A losing game that wins by remembering: Parrondo's history-dependent game with power-law memory

In Parrondo's paradox, two games of chance that are each losing become winning when they are alternated. In the history-dependent version of the paradox the second game looks at the outcomes of the last two rounds. Here the second game looks at its whole past instead, weighting the outcome of the round played $n$ steps earlier by $(n+1)^{-\gamma}$, the slowly fading memory typical of fractional-order systems. Two results follow. When the memory is long enough ($\gamma\lesssim1$) the history-dependent game is no longer losing: it wins on its own, without any alternation, so the paradox dissolves because one of its ingredients has changed sign. The reason is identified and tested with a control experiment: the game has a favourable and an unfavourable regime with opposite mean returns, and the unfavourable one produces streaks of losses that throw the memory deep into the favourable regime, where the player then stays longer. The second result concerns time. The gain of a player with power-law memory is not stationary: it decays as $t^{\gamma-1}$ with the number of rounds played, whereas with any memory of finite range, exponential or truncated, the gain settles to a constant. The decay follows from a bound on the memory-weighted average of the past gain, which cannot grow when the kernel is not summable, and it is the same law that governs the dying current of subdiffusive Brownian ratchets, of which Parrondo's games are the discrete image. The time after which the decay becomes visible grows very rapidly as $\gamma\to1$, so that for memory exponents close to one a gain that looks perfectly stationary over any practical game is still, in the long run, bound to vanish.