September  2020, 15(3): 519-542. doi: 10.3934/nhm.2020029

Kinetic modelling of multiple interactions in socio-economic systems

1. 

Department of Mathematics "F. Casorati", University of Pavia, Via A. Ferrata 5, 27100 Pavia, Italy

2. 

Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

Received  October 2019 Revised  March 2020 Published  September 2020

Unlike the classical kinetic theory of rarefied gases, where microscopic interactions among gas molecules are described as binary collisions, the modelling of socio-economic phenomena in a multi-agent system naturally requires to consider, in various situations, multiple interactions among the individuals. In this paper, we collect and discuss some examples related to economic and gambling activities. In particular, we focus on a linearisation strategy of the multiple interactions, which greatly simplifies the kinetic description of such systems while maintaining all their essential aggregate features, including the equilibrium distributions.

Citation: Giuseppe Toscani, Andrea Tosin, Mattia Zanella. Kinetic modelling of multiple interactions in socio-economic systems. Networks & Heterogeneous Media, 2020, 15 (3) : 519-542. doi: 10.3934/nhm.2020029
References:
[1]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29.  doi: 10.1137/120868748.  Google Scholar

[2]

M. Bisi, Some kinetic models for a market economy, Boll. Unione Mat. Ital., 10 (2017), 143-158.  doi: 10.1007/s40574-016-0099-4.  Google Scholar

[3]

M. BisiG. Spiga and G. Toscani, Kinetic models of conservative economies with wealth redistribution, Commun. Math. Sci., 7 (2009), 901-916.  doi: 10.4310/CMS.2009.v7.n4.a5.  Google Scholar

[4]

A. V. BobylevC. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized nonlinear kinetic Maxwell models, Comm. Math. Phys., 291 (2009), 599-644.  doi: 10.1007/s00220-009-0876-3.  Google Scholar

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A. V. Bobylev and Å. Windfall, Kinetic modeling of economic games with large number of participants, Kinet. Relat. Models, 4 (2011), 169-185.  doi: 10.3934/krm.2011.4.169.  Google Scholar

[6]

C. Brugna and G. Toscani, Kinetic models for goods exchange in a multi-agent market, Phys. A, 499 (2018), 362-375.  doi: 10.1016/j.physa.2018.02.070.  Google Scholar

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J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.  Google Scholar

[8]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[9]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[10]

G. Dimarco and G. Toscani, Kinetic modeling of alcohol consumption, J. Stat. Phys., 177 (2019), 1022-1042.  doi: 10.1007/s10955-019-02406-0.  Google Scholar

[11]

B. Düring, L. Pareschi and G. Toscani, Kinetic models for optimal control of wealth inequalities, Eur. Phys. J. B, 91 (2018), 12pp. doi: 10.1140/epjb/e2018-90138-1.  Google Scholar

[12]

M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys., 109 (2002), 407-432.  doi: 10.1023/A:1020437925931.  Google Scholar

[13]

U. GaribaldiE. Scalas and P. Viarengo, Statistical equilibrium in simple exchange games. Ⅱ. The redistribution game, Eur. Phys. J. B, 60 (2007), 241-246.  doi: 10.1140/epjb/e2007-00338-5.  Google Scholar

[14]

S. Guala, Taxes in a wealth distribution model by inelastically scattering of particles, Interdisciplinary Description Complex Syst., 7 (2009), 1-7.   Google Scholar

[15]

S. Gualandi and G. Toscani, Pareto tails in socio-economic phenomena: A kinetic description, Economics, 12 (2018), 1-17.  doi: 10.5018/economics-ejournal.ja.2018-31.  Google Scholar

[16]

S. Gualandi and G. Toscani, Human behavior and lognormal distribution. A kinetic description, Math. Models Methods Appl. Sci., 29 (2019), 717-753.  doi: 10.1142/S0218202519400049.  Google Scholar

[17]

D. J. Kuss and M. D. Griffiths, Online social networking and addiction – A review of the physchological literature, Int. J. Environ. Res. Public Health, 8 (2011), 3528-3552.  doi: 10.3390/ijerph8093528.  Google Scholar

[18] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[19]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[20]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 1-7.  doi: 10.1103/PhysRevE.69.046102.  Google Scholar

[21]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[22]

G. Toscani, Wealth redistribution in conservative linear kinetic models, Europhys. Lett. (EPL), 88 (2009). doi: 10.1209/0295-5075/88/10007.  Google Scholar

[23]

G. ToscaniC. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566.  doi: 10.1007/s10955-012-0653-0.  Google Scholar

[24]

G. ToscaniA. Tosin and M. Zanella, Opinion modeling on social media and marketing aspects, Phys. Rev. E, 98 (2018), 1-15.  doi: 10.1103/PhysRevE.98.022315.  Google Scholar

[25]

G. ToscaniA. Tosin and M. Zanella, Multiple-interaction kinetic modeling of a virtual-item gambling economy, Phys. Rev. E, 100 (2019), 1-16.  doi: 10.1103/PhysRevE.100.012308.  Google Scholar

[26]

X. Wang and M. Pleimling, Behavior analysis of virtual-item gambling, Phys. Rev. E, 98 (2018), 1-12.  doi: 10.1103/PhysRevE.98.012126.  Google Scholar

[27]

X. Wang and M. Pleimling, Online gambling of pure chance: Wager distribution, risk attitude, and anomalous diffusion, Sci. Rep., 9 (2019). doi: 10.1038/s41598-019-50168-2.  Google Scholar

show all references

References:
[1]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29.  doi: 10.1137/120868748.  Google Scholar

[2]

M. Bisi, Some kinetic models for a market economy, Boll. Unione Mat. Ital., 10 (2017), 143-158.  doi: 10.1007/s40574-016-0099-4.  Google Scholar

[3]

M. BisiG. Spiga and G. Toscani, Kinetic models of conservative economies with wealth redistribution, Commun. Math. Sci., 7 (2009), 901-916.  doi: 10.4310/CMS.2009.v7.n4.a5.  Google Scholar

[4]

A. V. BobylevC. Cercignani and I. M. Gamba, On the self-similar asymptotics for generalized nonlinear kinetic Maxwell models, Comm. Math. Phys., 291 (2009), 599-644.  doi: 10.1007/s00220-009-0876-3.  Google Scholar

[5]

A. V. Bobylev and Å. Windfall, Kinetic modeling of economic games with large number of participants, Kinet. Relat. Models, 4 (2011), 169-185.  doi: 10.3934/krm.2011.4.169.  Google Scholar

[6]

C. Brugna and G. Toscani, Kinetic models for goods exchange in a multi-agent market, Phys. A, 499 (2018), 362-375.  doi: 10.1016/j.physa.2018.02.070.  Google Scholar

[7]

J. A. CarrilloL. Pareschi and M. Zanella, Particle based gPC methods for mean-field models of swarming with uncertainty, Commun. Comput. Phys., 25 (2019), 508-531.  doi: 10.4208/cicp.oa-2017-0244.  Google Scholar

[8]

S. CordierL. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277.  doi: 10.1007/s10955-005-5456-0.  Google Scholar

[9]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[10]

G. Dimarco and G. Toscani, Kinetic modeling of alcohol consumption, J. Stat. Phys., 177 (2019), 1022-1042.  doi: 10.1007/s10955-019-02406-0.  Google Scholar

[11]

B. Düring, L. Pareschi and G. Toscani, Kinetic models for optimal control of wealth inequalities, Eur. Phys. J. B, 91 (2018), 12pp. doi: 10.1140/epjb/e2018-90138-1.  Google Scholar

[12]

M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails, J. Statist. Phys., 109 (2002), 407-432.  doi: 10.1023/A:1020437925931.  Google Scholar

[13]

U. GaribaldiE. Scalas and P. Viarengo, Statistical equilibrium in simple exchange games. Ⅱ. The redistribution game, Eur. Phys. J. B, 60 (2007), 241-246.  doi: 10.1140/epjb/e2007-00338-5.  Google Scholar

[14]

S. Guala, Taxes in a wealth distribution model by inelastically scattering of particles, Interdisciplinary Description Complex Syst., 7 (2009), 1-7.   Google Scholar

[15]

S. Gualandi and G. Toscani, Pareto tails in socio-economic phenomena: A kinetic description, Economics, 12 (2018), 1-17.  doi: 10.5018/economics-ejournal.ja.2018-31.  Google Scholar

[16]

S. Gualandi and G. Toscani, Human behavior and lognormal distribution. A kinetic description, Math. Models Methods Appl. Sci., 29 (2019), 717-753.  doi: 10.1142/S0218202519400049.  Google Scholar

[17]

D. J. Kuss and M. D. Griffiths, Online social networking and addiction – A review of the physchological literature, Int. J. Environ. Res. Public Health, 8 (2011), 3528-3552.  doi: 10.3390/ijerph8093528.  Google Scholar

[18] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013.   Google Scholar
[19]

L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74 (2018), 1575-1600.  doi: 10.1007/s10915-017-0510-z.  Google Scholar

[20]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 1-7.  doi: 10.1103/PhysRevE.69.046102.  Google Scholar

[21]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.  Google Scholar

[22]

G. Toscani, Wealth redistribution in conservative linear kinetic models, Europhys. Lett. (EPL), 88 (2009). doi: 10.1209/0295-5075/88/10007.  Google Scholar

[23]

G. ToscaniC. Brugna and S. Demichelis, Kinetic models for the trading of goods, J. Stat. Phys., 151 (2013), 549-566.  doi: 10.1007/s10955-012-0653-0.  Google Scholar

[24]

G. ToscaniA. Tosin and M. Zanella, Opinion modeling on social media and marketing aspects, Phys. Rev. E, 98 (2018), 1-15.  doi: 10.1103/PhysRevE.98.022315.  Google Scholar

[25]

G. ToscaniA. Tosin and M. Zanella, Multiple-interaction kinetic modeling of a virtual-item gambling economy, Phys. Rev. E, 100 (2019), 1-16.  doi: 10.1103/PhysRevE.100.012308.  Google Scholar

[26]

X. Wang and M. Pleimling, Behavior analysis of virtual-item gambling, Phys. Rev. E, 98 (2018), 1-12.  doi: 10.1103/PhysRevE.98.012126.  Google Scholar

[27]

X. Wang and M. Pleimling, Online gambling of pure chance: Wager distribution, risk attitude, and anomalous diffusion, Sci. Rep., 9 (2019). doi: 10.1038/s41598-019-50168-2.  Google Scholar

Figure 1.  The asymptotic wealth variance $ \Sigma^\infty $ vs. the taxation rate $ \alpha $ for both constant and $ \alpha $-dependent mean wealth $ m $ of the background
Figure 2.  Evolution at times $ t = 1,\,5,\,25 $ of the multiple-interaction model (8), (9) with $ N = 5 $, $ N = 100 $ and of its linearised version (12), (13)
Figure 3.  Comparison between the large time solution ($ t = 25 $) of the linearised model (12), (13) and the equilibrium distribution $ f^\infty $ (17) computed from the Fokker-Planck equation (16) in the quasi-invariant regime. Top row: $ \kappa = 0.1 $, bottom row: $ \kappa = 0.01 $. The right column displays the log-log plots of the graphs in the left column
Figure 4.  Comparison between the equilibrium distribution $ f^\infty $ (17) and the large time solution of the multiple-interaction model with $ N = 10^2 $ and $ N = 10^3 $ for $ \kappa = 0.1 $. The right panel is the log-log plot of the graph in the left panel, which gives a closer insight into the tails of the compared distributions
Figure 5.  Evolution at times $ t = 5,\,10,\,25 $ of the multiple-interaction model (18), (20) with $ N = 2 $, $ N = 5 $, $ N = 100 $ and of its linearised version (22), (23). The bottom row displays the log-log plots of the graphs in the top row to better appreciate the approximation of the tail of the distribution
Figure 6.  Left: comparison of the equilibrium distribution 30 and the large time distribution of the linearised model (22), (24) in the quasi-invariant limit. Right: comparison of the evolution of the energy of the multiple-interaction model (20) with $ N = 10 $ for decreasing $ \lambda $ and the solution to (28) obtained in the quasi-invariant limit
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