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 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.