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Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation

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  • Wang, Lu
  • Xie, Neng-gang
  • Zhu, Yong-fei
  • Ye, Ye
  • Meng, Rui

Abstract

In our previous study [Zhu et al., Quantum game interpretation for a special case of Parrondo’s paradox, Physica A 390 (2011) 579], the capital-dependent Parrondo’s game where one game depends on the capital modulus M=4 was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital. This paper presents a generalization of these results to even M greater than 4. An intuitive explanation for producing this phenomenon is that the discrete-time Markov chain of the game is divided into two completely unrelated inner and outer rings. The process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined. Quantum game theory is used to further analyze the phenomenon. The results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity.

Suggested Citation

  • Wang, Lu & Xie, Neng-gang & Zhu, Yong-fei & Ye, Ye & Meng, Rui, 2011. "Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4535-4542.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:23:p:4535-4542
    DOI: 10.1016/j.physa.2011.07.043
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    1. N. Masuda & N. Konno, 2004. "Subcritical behavior in the alternating supercritical Domany-Kinzel dynamics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 40(3), pages 313-319, August.
    2. Zhu, Yong-fei & Xie, Neng-gang & Ye, Ye & Peng, Fa-rui, 2011. "Quantum game interpretation for a special case of Parrondo’s paradox," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 579-586.
    3. Flitney, A.P. & Abbott, D., 2003. "Quantum models of Parrondo's games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 152-156.
    4. Flitney, A.P. & Ng, J. & Abbott, D., 2002. "Quantum Parrondo's games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 35-42.
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    Cited by:

    1. Wang, Lu & Zhu, Yong-fei & Ye, Ye & Meng, Rui & Xie, Neng-gang, 2012. "The coupling effect of the process sequence and the parity of the initial capital on Parrondo’s games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(21), pages 5197-5207.
    2. Song, Mi Jung & Lee, Jiyeon, 2021. "An approximation by Parrondo games of the Brownian ratchet," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).

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