Understanding Paradoxical Nature of Periodic Parrondo Game

That two losing (gambling) games, suitably combined, can result in a winning combination was shown by Juan M. R. Parrondo, and is known as the Parrondo paradox. We explored the periodic sequences of such games to see if it always holds, and why. A systematic study involving DTMC analysis and use of tree diagrams shows that a trade-off between the relative occupation of the capital being in a modulo state and the relative weight of tossing the coins of Parrondo games in that state stabilizes the probability of win for a given game sequence. Keywords-Parrondo game; discrete-time Markov chain (DTMC); tree diagrams; flashing ratchet.