Parrondo's effects with aperiodic protocols

In this work, we study the effectiveness of employing archetypal aperiodic sequencing—namely, Fibonacci, Thue–Morse, and Rudin–Shapiro—on the Parrondian effect. From a capital gain perspective, our results show that these series do yield a Parrondo's paradox with the Thue–Morse based strategy outperforming not only the other two aperiodic strategies but benchmark Parrondian games with random and periodical ( A A B B A A B B …) switching as well. The least performing of the three aperiodic strategies is the Rudin–Shapiro. To elucidate the underlying causes of these results, we analyze the cross correlation between the capital generated by the switching protocols and that of the isolated losing games. This analysis reveals that a strong anticorrelation with both isolated games is typically required to achieve a robust manifestation of Parrondo's effect. We also study the influence of the sequencing on the capital using the lacunarity and persistence measures. In general, we observe that the switching protocols tend to become less performing in terms of the capital as one increases the persistence and, thus, approaches the features of an isolated losing game. For the (log-)lacunarity, a property related to heterogeneity, we notice that for small persistence (less than 0.5), the performance increases with the lacunarity with a maximum around 0.4. In respect of this, our work shows that the optimization of a switching protocol is strongly dependent on a fine-tuning between persistence and heterogeneity.