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The coupling effect of the process sequence and the parity of the initial capital on Parrondo’s games

Author

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

Abstract

Based on the original Parrondo’s game and on the case where game A and game B are played randomly with modulo M=4, the processes of the game are divided into odd and even numbered plays, where the probability of playing game A in odd numbers is γ1 and the probability of playing game A in even numbers is γ2. By using the discrete time Markov chain, we find that the stationary probability distribution and mathematical expectation are not definite when γ1≠γ2 while they are definite when γ1=γ2. Meanwhile, we perform a more in-depth analysis. According to the residue values divided by an integer N, that is, 1,2,3,…,N−1,0, we divide the process of the game into 1,2,3,…,N−1, N times, where the probability of playing game A at each time is γi(i=1,2,…,N−1,N). The general conclusions we obtain through analysis are: (1) when the modulo M is odd, whatever odd or even number N is and whatever value γi is, the stationary probability distribution is definite and the profit of the game does not depend on the initial value; and (2) when the modulo M is even, if N is odd, then whatever value γi is, the stationary probability distribution is definite; if N is even, γ1=γ2=⋯=γN−1=γN must be satisfied and then the stationary probability distribution is definite; otherwise, the stationary probability distribution has infinite solutions and the profit of the game depends on the initial value.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:21:p:5197-5207
    DOI: 10.1016/j.physa.2012.06.008
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    References listed on IDEAS

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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. 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.
    3. 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.
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    Cited by:

    1. Cheong, Kang Hao & Soo, Wayne Wah Ming, 2013. "Construction of novel stochastic matrices for analysis of Parrondo’s paradox," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 4727-4738.

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