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...

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...

By using the discrete Markov chain method, Parrondo’s paradox is studied by means of theoretical analysis and computer simulation, built on the case of game AB played in alternation with modulus M=4. We find that such a case does not have a definite stationary probability distribution and that...