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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Harmer G. P. and Abbott D., Parrondo’s paradox. Statistical Science, 14(2):206–213, 1999.
Harmer G. P. and Abbott D., Parrondo’s paradox: losing strategies cooperate to win. Nature (London), 402:864, 1999.
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.
Astumian R. D. and Bier M., Fluctuation driven ratchets: Molecular motors. Physical Review Letters, 72(11):1766–1769, 1994.
Doering C. R., Randomly rattled ratchets. Nuovo Cimento, 17D(7–8):685–697, 1995.
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.
Harmer G. P., Abbott D., Taylor P. G. and Parrondo J. M. R., Parrondo’s games and Brownian ratchets. Chaos 11(3):705–714.
Parrondo J. M. R., Harmer G. P. and Abbott D., Newparadoxical games based on Brownian ratchets. Physical Review Letters, 85(24):5226–5229, 2000.
Costa A., Fackrell M. and Taylor P. G., Two issues surrounding Parrondo’s paradox. Birkhäuser Annals of Dynamic Games, This volume, 2004.
Doob J. L., Stochastic Processes. John Wiley & Sons, Inc., New York, 1953.
Onsager L., Reciprocal relations in irreversible processes I. Physical Review, 37:405–426, 1931.
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.
Pyke R., On random walks related to Parrondo’s games. Preprint math. PR/0206150, 2001.
Neuts M. F., Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The John Hopkins University Press, USA, 1981.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Boston
About this chapter
Cite this chapter
Harmer, G.P., Abbott, D., Parrondo, J.M.R. (2005). Parrondo’s Capital and History-Dependent Games. In: Nowak, A.S., Szajowski, K. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 7. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4429-6_33
Download citation
DOI: https://doi.org/10.1007/0-8176-4429-6_33
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4362-1
Online ISBN: 978-0-8176-4429-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)