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 ($AABBAABB\ldots$) 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.

翻译：本研究探讨了采用典型非周期序列（即斐波那契序列、图厄-莫尔斯序列和鲁丁-夏皮罗序列）对帕隆多效应的影响。从资本收益的角度来看，我们的结果表明这些序列确实能产生帕隆多悖论，其中基于图厄-莫尔斯序列的策略不仅优于另外两种非周期策略，也超越了采用随机切换和周期性切换（$AABBAABB\ldots$）的基准帕隆多博弈。三种非周期策略中表现最差的是鲁丁-夏皮罗序列。为阐明这些结果的根本原因，我们分析了切换协议产生的资本与独立亏损博弈资本之间的互相关性。该分析表明，通常需要与两个独立博弈均呈现强反相关性才能实现帕隆多效应的稳健显现。我们还通过空隙度和持续性度量研究了序列对资本的影响。总体而言，我们观察到随着持续性增加并趋近于独立亏损博弈的特征，切换协议在资本收益方面的表现往往趋于下降。对于与异质性相关的（对数）空隙度属性，我们注意到在低持续性（小于0.5）条件下，策略性能随空隙度增加而提升，并在0.4左右达到峰值。就此而言，我们的研究表明切换协议的优化强烈依赖于持续性与异质性之间的精细调谐。