Parrondo's effects with aperiodic protocols

TL;DR

This paper addresses whether aperiodic, deterministic switching protocols can realize Parrondo's paradox, where two losing games combine to win. It analyzes three archetypal aperiodic sequences—Fibonacci, Thue-Morse, and Rudin-Shapiro—playing between games A and B with biased coin probabilities $P_1=rac{1}{2}-\epsilon$, $P_2=rac{1}{10}-\epsilon$, and $P_3=rac{3}{4}-\epsilon$. The authors quantify performance via mean capital and examine structural features of sequences using cross-correlation $CC(S,X)$, lacunarity, and persistence, finding that strong anticorrelation with isolated losses is key to robust Parrondo effects, and that Thue-Morse provides the most favorable capital growth while Rudin-Shapiro lags. They show that increasing persistence generally reduces performance and that lacunarity interacts nonmonotonically with persistence to set optimal switching, highlighting a fine-tuned balance for protocol design. The results offer design principles for history-dependent switching strategies and connections between sequence structure and emergent paradoxical gains.

Abstract

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.

Figures (8)

• Figure 1: Autocorrelation Function (ACF) for various lags for the aperiodic sequences we used (Fb, TM, RS) considering $t_{max}=100$.
• Figure 2: Mean total capital versus time for different protocol considering $t_{\max}=100$.
• Figure 3: Barplot of the mean total capital at $t=200$ for the biasing parameter $\epsilon=\{0.005, 0.010, 0.015\}$ and different switching protocol.
• Figure 4: Pearson Cross-correlation $CC(S,X)$ between the capital of the switching protocol (S) with the game $X = \{A \ or \ B\}$. Results for $\epsilon=\{0.005, 0.010, 0.015\}$ and $t_{max}=200$. Similar findings are obtained if we compute $CC(S,X)$ using the Spearman and Kendall methods as shown in Appendix \ref{['app_extra_res']}.
• Figure 5: Mean total capital versus the Lacunarity and persistence measures of the input sequences used as a switching protocol considering $t_{max}=100$. For details on these measures see Appendix \ref{['app_lac_persis']}.