Parrondo's paradox as a framework for strategy switching and collective intelligence in complex systems Reply to comments on "Parrondo's paradox reveals counterintuitive wins in biology and decision making in society".

Organisms have evolved various mechanisms to cope with stochastic environmental fluctuations, and the survival strategies of being generalists and specialists are particularly crucial in this context. Generalists prioritize long-term population survival, providing equal benefits in both favorable and unfavorable environments. In contrast, specialists focus on short-term rapid growth, with benefits that differ significantly in different environmental conditions. In this paper we investigate the invading process of mutants in a finite wild-type population under fluctuating environmental conditions, aiming to elucidate the potential survival strategies that mutants can adopt to maximize their fixation probability. To this end, we construct and analyze a dynamic model in which the population dynamics is governed by a birth-death process, and the fluctuating environment is modeled by a Markov chain. We explore scenarios in which the mutant population adopts alternating strategies between specialist and generalist according to either responsive or stochastic switching schemes. Interestingly, we find that mutant populations adopting stochastic switching strategies achieve higher fixation probabilities than those using just pure strategies. This counterintuitive result is reminiscent of Parrondo's paradox in complex systems, where an unexpected victory arises from two individually losing strategies, provided that they are utilized in a particular or random manner. Our work reveals that in stochastic environments, survival in the evolutionary process depends not on the superiority of a specific strategy but on the population's adaptability.

Advancements in noisy intermediate-scale quantum (NISQ) computing are steadily pushing these systems toward outperforming classical supercomputers on specific well-defined computational tasks. In this work we explore and control quantum chaos in NISQ systems using discrete-time quantum walks (DTQWs) on cyclic graphs. To efficiently implement quantum walks on NISQ hardware, we employ the quantum Fourier transform to diagonalize the conditional shift operator, optimizing circuit depth and fidelity. We experimentally realize the transition from quantum chaos to order via DTQW dynamics on both odd and even cyclic graphs, specifically 3- and 4-cycle graphs, using the counterintuitive Parrondo paradox strategy across three different NISQ devices. While the 4-cycle graphs exhibit high-fidelity quantum evolution, the 3-cycle implementation shows significant fidelity improvement when augmented with dynamical decoupling pulses. Our results demonstrate a practical approach to probing and harnessing controlled chaotic dynamics on real quantum hardware, laying the groundwork for future quantum algorithms and cryptographic protocols based on quantum walks.

Maximum tolerated dose (MTD) and low-dose metronomic (LDM) schedules are widely used clinical strategies in cancer chemotherapy despite the fact that both approaches have inherent limitations. MTD often leads to drug resistance in tumors, whereas LDM usually results in the predominance of drug-sensitive or drug-resistant cancer cells, depending on the specific dose applied. To circumvent these unfavorable outcomes, we propose in this work a promising adaptive therapy strategy that alternates MTD and LDM schemes across the treatment cycles. By studying a three-component tumor system with replicator equations, we find that alternate administration of MTD and LDM at different lengths of the treatment cycle can significantly delay the development of drug-resistant phenotypes compared to when the two schemes are operated separately. This outcome recapitulates the weak form of the Parrondo's paradox, where appropriate combinations of two individually suboptimal strategies can produce an even more superior outcome (a more favorable therapeutic effect in our case). We also validate the feasibility of the proposed adaptive therapy strategy in spatially distributed tumor ecosystems by implementing agent-based simulations. Our findings offer potential clues to the challenges in state-of-the-art adaptive therapy methods, which often require precise regulation of drug dose and timing based on the proportions of drug-sensitive and drug-resistant cancer cells within a tumor.

The term Parrondo's paradox refers to the apparently paradoxical effect whereby two or more dynamics in which a given quantity decreases are combined in such a way that the same quantity increases in the resulting dynamics. We show that Parrondo's paradox can occur not only in the case of homogeneous but also in space-inhomogeneous one-dimensional discrete-time quantum walks. Moreover, we demonstrate that for such quantum walks Parrondo's paradox can affect the time evolution of quantum entanglement differently than it does in the case of homogeneous ones.

Corrigendum to “Designing a smart grid energy management with game theory and reinforcement learning using Parrondo's paradox” [Energy Rep. 13 (2025) 914–928

Designing a smart grid energy management with game theory and reinforcement learning using Parrondo's paradox

Reasonings on multiple strategies in differential systems: Comment on "Parrondo's paradox reveals counterintuitive wins in biology and decision making in society" by T. Wen & K.H. Cheong.

We develop a model for two-layer traffic flow in which a batch of n_{o} packets are dispatched from the source at regular intervals. This scenario is applicable in various communication and transportation systems, such as TCP congestion control mechanisms and congestion management through traffic lights. We demonstrate that implementing stochastic switching between the shortest-path and greedy approaches in the routing strategy of packet transmission results in a significant reduction in the total transmission weight when n_{o} exceeds a certain threshold. This phenomenon mirrors Parrondo's paradox, in which two games or strategies, individually yielding losses, produce a winning outcome or optimal results when combined. In addition, we observe that the influence of layer 1 on overall dynamics surpasses that of layer 2. Furthermore, we find that the impact of the structural characteristics of layers is more significant for switching probability γ<0.5.

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…) 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.

From Parrondo's paradox to collective intelligence: Comment on “Parrondo's paradox reveals counterintuitive wins in biology and decision making in society” by T. Wen & K.H. Cheong

A Parrondo paradox in susceptible-infectious-susceptible dynamics over periodic temporal networks

Modeling the architecture search process on a supernet and applying a differentiable method to find the importance of architecture are among the leading tools for differentiable neural architectures search (DARTS). One fundamental problem in DARTS is how to discretize or select a single-path architecture from the pretrained one-shot architecture. Previous approaches mainly exploit heuristic or progressive search methods for discretization and selection, which are not efficient and easily trapped by local optimizations. To address these issues, we formulate the task of finding a proper single-path architecture as an architecture game among the edges and operations with the strategies "keep" and "drop" and show that the optimal one-shot architecture is a Nash equilibrium of the architecture game. Then, we propose a novel and effective approach for discretizing and selecting a proper single-path architecture, which is based on extracting the single-path architecture that associates the maximal coefficient of the Nash equilibrium with the strategy "keep" in the architecture game. To further improve the efficiency, we employ a mechanism of entangled Gaussian representation of mini-batches, inspired by the classic Parrondo's paradox. If some mini-batch formed uncompetitive strategies, the entanglement of mini-batches would ensure the games be combined and, thus, turn into strong ones. We conduct extensive experiments on benchmark datasets and demonstrate that our approach is significantly faster than the state-of-the-art progressive discretizing methods while maintaining competitive performance with higher maximum accuracy.

Proof of a conjecture about Parrondo's paradox for two-armed slot machines

We propose a quantum game based on coin-based quantum walks. Given a quantum walk and a Hermitian operator on the coin-position composite space, winning this game involves choosing an initial coin state such that the given quantum walk leads to a composite state in which the expectation value of the given Hermitian operator is greater than a certain value. Parrondo's paradox is a phenomenon where a combination of losing strategies becomes a winning strategy. We give a deterministic scheme for identifying Parrondo's paradox in our game, in the sense that, given a collection of distinct quantum steps, we identify initial coin states which happen to be losing states for all quantum walks comprising solely of these steps individually, but turn out to be winning states for a quantum walk comprising of all the given steps taken in a sequence. Unlike traditional quantum steps that allow for equal magnitude forward and backward strides based on the outcome of the cointoss, the steps of the quantum walks employed here, though still contingent upon coin-toss, permit the strides to be of unequal magnitude, and not necessarily in opposite directions. We believe the results presented here will contribute to a deeper understanding of evolution of expectation values of observables in quantum walks, and facilitate the development of novel quantum algorithms.

Parrondo's paradox refers to the counterintuitive phenomenon whereby two losing strategies, when alternated in a certain manner, can result in a winning outcome. Understanding the optimal sequence in Parrondo's games is of significant importance for maximizing profits in various contexts. However, the current predefined sequences may not adapt well to changing environments, limiting their potential for achieving the best performance. We posit that the optimal strategy that determines which game to play should be learnable through experience. In this Letter, we propose an efficient and robust approach that leverages Q learning to adaptively learn the optimal sequence in Parrondo's games. Through extensive simulations of coin-tossing games, we demonstrate that the learned switching strategy in Parrondo's games outperforms other predefined sequences in terms of profit. Furthermore, the experimental results show that our proposed method can be easily adjusted to adapt to different cases of capital-dependent games and history-dependent games. Published by the American Physical Society 2024

Antagonistic coevolution between hosts and parasites substantially impacts community structure, with parasites displaying fluctuating selection or arms race dynamics during coevolution. The traditional matching alleles (MA) and gene-for-gene (GFG) models have been used to describe the dynamics and interaction of host-parasite coevolution, with these models assuming that parasites adopt a single strategy when competing with other parasites. We present a nonlinear dynamic population model that challenges this assumption, showing how a parasite that is disadvantaged under either the MA or the GFG model can win the competition by switching between the two losing strategies based on an external environmental cue, internal processes, or stochastic decision-making. This counterintuitive outcome is analogous to Parrondo's paradox, a game-theoretic concept that shows how alternating between two losing strategies can result in a winning outcome. Our numerical experiments support the validity of this model, suggesting that parasites can greatly benefit from maximum flexibility in their interactions with hosts. The flexibility of successful parasites puts an extra burden on the host defenses that have to adapt to different strategies of the parasites. These findings contribute to a deeper understanding of the coevolution of parasites and hosts, with broad implications for the evolution of complex ecological systems. Published by the American Physical Society 2024

The Parrondo paradox refers to a counterintuitive phenomenon where the combination of two losing strategies can result in a winning outcome. This paper examines the Parrondo paradox and its applications. The purpose of this study is to analyze the underlying mechanisms of the Parrondo paradox and explore its potential applications in various fields. In this research, a comprehensive analysis of existing literature and mathematical models is conducted to understand the theoretical foundations and practical implications of the Parrondo paradox. The results reveal that the emergence of the paradox is attributed to the interplay between deterministic and stochastic features in complex systems. By investigating the underlying mechanisms of the Parrondo paradox, this study contributes to a deeper understanding of complex systems and non-linear dynamics. Moreover, the applications of the Parrondo paradox are found in diverse fields such as finance, biology, and data analysis. It is concluded that the understanding of the Parrondo paradox can provide valuable insights for decision-making processes in dynamic and uncertain environments.

Influence analysis of network evolution on Parrondo effect

As a natural generalization of the discrete-time random walk, the continuous-time random walk (CTRW) has been applied to stochastic models with random dynamics in various fields. In this paper, we show that the deterministic alternation of two unbiased CTRWs can lead to a phenomenon similar to the Parrondo paradox, in which the asymptotic mean drift of the combined CTRW becomes positive or negative depending on the parameter values. This extends the case in which the paradox occurs due to the random combination of two CTRWs with memory shown by Montero [Phys. Rev. E 84 (2011) 051139].