Swarms dynamics approach to behavioral economy: Theoretical tools and price sequences

Nicola Bellomo; Sarah De Nigris; Damián Knopoff; Matteo Morini; Pietro Terna

doi:10.3934/nhm.2020022

Article Contents

2020, Volume 15, Issue 3: 353-368. Doi: 10.3934/nhm.2020022

This issue Previous Article Modelling pattern formation through differential repulsion Next Article Relative entropy method for the relaxation limit of hydrodynamic models

Swarms dynamics approach to behavioral economy: Theoretical tools and price sequences

1.
University of Granada, Departamento de Matemática Aplicada, 18071-Granada, Spain, Collegio Carlo Alberto, Torino, Italy, Politecnico Torino, Italy
2.
Joint Research Centre, European Commission, Ispra, VA, Italy
3.
Centro de Investigación y Estudios de Matemática (CONICET) and Famaf (UNC), Medina Allende s/n, 5000 Córdoba, Argentina
4.
Credimi S.p.A., Milano, MI, Italy
5.
University of Torino, Torino, Italy, Collegio Carlo Alberto, Torino, Italy

Received: December 2019

Revised: February 2020

Early access: September 2020

Published: September 2020

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Abstract

This paper presents a development of the mathematical theory of swarms towards a systems approach to behavioral dynamics of social and economical systems. The modeling approach accounts for the ability of social entities are to develop a specific strategy which is heterogeneously distributed by interactions which are nonlinearly additive. A detailed application to the modeling of the dynamics of prices in the interaction between buyers and sellers is developed to describe the predictive ability of the model.

Keywords:

Mathematics Subject Classification: 82D99, 91C99, 91D10.

Citation:

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Figure 1. 1.0, 5.0 ratios: 10/50 buyers (red) and 10/10 sellers (blue), mean price sequences; blue line hides in large part the red one

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Figure 2. 1.0, 5.0 ratio: 10/50 buyers (red) and 10/10 sellers (blue), zoom on individual price sequences. Y axes do not share the same scale

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Figure 3. 1.0, 5.0 ratio: 10/50 buyers (red) and 10/10 sellers (blue), standard deviation of mean prices within buyers and within sellers over time

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Figure 4. Buyers

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Figure 5. Sellers

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References

[1]	D. Acemoglu, D. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, Journal of European Economic Association, 9 (2011), 177-208. doi: 10.3386/w12748.
[2]	S.-M. Ahn, H.-O. Bae, S.-Y. Seung, Y. Kim and H. Lim, Application of flocking mechanisms to the modeling of stochastic volatily, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628. doi: 10.1142/S0218202513500176.
[3]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinetic & Related Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249.
[4]	G. Ajmone Marsan, N. Bellomo and L. Gibelli, Stochastic evolutionary differential games toward a systems theory of behavioral social dynamics, Math. Models Methods Appl. Sci., 26 (2016), 1051-1093. doi: 10.1142/S0218202516500251.
[5]	G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005. doi: 10.1142/S0218202519500374.
[6]	G. Albi, L. Pareschi, G. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 49-98.
[7]	H.-O. Bae, S.-Y. Cho, S.-H. Lee, J. Yoo and S.-B. Yun, A particle model for herding phenomena induced by dynamic market signals, Journal of Statistical Physics, 177 (2019), 365-398. doi: 10.1007/s10955-019-02371-8.
[8]	H.-O. Bae, S. -Y.Cho, J. Kim and S.-B. Yun, A kinetic description for the herding behavior in financial market, Journal of Statistical Physics, 176 (2019), 398-424. doi: 10.1007/s10955-019-02305-4.
[9]	K. D. Baley, Sociology and New Systems Theory - Toward a Theoretical Syntesis, Suny Press, 1994.
[10]	P. Ball, Why Society is a Complex Matter, Springer-Verlag, 2012. doi: 10.1007/978-3-642-29000-8.
[11]	M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proceedings of the Natural Academy of Sciences USA, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.
[12]	N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, Cham, 2017. doi: 10.1007/978-3-319-57436-3.
[13]	N. Bellomo, F. Colasuonno, D. Knopoff and J. Soler, From a systems theory of sociology to modeling the onset and evolution of criminality, Netw. Heterog. Media, 10 (2015), 421-441. doi: 10.3934/nhm.2015.10.421.
[14]	N. Bellomo, G. Dosi, D. A.Knopoff and M.E. Virgillito, From particles to firms: on the kinetic theory of climbing up evolutionary landscapes, Math. Models Methods Appl. Sci., 30 (2020), 1441-14060. doi: 10.1142/S021820252050027X.
[15]	N. Bellomo and S.-Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770. doi: 10.1142/S0218202517500154.
[16]	N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479. doi: 10.3934/krm.2013.6.459.
[17]	N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life "complexity" and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X.
[18]	J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Tipping Points: Modelling Social Problems and Health, Wiley, London, 2015. doi: 10.1002/9781118992005.
[19]	R. Boero, M. Morini, M. Sonnessa and P. Terna, Agent-based Models of the Economy From Theories to Applications, Palgrave Macmillan, 2015.
[20]	P. Bonacich and P. Lu, Introduction to Mathematical Sociology Princeton University Press, Princeton, NJ, 2012.
[21]	S. Bowles, A. Kirman and R. Sethi, Retrospectives: Friedrich hayek and the market algorithm, Journal of Economic Perspectives, 31 (2017), 215-230. doi: 10.1257/jep.31.3.215.
[22]	C. Brugna and G. Toscani, Kinetic models of opinion formation in the presence of personal conviction, Physical Review E, 92 (2015), 052818. doi: 10.1103/PhysRevE.92.052818.
[23]	C. Brugna and G. Toscani, Boltzmann-type models for price formation in the presence of behavioral aspects, Netw. Heterog. Media, 10 (2015), 543-557. doi: 10.3934/nhm.2015.10.543.
[24]	C. Brugna and G. Toscani, Kinetic models for goods exchange in a multi-agent market, Physica A, 499 (2018), 362-375. doi: 10.1016/j.physa.2018.02.070.
[25]	D. Burini, S. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Physics of Life Reviews, 16 (2016), 123-139.
[26]	D. Burini, L. Gibelli and N. Outada, A kinetic theory approach to the modeling of complex living systems, Active Particles, Advances in Theory, Models, and Applications, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 1 (2017), 229-258.
[27]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 853-862. doi: 10.1109/TAC.2007.895842.
[28]	A. Deaton, Measuring and understanding behavior, welfare, and poverty, American Economic Review, 106 (2016), 1221-1243. doi: 10.1257/aer.106.6.1221.
[29]	M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2361-2381. doi: 10.1142/S0218202514500237.
[30]	M. Dolfin and M. Lachowicz, Modeling opinion dynamics: How the network enhances consensus, Netw. Heterog. Media, 10 (2015), 877-896. doi: 10.3934/nhm.2015.10.877.
[31]	M. Dolfin, L. Leonida and N. Outada, Modeling human behaviour in economics and social science, Physics of Life Reviews, 22 (2017), 1-21.
[32]	M. Dolfin, D. Knopoff, L. Leonida and D. Maimone Ansaldo Patti, Escaping the trap of 'blocking': A kinetic model linking economic development and political competition, Kinet. Relat. Models, 10 (2017), 423-443. doi: 10.3934/krm.2017016.
[33]	G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modeling of socio-economic phenomena, Math. Models Methods Appl. Sci., 27 (2017), 115-158. doi: 10.1142/S0218202517400048.
[34]	S. Gächter and J. F. Schultz, Intrinsic honesty and the prevalence of rule violations across societies, Nature, 531(7595) (2017), 496-499.
[35]	S. Galam, Sociophysics. A Physicist's Modeling of Psycho-Political Phenomena, Understanding Complex Systems, Springer, New York, 2012. doi: 10.1007/978-1-4614-2032-3.
[36]	F. Gino and L. Pierce, The abundance effect: Unethical behavior in the presence of wealth, Organizational Behavior and Human Decision Processes, 109 (2009), 142-155.
[37]	H. Gintis, Game Theory Evolving, Second edition, Princeton University Press, Princeton NJ, 2009.
[38]	R. Hegselmann, Thomas C. Shelling and James M. Sakoda: The intellectual, technical and social history of a model, Journal of Artificial Societies and Social Simulation, 20 (2017). doi: 10.18564/jass.5311.
[39]	R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic and leaders, and other constant signals: A simple unifying model, Netw. Heterog. Media, 10 (2015), 477-509. doi: 10.3934/nhm.2015.10.477.
[40]	J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N.S.), 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.
[41]	A. Kirman, Complex Economics: Individual and Collective Rationality, Routledge, London, 2011. doi: 10.4324/9780203847497.
[42]	A. P. Kirman and J. B. Zimmermann, Economics with Heterogeneous Interacting Agents, Lecture Notes in Economics and Mathematical Systems, 503. Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56472-7.
[43]	D. Knopoff, On the modeling of migration phenomena on small networks, Math. Models Methods Appl. Sci., 23 (2013), 541-563. doi: 10.1142/S0218202512500558.
[44]	D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426. doi: 10.1142/S0218202513400137.
[45]	M. Mazzoli, M. Morini and P. Terna, Rethinking Macroeconomics with Endogenous Market Structure, Cambridge University Press, 2019. doi: 10.1017/9781108697019.
[46]	S. McQuade, B. Piccoli and N. Pouradier Duteil, Social dynamics models with time-varying influence, Math. Models Methods Appl. Sci., 29 (2019), 681-716. doi: 10.1142/S0218202519400037.
[47]	M. Nieddu, Brownian and More Complex Agents to Explain Markets Behavior, Master's thesis, University of Torino, 2018, https://terna.to.it/tesi/nieddu.pdf.
[48]	M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.
[49]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013.
[50]	L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A Boltzmann approach, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130396, 15 pp. doi: 10.1098/rsta.2013.0396.
[51]	B. Piccoli, N. Pouradier Duteil and E. Trelat, Sparse control of Hegselmann-Krause models: Black hole and declustering, SIAM Journal on Control and Optimization, 57 (2019), 2628-2659. doi: 10.1137/18M1168911.
[52]	P. K. Piff, D. M. Stancato, S. Coté, R. Mendoza-Denton and D. Keltner, Higher social class predicts increased unethical behavior, Proceedings of the Natural Academy of Sciences USA, 109 (2014), 4086-4091. doi: 10.1073/pnas.1118373109.
[53]	S. Salvi, Corruption corrupts: Society-level rule violations affect individuals' intrinsic honesty, Nature, 53 (2016), 456-457.
[54]	F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, Springer Series in Synergetics, Springer-Verlag, Berlin, 2003.
[55]	K. Sigmund, The Calculus of Selfishness, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2010. doi: 10.1515/9781400832255.
[56]	J. E. Stiglitz, Information and the change in the paradigm in economics, The American Economic Review, 92 (2009), 460-501. doi: 10.1017/CBO9780511754357.004.
[57]	N. N. Taleb, The Black Swan: The Impact of the Highly Improbable, Random House, New York City, 2007.
[58]	R. H. Thaler, Misbehaving: The Making of Behavioral Economics, W. W. Norton & Company, New York, 2015.
[59]	R. H. Thaler, Behavioral economics: Past, present, and future, The American Economic Review, 106 (2016), 1577-1600.
[60]	T. A. Weber, Price theory in economics, in Ö. Özer, and R. Phillips, The Oxford Handbook of Pricing Management, (2012), 281–340.
[61]	E.G. Weyl, Price theory, Journal of Economic Literature, 57 (2019), 329-384. doi: 10.1257/jel.20171321.