Unfair and Anomalous Evolutionary Dynamics from Fluctuating Payoffs

Perturbation-response dynamics of coupled nonlinear systems

How nonlinear systems dynamically respond to external perturbations essentially determines their function. Weak perturbations induce response dynamics near a stable operating point, often approximately characterized by linear response theory. However, stronger driving signals may induce genuinely nonlinear responses, including tipping transitions to qualitatively different dynamical states. Here, we analyze how inter-unit coupling impacts responses to periodic perturbations. We find that already in minimal systems of two identical and linearly coupled units, coupling impacts the dynamical responses in a distinct way. Any non-zero coupling strength extends the regime of non-tipping local responses relative to uncoupled units. Intriguingly, finite coupling may be more effective than infinitely strong coupling in keeping responses from tipping. Interestingly, already weak coupling may create novel response modes in strongly driven systems, implying the existence of multiple tipping points instead of only one. These results persist for systems of non-identical units, systems with nonlinear coupling, and larger networks of coupled units.

The evolution of social behaviors and risk preferences in settings with uncertainty

Humans update their social behavior in response to past experiences and changing environments. Behavioral decisions are further complicated by uncertainty in the outcome of social interactions. Faced with uncertainty, some individuals exhibit risk aversion while others seek risk. Attitudes toward risk may depend on socioeconomic status; and individuals may update their risk preferences over time, which will feedback on their social behavior. Here, we study how uncertainty and risk preferences shape the evolution of social behaviors. We extend the game-theoretic framework for behavioral evolution to incorporate uncertainty about payoffs and variation in how individuals respond to this uncertainty. We find that different attitudes toward risk can substantially alter behavior and long-term outcomes, as individuals seek to optimize their rewards from social interactions. In a standard setting without risk, for example, defection always overtakes a well-mixed population engaged in the classic Prisoner's Dilemma, whereas risk aversion can reverse the direction of evolution, promoting cooperation over defection. When individuals update their risk preferences along with their strategic behaviors, a population can oscillate between periods dominated by risk-averse cooperators and periods of risk-seeking defectors. Our analysis provides a systematic account of how risk preferences modulate, and even coevolve with, behavior in an uncertain social world.

Cooperation in a non-ergodic world on a network - insurance and beyond

Cooperation between individuals is emergent in all parts of society; yet, mechanistic reasons for this emergence are ill understood in the literature. A specific example of this is insurance. Recent work has, though, shown that assuming the risk individuals face is proportional to their wealth and optimizing the time average growth rate rather than the ensemble average results in a non-zero-sum game, where both parties benefit from cooperation through insurance contracts. In a recent paper, Peters and Skjold present a simple agent-based model and show how, over time, agents that enter into such cooperatives outperform agents that do not. Here, we extend this work by restricting the possible connections between agents via a lattice network. Under these restrictions, we still find that all agents profit from cooperating through insurance. We, though, further find that clusters of poor and rich agents emerge endogenously on the two-dimensional map and that wealth inequalities persist for a long duration, consistent with the phenomenon known as the poverty trap. By tuning the parameters that control the risk levels, we simulate both highly advantageous and extremely risky gambles and show that despite the qualitative shift in the type of risk, the findings are consistent.

Evolutionary game selection creates cooperative environments

The emergence of collective cooperation in competitive environments is a well-known phenomenon in biology, economics, and social systems. While most evolutionary game models focus on the evolution of strategies for a fixed game, how strategic decisions coevolve with the environment has so far mostly been overlooked. Here, we consider a game selection model where not only the strategies but also the game can change over time following evolutionary principles. Our results show that coevolutionary dynamics of games and strategies can induce novel collective phenomena, fostering the emergence of cooperative environments. When the model is taken on structured populations the architecture of the interaction network can significantly amplify pro-social behavior, with a critical role played by network heterogeneity and the presence of clustered groups of similar players, distinctive features observed in real-world populations. By unveiling the link between the evolution of strategies and games for different structured populations, our model sheds new light on the origin of social dilemmas ubiquitously observed in real-world social systems.

Hypochaos prevents tragedy of the commons in discrete-time eco-evolutionary game dynamics

While quite a few recent papers have explored game-resource feedback using the framework of evolutionary game theory, almost all the studies are confined to using time-continuous dynamical equations. Moreover, in such literature, the effect of ubiquitous chaos in the resulting eco-evolutionary dynamics is rather missing. Here, we present a deterministic eco-evolutionary discrete-time dynamics in generation-wise non-overlapping population of two types of harvesters-one harvesting at a faster rate than the other-consuming a self-renewing resource capable of showing chaotic dynamics. In the light of our finding that sometimes chaos is confined exclusively to either the dynamics of the resource or that of the consumer fractions, an interesting scenario is realized: The resource state can keep oscillating chaotically, and hence, it does not vanish to result in the tragedy of the commons-extinction of the resource due to selfish indiscriminate exploitation-and yet the consumer population, whose dynamics depends directly on the state of the resource, may end up being composed exclusively of defectors, i.e., high harvesters. This appears non-intuitive because it is well known that prevention of tragedy of the commons usually requires substantial cooperation to be present.

Structure and function in artificial, zebrafish and human neural networks

Network science provides a set of tools for the characterization of the structure and functional behavior of complex systems. Yet a major problem is to quantify how the structural domain is related to the dynamical one. In other words, how the diversity of dynamical states of a system can be predicted from the static network structure? Or the reverse problem: starting from a set of signals derived from experimental recordings, how can one discover the network connections or the causal relations behind the observed dynamics? Despite the advances achieved over the last two decades, many challenges remain concerning the study of the structure-dynamics interplay of complex systems. In neuroscience, progress is typically constrained by the low spatio-temporal resolution of experiments and by the lack of a universal inferring framework for empirical systems. To address these issues, applications of network science and artificial intelligence to neural data have been rapidly growing. In this article, we review important recent applications of methods from those fields to the study of the interplay between structure and functional dynamics of human and zebrafish brain. We cover the selection of topological features for the characterization of brain networks, inference of functional connections, dynamical modeling, and close with applications to both the human and zebrafish brain. This review is intended to neuroscientists who want to become acquainted with techniques from network science, as well as to researchers from the latter field who are interested in exploring novel application scenarios in neuroscience.

Stochastic replicator dynamics and evolutionary stability

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equation for the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

Emergence of synergistic and competitive pathogens in a coevolutionary spreading model

Cooperation and competition between pathogens can alter the amount of individuals affected by a coinfection. Nonetheless, the evolution of the pathogens' behavior has been overlooked. Here, we consider a coevolutionary model where the simultaneous spreading is described by a two-pathogen susceptible-infected-recovered model in an either synergistic or competitive manner. At the end of each epidemic season, the pathogens species reproduce according to their fitness that, in turn, depends on the payoff accumulated during the spreading season in a hawk-and-dove game. This coevolutionary model displays a rich set of features. Specifically, the evolution of the pathogens' strategy induces abrupt transitions in the epidemic prevalence. Furthermore, we observe that the long-term dynamics results in a single, surviving pathogen species, and that the cooperative behavior of pathogens can emerge even under unfavorable conditions.

Stochastic evolutionary stability in matrix games with random payoffs

Evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behavior, but also widely used in economics and social sciences. Recently, in order to reveal the stochastic dynamical properties of evolutionary games in randomly fluctuating environments, the concept of stochastic evolutionary stability based on conditions for stochastic local stability for a fixation state was developed in the context of a symmetric matrix game with two phenotypes and random payoffs in pairwise interactions [Zheng et al., Phys. Rev. E 96, 032414 (2017)2470-004510.1103/PhysRevE.96.032414]. In this paper, we extend this study to more general situations, namely, multiphenotype symmetric as well as asymmetric matrix games with random payoffs. Conditions for stochastic local stability and stochastic evolutionary stability are established. Conditions for a fixation state to be stochastically unstable and almost everywhere stochastically unstable are distinguished in a multiphenotype setting according to the initial population state. Our results provide some alternative perspective and a more general theoretical framework for a better understanding of the evolution of animal behavior in a stochastic environment.

Criticality and Griffiths phases in random games with quenched disorder

The perceived risk and reward for a given situation can vary depending on resource availability, accumulated wealth, and other extrinsic factors such as individual backgrounds. Based on this general aspect of everyday life, here we use evolutionary game theory to model a scenario with randomly perturbed payoffs in a prisoner's dilemma game. The perception diversity is modeled by adding a zero-average random noise in the payoff entries and a Monte Carlo simulation is used to obtain the population dynamics. This payoff heterogeneity can promote and maintain cooperation in a competitive scenario where only defectors would survive otherwise. In this work, we give a step further, understanding the role of heterogeneity by investigating the effects of quenched disorder in the critical properties of random games. We observe that payoff fluctuations induce a very slow dynamic, making the cooperation decay behave as power laws with varying exponents, instead of the usual exponential decay after the critical point, showing the emergence of a Griffiths phase. We also find a symmetric Griffiths phase near the defector's extinction point when fluctuations are present, indicating that Griffiths phases may be frequent in evolutionary game dynamics and play a role in the coexistence of different strategies.

Decisive conditions for strategic vaccination against SARS-CoV-2

While vaccines against severe acute respiratory syndrome coronavirus (SARS-CoV-2) are being administered, in many countries it may still take months until their supply can meet demand. The majority of available vaccines elicit strong immune responses when administered as prime-boost regimens. Since the immunological response to the first ("prime") dose may provide already a substantial reduction in infectiousness and protection against severe disease, it may be more effective-under certain immunological and epidemiological conditions-to vaccinate as many people as possible with only one dose instead of administering a person a second ("booster") dose. Such a vaccination campaign may help to more effectively slow down the spread of SARS-CoV-2 and reduce hospitalizations and fatalities. The conditions that make prime-first vaccination favorable over prime-boost campaigns, however, are not well understood. By combining epidemiological modeling, random-sampling techniques, and decision tree learning, we find that prime-first vaccination is robustly favored over prime-boost vaccination campaigns even for low single-dose efficacies. For epidemiological parameters that describe the spread of coronavirus disease 2019 (COVID-19), recent data on new variants included, we show that the difference between prime-boost and single-shot waning rates is the only discriminative threshold, falling in the narrow range of 0.01-0.02  day^-1 below which prime-first vaccination should be considered.

Decisive Conditions for Strategic Vaccination against SARS-CoV-2

While vaccines against SARS-CoV-2 are being administered, in most countries it may still take months until their supply can meet demand. The majority of available vaccines elicits strong immune responses when administered as prime-boost regimens. Since the immunological response to the first ("prime") injection may provide already a substantial reduction in infectiousness and protection against severe disease, it may be more effective-under certain immunological and epidemiological conditions-to vaccinate as many people as possible with only one shot, instead of administering a person a second ("boost") shot. Such a vaccination campaign may help to more effectively slow down the spread of SARS-CoV-2, reduce hospitalizations, and reduce fatalities, which is our objective. Yet, the conditions which make single-dose vaccination favorable over prime-boost administrations are not well understood. By combining epidemiological modeling, random sampling techniques, and decision tree learning, we find that single-dose vaccination is robustly favored over prime-boost vaccination campaigns, even for low single-dose efficacies. For realistic scenarios and assumptions for SARS-CoV-2, recent data on new variants included, we show that the difference between prime-boost and single-shot waning rates is the only discriminative threshold, falling in the narrow range of 0.01-0.02 day-1 below which single-dose vaccination should be considered.

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.

Strategy equilibrium in dilemma games with off-diagonal payoff perturbations

We analyze the strategy equilibrium of dilemma games considering a payoff matrix affected by small and random perturbations on the off-diagonal. Notably, a recent work [Proc. R. Soc. A 476, 20200116 (2020)1364-502110.1098/rspa.2020.0116] reported that while cooperation is sustained by perturbations acting on the main diagonal, a less clear scenario emerges when perturbations act on the off-diagonal. Thus, the second case represents the core of this investigation, aimed at completing the description of the effects that payoff perturbations have on the dynamics of evolutionary games. Our results, achieved by analyzing the proposed model under a variety of configurations as different update rules, suggest that off-diagonal perturbations actually constitute a nontrivial form of noise. In particular, the most interesting effects are detected near the phase transition, as perturbations tend to move the strategy distribution towards nonordered states of equilibrium, supporting cooperation when defection is pervading the population, and supporting defection in the opposite case. To conclude, we identified a form of noise that, under controlled conditions, could be used to enhance cooperation and greatly delay its extinction.

Heterogeneity in evolutionary games: an analysis of the risk perception

In this study, we analyse the relationship between heterogeneity and cooperation. Previous investigations suggest that this relation is non-trivial, as some authors found that heterogeneity sustains cooperation, while others obtained different results. Among the possible forms of heterogeneity, we focus on the individual perception of risks and rewards related to a generic event, which can appear in a number of social and biological systems. The modelling approach is based on the framework of evolutionary game theory. To represent this kind of heterogeneity, we implement small and local perturbations on the pay-off matrix of simple two-strategy games, such as the Prisoner's Dilemma. So, while usually the pay-off is considered to be a global and time-invariant structure, i.e. it is the same for all individuals of a population at any time, in our model its value is continuously affected by small variations, in both time and space (i.e. position on a lattice). We found that such perturbations can be beneficial or detrimental to cooperation, depending on their setting. Notably, cooperation is strongly supported when perturbations act on the main diagonal of the pay-off matrix, whereas when they act on the off-diagonal the resulting effect is more difficult to quantify. To conclude, the proposed model shows a rich spectrum of possible equilibria, whose interpretation might offer insights and enrich the description of several systems.

Adaptive Bet-Hedging Revisited: Considerations of Risk and Time Horizon

Models of adaptive bet-hedging commonly adopt insights from Kelly’s famous work on optimal gambling strategies and the financial value of information. In particular, such models seek evolutionary solutions that maximize long-term average growth rate of lineages, even in the face of highly stochastic growth trajectories. Here, we argue for extensive departures from the standard approach to better account for evolutionary contingencies. Crucially, we incorporate considerations of volatility minimization, motivated by interim extinction risk in finite populations, within a finite time horizon approach to growth maximization. We find that a game-theoretic competitive optimality approach best captures these additional constraints and derive the equilibria solutions under straightforward fitness payoff functions and extinction risks. We show that for both maximal growth and minimal time relative payoffs, the log-optimal strategy is a unique pure strategy symmetric equilibrium, invariant with evolutionary time horizon and robust to low extinction risks.

Weak selection can filter environmental noise in the evolution of animal behavior

Weak selection is an important assumption in theoretical evolutionary biology, but its biological significance remains unclear. In this study, we investigate the effect of weak selection on stochastic evolutionary stability in a two-phenotype evolutionary game dynamics with a random payoff matrix assuming an infinite, well-mixed population undergoing discrete, nonoverlapping generations. We show that, under weak selection, both stochastic local stability and stochastic evolutionary stability in this system depend on the means of the random payoffs but not on their variances. Moreover, although stochastic local stability or instability of an equilibrium may not depend on environmental noise if selection is weak enough, the growth rate near an equilibrium not only depends on environmental noise, but can even be enhanced by environmental noise if selection is weak. This is the case, for instance, when the variances of the random payoffs as well as the covariances are equal. These results suggest that natural selection could be able to filter (or resist) the effect of environmental noise on the evolution of animal behavior if selection is weak.

Cooperation dynamics in networked geometric Brownian motion

Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing a geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e., is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.