September  2020, 15(3): 353-368. doi: 10.3934/nhm.2020022

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 Published  September 2020

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.

Citation: Nicola Bellomo, Sarah De Nigris, Damián Knopoff, Matteo Morini, Pietro Terna. Swarms dynamics approach to behavioral economy: Theoretical tools and price sequences. Networks & Heterogeneous Media, 2020, 15 (3) : 353-368. doi: 10.3934/nhm.2020022
References:
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G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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.  Google Scholar

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H.-O. BaeS.-Y. ChoS.-H. LeeJ. 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.  Google Scholar

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H.-O. BaeS. -Y.ChoJ. 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.  Google Scholar

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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.  Google Scholar

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D. BuriniL. 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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. DolfinL. Leonida and N. Outada, Modeling human behaviour in economics and social science, Physics of Life Reviews, 22 (2017), 1-21.   Google Scholar

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M. DolfinD. KnopoffL. 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.  Google Scholar

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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.  Google Scholar

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show all references

References:
[1]

D. AcemogluD. Ticchi and A. Vindigni, Emergence and persistence of inefficient states, Journal of European Economic Association, 9 (2011), 177-208.  doi: 10.3386/w12748.  Google Scholar

[2]

S.-M. AhnH.-O. BaeS.-Y. SeungY. 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.  Google Scholar

[3]

G. Ajmone MarsanN. 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.  Google Scholar

[4]

G. Ajmone MarsanN. 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.  Google Scholar

[5]

G. AlbiN. BellomoL. FermoS.-Y. HaJ. KimL. PareschiD. 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.  Google Scholar

[6]

G. AlbiL. PareschiG. 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.   Google Scholar

[7]

H.-O. BaeS.-Y. ChoS.-H. LeeJ. 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.  Google Scholar

[8]

H.-O. BaeS. -Y.ChoJ. 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.  Google Scholar

[9] K. D. Baley, Sociology and New Systems Theory - Toward a Theoretical Syntesis, Suny Press, 1994.   Google Scholar
[10]

P. Ball, Why Society is a Complex Matter, Springer-Verlag, 2012. doi: 10.1007/978-3-642-29000-8.  Google Scholar

[11]

M. BalleriniN. CabibboR. CandelierA. CavagnaE. CisbaniI. GiardinaV. LecomteA. OrlandiG. ParisiA. ProcacciniM. 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.  Google Scholar

[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.  Google Scholar

[13]

N. BellomoF. ColasuonnoD. 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.  Google Scholar

[14]

N. BellomoG. DosiD. 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.  Google Scholar

[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.  Google Scholar

[16]

N. BellomoM. 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.  Google Scholar

[17]

N. BellomoD. 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.  Google Scholar

[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.  Google Scholar

[19]

R. Boero, M. Morini, M. Sonnessa and P. Terna, Agent-based Models of the Economy From Theories to Applications, Palgrave Macmillan, 2015. Google Scholar

[20]

P. Bonacich and P. Lu, Introduction to Mathematical Sociology Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[21]

S. BowlesA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

D. BuriniS. De Lillo and L. Gibelli, Stochastic differential "nonlinear" games modeling collective learning dynamics, Physics of Life Reviews, 16 (2016), 123-139.   Google Scholar

[26]

D. BuriniL. 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.   Google Scholar

[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.  Google Scholar

[28]

A. Deaton, Measuring and understanding behavior, welfare, and poverty, American Economic Review, 106 (2016), 1221-1243.  doi: 10.1257/aer.106.6.1221.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. DolfinL. Leonida and N. Outada, Modeling human behaviour in economics and social science, Physics of Life Reviews, 22 (2017), 1-21.   Google Scholar

[32]

M. DolfinD. KnopoffL. 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.  Google Scholar

[33]

G. FurioliA. PulvirentiE. 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.  Google Scholar

[34]

S. Gächter and J. F. Schultz, Intrinsic honesty and the prevalence of rule violations across societies, Nature, 531(7595) (2017), 496-499.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[37]

H. Gintis, Game Theory Evolving, Second edition, Princeton University Press, Princeton NJ, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

A. Kirman, Complex Economics: Individual and Collective Rationality, Routledge, London, 2011. doi: 10.4324/9780203847497.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45] M. MazzoliM. Morini and P. Terna, Rethinking Macroeconomics with Endogenous Market Structure, Cambridge University Press, 2019.  doi: 10.1017/9781108697019.  Google Scholar
[46]

S. McQuadeB. 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.  Google Scholar

[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. Google Scholar

[48]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.  Google Scholar

[49]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013. Google Scholar

[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.  Google Scholar

[51]

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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
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
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
Figure 4.  Buyers
Figure 5.  Sellers
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