Statistical Science

Parrondo's paradox

D. Abbott and G. P. Harmer

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We introduce Parrondo's paradox that involves games of chance. We consider two fair gambling games, A and B, both of which can be made to have a losing expectation by changing a biasing parameter $\epsilon$. When the two games are played in any alternating order, a winning expectation is produced, even though A and B are now losing games when played individually. This strikingly counter­intuitive result is a consequence of discrete­time Markov chains and we develop a heuristic explanation of the phenomenon in terms of a Brownian ratchet model. As well as having possible applications in electronic signal processing, we suggest important applications in a wide range of physical processes, biological models, genetic models and sociological models. Its impact on stock market models is also an interesting open question.

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Statist. Sci., Volume 14, Number 2 (1999), 206-213.

First available in Project Euclid: 24 December 2001

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Gambling paradox Brownian ratchet noise


Harmer, G. P.; Abbott, D. Parrondo's paradox. Statist. Sci. 14 (1999), no. 2, 206--213. doi:10.1214/ss/1009212247.

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  • ABBOTT, D., DAVIS, B. R. and PARRONDO, J. M. R. 1999. The problem of detailed balance for the Feynman-Smoluchowski Z. engine FSE and the multiple pawl paradox. In Proceedings of the Second International Conference on Unsolved ProbZ lems of Noise and Fluctuations D. Abbott and L. B. Kiss,. eds. American Institute of Physics, Adelaide, Australia. To appear. Z.
  • ASTUMIAN, R. D. and BIER, M. 1994. Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett. 72 1766 1769. Z.
  • ASTUMIAN, R. D. 1997. Thermodynamics and kinetics of a Brownian motor. Science 276 917 922. Z.
  • BERDICHEVSKY, V. and GITTERMAN, M. 1998. Stochastic resonance and ratchets new manifestations. Phys. A 249 88 95.Z.
  • BIER, M. 1997a. Brownian ratchets in physics and biology. Contemp. Phys. 38 371 379. Z.
  • BIER, M. 1997b. A motor protein model and how it relates to stochastic resonance, Feynman's ratchet, and Maxwell's demon. In Stochastic Dynamics 386 81 87. Springer, Berlin. Z.
  • DOERING, C. R. 1995. Randomly rattled ratchets. Nuovo Cimento 17D 685 697.
  • FAUCHEUX, L. P., BOURDIEU, L. S., KAPLAN, P. D. and LIBCHABER, Z. A. J. 1995. Optical thermal ratchet. Phys. Rev. Lett. 74 1504 1509. Z.
  • FEYNMAN, R. P., LEIGHTON, R. B. and SANDS, M. 1963. The Feynman Lectures on Physics 1 46.1 46.9. Addison-Wesley, Reading, MA. Z.
  • GAMMAITONI, L., HANGGI, P., JUNG, P. and MARCHESONI, F. 1998. ¨ Stochastic resonance. Rev. Modern Phys. 70 223 287. Z.
  • GRIMMETT, G. R. and STIRZAKER, D. R. 1982. Probability and Random Processes. Oxford Univ. Press. Z.
  • HANGGI, P. and BARTUSSEK, R. 1996. Brownian rectifiers: how ¨ to convert Brownian motion into directed transport. Nonlinear Physics of Complex Systems Current Status and Future Trends. Lecture Notes in Phys. 476 294 308. Springer, Berlin. Z.
  • HUGHES, B. D. 1995. Random Walks and Random Variables 1. Oxford Univ. Press. Z.
  • MAGNASCO, M. O. 1993. Forced thermal ratchets. Phys. Rev. Lett. 71 1477 1481. Z.
  • PARRONDO, J. M. R. 1997. Universidad Complutense, Madrid, Spain. Private communication. Z.
  • PARRONDO, J. M. R. and ESPANOL, P. 1996. Criticism of Feyn man's analysis of the ratchet as an engine. Amer. J. Phys. 64 1125 1130. Z.
  • ROUSSELET, J., SALOME, L., AJDARI, A. and PROST, J. 1994. Directional motion of Brownian particles induced by a periodic asymmetric potential. Nature 370 446 448. Z.
  • SHLESINGER, M. F. 1996. A brief history of random processes. In Proceedings of the First International Conference on UnZ solved Problems of Noise C. R. Doering, L. B. Kiss and M. F.. Shlesinger, eds. 3 10. World Scientific, Szeged, Hungary. Z.
  • VON SMOLUCHOWSKI, M. 1912. Experimentall nachweisbare, der ublichen Thermodynamic widersprechende Molekular¨ phanomene. Physikalische Zeitschrift 13 1069 1080. ¨