American Institute of Mathematical Sciences

Advanced
March  2011, 4(1): 169-185. doi: 10.3934/krm.2011.4.169

Kinetic modeling of economic games with large number of participants

Alexander Bobylev 1, and  Åsa Windfäll 1,

1. 

Department of Mathematics, Karlstad University, SE-651 88 Karlstad

Received  October 2010 Revised  December 2010 Published  January 2011

We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters $\{\gamma, q ,s\}$ where $0<\gamma\leq 1$ is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.
Keywords: Maxwell models, distribution of wealth, market economy., self-similar solutions.
Mathematics Subject Classification: 82C4.
Citation: Alexander Bobylev, Åsa Windfäll. Kinetic modeling of economic games with large number of participants. Kinetic & Related Models, 2011, 4 (1) : 169-185. doi: 10.3934/krm.2011.4.169
References:
[1]

F. Bassetti and G. Toscani, Explicit equilibria in a kinetic model of gambling,, Phys. Rev. E, 81 (2010).   Google Scholar

[2]

A. V. Bobylev, J. A. Carillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions,, J. Stat. Phys., 98 (2000), 743.  doi: 10.1023/A:1018627625800.  Google Scholar

[3]

A. V. Bobylev and C. Cercignani, Self-similar solutions for the Boltzmann equation with inelastic and elastic interactions,, J. Stat. Phys., 110 (2003), 333.  doi: 10.1023/A:1021031031038.  Google Scholar

[4]

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

[5]

A. V. Bobylev, C. Cercignani and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials,, J. Stat. Phys., 111 (2003), 403.  doi: 10.1023/A:1022273528296.  Google Scholar

[6]

L. Boltzmann, "Populäre Schriften,'', J.A. Barth, (1905).   Google Scholar

[7]

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

[8]

D. Matthes and G. Toscani, Analysis of a model for wealth redistribution,, Kinet. Relat. Models, 1 (2008), 1.   Google Scholar

[9]

G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences,'', Birkhauser, (2010).   Google Scholar

[10]

I. Shafarevish, "Socialism as Phenomenon of World History'' (in Russian),, YMCA-Press, (1977).   Google Scholar

[11]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy,, Phys. Rev. E, 69 (2004).   Google Scholar

[12]

I. Wright, The social architecture of capitalism,, Physica A, 346 (2005), 589.  doi: 10.1016/j.physa.2004.08.006.  Google Scholar

show all references

References:
[1]

F. Bassetti and G. Toscani, Explicit equilibria in a kinetic model of gambling,, Phys. Rev. E, 81 (2010).   Google Scholar

[2]

A. V. Bobylev, J. A. Carillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions,, J. Stat. Phys., 98 (2000), 743.  doi: 10.1023/A:1018627625800.  Google Scholar

[3]

A. V. Bobylev and C. Cercignani, Self-similar solutions for the Boltzmann equation with inelastic and elastic interactions,, J. Stat. Phys., 110 (2003), 333.  doi: 10.1023/A:1021031031038.  Google Scholar

[4]

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

[5]

A. V. Bobylev, C. Cercignani and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials,, J. Stat. Phys., 111 (2003), 403.  doi: 10.1023/A:1022273528296.  Google Scholar

[6]

L. Boltzmann, "Populäre Schriften,'', J.A. Barth, (1905).   Google Scholar

[7]

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

[8]

D. Matthes and G. Toscani, Analysis of a model for wealth redistribution,, Kinet. Relat. Models, 1 (2008), 1.   Google Scholar

[9]

G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences,'', Birkhauser, (2010).   Google Scholar

[10]

I. Shafarevish, "Socialism as Phenomenon of World History'' (in Russian),, YMCA-Press, (1977).   Google Scholar

[11]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy,, Phys. Rev. E, 69 (2004).   Google Scholar

[12]

I. Wright, The social architecture of capitalism,, Physica A, 346 (2005), 589.  doi: 10.1016/j.physa.2004.08.006.  Google Scholar

[1]

Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[2]

Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[3]

Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[4]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[5]

Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[6]

Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[7]

Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[8]

Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[9]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[10]

Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471
[11]

K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51
[12]

Meiyue Jiang, Juncheng Wei. $2\pi$-Periodic self-similar solutions for the anisotropic affine curve shortening problem II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 785-803. doi: 10.3934/dcds.2016.36.785
[13]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
[14]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[15]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
[16]

Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323
[17]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[18]

Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
[19]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
[20]

G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences