Occurrence of complementary processes in Parrondo's paradox

Parrondo's paradox involves two losing processes producing a winning outcome. We analyze the paradox with an original and novel method in which we start with one process and seek to construct a complementary process to achieve the paradox. We then derive a general condition for the classical Parrondo game to have a complementary process. Numerical simulation predicts that approximately two-thirds of such losing games satisfy the required condition. This suggests the common occurrence of the paradox, indicative of many potentially undiscovered applications in real-life scenarios involving stochastic processes.