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首页> 外文期刊>Physica, A. Statistical mechanics and its applications >The coupling effect of the process sequence and the parity of the initial capital on Parrondo's games
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The coupling effect of the process sequence and the parity of the initial capital on Parrondo's games

机译：过程序列和初始资金的均等性对Parrondo游戏的耦合作用

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摘要

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 γ _1 and the probability of playing game A in even numbers is γ _2. By using the discrete time Markov chain, we find that the stationary probability distribution and mathematical expectation are not definite when γ _1≠γ _2 while they are definite when γ _1=γ _2. 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 γ _i(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 γ _i 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 γ i is, the stationary probability distribution is definite; if N is even, γ _1=γ _2=?=γ _(N-1)=γ _N 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.
机译：根据原始的Parrondo游戏，并且以模M = 4随机进行游戏A和游戏B的情况，游戏过程分为奇数和偶数游戏，其中以奇数进行游戏A的概率为γ_1，并且参加偶数游戏A的概率为γ_2。通过使用离散时间马尔可夫链，我们发现当γ_1≠γ_2时，平稳概率分布和数学期望不是确定的，而当γ_1 =γ_2时，它们是确定的。同时，我们进行了更深入的分析。根据残值除以整数N，即1,2,3，？，N-1,0，我们将游戏过程分为1,2,3，？，N-1，N次，其中每次玩游戏A的概率为γ_i（i = 1,2，？，N-1，N）。我们通过分析得出的一般结论是：（1）当模M为奇数时，无论奇数N等于奇数N还是γ_i等于什么值，固定概率分布都是确定的，博弈的利润不取决于初始值; （2）当模M为偶数时，如果N为奇数，则无论γi为多少，平稳概率分布都是确定的;如果N是偶数，则必须满足γ_1=γ_2=？=γ_（N-1）=γ_N，则固定概率分布是确定的。否则，平稳概率分布具有无限解，并且博弈的利润取决于初始值。

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