Statistical physics of the spatial Prisoner’s Dilemma with memory-aware agents

  • Marco Alberto Javarone
Regular Article


We introduce an analytical model to study the evolution towards equilibrium in spatial games, with ‘memory-aware’ agents, i.e., agents that accumulate their payoff over time. In particular, we focus our attention on the spatial Prisoner’s Dilemma, as it constitutes an emblematic example of a game whose Nash equilibrium is defection. Previous investigations showed that, under opportune conditions, it is possible to reach, in the evolutionary Prisoner’s Dilemma, an equilibrium of cooperation. Notably, it seems that mechanisms like motion may lead a population to become cooperative. In the proposed model, we map agents to particles of a gas so that, on varying the system temperature, they randomly move. In doing so, we are able to identify a relation between the temperature and the final equilibrium of the population, explaining how it is possible to break the classical Nash equilibrium in the spatial Prisoner’s Dilemma when considering agents able to increase their payoff over time. Moreover, we introduce a formalism to study order-disorder phase transitions in these dynamics. As result, we highlight that the proposed model allows to explain analytically how a population, whose interactions are based on the Prisoner’s Dilemma, can reach an equilibrium far from the expected one; opening also the way to define a direct link between evolutionary game theory and statistical physics.


Statistical and Nonlinear Physics 


  1. 1.
    M. Perc, P. Grigolini, Chaos Solitons Fractals 56, 1 (2013)CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life (Harvard University Press, 2006)Google Scholar
  3. 3.
    M. Tomassini, Introduction to evolutionary game theory, in Proc. Conf. on Genetic and evolutionary computation companion (2014)Google Scholar
  4. 4.
    P.C. Julia, J. Gomez-Gardenes, A. Traulsen, Y. Moreno, New J. Phys. 11, 083031 (2009)CrossRefGoogle Scholar
  5. 5.
    L.M. Floria, C. Gracia-Lazaro, J. Gomez-Gardenes, Y. Moreno, Phys. Rev. E 79, 026106 (2009)CrossRefADSGoogle Scholar
  6. 6.
    J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems (Cambridge University Press, 1988)Google Scholar
  7. 7.
    A.M. Colman, Game Theory and Its Applications (Digital Printing, 2008)Google Scholar
  8. 8.
    M. Perc, A. Szolnoki, Phys. Rev. E 77, 011904 (2008)CrossRefADSGoogle Scholar
  9. 9.
    A. Szolnoki, M. Perc, J. R. Soc. Interface 12, 20141299 (2015)CrossRefGoogle Scholar
  10. 10.
    Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3, 1183 (2013)ADSGoogle Scholar
  11. 11.
    A. Szolnoki, N.-G. Xie, C. Wang, M. Perc, Europhys. Lett. 96, 38002 (2011)CrossRefADSGoogle Scholar
  12. 12.
    M. Perc, A. Szolnoki, New J. Phys. 14, 043013 (2012)CrossRefADSGoogle Scholar
  13. 13.
    D. Friedman, J. Evol. Econ. 8, 15 (1998)CrossRefGoogle Scholar
  14. 14.
    S. Schuster, L. de Figueiredo, A. Schroeter, C. Kaleta, BioSystems 105, 147 (2011)CrossRefGoogle Scholar
  15. 15.
    E. Frey, Physica A 389, 4265 (2010)CrossRefADSMathSciNetMATHGoogle Scholar
  16. 16.
    F. Fu, D.I. Rosenbloom, L. Wang, M.A. Nowak, Proc. R. Soc. B 278, 42 (2011)CrossRefGoogle Scholar
  17. 17.
    E. Lieberman, C. Hauert, M.A. Nowak, Nature 433, 312 (2005)CrossRefADSGoogle Scholar
  18. 18.
    S. Galam, B. Walliser, Physica A 389, 481 (2010)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    S. Meloni, A. Buscarino, L. Fortuna, M. Frasca, J. Gomez-Gardenes, V. Latora, Y. Moreno, Phys. Rev. E 79, 067101 (2009)CrossRefADSGoogle Scholar
  20. 20.
    A. Antonioni, M. Tomassini, P. Buesser, J. Theor. Biol. 344, 40 (2014)CrossRefGoogle Scholar
  21. 21.
    M. Tomassini, A. Antonioni, J. Theor. Biol. 364, 154 (2015)CrossRefGoogle Scholar
  22. 22.
    A. Antonioni, M. Tomassini, A. Sanchez, Sci. Rep. 5, 10282 (2015)CrossRefADSGoogle Scholar
  23. 23.
    M. Perc, J. Gomez-Gardenes, A. Szolnoki, L.M. Floria, Y. Moreno, J. R. Soc. Interface 10, 20120997 (2013)CrossRefGoogle Scholar
  24. 24.
    M.A. Javarone, A.E. Atzeni, Comput. Soc. Netw. 2, 15 (2015)CrossRefGoogle Scholar
  25. 25.
    M.A. Javarone, A.E. Atzeni, S. Galam, Lect. Notes Comput. Sci. 9028, 155 (2015)CrossRefGoogle Scholar
  26. 26.
    M.A. Nowak, Science 314, 1560 (2006)CrossRefADSGoogle Scholar
  27. 27.
    G. Szabo, G. Fath, Phys. Rep. 446, 97 (2007)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    M.A. Nowak, R.M. May, Nature 359, 826 (1992)CrossRefADSGoogle Scholar
  29. 29.
    C. Hauert, G. Szabo, Am. J. Phys. 73, 405 (2005)CrossRefADSMathSciNetMATHGoogle Scholar
  30. 30.
    K. Huang, Statistical Mechanics, 2nd edn. (Wiley, 1987)Google Scholar
  31. 31.
    A. Szolnoki, G. Szabo, M. Perc, Phys. Rev. E 83, 0361101 (2011)CrossRefGoogle Scholar
  32. 32.
    A. Szolnoki, M. Perc, Europhys. Lett. 92, 38003 (2010)CrossRefADSGoogle Scholar
  33. 33.
    M.A. Javarone, Europhys. Lett. 110, 58003 (2015)CrossRefADSGoogle Scholar
  34. 34.
    M. Mobilia, S. Redner, Phys. Rev. E 68, 046106 (2003)CrossRefADSGoogle Scholar
  35. 35.
    A. Barra, J. Stat. Phys. 132, 787 (2008)CrossRefADSMathSciNetMATHGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CagliariCagliariItaly
  2. 2.DUMAS – Department of Humanities and Social Sciences, University of SassariSassariItaly

Personalised recommendations