Analytical Expressions for Parrondo Games

We present some new analytical expressions for the so-called Parrondo effect, where simple coin-flipping games with negative expected value are combined into a winning game. Parrondo games are state-dependent. By identifying the game state variable, we can compute the stationary game state probabilities. Mixing losing games increases the probability of entering into a game state with positive value, as can be seen clearly from our analytical expressions for the Parrondo game value.