Kinetic modeling of economic games with large number of participants

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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

Abstract
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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 and 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).
[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), 743-773. doi: 10.1023/A:1018627625800.
[3]	A. V. Bobylev and C. Cercignani, Self-similar solutions for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375. doi: 10.1023/A:1021031031038.
[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), 599-644. doi: 10.1007/s00220-009-0876-3.
[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), 403-417. doi: 10.1023/A:1022273528296.
[6]	L. Boltzmann, "Populäre Schriften,'' J.A. Barth, Leipzig, 1905.
[7]	S. Cordier, L. 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.
[8]	D. Matthes and G. Toscani, Analysis of a model for wealth redistribution, Kinet. Relat. Models, 1 (2008), 1-27.
[9]	G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences,'' Birkhauser, (to appear in 2010).
[10]	I. Shafarevish, "Socialism as Phenomenon of World History'' (in Russian), YMCA-Press, Paris, 1977.
[11]	F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004).
[12]	I. Wright, The social architecture of capitalism, Physica A, 346 (2005), 589-620. doi: 10.1016/j.physa.2004.08.006.

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References:

[1]	F. Bassetti and G. Toscani, Explicit equilibria in a kinetic model of gambling, Phys. Rev. E, 81 (2010).
[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), 743-773. doi: 10.1023/A:1018627625800.
[3]	A. V. Bobylev and C. Cercignani, Self-similar solutions for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375. doi: 10.1023/A:1021031031038.
[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), 599-644. doi: 10.1007/s00220-009-0876-3.
[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), 403-417. doi: 10.1023/A:1022273528296.
[6]	L. Boltzmann, "Populäre Schriften,'' J.A. Barth, Leipzig, 1905.
[7]	S. Cordier, L. 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.
[8]	D. Matthes and G. Toscani, Analysis of a model for wealth redistribution, Kinet. Relat. Models, 1 (2008), 1-27.
[9]	G. Naldi, L. Pareschi and G. Toscani (Eds.), "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences,'' Birkhauser, (to appear in 2010).
[10]	I. Shafarevish, "Socialism as Phenomenon of World History'' (in Russian), YMCA-Press, Paris, 1977.
[11]	F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004).
[12]	I. Wright, The social architecture of capitalism, Physica A, 346 (2005), 589-620. doi: 10.1016/j.physa.2004.08.006.

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