A Game Theoretic Model of Wealth Distribution

  • Juan Pablo Pinasco
  • Mauro Rodríguez Cartabia
  • Nicolas Saintier


In this work, we consider an agent-based model in order to study the wealth distribution problem where the interchange is determined with a symmetric zero-sum game. Simultaneously, the agents update their way of play trying to learn the optimal one. Here, the agents use mixed strategies. We study this model using both simulations and theoretical tools. We derive the equations for the learning mechanism, and we show that the mean strategy of the population satisfies an equation close to the classical replicator equation. Concerning the wealth distribution, there are two interesting situations depending on the equilibrium of the game. For pure strategies equilibria, the wealth distribution is fixed after some transient time, and those players which initially were close to the optimal strategy are richer. When the game has an equilibrium in mixed strategies, the stationary wealth distribution is close to a Gamma distribution with variance depending on the coefficients of the game matrix. We compute theoretically their second moment in this case.


Wealth distribution Evolutionary games Agent-based models 

Mathematics Subject Classification

91A22 91B69 91B60 



This paper was partially supported by Grants UBACyT 20020130100283BA, CONICET PIP 11220150100032CO and ANPCyT PICT 2012-0153. J. P. Pinasco and N. Saintier members of CONICET. M. Rodríguez Cartabia is a Fellow of CONICET.


  1. 1.
    Boccabella A, Natalini R, Pareschi L (2011) On a continuous mixed strategy model for evolutionary game-theory. Kinet Relat Models 4:187–213MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bournez O, Chalopin J, Cohen J, Koegler X, Rabie M (2011) Computing with Pavlovian populations. In: International conference on principles of distributed systems. Springer, Berlin, pp 409-420Google Scholar
  3. 3.
    Chakrabarti AS, Chakrabarti BK (2009) Microeconomics of the ideal gas like market models. Phys A Stat Mech Appl 388:4151–4158CrossRefGoogle Scholar
  4. 4.
    Chakrabarti BK, Chakraborti A, Chakravarty SR, Chatterjee A (2013) Econophysics of income and wealth distributions. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  5. 5.
    Chakraborti A, Chakrabarti BK (2000) Statistical mechanics of money: how saving propensity affects its distribution. Eur Phys J B 17:167CrossRefGoogle Scholar
  6. 6.
    Chatterjee A, Chakrabarti BK, Manna SS (2004) Pareto law in a kinetic model of market with random saving propensity. Phys A Stat Mech Appl 335:155–163MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cressman R, Tao Y (2014) The replicator equation and other game dynamics. Proc Natl Acad Sci USA 111:10810–10817MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
  9. 9.
    Friedman D (1998) On economic applications of evolutionary game theory. J Evol Econ 8:15–43CrossRefGoogle Scholar
  10. 10.
    Gupta AK (2006) Money exchange model and a general outlook. Phys A Stat Mech Appl 359:634–640CrossRefGoogle Scholar
  11. 11.
    Hofbauer J, Schuster P, Sigmund K (1979) A note on evolutionarily stable strategies and game dynamics. J Theor Biol 81:609–612CrossRefGoogle Scholar
  12. 12.
    Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  13. 13.
    Iriberri N, Uriarte JR (2012) Minority language and the stability of bilingual equilibria. Rational Soc 24:442–462CrossRefGoogle Scholar
  14. 14.
    Kraines D, Kraines V (1989) Pavlov and the prisoner’s dilemma. Theory Decis 26:47–79MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pareschi L, Toscani G (2006) Self-similarity and power-like tails in nonconservative kinetic models. J Stat Phys 124:747–779MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pareto V (1897) Cours d’economie politique. Rouge, LausanneGoogle Scholar
  17. 17.
    Patriarca M, Chakraborti A, Germano G (2006) Influence of saving propensity on the power law tail of wealth distribution. Phys A Stat Mech Appl 369(2):723–736CrossRefGoogle Scholar
  18. 18.
    Patriarca AM, Kaski K, Chakraborti K (2004) Statistical model with a standard \(\Gamma \) distribution. Phys Rev E 70:016104CrossRefGoogle Scholar
  19. 19.
    Patriarca M, Castelló X, Uriarte J, Eguíluz V, San Miguel M (2012) Modeling two-language competition dynamics. Adv Comput Syst 15:1250048MathSciNetCrossRefGoogle Scholar
  20. 20.
    Repetowicz P, Hutzler S, Richmond P (2005) Dynamics of money and income distributions. Phys A Stat Mech Appl 356:641–654MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sandholm WH (2010) Population games and evolutionary dynamics. MIT Press, CambridgeMATHGoogle Scholar
  22. 22.
    Silva C, Pereira W, Knotek J, Campos P (2011) Evolutionary dynamics of the spatial prisoners dilemma with single and multi-behaviors: a multi-agent application. In: Peixoto MM, Pinto AA, Rand DA (eds) Chapter 49 in dynamics, games and science II DYNA 2008, in honor of Mauricio Peixoto and David Rand. Springer, BerlinGoogle Scholar
  23. 23.
    Silver J, Slud E (2002) Statistical equilibrium wealth distributions in an exchange economy with stochastic preferences. J Econ Theory 106:417–435MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Szilagyi MN (2015) A general N-person game solver for Pavlovian agents. Complex Syst 24:261–274MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Taylor PD, Jonker L (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zeeman EC (1980) Population dynamics from game theory. Lecture Notes in Mathematics. In: Nitecki A, Robinson C (eds) Proceedings of an international conference on global theory of dynamical systems, vol 819. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Juan Pablo Pinasco
    • 1
  • Mauro Rodríguez Cartabia
    • 1
  • Nicolas Saintier
    • 1
  1. 1.Departamento de Matemática, FCENUniversidad de Buenos Aires, Instituto de Matemática Santaló (IMAS), UBA-CONICETBuenos AiresArgentina

Personalised recommendations