Abstract
Parrondo’s paradox refers to the counter-intuitive situation where a winning strategy results from a suitable combination of losing ones. Simple stochastic games exhibiting this paradox have been introduced around the turn of the millennium. The common setting of these Parrondo games is that two rules, A and B, are played at discrete time steps, following either a periodic pattern or an aperiodic one, be it deterministic or random. These games can be mapped onto 1D random walks. In capital-dependent games, the probabilities of moving right or left depend on the walker’s position modulo some integer K. In history-dependent games, each step is correlated with the Q previous ones. In both cases the gain identifies with the velocity of the walker’s ballistic motion, which depends non-linearly on model parameters, allowing for the possibility of Parrondo’s paradox. Calculating the gain involves products of non-commuting Markov matrices, which are somehow analogous to the transfer matrices used in the physics of 1D disordered systems. Elaborating upon this analogy, we study a paradigmatic Parrondo game of each class in the neutral situation where each rule, when played alone, is fair. The main emphasis of this systematic approach is on the dependence of the gain on the remaining parameters and, above all, on the game, i.e., the rule pattern, be it periodic or aperiodic, deterministic or random. One of the most original sides of this work is the identification of weak-contrast regimes for capital-dependent and history-dependent Parrondo games, and a detailed quantitative investigation of the gain in the latter scaling regimes.
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References
G.P. Harmer, D. Abbott, Nature 402, 864 (1999)
P.V.E. McClintock, Nature 401, 23 (1999)
G.P. Harmer, D. Abbott, Stat. Sci. 14, 206 (1999)
G.P. Harmer, D. Abbott, P.G. Taylor, Proc. R. Soc. London A 456, 247 (2000)
J.M.R. Parrondo, G.P. Harmer, D. Abbott, Phys. Rev. Lett. 85, 5226 (2000)
G.P. Harmer, D. Abbott, P.G. Taylor, J.M.R. Parrondo, Chaos 11, 705 (2001)
G.P. Harmer, D. Abbott, Fluct. Noise Lett. 2, R71 (2002)
J.M.R. Parrondo, L. Dinis, Contemp. Phys. 45, 147 (2004)
D. Abbott, Fluct. Noise Lett. 9, 129 (2010)
R.P. Feynman, R.B. Leighton, M. Sands, inFeynman Lectures on Physics (Addison-Wesley, Reading, MA, 1966), Vol. I, Chap. 46
A. Ajdari, J. Prost, C.R. Acad. Sci. Paris, Ser. II 315, 1635 (1992)
M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993)
R.D. Astumian, M. Bier, Phys. Rev. Lett. 72, 1766 (1994)
F. Jülicher, A. Ajdari, J. Prost, Rev. Mod. Phys. 69, 1269 (1997)
P. Reimann, Phys. Rep. 361, 57 (2002)
B. Cleuren, C. Van den Broeck, Europhys. Lett. 67, 151 (2004)
C. Wang, N.G. Xie, L. Wang, Y. Ye, G. Xu, Fluct. Noise Lett. 10, 147 (2011)
R.J. Kay, N.F. Johnson, Phys. Rev. E 67, 056128 (2003)
S.N. Ethier, J. Lee, Electron. J. Probab. 14, 1827 (2009)
P. Bougerol, J. Lacroix,Products of Random Matrices, with Applications to Schrödinger Operators (Birkhäuser, Boston, 1985)
A. Crisanti, G. Paladin, A. Vulpiani,Products of Random Matrices in Statistical Physics, Springer Series in Solid-State Sciences (Springer, Berlin, 1992)
J.M. Luck,Systèmes désordonnés unidimensionnels (Collection Aléa, Saclay, 1992)
J.B. Pendry, Adv. Phys. 43, 461 (1994)
A. Comtet, C. Texier, Y. Tourigny, J. Phys. A 46, 254003 (2013)
A. Comtet, Y. Tourigny, inStochastic Processes and RandomMatrices, edited by G. Schehr, A. Altland, Y.V. Fyodorov, N. O’Connell, L.F. Cugliandolo (Oxford University Press, Oxford, 2017)
J.L. Doob,Stochastic Processes (Wiley, New York, 1953)
W. Feller,An Introduction to Probability Theory and its Applications (Wiley, New York, 1968)
S. Karlin, H.M. Taylor,A First Course in Stochastic Processes (Academic Press, New York, 1975)
N.G. vanKampen,Stochastic Processes in Physics and Chemistry (North-, Amsterdam, 1992)
F.P. Kelly,Reversibility and Stochastic Networks (Wiley, Chichester, 1979)
D. Stirzaker,Stochastic Processes and Models (Oxford University Press, Oxford, 2005)
G.C. Crisan, E. Nechita, M. Talmaciu, Fluct. Noise Lett. 7, C19 (2007)
L. Dinis, Phys. Rev. E 77, 021124 (2008)
T.W. Tang, A. Allison, D. Abbott, Fluct. Noise Lett. 4, L585 (2004)
N.G. de Bruijn, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84, 27 (1981)
D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53, 1951 (1984)
C. Janot,Quasicrystals: A Primer (Oxford University Press, Oxford, 1992)
M. Senechal,Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995)
E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376, 225 (2003)
E. Maciá, Rep. Prog. Phys. 69, 397 (2006)
K.H. Cheong, J.M. Koh, M.C. Jones, BioEssays 41, 1900027 (2019)
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Contribution to the Topical Issue “Recent Advances in the Theory of Disordered Systems”, edited by Ferenc Iglói and Heiko Rieger.
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Luck, JM. Parrondo games as disordered systems. Eur. Phys. J. B 92, 180 (2019). https://doi.org/10.1140/epjb/e2019-100259-4
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DOI: https://doi.org/10.1140/epjb/e2019-100259-4