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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 62 open journal systems 


Limit theorems for Parrondo's paradox

S N Ethier, University of Utah
Jiyeon Lee, Yeungnam University


Abstract
That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.


Full text: PDF

Pages: 1827-1862

Published on: September 2, 2009
Bibliography
  1. Abbott, Derek (2009) Developments in Parrondo's paradox. In: In, V., Longhini, P., and Palacios, A. (eds.) Applications of Nonlinear Dynamics: Model and Design of Complex Systems. Series: Understanding Complex Systems. Springer-Verlag, Berlin, 307–321.
  2. Ajdari, A. and Prost, J. (1992) Drift induced by a spatially periodic potential of low symmetry: Pulsed dielectrophoresis. Comptes Rendus de l'Académie des Sciences, Série 2 315 (13) 1635–1639.
  3. Behrends, Ehrhard (2002) Parrondo's paradox: A priori and adaptive strategies. Preprint A-02-09. ftp://ftp.math.fu-berlin.de/pub/math/publ/pre/2002/index.html.
  4. Berresford, Geoffrey C. and Rockett, Andrew M. (2003) Parrondo's paradox. Int. J. Math. Math. Sci. 2003 (62) 3957–3962. MR2036089 (2004j:91068).
  5. Billingsley, Patrick (1995) Probability and Measure, third edition. John Wiley & Sons Inc., New York. MR1324786 (95k:60001).
  6. Bradley, Richard C. (2007) Introduction to Strong Mixing Conditions, Volume 1. Kendrick Press, Heber City, UT. MR2325294 (2009f:60002a).
  7. Cleuren, B. and Van den Broeck, C. (2004) Primary Parrondo paradox. Europhys. Lett. 67 (2) 151–157.
  8. Costa, Andre, Fackrell, Mark, and Taylor, Peter G. (2005) Two issues surrounding Parrondo's paradox. In: Nowak, A. S. and Szajowski, K. (eds.) Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control, Annals of the International Society of Dynamic Games 7, Birkhäuser, Boston, 599–609. MR2104716 (2005h:91066).
  9. Durrett, Richard (1996) Probability: Theory and Examples, second edition. Duxbury Press, Belmont, CA. MR1609153 (98m:60001).
  10. Ekhad, Shalosh B. and Zeilberger, Doron (2000) Remarks on the Parrondo paradox. The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger. http://www.math.rutgers.edu/~zeilberg/pj.html.
  11. Epstein, Richard A. (2007) Parrondo's principle: An overview. In: Ethier, S. N. and Eadington, W. R. (eds.) Optimal Play: Mathematical Studies of Games and Gambling. Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, 471–492.
  12. Harmer, Gregory P. and Abbott, Derek (2002) A review of Parrondo's paradox. Fluct. Noise Lett. 2 (2) R71–R107.
  13. Kay, Roland J. and Johnson, Neil F. (2003) Winning combinations of history-dependent games. Phys. Rev. E 67 (5) 056128. arXiv:cond-mat/0207386.
  14. Kemeny, John G. and Snell, J. Laurie (1960) Finite Markov Chains. D. Van Nostrand Company, Inc., Princeton, NJ. MR0115196 (22 #5998).
  15. Key, Eric S., Klosek, Malgorzata M., and Abbott, Derek (2006) On Parrondo's paradox: How to construct unfair games by composing fair games. ANZIAM J. 47 (4) 495–511. MR2234017. arXiv:math/0206151.
  16. Moraal, Hendrik (2000) Counterintuitive behaviour in games based on spin models. J. Phys. A: Math. Gen. 33 L203–L206. MR1778614 (2001d:91049).
  17. Parrondo, Juan M. R. (1996) Efficiency of Brownian motors. Presented at the Workshop of the EEC HC&M Network on Complexity and Chaos, Institute for Scientific Interchange Foundation, Torino, Italy.
  18. Parrondo, Juan M. R., Harmer, Gregory P., and Abbott, Derek (2000) New paradoxical games based on Brownian ratchets. Phys. Rev. Lett. 85 (24) 5226–5229. arXiv:cond-mat/0003386.
  19. Parrondo, J. M. R. and Dínis, Luis (2004) Brownian motion and gambling: From ratchets to paradoxical games. Contemp. Phys. 45 (2) 147–157.
  20. Percus, Ora E. and Percus, Jerome K. (2002) Can two wrongs make a right? Coin-tossing games and Parrondo's paradox. Math. Intelligencer 24 (3) 68–72. MR1927042 (2003f:91025).
  21. Philips, Thomas K. and Feldman, Andrew B. (2004) Parrondo's paradox is not paradoxical. Social Sciences Research Network. http://ssrn.com/abstract=581521.
  22. Pyke, Ronald (2003) On random walks and diffusions related to Parrondo's games. In: Moore, M., Froda, S., and Léger, C. (eds.) Mathematical Statistics and Applications: Festschrift for Constance Van Eeden. Institute of Mathematical Statistics, Lecture Notes–Monograph Series 42, Beachwood, OH, 185–216. MR2138293 (2006e:60062). arXiv:math/0206150.
  23. Rahmann, Sven (2002) Optimal adaptive strategies for games of the Parrondo type. Preprint. http://gi.cebitec.uni-bielefeld.de/people/rahmann/parrondo/rahmann-report.pdf.
















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Electronic Journal of Probability. ISSN: 1083-6489