Developments in Parrondo’s Paradox

  • Derek Abbott
Part of the Understanding Complex Systems book series (UCS)


Parrondo’s paradox is the well-known counterintuitive situation where individually losing strategies or deleterious effects can combine to win. In 1996, Parrondo’s games were devised illustrating this effect for the first time in a simple coin tossing scenario. It turns out that, by analogy, Parrondo’s original games are a discrete-time, discrete-space version of a flashing Brownian ratchet—this was later formally proven via discretization of the Fokker-Planck equation. Over the past ten years, a number of authors have pointed to the generality of Parrondian behavior, and many examples ranging from physics to population genetics have been reported. In its most general form, Parrondo’s paradox can occur where there is a nonlinear interaction of random behavior with an asymmetry, and can be mathematically understood in terms of a convex linear combination. Many effects, where randomness plays a constructive role, such as stochastic resonance, volatility pumping, the Brazil nut paradox etc., can all be viewed as being in the class of Parrondian phenomena. We will briefly review Parrondo’s paradox, its recent developments, and its connection to related phenomena. In particular, we will review in detail a new form of Parrondo’s paradox: the Allison mixture—this is where random sequences with zero autocorrelation can be randomly mixed, paradoxically producing a sequence with non-zero autocorrelation. The equations for the autocorrelation have been previously analytically derived, but, for the first time, we will now give a complete physical picture that explains this phenomenon.


Stochastic Resonance Noise Letter Convex Linear Combination Brownian Motor Brownian Ratchet 
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., Losing strategies can win by Parrondo’s paradox, Nature 402 864 (1999).CrossRefGoogle Scholar
  2. 2.
    Arena, P., Fazzino, S., Fortuna, L., and Maniscalco, P., Game theory and non-linear dynamics: the Parrondo Paradox case study, Chaos, Solitons & Fractals 17(2–3) 545–555 (2003).zbMATHCrossRefGoogle Scholar
  3. 3.
    Behrends, E., The mathematical background of Parrondo’s paradox, Proc. SPIE Noise in Complex Systems and Stochastic Dynamics II, Maspalomas, Spain, Ed: Zoltan Gingl, 5471 510–517 (2004).Google Scholar
  4. 4.
    von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, Princeton University Press, New York, (1954).Google Scholar
  5. 5.
    Blackwell, D. and Girshick, M. A., Theory of Games and Statistical Decisions, John Wiley & Sons, New York (1954).zbMATHGoogle Scholar
  6. 6.
    Behrends, E., Parrondo’s paradox: a priori and adaptive strategies, Preprint: A-02-09, (2002).Google Scholar
  7. 7.
    Groeber, P., On Parrondo’s games as generalized by Behrends, Lecture Notes in Control and Information Sciences, 341 223–230 (2006).CrossRefMathSciNetGoogle Scholar
  8. 8.
    Abbott, D., Davies, P. C. W., and Shalizi, C. R., Order from disorder: the role of noise in creative processes: A special issue on game theory and evolutionary processes–overview, Fluctuation and Noise Letters, 2 C1–C12 (2002).CrossRefMathSciNetGoogle Scholar
  9. 9.
    Allison, A., Pearce, C. E. M., and Abbott, D., Finding keywords amongst noise: Automatic text classification without parsing, Proc. SPIE Noise and Stochastics in Complex Systems and Finance, Florence, Italy, Eds: János Kertész, Stefan Bornholdt, and Rosario N. Mantegna 6601 660113 (2007).Google Scholar
  10. 10.
    Davies, P. C. W., Physics and life: The Abdus Salam Memorial Lecture, Sixth Trieste Conference on Chemical Evolution, Trieste, Italy, Eds: J. Chela-Flores, T. Tobias, and F. Raulin, Kluwer Academic Publishers 13–20 (2001).Google Scholar
  11. 11.
    Harmer, G. P. and Abbott, D., Parrondo’s paradox, Statistical Science 14 206–213 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Parrondo, J. M. R., How to cheat a bad mathematician, in EEC HC&M Network on Complexity and Chaos (#ERBCHRX-CT940546), ISI, Torino, Italy (1996), Unpublished.Google Scholar
  13. 13.
    Adjari, A. and Prost, J., Drift induced by a periodic potential of low symmetry: Pulsed dielectrophoresis, C. R. Acad. Science Paris, Série II, 315 1635–1639 (1993).Google Scholar
  14. 14.
    Johnson, N. F., Jeffries, P., and Hui, P. M., Financial Market Complexity, Oxford University Press, Oxford (2003).Google Scholar
  15. 15.
    Lee, C. F., Johnson, N. F., Rodriguez, F., and Quiroga, L., Quantum coherence, correlated noise and Parrondo games, Fluctuation and Noise Letters 2(4) L293–L297 (2002).CrossRefMathSciNetGoogle Scholar
  16. 16.
    Flitney, A. P. and Abbott, D., Quantum Parrondo games, Physica A 314(1–4) 35–42 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Meyer, D. A. and Blumer, H., Quantum Parrondo games: biased and unbiased, Fluctuation and Noise Letters 2(4) L257–L262 (2002).CrossRefMathSciNetGoogle Scholar
  18. 18.
    Wolf, D. M., Vazirani, V. V., and Arkin, A. P., Diversity in times of adversity: Probabilistic strategies in microbial survival games, Journal of Theoretical Biology 234 227–253 (2005).CrossRefMathSciNetGoogle Scholar
  19. 19.
    Reed, F. A., Two-locus epistasis with sexually antagonistic selection: A genetic Parrondo’s paradox, Genetics, 176, 1923–1929 (2007).CrossRefGoogle Scholar
  20. 20.
    Masuda, N. and Konno, N., Subcritical behavior in the alternating supercritical Domany-Kinzel dynamics, European Physical Journal B 40 313–319 (2004).Google Scholar
  21. 21.
    Harmer, G. P. and Abbott, D., A review of Parrondo’s paradox, Fluctuation and Noise Letters, 2(2) R71–R107 (2002).CrossRefGoogle Scholar
  22. 22.
    Pinsky, R. and Scheutzow, M., Some remarks and examples concerning the transient and recurrence of random diffusions, Annales de l’Institut Henri Poincaré—Probabilités et Statistiques 28 519–536 (1992).zbMATHMathSciNetGoogle Scholar
  23. 23.
    Maslov, S. and Zhang, Y., Optimal investment strategy for risky assets, International Journal of Theoretical and Applied Finance 1 377–387 (1998).zbMATHCrossRefGoogle Scholar
  24. 24.
    Westerhoff, H. V., Tsong, T. Y., Chock, P. B., Chen Y., and Astumian, R. D., How enzymes can capture and transmit free energy contained in an oscillating electric field, Proceedings of the National Academy of Science 83 4734–4738 (1986).CrossRefGoogle Scholar
  25. 25.
    Key, E. S., Computable examples of the maximal Lyapunov exponent, Probability Theory and Related Fields 75 97–107 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Abbott, D., Overview: Unsolved problems of noise and fluctuations, Chaos, 11 526–538 (2001).zbMATHCrossRefGoogle Scholar
  27. 27.
    Luenberger, D. G., Investment Science, Oxford University Press, Oxford (1997).Google Scholar
  28. 28.
    Rosato, A., Strandburg, K. J., Prinz F., and Swendsen, R. H., Why the Brazil nuts are on top: Size segregation of particulate matter by shaking, Physical Review Letters 58 1038–1040 (1987).CrossRefMathSciNetGoogle Scholar
  29. 29.
    Allison, A. and Abbott, D., The physical basis for Parrondo’s games, Fluctuation and Noise Letters, 2(4) L327–L341 (2002).CrossRefMathSciNetGoogle Scholar
  30. 30.
    Toral, R., Amengual, P., and Mangioni, S., Parrondo’s games as a discrete ratchet, Physica A, 327(1–2) 105–110 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Amengual, P., Allison, A., Toral, R., and Abbott, D., Discrete-time ratchets, the Fokker-Planck equation and Parrondo’s paradox, Proceedings of the Royal Society London A, 460(2048), 2269–2284 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Harmer, G. P., Abbott, D., and Taylor, P. G., The paradox of Parrondo’s games, Proceedings of the Royal Society London A 456 247–259 (2000).zbMATHMathSciNetGoogle Scholar
  33. 33.
    Key, E. S., Kłosek, M. M., Abbott, D., On Parrondo’s paradox: how to construct unfair games by composing fair games, ANZIAM J. 47, 495–511 (2006).zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Allison, A. and Abbott, D., Stochastic resonance in a Brownian ratchet, Fluctuation and Noise Letters 1(4) L239–L244 (2001).CrossRefMathSciNetGoogle Scholar
  35. 35.
    Moraal, H., Counterintuitive behaviour in games based on spin models, Journal of Physics A, 33 L203–L206 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Costa, A., Fackrell, M., and Taylor, P. G., Two issues surrounding Parrondo’s paradox, Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Eds: Andrzej S. Nowak and Krzysztof Szajowski, 7 599–609 (2005).MathSciNetGoogle Scholar
  37. 37.
    Parrondo, J. M. R., Harmer, G. P., and Abbott, D., New paradoxical games based on Brownian ratchets, Physical Review Letters 85 5226–5229 (2000).CrossRefGoogle Scholar
  38. 38.
    Kay, R. J. and Johnson, N. F., Winning combinations of history-dependent games, Physical Review E 67 056128 (2003).Google Scholar
  39. 39.
    Toral, R., Cooperative Parrondo’s games, Fluctuation and Noise Letters 1 L7–L12 (2001).CrossRefMathSciNetGoogle Scholar
  40. 40.
    Allison, A. and Abbott, D., Control systems with stochastic feedback, Chaos 11 715–724 (2001).zbMATHCrossRefGoogle Scholar
  41. 41.
    Bishop, C. M., Neural Networks for Pattern Recognition, Oxford Press Oxford Chapter 9, 346–349 (1996).zbMATHGoogle Scholar
  42. 42.
    Van den Broeck, C., Reimann P., Kawai, R., and Hänggi, P., Coupled Brownian motors, Lecture Notes in Physics: Statistical Mechanics of Biocomplexity, Eds: D. Reguera, M. Rubi, and J. M. G. Vilar, 527 Springer-Verlag: Berlin, Heidelberg, New York, 93–111 (1999).Google Scholar
  43. 43.
    Onsager, L., Reciprocal relations in irreversible processes I, Physical Review 37 405–426 (1931).zbMATHCrossRefGoogle Scholar
  44. 44.
    Onsager, L., Reciprocal relations in irreversible processes II, Physical Review 38 2265 (1931).Google Scholar
  45. 45.
    Cleuren, B. and Van den Broeck C., Random walks with absolute negative mobility, Physical Review E, 64 030101 (2002).CrossRefGoogle Scholar
  46. 46.
    Di Crescenzo, A., A Parrondo paradox in reliability theory, The Mathematical Scientist 32(1) 17–22 arXiv:math/0602308v2 (2007).zbMATHMathSciNetGoogle Scholar
  47. 47.
    Kocarev, L. and Tasev, Z., Lyanpunov exponents, noise-induced synchronization, and Parrondo’s paradox, Physical Review E 65 046215 (2002).CrossRefMathSciNetGoogle Scholar
  48. 48.
    Buceta, J., Lindenberg, K., and Parrondo, J. M. R., Pattern formation induced by nonequilibrium global alternation of dynamics, Physical Review E 66 036216 (2002).CrossRefMathSciNetGoogle Scholar
  49. 49.
    Almeida, J., Peralta-Salas, D., and Romera, M., Can two chaotic systems give rise to order? Physica D 200 124–132 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Boyarsky, A., Góra, P., and Shafiqul Islam, Md., Randomly chosen chaotic maps can give rise to nearly ordered behavior, Physica D 210 284–294 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Harmer, G. P., Abbott, D., Taylor, P. G., and Parrondo, J. M. R., Parrondo’s games and Brownian ratchets, Chaos 11 705–714 (2001).zbMATHCrossRefGoogle Scholar
  52. 52.
    Atkinson, D. and Peijnenburg, J., Acting rationally with irrational strategies: Applications of the Parrondo effect, Reasoning, Rationality, Probability, Eds: Maria Carla Galavotti, Roberto Scazzieri, and Patrick Suppes, CSLI Publications, Stanford (2007).Google Scholar
  53. 53.
    Diamond, J. M., Why Sex is Fun?: The Evolution of Human Sexuality, Harper Collins, New York (1997).Google Scholar
  54. 54.
    Arizmendi, C. M., Paradoxical way for losers in a dating game, Proceedings of the AIP Nonequilibrium Statistical Mechaniucs and Nonliear Physics: XV Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics, Mar del Plata, Argentina, 4–8 December, 2006, Eds: Orazio Descalzi, Osvaldo A. Rosso, and Hilda A. Larrondo, 913, 20–25 arXiv:physics/0703189v1 (2007).Google Scholar
  55. 55.
    Satinover, J. B. and Sornette, D., ‘Illusion of control’ in time-horizon minority and Parrondo games, The European Physical Journal B 60(3) 369–384 (2007).CrossRefMathSciNetGoogle Scholar
  56. 56.
    Satinover, J. B. and Sornette, D., Illusion of control in a Brownian game, Physica A 386(1) 339–344 (2007).CrossRefGoogle Scholar
  57. 57.
    Boman, M., Johansson, S. J., and Lyback, D., Parrondo strategies for artificial traders, in Intelligent Agent Technology: Research and Development, Eds: Ning Zhong, Jiming Liu, Setsuo Ohsuga, Jeffrey Bradshaw, World Scientific, 150–159 arXiv:cs.ce/0204051 (2001).Google Scholar
  58. 58.
    Wah-Sui Almberg, W-S. and Boman, M., An active agent portfolio management algorithm, Artificial Intelligence and Computer Science, Ed: Susan Shannon, Nova Science Publishers, Inc. Hauppauge NY Chapter 4, 123–134 (2005).Google Scholar
  59. 59.
    Fernholz, R. and Shay, B., Stochastic portfolio theory and stock market equilibrium, Journal of Finance, 37 615–624 (1982).CrossRefGoogle Scholar
  60. 60.
    Cover, T. M. and Ordentlich, E., Universal portfolios with side information, IEEE Transactions on Information Theory 42(2), 348–363 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Dempster, M. A. H. and Evstigneev, I. G., Volatility-induced financial growth, Quantitative Finance 7(2) 151–160 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Heath, D., Kinderlehrer, D., and Kowalczyk, M., Discrete and continuous ratchets: From coin toss to molecular motor, Discrete and Continuous Dynamical Systems—Series B, 2 153–167 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Amengual, P., Toral R., Allison, A., and Abbott, D., Efficiency of discrete-time ratchets, arXiv:cond-mat/0410173 (2004).Google Scholar
  64. 64.
    Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics, 1 46.1–46.9 Addison-Wesley, Reading, MA (1963).Google Scholar
  65. 65.
    Chaitin, G. J., The Unknowable, Springer-Verlag Berlin (1999).zbMATHGoogle Scholar
  66. 66.
    Pearce, C. E. M., Allison, A., and Abbott, D., Perturbing singular systems and the correlating of uncorrelated random sequences, Proceedings of the AIP International Conference on Numerical Analysis and Applied Mathematics, Corfu, Greece, Eds: Theodore E. Simos, George Psihoyios, and Ch. Tsitouras, 936 699 (2007).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Derek Abbott
    • 1
  1. 1.Centre for Biomedical Engineering (CBME) and School of Electrical & Electronic EngineeringThe University of AdelaideAdelaideAustralia

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