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

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.