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Parrondo’s paradox from classical to quantum: A review

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Abstract

Two losing games can be combined in a certain manner to give a winning outcome—this is known as Parrondo’s paradox. Parrondo’s paradox has found its applications across different disciplines such as physics, biology and finance, amongst others. At the turn of the millennium, there has been immense attention on the quantum Parrondo’s games as classical games are simulated using quantum notation. This review paper traces the construction of quantum Parrondo’s games from classical capital-dependent, history-dependent and cooperative Parrondo’s games. Directions for future research will also be discussed.

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References

  1. Parrondo, J.M., Español, P.: Criticism of Feynman’s analysis of the ratchet as an engine. Am. J. Phys. 64, 1125–1130 (1996)

    Google Scholar 

  2. Harmer, G.P., Abbott, D., Taylor, P.G., Parrondo, J.M.: Brownian ratchets and Parrondo’s games. Chaos Interdiscip. J. Nonlinear Sci. 11, 705–714 (2001)

    MATH  Google Scholar 

  3. Amengual, P., Allison, A., Toral, R., Abbott, D.: Discrete-time ratchets, the fokker-planck equation and Parrondo’s paradox. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460, 2269–2284 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Feynman, R.P., Leighton, R.B., Sands, M., Gottlieb, M.A., Leighton, R.: The Feynman Lectures on Physics Vol. 1, Chapter 46: Ratchet and Pawl, vol. 1. Addison-Wesley, Boston (1964)

    Google Scholar 

  5. Cheong, K.H., Koh, J.M., Jones, M.C.: Paradoxical survival Examining the Parrondo effect across biology. BioEssays 41, 1900027 (2019). https://doi.org/10.1002/bies.201900027

    Article  Google Scholar 

  6. Reed, F.A.: Two-locus epistasis with sexually antagonistic selection: a genetic Parrondo’s paradox. Genetics 176, 1923–1929 (2007)

    Google Scholar 

  7. Cheong, K.H., Koh, J.M., Jones, M.C.: Multicellular survival as a consequence of Parrondo’s paradox. Proc. Natl. Acad. Sci. 115, E5258–E5259 (2018)

    Google Scholar 

  8. Koh, J.M., Xie, N.-G., Cheong, K.H.: Nomadic-colonial switching with stochastic noise: subsidence-recovery cycles and long-term growth. Nonlinear Dyn. 94, 1467–1477 (2018). https://doi.org/10.1007/s11071-018-4436-2

    Article  Google Scholar 

  9. Tan, Z.-X., Cheong, K.H.: Periodic habitat destruction and migration can paradoxically enable sustainable territorial expansion. Nonlinear Dyn. 98, 1–13 (2019)

    MATH  Google Scholar 

  10. Cheong, K.H., Koh, J.M., Jones, M.C.: Do arctic hares play Parrondo’s games? Fluct. Noise Lett. 18, 1971001 (2019). https://doi.org/10.1142/S0219477519710019

    Article  Google Scholar 

  11. Harmer, G.P., Abbott, D.: Game theory: losing strategies can win by Parrondo’s paradox. Nature 402, 864 (1999)

    Google Scholar 

  12. Arena, P., Fazzino, S., Fortuna, L., Maniscalco, P.: Game theory and non-linear dynamics: the Parrondo paradox case study. Chaos Solitons Fractals 17, 545–555 (2003)

    MATH  Google Scholar 

  13. Dinís, L., Parrondo, J.M.: Inefficiency of voting in Parrondo games. Phys. A Stat. Mech. Appl. 343, 701–711 (2004)

    MathSciNet  Google Scholar 

  14. Ethier, S., Lee, J., et al.: Parrondo’s paradox via redistribution of wealth. Electron. J. Probab. 17, 1–21 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Toral, R.: Capital redistribution brings wealth by Parrondo’s paradox. Fluct. Noise Lett. 2, L305–L311 (2002)

    MathSciNet  Google Scholar 

  16. Koh, J.M., Cheong, K.H.: Emergent preeminence of selfishness: an anomalous Parrondo perspective. Nonlinear Dyn 98, 943–951 (2019). https://doi.org/10.1007/s11071-019-05237-6

    Article  Google Scholar 

  17. Koh, J.M., Cheong, K.H.: New doubly-anomalous Parrondo’s games suggest emergent sustainability and inequality. Nonlinear Dyn. 96, 257–266 (2019)

    Google Scholar 

  18. Harmer, G.P., Abbott, D., Taylor, P.G., Pearce, C.E., Parrondo, J.M.: Information entropy and Parrondo’s discrete-time ratchet. In: AIP Conference Proceedings, vol. 502, pp. 544–549 (AIP) (2000)

  19. Pearce, C.E.: Entropy, markov information sources and Parrondo games. In: AIP Conference Proceedings, vol. 511, pp. 207–212 (AIP) (2000)

  20. Osipovitch, D.C., Barratt, C., Schwartz, P.M.: Systems chemistry and Parrondo’s paradox: computational models of thermal cycling. New J. Chem. 33, 2022–2027 (2009)

    Google Scholar 

  21. Kocarev, L., Tasev, Z.: Lyapunov exponents, noise-induced synchronization, and Parrondo’s paradox. Phys. Rev. E 65, 046215 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Chang, C.-H., Tsong, T.Y.: Truncation and reset process on the dynamics of Parrondo’s games. Phys. Rev. E 67, 025101 (2003)

    Google Scholar 

  23. Allison, A., Abbott, D.: Control systems with stochastic feedback. Chaos Interdiscip. J. Nonlinear Sci. 11, 715–724 (2001)

    MATH  Google Scholar 

  24. Di Crescenzo, A.: A Parrondo paradox in reliability theory. Preprint arXiv:math/0602308 (2006)

  25. Koh, J.M., Cheong, K.H.: Automated electron-optical system optimization through switching levenberg–marquardt algorithms. J. Electron Spectrosc. Relat. Phenom. 227, 31–39 (2018)

    Google Scholar 

  26. Cheong, K.H., Koh, J.M.: A hybrid genetic-Levenberg Marquardt algorithm for automated spectrometer design optimization. Ultramicroscopy 202, 100–106 (2019)

    Google Scholar 

  27. Gunn, L.J., Allison, A., Abbott, D.: Allison mixtures: where random digits obey thermodynamic principles. Int. J. Mod. Phys. Conf. Ser. 33, 1460360 (2014)

    Google Scholar 

  28. Soo, W.W.M., Cheong, K.H.: Parrondo’s paradox and complementary Parrondo processes. Phys. A Stat. Mech. Appl. 392, 17–26 (2013)

    MathSciNet  Google Scholar 

  29. Soo, W.W.M., Cheong, K.H.: Occurrence of complementary processes in Parrondo’s paradox. Phys. A Stat. Mech. Appl. 412, 180–185 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Cheong, K.H., Soo, W.W.M.: Construction of novel stochastic matrices for analysis of Parrondo’s paradox. Phys. A Stat. Mech. Appl. 392, 4727–4738 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Cheong, K.H., Saakian, D.B., Zadourian, R.: Allison mixture and the two-envelope problem. Phys. Rev. E 96, 062303 (2017)

    Google Scholar 

  32. Ye, Y., Cheong, K.H., Cen, Y.-W., Xie, N.-G.: Effects of behavioral patterns and network topology structures on Parrondo’s paradox. Sci. Rep. 6, 37028 (2016). https://doi.org/10.1038/srep37028

    Article  Google Scholar 

  33. Ye, Y., et al.: Passive network evolution promotes group welfare in complex networks. Chaos, Solitons & Fractals 130, 109464 (2020). https://doi.org/10.1016/j.chaos.2019.109464

    Article  MathSciNet  Google Scholar 

  34. Ye, Y., et al.: Ratcheting based on neighboring niches determines lifestyle. Nonlinear Dyn. 98, 1821–1830 (2019)

    MATH  Google Scholar 

  35. Cheong, K.H., Tan, Z.X., Ling, Y.H.: A time-based switching scheme for nomadic-colonial alternation under noisy conditions. Commun. Nonlinear Sci. Numer. Simul. 60, 107–114 (2018)

    MathSciNet  Google Scholar 

  36. Cheong, K.H., Tan, Z.X., Xie, N.-G., Jones, M.C.: A paradoxical evolutionary mechanism in stochastically switching environments. Sci. Rep. 6, 34889 (2016). https://doi.org/10.1038/srep34889

    Article  Google Scholar 

  37. Tan, Z.X., Cheong, K.H.: Nomadic-colonial life strategies enable paradoxical survival and growth despite habitat destruction. eLife 6, e21673 (2017)

    Google Scholar 

  38. Tan, Z.-X., Koh, J.M., Koonin, E.V., Cheong, K.H.: Predator dormancy is a stable adaptive strategy due to Parrondo’s paradox. Adv. Sci. (2019). https://doi.org/10.1002/advs.201901559

    Article  Google Scholar 

  39. Harmer, G.P., Abbott, D.: A review of Parrondo’s paradox. Fluct. Noise Lett. 2, R71–R107 (2002)

    Google Scholar 

  40. Abbott, D.: Developments in Parrondo’s paradox. In: In, V., Longhini, P., Palacios, A. (eds.) Applications of Nonlinear Dynamics, pp. 307–321. Springer, Berlin, Heidelberg (2009)

    Google Scholar 

  41. Abbott, D.: Asymmetry and disorder: a decade of Parrondo’s paradox. Fluct. Noise Lett. 9, 129–156 (2010)

    MathSciNet  Google Scholar 

  42. Cheong, K.H., et al.: Paradoxical simulations to enhance education in mathematics. IEEE Access 7, 17941–17950 (2019)

    Google Scholar 

  43. Osborne, T.: Advanced quantum theory, lecture 1 (2016). https://www.youtube.com/watch?v=Og13-bSF9kA. Acce-ssed 1 Oct 2019

  44. Gomes, L.: Quantum computing: both here and not here. IEEE Spectr. 55, 42–47 (2018)

    Google Scholar 

  45. Neill, C., et al.: A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018)

    MathSciNet  Google Scholar 

  46. Arute, F., et al.: Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019)

    Google Scholar 

  47. Flitney, A., Abbott, D.: Quantum models of Parrondo’s games. Phys. A Stat. Mech. Appl. 324, 152–156 (2003)

    MathSciNet  MATH  Google Scholar 

  48. Bravyi, S., Gosset, D., Koenig, R.: Quantum advantage with shallow circuits. Science 362, 308–311 (2018)

    MathSciNet  MATH  Google Scholar 

  49. Harmer, G.P., Abbott, D., et al.: Parrondo’s paradox. Stat. Sci. 14, 206–213 (1999)

    MathSciNet  MATH  Google Scholar 

  50. Harmer, G.P., Abbott, D., Taylor, P.G., Parrondo, J.M.: Parrondo’s paradoxical games and the discrete Brownian ratchet. In: AIP Conference Proceedings, vol. 511, pp. 189–200. (AIP) (2000)

  51. Meyer, D.A., Blumer, H.: Parrondo games as lattice gas automata. J. Stat. Phys. 107, 225–239 (2002)

    MATH  Google Scholar 

  52. Meyer, D.A.: Noisy quantum Parrondo games. In: Abbott, D., Shapiro, J.H., Yamamoto, Y. (eds.) Fluctuations and Noise in Photonics and Quantum Optics, vol. 5111, pp. 344–350. International Society for Optics and Photonics, Bellingham, WA, USA (2003)

    Google Scholar 

  53. Gawron, P., Miszczak, J.A.: Quantum implementation of Parrondo’s paradox. Fluct. Noise Lett. 5, L471–L478 (2005)

    Google Scholar 

  54. Košík, J., Miszczak, J., Bužek, V.: Quantum Parrondo’s game with random strategies. J. Mod. Opt. 54, 2275–2287 (2007)

    MATH  Google Scholar 

  55. Flitney, A.P.: Quantum Parrondo’s games using quantum walks. arXiv preprint arXiv:1209.2252 (2012)

  56. Tregenna, B., Flanagan, W., Maile, R., Kendon, V.: Controlling discrete quantum walks: coins and initial states. New J. Phys. 5, 83 (2003)

    Google Scholar 

  57. Parrondo, J.M.R., Harmer, G.P., Abbott, D.: New paradoxical games based on brownian ratchets. Phys. Rev. Lett. 85, 5226–5229 (2000)

    Google Scholar 

  58. Harmer, G.P., Abbott, D., Parrondo, J.M.: Parrondo’s capital and history-dependent games. In: Nowak, A.S., Szajowski, K. (eds.) Advances in Dynamic Games, pp. 635–648. Springer Basel, Switzerland (2005)

    MATH  Google Scholar 

  59. Flitney, A.P., Ng, J., Abbott, D.: Quantum Parrondo’s games. Phys. A Stat. Mech. Appl. 314, 35–42 (2002)

    MathSciNet  MATH  Google Scholar 

  60. Flitney, A.P., Abbott, D., Johnson, N.F.: Quantum walks with history dependence. J. Phys. A Math. Gen. 37, 7581 (2004)

    MathSciNet  MATH  Google Scholar 

  61. Khan, S., Ramzan, M., Khan, M.: Quantum Parrondo’s games under decoherence. Int. J. Theor. Phys. 49, 31 (2010)

    MathSciNet  MATH  Google Scholar 

  62. Bleiler, S.A., Khan, F.S.: Properly quantized history-dependent Parrondo games, markov processes, and multiplexing circuits. Phys. Lett. A 375, 1930–1943 (2011)

    MathSciNet  MATH  Google Scholar 

  63. Bleiler, S.A.: A formalism for quantum games and an application. arXiv preprint arXiv:0808.1389 (2008)

  64. Toral, R.: Cooperative Parrondo’s games. Fluct. Noise Lett. 1, L7–L12 (2001)

    MathSciNet  Google Scholar 

  65. Mihailović, Z., Rajkoić, M.: One dimensional asynchronous cooperative Parrondo’s games. Fluct. Noise Lett. 3, L389–L398 (2003)

    MathSciNet  Google Scholar 

  66. Mihailović, Z., Rajkoić, M.: Synchronous cooperative Parrondo’s games. Fluct. Noise Lett. 3, L399–L406 (2003)

    MathSciNet  Google Scholar 

  67. Mihailović, Z., Rajković, M.: Cooperative Parrondo’s games on a two-dimensional lattice. Phys. A Stat. Mech. Appl. 365, 244–251 (2006)

    Google Scholar 

  68. Ethier, S., Lee, J.: Parrondo games with spatial dependence II. Fluct. Noise Lett. 11, 1250030 (2012)

    Google Scholar 

  69. Bulger, D., Freckleton, J., Twamley, J.: Position-dependent and cooperative quantum Parrondo walks. New J. Phys. 10, 093014 (2008)

    Google Scholar 

  70. Pawela, Ł., Sładkowski, J.: Cooperative quantum Parrondo’s games. Phys. D Nonlinear Phenom. 256, 51–57 (2013)

    MathSciNet  MATH  Google Scholar 

  71. Si, T.: An optical model for implementing Parrondo’s game and designing stochastic game with long-term memory. Chaos Solitons Fractals 45, 1430–1436 (2012)

    MathSciNet  MATH  Google Scholar 

  72. Chandrashekar, C., Banerjee, S.: Parrondo’s game using a discrete-time quantum walk. Phys. Lett. A 375, 1553–1558 (2011)

    MathSciNet  MATH  Google Scholar 

  73. Li, M., Zhang, Y.-S., Guo, G.-C.: Quantum Parrondo’s games constructed by quantum random walks. Fluct. Noise Lett. 12, 1350024 (2013)

    Google Scholar 

  74. Rajendran, J., Benjamin, C.: Playing a true Parrondo’s game with a three-state coin on a quantum walk. EPL (Europhys. Lett.) 122, 40004 (2018)

    Google Scholar 

  75. Machida, T., Grünbaum, F.A.: Some limit laws for quantum walks with applications to a version of the Parrondo paradox. Quantum Inf. Process. 17, 241 (2018)

    MathSciNet  MATH  Google Scholar 

  76. Li, R., Zhu, Y.-F., Guo, J.-Y., Wang, L., Xie, N.-G.: The quantum game interpretation for a special phenomenon of Parrondo’s paradox. Proc. Eng. 15, 3715–3722 (2011)

    Google Scholar 

  77. Chen, L., Li, C.-F., Gong, M., Guo, G.-C.: Quantum Parrondo game based on a quantum ratchet effect. Phys. A Stat. Mech. Appl. 389, 4071–4074 (2010)

    Google Scholar 

  78. Piotrowski, E.W., Sładkowski, J.: Quantum game theoretical frameworks in economics. In: Haven, E., Khrennikov, A. (eds.) The Palgrave Handbook of Quantum Models in Social Science, pp. 39–57. Springer, London, England, United Kingdom (2017)

    MATH  Google Scholar 

  79. Lee, C.F., Johnson, N.: Parrondo games and quantum algorithms. arXiv preprint arXiv:quant-ph/0203043 (2002)

  80. Ng, J., Abbott, D.: Introduction to quantum games and a quantum Parrondo game. In: Křivan, V., Zaccour, G. (eds.) Advances in Dynamic Games, pp. 649–665. Springer Basel, Switzerland (2005)

    MATH  Google Scholar 

  81. Zhu, Y.-F., Xie, N.-G., Ye, Y., Peng, F.-R.: Quantum game interpretation for a special case of Parrondo’s paradox. Phys. A Stat. Mech. Appl. 390, 579–586 (2011)

    MathSciNet  Google Scholar 

  82. Wang, L., Xie, N.-G., Zhu, Y.-F., Ye, Y., Meng, R.: Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation. Phys. A Stat. Mech. Appl. 390, 4535–4542 (2011)

    MathSciNet  Google Scholar 

  83. Piotrowski, E.W., Sladkowski, J.: The next stage: quantum game theory. arXiv preprint arXiv:quant-ph/0308027 (2003)

  84. Flitney, A.P.: Aspects of quantum game theory. Ph.D. thesis, The University of Adelaide (2005)

  85. Khan, F.S.: Quantum multiplexers, Parrondo games, and proper quantization. arXiv preprint arXiv:0906.0645 (2009)

  86. Flitney, A.P., Abbott, D.: Quantum games with decoherence. J. Phys. A Math. Gen. 38, 449 (2004)

    MathSciNet  MATH  Google Scholar 

  87. Banerjee, S., Chandrashekar, C., Pati, A.K.: Enhancement of geometric phase by frustration of decoherence: a Parrondo-like effect. Phys. Rev. A 87, 042119 (2013)

    Google Scholar 

  88. Lee, C.F., Johnson, N.F., Rodriguez, F., Quiroga, L.: Quantum coherence, correlated noise and Parrondo games. Fluct. Noise Lett. 2, L293–L298 (2002)

    MathSciNet  Google Scholar 

  89. Dinis, L.: Optimal sequence for Parrondo games. Phys. Rev. E 77, 021124 (2008)

    MathSciNet  Google Scholar 

  90. Rajendran, J., Benjamin, C.: Implementing Parrondo’s paradox with two-coin quantum walks. R. Soc. Open Sci. 5, 171599 (2018)

    Google Scholar 

  91. Xu, K., Huang, L., Yang, W.: Influence of entanglement in quantum Parrondo game. Int. J. Quantum Inf. 6, 1203–1212 (2008)

    MATH  Google Scholar 

  92. Tang, T.W., Allison, A., Abbott, D.: Investigation of chaotic switching strategies in Parrondo’s games. Fluct. Noise Lett. 4, L585–L596 (2004)

    Google Scholar 

  93. Canovas, J.S., Munoz, M.: Revisiting Parrondo’s paradox for the logistic family. Fluct. Noise Lett. 12, 1350015 (2013)

    Google Scholar 

  94. Buceta, J., Lindenberg, K., Parrondo, J.: Stationary and oscillatory spatial patterns induced by global periodic switching. Phys. Rev. Lett. 88, 024103 (2001)

    Google Scholar 

  95. Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Phys. D Nonlinear Phenom. 200, 124–132 (2005)

    MathSciNet  MATH  Google Scholar 

  96. Sagués, F., Sancho, J.M., García-Ojalvo, J.: Spatiotemporal order out of noise. Rev. Mod. Phys. 79, 829 (2007)

    Google Scholar 

  97. Danca, M.-F., Fečkan, M., Romera, M.: Generalized form of Parrondo’s paradoxical game with applications to chaos control. Int. J. Bifurc. Chaos 24, 1450008 (2014)

    MathSciNet  MATH  Google Scholar 

  98. Romera, M., Small, M., Danca, M.-F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192, 283–297 (2007)

    MathSciNet  MATH  Google Scholar 

  99. Mendoza, S.A., Matt, E.W., Guimarães-Blandón, D.R., Peacock-López, E.: Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems. Chaos Solitons Fractals 106, 86–93 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project was funded by the Singapore University of Technology and Design Start-up Research Grant (SRG SCI 2019 142).

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  1. Science and Math Cluster, Singapore University of Technology and Design, 8 Somapah Road, Singapore, 487372, Singapore

    Joel Weijia Lai & Kang Hao Cheong

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  1. Joel Weijia Lai
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Lai, J.W., Cheong, K.H. Parrondo’s paradox from classical to quantum: A review. Nonlinear Dyn 100, 849–861 (2020). https://doi.org/10.1007/s11071-020-05496-8

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