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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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    References listed on IDEAS

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    1. Basak, Suleyman & Shapiro, Alexander, 2001. "Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 371-405.
    2. Susanne Emmer & Claudia Klüppelberg & Ralf Korn, 2001. "Optimal Portfolios with Bounded Capital at Risk," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 365-384.
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    Cited by:

    1. Costas Efthimiou & Adam Wearne, 2016. "Household Income Distribution in the USA," Papers 1602.06234, arXiv.org.
    2. Giuseppe Toscani, 2016. "Kinetic and mean field description of Gibrat's law," Papers 1606.04796, arXiv.org.
    3. Marco Torregrossa & Giuseppe Toscani, 2017. "Wealth distribution in presence of debts. A Fokker--Planck description," Papers 1709.09858, arXiv.org.
    4. 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).
    5. 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.

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