A Paradoxical Evolutionary Mechanism in Stochastically Switching Environments

Choice anticipation as gated accumulation of sensory predictions

Predictions are combined with sensory information when making choices. Accumulator models have conceptualized predictions as trial-by-trial updates to a baseline evidence level. These models have been successful in explaining the influence of choice history across-trials, however, they do not account for how sensory information is transformed into choice evidence. Here, we derive a gated accumulator that models the onset of evidence accumulation as a combination of delayed sensory information and a prediction of sensory timing. To test how delays interact with predictions, we designed a free-choice saccade task where participants directed eye movements to either of two targets that appeared with variable delays and asynchronies. Despite instructions not to anticipate, participants responded before target onset on some trials. We reasoned that anticipatory responses reflected a trade-off between inhibiting and facilitating the onset of evidence accumulation via a gating mechanism as target appearance became more likely. We then found that anticipatory responses were more likely following repeated choices, suggesting that the balance between anticipatory and sensory responses was driven by a prediction of sensory timing. By fitting the gated accumulator model to the data, we found that variance in within-trial fluctuations in baseline evidence best explained the joint increase of anticipatory responses and faster sensory-guided responses with longer delays. Thus, we conclude that a prediction of sensory timing is involved in balancing the costs of anticipation with lowering the amount of accumulated evidence required to trigger saccadic choice.NEW & NOTEWORTHY Evidence accumulation models are used to study how recent history impacts the processes underlying how we make choices. Biophysical evidence suggests that the accumulation of evidence is gated, however, classic accumulator models do not account for this. In this work, we show that predictions of the timing of sensory information are important in controlling how evidence accumulation is gated and that signatures of these predictions can be detected even in randomized task environments.

Parrondo's paradox reveals counterintuitive wins in biology and decision making in society

Parrondo's paradox refers to the paradoxical phenomenon of combining two losing strategies in a certain manner to obtain a winning outcome. It has been applied to uncover unexpected outcomes across various disciplines, particularly at different spatiotemporal scales within ecosystems. In this article, we provide a comprehensive review of recent developments in Parrondo's paradox within the interdisciplinary realm of the physics of life, focusing on its significant applications across biology and the broader life sciences. Specifically, we examine its relevance from genetic pathways and phenotypic regulation, to intercellular interaction within multicellular organisms, and finally to the competition between populations and species in ecosystems. This phenomenon, spanning multiple biological domains and scales, enhances our understanding of the unified characteristics of life and reveals that adaptability in a drastically changing environment, rather than the inherent excellence of a trait, underpins survival in the process of evolution. We conclude by summarizing our findings and discussing future research directions that hold promise for advancing the field.

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 (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.

Noise-induced Parrondo's paradox in discrete-time quantum walks

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 noise can induce Parrondo's paradox in one-dimensional discrete-time quantum walks with deterministic periodic as well as aperiodic sequences of two-state quantum coins where this paradox does not occur in the absence of noise. Moreover, we show how the noise-induced Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

Optimizing crop rotations via Parrondo's paradox for sustainable agriculture

Crop rotation, a sustainable agricultural technique, has been at humanity's disposal since time immemorial and is practised globally. Switching between cover crops and cash crops helps avoid the adverse effects of intensive farming. Determining the optimum cash-cover rotation schedule for maximizing yield has been tackled on multiple fronts by agricultural scientists, economists, biologists and computer scientists, to name a few. However, considering the uncertainty due to diseases, pests, droughts, floods and impending effects of climate change is essential when designing rotation strategies. Analysing this time-tested technique of crop rotations with a new lens of Parrondo's paradox allows us to optimally use the rotation technique in synchrony with uncertainty. While previous approaches are reactive to the diversity of crop types and environmental uncertainties, we make use of the said uncertainties to enhance crop rotation schedules. We calculate optimum switching probabilities in a randomized cropping sequence and suggest optimum deterministic sequences and judicious use of fertilizers. Our methods demonstrate strategies to enhance crop yield and the eventual profit margins for farmers. Conforming to translational biology, we extend Parrondo's paradox, where two losing situations can be combined eventually into a winning scenario, to agriculture.

Parrondo's paradox in quantum walks with three coins

Parrondo's paradox refers to the apparently paradoxical effect whereby a certain combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's paradox can occur not only in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of two quantum coins but also in the case of one-dimensional discrete-time quantum walks with a deterministic sequence of three quantum coins. Moreover, we show how Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

Parrondo's paradox in quantum walks with deterministic aperiodic sequence of coins

Parrondo's effect is a well-known apparent paradox where a combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's effect can occur not only in the case of one-dimensional discrete quantum walks with random or deterministic periodic sequence of two- or multistate quantum coins but also in the case of one-dimensional discrete quantum walks with deterministic aperiodic sequence of two-state quantum coins. Moreover, we show how Parrondo's effect affects the time evolution of the walker-coin quantum entanglement.

An alternating active-dormitive strategy enables disadvantaged prey to outcompete the perennially active prey through Parrondo's paradox

BACKGROUND

Dormancy is widespread in nature, but while it can be an effective adaptive strategy in fluctuating environments, the dormant forms are costly due to the inability to breed and the relatively high energy consumption. We explore mathematical models of predator-prey systems, in order to assess whether dormancy can be an effective adaptive strategy to outcompete perennially active (PA) prey, even when both forms of the dormitive prey (active and dormant) are individually disadvantaged.

RESULTS

We develop a dynamic population model by introducing an additional dormitive prey population to the existing predator-prey model which can be active (active form) and enter dormancy (dormant form). In this model, both forms of the dormitive prey are individually at a disadvantage compared to the PA prey and thus would go extinct due to their low growth rate, energy waste on the production of dormant prey, and the inability of the latter to grow autonomously. However, the dormitive prey can paradoxically outcompete the PA prey with superior traits and even cause its extinction by alternating between the two losing strategies. We observed higher fitness of the dormitive prey in rich environments because a large predator population in a rich environment cannot be supported by the prey without adopting an evasive strategy, that is, dormancy. In such environments, populations experience large-scale fluctuations, which can be survived by dormitive but not by PA prey.

CONCLUSION

We show that dormancy can be an effective adaptive strategy to outcompete superior prey, recapitulating the game-theoretic Parrondo's paradox, where two losing strategies combine to achieve a winning outcome. We suggest that the species with the ability to switch between the active and dormant forms can dominate communities via competitive exclusion.

Making the most of potential: potential games and genotypic convergence

We consider genotypic convergence of populations and show that under fixed fitness asexual and haploid sexual populations attain monomorphic convergence (even under genetic linkage between loci) to basins of attraction with locally exponential convergence rates; the same convergence obtains in single locus diploid sexual reproduction but to polymorphic populations. Furthermore, we show that there is a unified theory underlying these convergences: all of them can be interpreted as instantiations of players in a potential game implementing a multiplicative weights updating algorithm to converge to equilibrium, making use of the Baum-Eagon Theorem. To analyse varying environments, we introduce the concept of 'virtual convergence', under which, even if fixation is not attained, the population nevertheless achieves the fitness growth rate it would have had under convergence to an optimal genotype. Virtual convergence is attained by asexual, haploid sexual and multi-locus diploid reproducing populations, even if environments vary arbitrarily. We also study conditions for true monomorphic convergence in asexually reproducing populations in varying environments.

Does Cancer Biology Rely on Parrondo's Principles?

Simple Summary

Parrondo’s paradox, whereby losing strategies or deleterious effects can combine to provide a winning outcome, has been increasingly applied by biologists to explain complex adaptations in many living systems. Here, we suggest that considering this paradox in oncology, particularly in relation to the phenotypic diversity of malignant cells, could also be a promising approach to understand several puzzling aspects of cancer biology. For example, the high genetic and epigenetic instability of cancer cells, their metastatic behavior and their capacity to enter dormancy could be explained by Parrondo’s theory. We also discuss the relevance of Parrondo’s paradox in a therapeutical framework using different examples. This work provides a compelling argument that the traditional separation between medicine and other disciplines remains a fundamental limitation that needs to be overcome if complex processes, such as oncogenesis, are to be completely understood.

Abstract

Many aspects of cancer biology remain puzzling, including the proliferative and survival success of malignant cells in spite of their high genetic and epigenetic instability as well as their ability to express migrating phenotypes and/or enter dormancy despite possible fitness loss. Understanding the potential adaptive value of these phenotypic traits is confounded by the fact that, when considered separately, they seem to be rather detrimental at the cell level, at least in the short term. Here, we argue that cancer’s biology and success could frequently be governed by processes underlying Parrondo’s paradox, whereby combinations of intrinsically losing strategies may result in winning outcomes. Oncogenic selection would favor Parrondo’s dynamics because, given the environmental adversity in which malignant cells emerge and evolve, alternating between various less optimal strategies would represent the sole viable option to counteract the changing and deleterious environments cells are exposed to during tumorigenesis. We suggest that malignant processes could be viewed through this lens, and we discuss how Parrondo’s principles are also important when designing therapies against cancer.

Generalized Solutions of Parrondo's Games

In game theory, Parrondo's paradox describes the possibility of achieving winning outcomes by alternating between losing strategies. The framework had been conceptualized from a physical phenomenon termed flashing Brownian ratchets, but has since been useful in understanding a broad range of phenomena in the physical and life sciences, including the behavior of ecological systems and evolutionary trends. A minimal representation of the paradox is that of a pair of games played in random order; unfortunately, closed-form solutions general in all parameters remain elusive. Here, we present explicit solutions for capital statistics and outcome conditions for a generalized game pair. The methodology is general and can be applied to the development of analytical methods across ratchet-type models, and of Parrondo's paradox in general, which have wide-ranging applications across physical and biological systems.

Cooperate or Not Cooperate in Predictable but Periodically Varying Situations? Cooperation in Fast Oscillating Environment

In this work, the cooperation problem between two populations in a periodically varying environment is discussed. In particular, the two-population prisoner's dilemma game with periodically oscillating payoffs is discussed, such that the time-average of these oscillations over the period of environmental variations vanishes. The possible overlaps of these oscillations generate completely new dynamical effects that drastically change the phase space structure of the two-population evolutionary dynamics. Due to these effects, the emergence of some level of cooperators in both populations is possible under certain conditions on the environmental variations. In the domain of stable coexistence the dynamics of cooperators in each population form stable cycles. Thus, the cooperators in each population promote the existence of cooperators in the other population. However, the survival of cooperators in both populations is not guaranteed by a large initial fraction of them.

Parrondo effect in quantum coin-toss simulations

Game A + Game B = Game C. Parrondo's games follow this basic structure where A and B are losing games and C is a winning game-a phenomenon called Parrondo's paradox. These games can take on a wider class of definitions and exhibit these paradoxical results. In this paper, we show three paradoxical cases. (i) The successive "tossing" of a single fair quantum coin gives a biased result, a previously known result. (ii) The random tossing of two quantum coins, each with successive biased expectations, gives an average random walk position of approximately zero. (iii) The sequential periodic tossing of two quantum coins, each with successive negative biased expectations, gives an average random walk with positive expectation. Using these results, we then propose a protocol for identifying and classifying quantum operations that span the same Hilbert space for a two-level quantum system.

Paradoxical Survival: Examining the Parrondo Effect across Biology

Parrondo's paradox, in which losing strategies can be combined to produce winning outcomes, has received much attention in mathematics and the physical sciences; a plethora of exciting applications has also been found in biology at an astounding pace. In this review paper, the authors examine a large range of recent developments of Parrondo's paradox in biology, across ecology and evolution, genetics, social and behavioral systems, cellular processes, and disease. Intriguing connections between numerous works are identified and analyzed, culminating in an emergent pattern of nested recurrent mechanics that appear to span the entire biological gamut, from the smallest of spatial and temporal scales to the largest-from the subcellular to the complete biosphere. In analyzing the macro perspective, the pivotal role that the paradox plays in the shaping of biological life becomes apparent, and its identity as a potential universal principle underlying biological diversity and persistence is uncovered. Directions for future research are also discussed in light of this new perspective.

Predator Dormancy is a Stable Adaptive Strategy due to Parrondo's Paradox

Many predators produce dormant offspring to escape harsh environmental conditions, but the evolutionary stability of this adaptation has not been fully explored. Like seed banks in plants, dormancy provides a stable competitive advantage when seasonal variations occur, because the persistence of dormant forms under harsh conditions compensates for the increased cost of producing dormant offspring. However, dormancy also exists in environments with minimal abiotic variation-an observation not accounted for by existing theory. Here it is demonstrated that dormancy can out-compete perennial activity under conditions of extensive prey density fluctuation caused by overpredation. It is shown that at a critical level of prey density fluctuations, dormancy becomes an evolutionarily stable strategy. This is interpreted as a manifestation of Parrondo's paradox: although neither the active nor dormant forms of a dormancy-capable predator can individually out-compete a perennially active predator, alternating between these two losing strategies can paradoxically result in a winning strategy. Parrondo's paradox may thus explain the widespread success of quiescent behavioral strategies such as dormancy, suggesting that dormancy emerges as a natural evolutionary response to the self-destructive tendencies of overpredation and related biological phenomena.

Solution of the Crow-Kimura model with changing population size and Allee effect

The Crow-Kimura model is commonly used in the modeling of genetic evolution in the presence of mutations and associated selection pressures. We consider a modified version of the Crow-Kimura model, in which population sizes are not fixed and Allee saturation effects are present. We demonstrate the evolutionary dynamics in this system through an analytical approach, examining both symmetric and single-peak fitness landscape cases. Especially interesting are the dynamics of the populations near extinction. A special version of the model with saturation and degradation on the single-peak fitness landscape is investigated as a candidate of the Allee effect in evolution, revealing reduction tendencies of excessively large populations, and extinction tendencies for small populations. The analytical solutions for these dynamics are presented with accuracy O(1/N), where N is the number of nucleotides in the genome.

Allison mixture and the two-envelope problem

In the present study, we have investigated the Allison mixture, a variant of the Parrondo's games where random mixing of two random sequences creates autocorrelation. We have obtained the autocorrelation function and mutual entropy of two elements. Our analysis shows that the mutual information is nonzero even if two distributions have identical average values. We have also considered the two-envelope problem and solved for its exact probability distribution.

Nomadic-colonial life strategies enable paradoxical survival and growth despite habitat destruction

Organisms often exhibit behavioral or phenotypic diversity to improve population fitness in the face of environmental variability. When each behavior or phenotype is individually maladaptive, alternating between these losing strategies can counter-intuitively result in population persistence-an outcome similar to the Parrondo's paradox. Instead of the capital or history dependence that characterize traditional Parrondo games, most ecological models which exhibit such paradoxical behavior depend on the presence of exogenous environmental variation. Here we present a population model that exhibits Parrondo's paradox through capital and history-dependent dynamics. Two sub-populations comprise our model: nomads, who live independently without competition or cooperation, and colonists, who engage in competition, cooperation, and long-term habitat destruction. Nomads and colonists may alternate behaviors in response to changes in the colonial habitat. Even when nomadism and colonialism individually lead to extinction, switching between these strategies at the appropriate moments can paradoxically enable both population persistence and long-term growth.

Effects of behavioral patterns and network topology structures on Parrondo's paradox

A multi-agent Parrondo's model based on complex networks is used in the current study. For Parrondo's game A, the individual interaction can be categorized into five types of behavioral patterns: the Matthew effect, harmony, cooperation, poor-competition-rich-cooperation and a random mode. The parameter space of Parrondo's paradox pertaining to each behavioral pattern, and the gradual change of the parameter space from a two-dimensional lattice to a random network and from a random network to a scale-free network was analyzed. The simulation results suggest that the size of the region of the parameter space that elicits Parrondo's paradox is positively correlated with the heterogeneity of the degree distribution of the network. For two distinct sets of probability parameters, the microcosmic reasons underlying the occurrence of the paradox under the scale-free network are elaborated. Common interaction mechanisms of the asymmetric structure of game B, behavioral patterns and network topology are also revealed.