A new generalization of Parrondo's games to three players and its application in genetic switches

Parrondo's paradox indicates a paradoxical situation in which a winning expectation may occur in sequences of losing games. There are many versions of the original Parrondo's games in the literature, but the games are played by two players in all of them. We introduce a new extended version of games played by three players and a three-sided biased dice instead of two players and a biased coin in this work. In the first step, we find the part of the parameters space where the games are played fairly. After adding noise to fair probabilities, we combine two games randomly, periodically, and nonlinearly and obtain the conditions under which the paradox can occur. This generalized model can be applied in all science and engineering fields. It can also be used for genetic switches. Genetic switches are often made by two reactive elements, but the existence of more elements can lead to more existing decisions for cells. Each genetic switch can be considered a game in which the reactive elements compete to increase their molecular concentrations. We present three genetic networks based on a new generalized Parrondo's games model, consisting of two noisy genetic switches. The combination of them can increase network robustness to noise. Each switch can also be used as an initial pattern to construct a synthetic switch to change undesirable cells' fate.