Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T21:25:20.234Z Has data issue: false hasContentIssue false

How strong can the Parrondo effect be?

Published online by Cambridge University Press:  11 December 2019

S. N. Ethier*
Affiliation:
University of Utah
Jiyeon Lee*
Affiliation:
Yeungnam University
*
*Postal address: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA.
**Postal address: Department of Statistics, Yeungnam University, 280 Daehak-Ro, Gyeongsan, Gyeongbuk 38541, South Korea. Email address: [email protected]

Abstract

Parrondo’s coin-tossing games were introduced as a toy model of the flashing Brownian ratchet in statistical physics but have emerged as a paradigm for a much broader phenomenon that occurs if there is a reversal in direction in some system parameter when two similar dynamics are combined. Our focus here, however, is on the original Parrondo games, usually labeled A and B. We show that if the parameters of the games are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games A and B are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but can also be arbitrarily close to 1 (i.e. 100%).

Type
Research Papers
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, D. (2010). Asymmetry and disorder: A decade of Parrondo’s paradox. Fluct. Noise Lett. 9, 129156.CrossRefGoogle Scholar
Ajdari, A. and Prost, J. (1992). Drift induced by a spatially periodic potential of low symmetry: Pulsed dielectrophoresis. C. R. Acad. Sci., Série 2 315, 16351639.Google Scholar
Behrends, E. (2004). Mathematical background of Parrondo’s paradox. In Noise in Complex Systems and Stochastic Dynamics II, ed. Gingl, Z., Sancho, J. M., Schimansky-Geier, L., and Kertesz, J., Vol. 5471 of Proc. SPIE Series, Society of Photo-optical Instrumentation Engineers, Bellingham, WA, pp. 510517.CrossRefGoogle Scholar
Cheong, K. H., Koh, J. M. and Jones, M. C. (2019). Paradoxical survival: Examining the Parrondo effect across biology. BioEssays 41, 1900027.CrossRefGoogle ScholarPubMed
Costa, A., Fackrell, M. and Taylor, P. G. (2005). Two issues surrounding Parrondo’s paradox. In Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, ed. Nowak, A. S. and Szajowski, K., Vol. 7 of Annals of the International Society of Dynamic Games, Birkhäuser, Boston, pp. 599609.CrossRefGoogle Scholar
Di Crescenzo, A. (2007). A Parrondo paradox in reliability theory. Math. Scientist 32, 1722.Google Scholar
Dinis, L. (2008). Optimal sequence for Parrondo games. Phys. Rev. E 77, 021124.CrossRefGoogle ScholarPubMed
Ethier, S. N. and Lee, J. (2009). Limit theorems for Parrondo’s paradox. Electron. J. Prob. 14, 18271862.CrossRefGoogle Scholar
Ethier, S. N. and Lee, J. (2010). A Markovian slot machine and Parrondo’s paradox. Ann. Appl. Prob. 20, 10981125.CrossRefGoogle Scholar
Ethier, S. N. and Lee, J. (2018). The flashing Brownian ratchet and Parrondo’s paradox. R. Soc. Open Sci. 5, 171685.CrossRefGoogle ScholarPubMed
Harmer, G. P. and Abbott, D. (1999). Parrondo’s paradox. Statist. Sci. 14, 206213.Google Scholar
Harmer, G. P. and Abbott, D. (2002). A review of Parrondo’s paradox. Fluct. Noise Lett. 2, R71R107.CrossRefGoogle Scholar
Harmer, G. P., Abbott, D. and Taylor, P. G. (2000a). The paradox of Parrondo’s games. Proc. R. Soc. Lond. Ser. A 456, 247259.CrossRefGoogle Scholar
Harmer, G. P. (2000b). Information entropy and Parrondo’s discrete-time ratchet. In Proc. Stochastic and Chaotic Dynamics in the Lakes: STOCHAOS: Ambleside, Cumbria, UK, August 1999, ed. Broomhead, D. S., Luchinskaya, E. A., McClintock, P. V. E., and Mullin, T., Vol. 502 of AIP Conference Proceedings, American Institute of Physics, Melville, NY, pp. 544549.CrossRefGoogle Scholar
Kay, R. J. and Johnson, N. F. (2003). Winning combinations of history-dependent games. Phys. Rev. E 67, 056128.CrossRefGoogle ScholarPubMed
Key, E. S., Kłosek, M. M. and Abbott, D. (2006). On Parrondo’s paradox: How to construct unfair games by composing fair games. ANZIAM J. 47, 495512.CrossRefGoogle Scholar
Machida, T. and Grünbaum, F. A. (2018). Some limit laws for quantum walks with applications to a version of the Parrondo paradox. Quantum Inf. Process. 17, 241.CrossRefGoogle Scholar
Marzuoli, A. (2009). Toy models in physics and the reasonable effectiveness of mathematics. Scientifica Acta 3, 1324.Google Scholar
Miles, C. E., Lawley, S. D. and Keener, J. P. (2018). Analysis of nonprocessive molecular motor transport using renewal reward theory. SIAM J. Appl. Math. 78, 25112532.CrossRefGoogle Scholar
Percus, O. E. and Percus, J. K. (2002). Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox. Math. Intelligencer 24, 3-68–3-72.CrossRefGoogle Scholar
Pyke, R. (2003). On random walks and diffusions related to Parrondo’s games. In Mathematical Statistics and Applications: Festschrift for Constance Van Eeden, ed. Moore, M., Froda, S., and Léger, C., Vol. 42 of IMS Lecture Notes–Monograph Series, Institute of Mathematical Statistics, Beachwood, OH, pp. 185216.Google Scholar
Rémillard, B. and Vaillancourt, J. (2019). Combining losing games into a winning game. Fluct. Noise Lett. 18, 1950003.CrossRefGoogle Scholar
Wang, L. (2011). Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation. Phys. A 390, 45354542.CrossRefGoogle Scholar
Zhu, Y.-F., Xie, N.-G., Ye, Y. and Peng, F.-R. (2011). Quantum game interpretation for a special case of Parrondo’s paradox. Phys. A 390, 579586.CrossRefGoogle Scholar