State-Space Visualization and Fractal Properties of Parrondo’s Games

  • Andrew Allison
  • Derek Abbott
  • Charles Pearce
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 7)


Parrondo’s games are essentially Markov games. They belong to the same class as Snakes and Ladders. The important distinguishing feature of Parrondo’s games is that the transition probabilities may vary in time. It is as though “snakes,” “ladders” and “dice” were being added and removed while the game was still in progress. Parrondo’s games are not homogeneous in time and do not necessarily settle down to an equilibrium. They model non-equilibrium processes in physics.

We formulate Parrondo’s games as an inhomogeneous sequence of Markov transition operators, with rewards. Parrondo’s “paradox” is shown to be equivalent to saying that the expected value of the reward, from the whole process, is not a linear function of the Markov operators. When we say that a game is “winning” or “losing” then we must be careful to include the whole process in our definition of the word “game.” Specifically, we must include the time varying probability vector in our calculations.We give practical rules for calculating the expected value of the return from sequences of Parrondo’s games. We include a worked example and a comparison between the theory and a simulation.

We apply visualization techniques, from physics and engineering, to an inhomogeneous Markov process and show that the limiting set or “attractor” of this process has fractal geometry. This is in contrast to the relevant theory for homogeneous Markov processes where the stable, equilibrium limiting set is a single point in the state space.We show histograms of simulations and describe methods for calculating the capacity dimension and the moments of the fractal attractors. We indicate how to construct optimal forms of Parrondo’s games and describe a symmetrical family of games which includes the optimal form, as a limiting case.We investigate the fractal geometry of the attractors for this symmetrical family of games. The resulting geometry is very interesting, even beautiful.


Fractal Property Probability Vector Iterate Function System Markov Operator Homogeneous Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Harmer G. P. and Abbott D., Parrondo’s paradox: losing strategies cooperate to win. Nature (London), 402, 864 (1999).CrossRefGoogle Scholar
  2. [2]
    Harmer G. P., Abbott D. and Taylor P. G., The paradox of Parrondo’s games Proc. Royal Soc., Series A, (Math. Phys. and Eng. Science), 456(99), 247–259 (2000).zbMATHMathSciNetGoogle Scholar
  3. [3]
    Harmer G. P., Abbott D., Taylor P. G. and Parrondo J. M. R., “Parrondo’s Paradoxical Games and the Discrete Brownian Ratchet,” in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations (UPoN’99) D. Abbott and L. B. Kish editors, vol. 511, American Inst. Phys., 2000, pp. 189–200.Google Scholar
  4. [4]
    Harmer G. P. and Abbott D., Parrondo’s paradox. Statistical Science, 14(2), 206–213, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Pearce C. E. M., “Parrondo’s paradoxical games,” in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations (UPoN’99) D. Abbott and L. B. Kish editors, vol. 511, American Inst. Phys., 2000, pp. 201–206.Google Scholar
  6. [6]
    Pearce C. E. M., “Entropy, Markov Information Sources and Parrondo’s Games,” in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations (UPoN’99) D. Abbott and L. B. Kish editors, vol. 511, American Inst. Phys., 2000, pp. 207–212.Google Scholar
  7. [7]
    Harmer G. P. and Abbott D., A review of Parrondo’s paradox, Fluctuation and Noise Letters, 2, R71–R107 (2002).CrossRefGoogle Scholar
  8. [8]
    Meyer D. A. and Blumer H., Parrondo’s games as lattice gas automata, J. Stat. Phys., 107, 225–239 (2002).zbMATHCrossRefGoogle Scholar
  9. [9]
    Moraal H., Counterintuitive behaviour in games based on spin models, J. Phys. A, Math. Gen., 33, L203–L206 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Davies P. C. W., “Physics and Life, The Abdus Salam Memorial Lecture,” in Sixth Trieste Conference on Chemical Evolution, Trieste, Italy, eds.: J. Chela-Flores and T. Tobias and F. Raulin., Kluwer Academic Publishers (2001), 13–20.Google Scholar
  11. [11]
    Toral R., Cooperative Parrondo’s games, Fluctuation and Noise Letters, 1, L7–L12 (2001).CrossRefMathSciNetGoogle Scholar
  12. [12]
    McClintock P. V. E., Unsolved problems of noise, Nature (London), 401, 23 (1999).CrossRefGoogle Scholar
  13. [13]
    Borisenko A. I., and Tarapov I. E., Vector and Tensor Analysis, with Applications, Dover Publications, Inc., 1968.Google Scholar
  14. [14]
    McCoy N. H., Fundamentals of Abstract Algebra, Allyn and Bacon, Inc., 1972.Google Scholar
  15. [15]
    DeRusso P. M., Roy R. J., and Close C. M., State Variables for Engineers, John Wiley and Sons, Inc., 1965.Google Scholar
  16. [16]
    Karlin S., and Taylor, H. M., A First Course in Stochastic Processes, Academic Press, 1975.Google Scholar
  17. [17]
    Karlin S., and Taylor, H. M., An Introduction to Stochastic Modeling, Academic Press, 1998.Google Scholar
  18. [18]
    Norris J. R., Markov chains, Cambridge University Press, 1997.Google Scholar
  19. [19]
    Yates R. D., and Goodman, D. J., Probability and Stochastic Processes, John Wiley and Sons Inc., 1999.Google Scholar
  20. [20]
    Diacu F., and Holmes P., Celestial Encounters, Princeton University Press, 1996, chap. 1.Google Scholar
  21. [21]
    Baierlein R., Thermal Physics, Cambridge University Press, 1999, chap. 13.Google Scholar
  22. [22]
    Lanczos C., The Variational Principles of Mechanics, Dover Publications Inc., 1949, chap. 1.Google Scholar
  23. [23]
    Barnsley M., Fractals Everywhere, Academic Press, 1988.Google Scholar
  24. [24]
    Middlebrook R. D., and Ćuk, S. A., “A general unified approach to modelling switching converter power stages,” in IEEE Power Electronics Specialist’s Conf. Rec, 1976, pp. 18–34, in Proceedings of IEEE Power Electronics Specialist’s Conf. Rec.Google Scholar
  25. [25]
    Howard R. A., Dynamic Programming and Markov Processes, John Wiley and Sons Inc., 1960.Google Scholar

Copyright information

© Birkhäuser Boston 2005

Authors and Affiliations

  • Andrew Allison
    • 1
  • Derek Abbott
    • 1
  • Charles Pearce
    • 2
  1. 1.Centre for Biomedical Engineering (CBME), Department of Electrical and Electronic EngineeringThe University of AdelaideAustralia
  2. 2.Department of Applied MathematicsUniversity of AdelaideAustralia

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