Parrondo games as disordered systems

  • Jean-Marc LuckEmail author
Regular Article
Part of the following topical collections:
  1. Topical issue: Recent Advances in the Theory of Disordered Systems


Parrondo’s paradox refers to the counter-intuitive situation where a winning strategy results from a suitable combination of losing ones. Simple stochastic games exhibiting this paradox have been introduced around the turn of the millennium. The common setting of these Parrondo games is that two rules, A and B, are played at discrete time steps, following either a periodic pattern or an aperiodic one, be it deterministic or random. These games can be mapped onto 1D random walks. In capital-dependent games, the probabilities of moving right or left depend on the walker’s position modulo some integer K. In history-dependent games, each step is correlated with the Q previous ones. In both cases the gain identifies with the velocity of the walker’s ballistic motion, which depends non-linearly on model parameters, allowing for the possibility of Parrondo’s paradox. Calculating the gain involves products of non-commuting Markov matrices, which are somehow analogous to the transfer matrices used in the physics of 1D disordered systems. Elaborating upon this analogy, we study a paradigmatic Parrondo game of each class in the neutral situation where each rule, when played alone, is fair. The main emphasis of this systematic approach is on the dependence of the gain on the remaining parameters and, above all, on the game, i.e., the rule pattern, be it periodic or aperiodic, deterministic or random. One of the most original sides of this work is the identification of weak-contrast regimes for capital-dependent and history-dependent Parrondo games, and a detailed quantitative investigation of the gain in the latter scaling regimes.

Graphical abstract


  1. 1.
    G.P. Harmer, D. Abbott, Nature 402, 864 (1999) ADSCrossRefGoogle Scholar
  2. 2.
    P.V.E. McClintock, Nature 401, 23 (1999) ADSCrossRefGoogle Scholar
  3. 3.
    G.P. Harmer, D. Abbott, Stat. Sci. 14, 206 (1999) CrossRefGoogle Scholar
  4. 4.
    G.P. Harmer, D. Abbott, P.G. Taylor, Proc. R. Soc. London A 456, 247 (2000) ADSCrossRefGoogle Scholar
  5. 5.
    J.M.R. Parrondo, G.P. Harmer, D. Abbott, Phys. Rev. Lett. 85, 5226 (2000) ADSCrossRefGoogle Scholar
  6. 6.
    G.P. Harmer, D. Abbott, P.G. Taylor, J.M.R. Parrondo, Chaos 11, 705 (2001) ADSCrossRefGoogle Scholar
  7. 7.
    G.P. Harmer, D. Abbott, Fluct. Noise Lett. 2, R71 (2002) CrossRefGoogle Scholar
  8. 8.
    J.M.R. Parrondo, L. Dinis, Contemp. Phys. 45, 147 (2004) ADSCrossRefGoogle Scholar
  9. 9.
    D. Abbott, Fluct. Noise Lett. 9, 129 (2010) CrossRefGoogle Scholar
  10. 10.
    R.P. Feynman, R.B. Leighton, M. Sands, inFeynman Lectures on Physics (Addison-Wesley, Reading, MA, 1966), Vol. I, Chap. 46 Google Scholar
  11. 11.
    A. Ajdari, J. Prost, C.R. Acad. Sci. Paris, Ser. II 315, 1635 (1992) Google Scholar
  12. 12.
    M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993) ADSCrossRefGoogle Scholar
  13. 13.
    R.D. Astumian, M. Bier, Phys. Rev. Lett. 72, 1766 (1994) ADSCrossRefGoogle Scholar
  14. 14.
    F. Jülicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997) ADSCrossRefGoogle Scholar
  15. 15.
    P. Reimann, Phys. Rep. 361, 57 (2002) ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    B. Cleuren, C. Van den Broeck, Europhys. Lett. 67, 151 (2004) ADSCrossRefGoogle Scholar
  17. 17.
    C. Wang, N.G. Xie, L. Wang, Y. Ye, G. Xu, Fluct. Noise Lett. 10, 147 (2011) CrossRefGoogle Scholar
  18. 18.
    R.J. Kay, N.F. Johnson, Phys. Rev. E 67, 056128 (2003) ADSCrossRefGoogle Scholar
  19. 19.
    S.N. Ethier, J. Lee, Electron. J. Probab. 14, 1827 (2009) MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Bougerol, J. Lacroix,Products of Random Matrices, with Applications to Schrödinger Operators (Birkhäuser, Boston, 1985) Google Scholar
  21. 21.
    A. Crisanti, G. Paladin, A. Vulpiani,Products of Random Matrices in Statistical Physics, Springer Series in Solid-State Sciences (Springer, Berlin, 1992) Google Scholar
  22. 22.
    J.M. Luck,Systèmes désordonnés unidimensionnels (Collection Aléa, Saclay, 1992) Google Scholar
  23. 23.
    J.B. Pendry, Adv. Phys. 43, 461 (1994) ADSCrossRefGoogle Scholar
  24. 24.
    A. Comtet, C. Texier, Y. Tourigny, J. Phys. A 46, 254003 (2013) ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A. Comtet, Y. Tourigny, inStochastic Processes and RandomMatrices, edited by G. Schehr, A. Altland, Y.V. Fyodorov, N. O’Connell, L.F. Cugliandolo (Oxford University Press, Oxford, 2017) Google Scholar
  26. 26.
    J.L. Doob,Stochastic Processes (Wiley, New York, 1953) Google Scholar
  27. 27.
    W. Feller,An Introduction to Probability Theory and its Applications (Wiley, New York, 1968) Google Scholar
  28. 28.
    S. Karlin, H.M. Taylor,A First Course in Stochastic Processes (Academic Press, New York, 1975) Google Scholar
  29. 29.
    N.G. vanKampen,Stochastic Processes in Physics and Chemistry (North-, Amsterdam, 1992) Google Scholar
  30. 30.
    F.P. Kelly,Reversibility and Stochastic Networks (Wiley, Chichester, 1979) Google Scholar
  31. 31.
    D. Stirzaker,Stochastic Processes and Models (Oxford University Press, Oxford, 2005) Google Scholar
  32. 32.
    G.C. Crisan, E. Nechita, M. Talmaciu, Fluct. Noise Lett. 7, C19 (2007) CrossRefGoogle Scholar
  33. 33.
    L. Dinis, Phys. Rev. E 77, 021124 (2008) ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    T.W. Tang, A. Allison, D. Abbott, Fluct. Noise Lett. 4, L585 (2004) CrossRefGoogle Scholar
  35. 35.
    N.G. de Bruijn, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84, 27 (1981) CrossRefGoogle Scholar
  36. 36.
    D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984) ADSCrossRefGoogle Scholar
  37. 37.
    C. Janot,Quasicrystals: A Primer (Oxford University Press, Oxford, 1992) Google Scholar
  38. 38.
    M. Senechal,Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995) Google Scholar
  39. 39.
    E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376, 225 (2003) ADSCrossRefGoogle Scholar
  40. 40.
    E. Maciá, Rep. Prog. Phys. 69, 397 (2006) ADSCrossRefGoogle Scholar
  41. 41.
    K.H. Cheong, J.M. Koh, M.C. Jones, BioEssays 41, 1900027 (2019) CrossRefGoogle Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRSGif-sur-YvetteFrance

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