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Explicit equilibria in a kinetic model of gambling

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  • Federico Bassetti
  • Giuseppe Toscani

Abstract

We introduce and discuss a nonlinear kinetic equation of Boltzmann type which describes the evolution of wealth in a pure gambling process, where the entire sum of wealths of two agents is up for gambling, and randomly shared between the agents. For this equation the analytical form of the steady states is found for various realizations of the random fraction of the sum which is shared to the agents. Among others, Gibbs distribution appears as steady state in case of a uniformly distributed random fraction, while Gamma distribution appears for a random fraction which is Beta distributed. The case in which the gambling game is only conservative-in-the-mean is shown to lead to an explicit heavy tailed distribution.

Suggested Citation

  • Federico Bassetti & Giuseppe Toscani, 2010. "Explicit equilibria in a kinetic model of gambling," Papers 1002.3689, arXiv.org.
    • Handle: RePEc:arx:papers:1002.3689
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    Cited by:

    1. Giuseppe Toscani & Andrea Tosin & Mattia Zanella, 2019. "Multiple-interaction kinetic modelling of a virtual-item gambling economy," Papers 1904.07660, arXiv.org.
    2. Gualandi, Stefano & Toscani, Giuseppe, 2019. "Size distribution of cities: A kinetic explanation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 221-234.
    3. Giuseppe Toscani, 2016. "Kinetic and mean field description of Gibrat's law," Papers 1606.04796, arXiv.org.
    4. Max Greenberg & H. Oliver Gao, 2024. "Twenty-five years of random asset exchange modeling," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 97(6), pages 1-27, June.
    5. Gualandi, Stefano & Toscani, Giuseppe, 2017. "Pareto tails in socio-economic phenomena: A kinetic description," Economics Discussion Papers 2017-111, Kiel Institute for the World Economy (IfW Kiel).
    6. Gualandi, Stefano & Toscani, Giuseppe, 2018. "Pareto tails in socio-economic phenomena: A kinetic description," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 12, pages 1-17.
    7. G. Dimarco & L. Pareschi & G. Toscani & M. Zanella, 2020. "Wealth distribution under the spread of infectious diseases," Papers 2004.13620, arXiv.org.
    8. Andrea Medaglia & Andrea Tosin & Mattia Zanella, 2022. "Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-30, August.
    9. Bassetti, Federico & Ladelli, Lucia, 2023. "Central limit theorem in uniform metrics for generalized Kac equations," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    10. Düring, Bertram & Georgiou, Nicos & Merino-Aceituno, Sara & Scalas, Enrico, 2022. "Continuum and thermodynamic limits for a simple random-exchange model," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 248-277.
    11. Toscani, Giuseppe, 2016. "Kinetic and mean field description of Gibrat’s law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 802-811.
    12. Costas Efthimiou & Adam Wearne, 2016. "Household Income Distribution in the USA," Papers 1602.06234, arXiv.org.
    13. Marco Torregrossa & Giuseppe Toscani, 2017. "Wealth distribution in presence of debts. A Fokker--Planck description," Papers 1709.09858, arXiv.org.
    14. Luquini, Evandro & Montagna, Guido & Omar, Nizam, 2020. "Fusing non-conservative kinetic market models and evolutionary computing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).

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