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Parrondo’s Capital and History-Dependent Games

  • Gregory P. Harmer
  • Derek Abbott
  • Juan M. R. Parrondo
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 7)

Abstract

It has been shown that it is possible to construct two games that when played individually lose, but alternating randomly or deterministically between them can win. This apparent paradox has been dubbed “Parrondo’s paradox.” The original games are capital-dependent, which means that the winning and losing probabilities depend on how much capital the player currently has. Recently, new games have been devised, that are not capital-dependent, but historydependent. We present some analytical results using discrete-time Markovchain theory, which is accompanied by computer simulations of the games.

Keywords

Transition Matrix Physical Review Letter Original Game Markov Chain Theory 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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References

  1. [1]
    Harmer G. P. and Abbott D., Parrondo’s paradox. Statistical Science, 14(2):206–213, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Harmer G. P. and Abbott D., Parrondo’s paradox: losing strategies cooperate to win. Nature (London), 402:864, 1999.CrossRefGoogle Scholar
  3. [3]
    Adjari A. and Prost J., Drift induced by a periodic potential of low symmetry: pulsed dielectrophoresis. C. R. Academy of Science Paris, Série II, 315:1635–1639, 1992.Google Scholar
  4. [4]
    Astumian R. D. and Bier M., Fluctuation driven ratchets: Molecular motors. Physical Review Letters, 72(11):1766–1769, 1994.CrossRefGoogle Scholar
  5. [5]
    Doering C. R., Randomly rattled ratchets. Nuovo Cimento, 17D(7–8):685–697, 1995.CrossRefGoogle Scholar
  6. [6]
    Harmer G. P., Abbott D., Taylor P. G. and Parrondo J. M. R., Parrondo’s paradoxical games and the discrete Brownian ratchet. In D. Abbott and L. B. Kish, editors, Second International Conference on Unsolved Problems of Noise and Fluctuations, volume 511, pages 189–200, Adelaide, Australia, American Institute of Physics, 2000.Google Scholar
  7. [7]
    Harmer G. P., Abbott D., Taylor P. G. and Parrondo J. M. R., Parrondo’s games and Brownian ratchets. Chaos 11(3):705–714.Google Scholar
  8. [8]
    Parrondo J. M. R., Harmer G. P. and Abbott D., Newparadoxical games based on Brownian ratchets. Physical Review Letters, 85(24):5226–5229, 2000.CrossRefGoogle Scholar
  9. [9]
    Costa A., Fackrell M. and Taylor P. G., Two issues surrounding Parrondo’s paradox. Birkhäuser Annals of Dynamic Games, This volume, 2004.Google Scholar
  10. [10]
    Doob J. L., Stochastic Processes. John Wiley & Sons, Inc., New York, 1953.zbMATHGoogle Scholar
  11. [11]
    Onsager L., Reciprocal relations in irreversible processes I. Physical Review, 37:405–426, 1931.zbMATHCrossRefGoogle Scholar
  12. [12]
    Pearce C. E. M., Entropy, Markov information sources and Parrondo games. In D. Abbott and L. B. Kish, editors, Second International Conference on Unsolved Problems of Noise and Fluctuations, volume 511, pages 207–212, Adelaide, Australia, American Institute of Physics, 2000.Google Scholar
  13. [13]
    Pyke R., On random walks related to Parrondo’s games. Preprint math. PR/0206150, 2001.Google Scholar
  14. [14]
    Neuts M. F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The John Hopkins University Press, USA, 1981.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 2005

Authors and Affiliations

  • Gregory P. Harmer
    • 1
  • Derek Abbott
    • 1
  • Juan M. R. Parrondo
    • 2
  1. 1.Centre for Biomedical Engineering (CBME), Department of Electrical and Electronic EngineeringThe University of AdelaideAustralia
  2. 2.Departamento de Física Atómica, Molecular y NuclearUniversidad Complutense de MadridSpain

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