & Business Media.CrossRefGoogle Scholar
Hilbert, D. (1902). Mathematical problems. Bulletin American Mathematical Society, 8, 437–479.CrossRefGoogle Scholar
Jager, E., & Segel, L. A. (1992). On the distribution of dominance in populations of social organisms. SIAM Journal on Applied Mathematics, 52(5), 1442–1468.CrossRefGoogle Scholar
Jovanovic, F., & Le Gall, P. (2021). Mathematical analogies: An engine for understanding the transfers between economics and physics. History of Economics Review, 79(1), 18–38.CrossRefGoogle Scholar
Kant, I. (2000). Critique of the power of judgment. Cambridge University Press.CrossRefGoogle Scholar
Knopoff, D. (2014). On a mathematical theory of complex systems on networks with application to opinion formation. Mathematical Models and Methods Applied Sciences, 24(405–426).CrossRefGoogle Scholar
Knopoff, D., Secchini, V., & Terna, P. (2020). Cherry picking: Consumer choices in swarm dynamics, considering price and quality of goods. Symmetry, 12(11), 1912.CrossRefGoogle Scholar
Lachowicz, M., Matusik, M., & Topolski, K. A. (2024). Population of entities with three individual states and asymmetric interactions. Applied Mathematics and Computation, 464.CrossRefGoogle Scholar
May, R. M. (2004). Uses and abuses of mathematics in biology. Science, 303(5659), 790–793.CrossRefGoogle ScholarPubMed
Mayr, E. (1981). La biologie de l’évolution. Paris: Hermann.Google Scholar
Mazzoli, M., Morini, M., & Terna, P. (2019). Rethinking macroeconomics with endogenous market structure. Cambridge University Press.CrossRefGoogle Scholar
Nash, J. (1951). Non-cooperative games. Annals of Mathematics, 286–295.CrossRefGoogle Scholar
Nash, J. (1996). Essays on game theory. Cheltenham, UK: Elgar.CrossRefGoogle Scholar
Pareschi, L., & Toscani, G. (2013). Interacting multiagent systems: Kinetic equations and Monte Carlo methods. Oxford: Oxford University Press.Google Scholar
Paveri-Fontana, S. L. (1975). On Boltzmann-like treatments for traffic flow. Transportation Research, 9(4), 225–235.CrossRefGoogle Scholar
Perlovsky, L., & Schoeller, F. (2019). Unconscious emotions of human learning. Physics of Life Reviews, 31, 257–262.CrossRefGoogle ScholarPubMed
Prigogine, I., & Herman, R. (1971). Kinetic theory of vehicular traffic. New York: Elsevier.Google Scholar
Reed, M. C. (2004). Why is mathematical biology so hard. Notices of the AMS, 51(3), 338–342.Google Scholar
Salomon, G., & Perkins, D. N. (1998). Individual and social aspects of learning. Review of Research in Education, 23(1), 1–24.Google Scholar
Schoeller, F., Perlovsky, L., & Arseniev, D. (2018). Physics of mind: Experimental confirmations of theoretical predictions. Physics of Life Reviews, 25, 45–68.CrossRefGoogle ScholarPubMed
Schumpeter, J. A. (1947). Capitalism, socialism and democracy. Taylor and Francis.Google Scholar
Simon, H. A. (1978). Rational decision making in business organizations. Nobel Memorial Lecture, 8 December, 69(4), 493–513. Retrieved from www.nobelprize.org/uploads/2018/06/simon-lecture.pdfGoogle Scholar
Simon, H. A. (2019). The sciences of the artificial, third edition. Massachusetts Institute of Technology Press.Google Scholar
Stiglitz, J. E. (2010). Freefall: America, free markets, and the sinking of the world economy. W. W. Norton & Company.Google Scholar
Taleb, N. N. (2007). The black swan: The impact of the highly improbable. Random house.Google Scholar
Thaler, R. H. (2016). Behavioral economics: Past, present, and future. American Economic Review, 106(7), 1577–1600.CrossRefGoogle Scholar
Thaler, R. H., & Sunstein, C. R. (2009). Nudge: Improving decisions about health, wealth, and happiness. Penguin.Google Scholar
Zhang, W.-B. (1991). Synergetic economics. Heidelberg: Springer.CrossRefGoogle Scholar
Zhang, W.-B. (2023). Chaos, complexity, and nonlinear economic theory. Singapore, London: World Scientific.CrossRefGoogle Scholar

Save element to Kindle

To save this element to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Active Particles Methods in Economics
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Active Particles Methods in Economics
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Active Particles Methods in Economics
Available formats
×