Universality of weak selection

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.

Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

Evolutionary dynamics on sequential temporal networks

Population structure is a well-known catalyst for the evolution of cooperation and has traditionally been considered to be static in the course of evolution. Conversely, real-world populations, such as microbiome communities and online social networks, frequently show a progression from tiny, active groups to huge, stable communities, which is insufficient to be captured by constant structures. Here, we propose sequential temporal networks to characterize growing networked populations, and we extend the theory of evolutionary games to these temporal networks with arbitrary structures and growth rules. We derive analytical rules under which a sequential temporal network has a higher fixation probability for cooperation than its static counterpart. Under neutral drift, the rule is simply a function of the increment of nodes and edges in each time step. But if the selection is weak, the rule is related to coalescence times on networks. In this case, we propose a mean-field approximation to calculate fixation probabilities and critical benefit-to-cost ratios with lower calculation complexity. Numerical simulations in empirical datasets also prove the cooperation-promoting effect of population growth. Our research stresses the significance of population growth in the real world and provides a high-accuracy approximation approach for analyzing the evolution in real-life systems.

The effect of environmental information on evolution of cooperation in stochastic games

Many human interactions feature the characteristics of social dilemmas where individual actions have consequences for the group and the environment. The feedback between behavior and environment can be studied with the framework of stochastic games. In stochastic games, the state of the environment can change, depending on the choices made by group members. Past work suggests that such feedback can reinforce cooperative behaviors. In particular, cooperation can evolve in stochastic games even if it is infeasible in each separate repeated game. In stochastic games, participants have an interest in conditioning their strategies on the state of the environment. Yet in many applications, precise information about the state could be scarce. Here, we study how the availability of information (or lack thereof) shapes evolution of cooperation. Already for simple examples of two state games we find surprising effects. In some cases, cooperation is only possible if there is precise information about the state of the environment. In other cases, cooperation is most abundant when there is no information about the state of the environment. We systematically analyze all stochastic games of a given complexity class, to determine when receiving information about the environment is better, neutral, or worse for evolution of cooperation.

Weak selection and stochastic evolutionary stability in a stochastic replicator dynamics

It is a very challenging problem whether natural selection is able to effectively resist the continuous disturbance of environmental noise such that the direction or outcome of evolution determined by the deterministic selection pressure will not be changed. By analyzing the impact of weak selection on the evolutionary stability of a stochastic replicator dynamics with n possible pure strategies, we found that the weak selection is able to enhance the evolutionary stability, that is, under weak selection, the stochastic evolutionary stability of the system is determined by the mean payoff matrix. This finding strongly implies that the weak selection should be regarded as an important mechanism to ensure evolutionary stability in stochastic environments.

Evolutionary dynamics under partner preferences

The fact that people often have preference rankings for their partners is a distinctive aspect of human behavior. Little is known, however, about how this talent as a powerful force shapes human behavioral traits, including those which should not have been favored by selection, such as cooperation in social dilemma situations. Here we propose a dynamic model in which network-structured individuals can switch their interaction partners within neighborhoods based on their preferences. For the partner switching, we propose two interruption regimes: dictatorial regime and negotiating regime. In the dictatorial regime, focal individuals are able to suspend interactions out of preferences unilaterally. In the negotiating regime, either focal individuals or the associated partners agree to suspend, then these interactions can be successfully suspended. We investigate the evolution of cooperation under both preference-driven partner switching regimes in the context of both the weakened variant of the donation game and the standard one. Specifically, we theoretically approximate the critical conditions for cooperation to be favored by weak selection in the weakened donation game where cooperators bear a unit cost to provide a benefit for each active neighbor and simulate the evolutionary dynamics of cooperation in the standard donation game to test the robustness of the analytical results. Under dictatorial regime, selection of cooperation becomes harder when individuals have preferences for either cooperator or defector partners, implying that the expulsion of defectors by cooperators is overwhelmed by the chasing of defectors towards cooperators. Under negotiating regime, both preferences for cooperator and defector partners can significantly favor the evolution of cooperation, yet underlying mechanisms differ greatly. For preferences over cooperator partners, cooperator-cooperator interaction relationships are reinforced and the associated mutual reciprocity can resist and assimilate defectors. For preferences over defector partners, defector-defector interaction relationships are anchored, weakening defectors' exploitation over cooperators. Cooperators are thus offered much time space to interact among cospecies and spread. Our work may help better understand the critical role of preference-based adaptive partner switching in promoting the evolution of cooperation.

Further Quantifying the Niche-Neutral Continuum of Human Digestive Tract Microbiomes with Near Neutral Model and Stochasticity Analysis

It is postulated that the human digestive tract (DT) from mouth to intestine is differentiated into diverse niches. For example, Segata et al. discovered that the microbiomes of diverse habitats along the DT could be distinguished as 4 types (niches) including (i) stool; (ii) sub-gingival plaques (SubP) and supra-gingival plaques (SupP); (iii) tongue dorsum (TD), throat (TH), palatine tonsils (PT), and saliva (Sal); and (iv) hard palate (HP) and buccal mucosa (BM), and keratinized gingiva (KG). These niches are different not only in composition, but also in metabolic potentials. In a previous study, we applied Harris et al's multi-site neutral and Tang and Zhou's niche-neutral hybrid models to characterize the DT niches discovered by Segata et al. Here, we complement the previous study by applying Sloan's near-neural model and Ning et al's stochasticity analysis framework to quantify the niche-neutral continuum of the DT microbiome distribution to shed light on the possible ecological/evolutionary mechanism that shapes the continuum. Overall but excluding the stool site, the proportion of neutral OTUs (46%) is slightly higher than that of the positive selection (38%), but significantly higher than negative selection (15%). The gut (stool) exhibited 3 to 12 times lower neutrality than other DT sites. The analysis also cross-verified our previous hypothesis that the KG (keratinized gingiva) is of distinct assembly dynamics in the DT microbiome, should be treated as a fifth niche. Our findings offer new insight on the long-standing debate concerning whether a minimum of 2-mm of KG width is necessary for marginal periodontal health.

Coevolution of vaccination behavior and perceived vaccination risk can lead to a stag-hunt-like game

Voluntary vaccination is effective to prevent infectious diseases from spreading. Both vaccination behavior and cognition of the vaccination risk play important roles in individual vaccination decision making. However, it is not clear how the coevolution of the two shapes population-wide vaccination behavior. We establish a coupled dynamics of epidemic, vaccination behavior, and perceived vaccination risk with three different time scales. We assume that the increase of vaccination level inhibits the rise of perceived vaccination risk, and the increase of perceived vaccination risk inhibits the rise of vaccination level. It is shown that the resulting vaccination behavior is similar to the stag-hunt game, provided that the basic reproductive ratio is moderate and that the epidemic dynamics evolves sufficiently fast. This is in contrast with the previous view that vaccination is a snowdriftlike game. And we find that epidemic breaks out repeatedly and eventually leads to vaccine scares if these three dynamics evolve on a similar time scale. Furthermore, we propose some ways to promote vaccination behavior, such as controlling side-effect bias and perceived vaccination costs. Our work sheds light on epidemic control via vaccination by taking into account the coevolutionary dynamics of cognition and behavior.

Evolutionary stability is sensitive on the conflict between reproduction and survival: proofs

Evolutionary game theory is a powerful mathematical framework to study how phenotypes evolve by natural selection. Both birth and death are key in classic models in evolutionary games. The conflict between the two is fundamental in life history theory. The conflict between birth and death has been shown to change the evolutionary outcome for continuous traits. However, it is not clear how the conflict reshapes the evolutionary outcome for discrete strategies. An allocation model is proposed, in which part of the payoff is mapped to reproduction and the rest is mapped to illness. For non-evolving allocation, it is proved that the allocation does not change the fixation probability if and only if the illness is an inverse exponential function and the product of reproduction function and illness function is a constant. The necessary and sufficient condition implies that the allocation dramatically alters the evolutionary stability for a wide class of evolutionary processes. This is also verified by alternative construction proofs and numerical examples. Furthermore, the illness and reproduction function also ensures that every allocation is a neutral stable regime, if the allocation evolves to maximize the invasion probability. A deviation can lead to a non-trivial evolutionary branching. These results explicitly show that the reproduction and illness functions are restrictive to ensure the invariance of evolutionary outcome. Thus it implies that the demographic and life history need to be considered together to understand patterns of evolutionary dynamics.

The role of evolutionary game theory in spatial and non-spatial models of the survival of cooperation in cancer: a review

Evolutionary game theory (EGT) is a branch of mathematics which considers populations of individuals interacting with each other to receive pay-offs. An individual’s pay-off is dependent on the strategy of its opponent(s) as well as on its own, and the higher its pay-off, the higher its reproductive fitness. Its offspring generally inherit its interaction strategy, subject to random mutation. Over time, the composition of the population shifts as different strategies spread or are driven extinct. In the last 25 years there has been a flood of interest in applying EGT to cancer modelling, with the aim of explaining how cancerous mutations spread through healthy tissue and how intercellular cooperation persists in tumour-cell populations. This review traces this body of work from theoretical analyses of well-mixed infinite populations through to more realistic spatial models of the development of cooperation between epithelial cells. We also consider work in which EGT has been used to make experimental predictions about the evolution of cancer, and discuss work that remains to be done before EGT can make large-scale contributions to clinical treatment and patient outcomes.

Weak Selection and the Separation of Eco-evo Time Scales using Perturbation Analysis

We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka-Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for [Formula: see text] and 2, that the error scales according to [Formula: see text], where [Formula: see text] is the strength of selection and k is the order of the approximation.

Evolution of cooperation with contextualized behavior

How do networks of social interaction govern the emergence and stability of prosocial behavior? Theoretical studies of this question typically assume unconditional behavior, meaning that an individual either cooperates with all opponents or defects against all opponents-an assumption that produces a pessimistic outlook for the evolution of cooperation, especially in highly connected populations. Although these models may be appropriate for simple organisms, humans have sophisticated cognitive abilities that allow them to distinguish between opponents and social contexts, so they can condition their behavior on the identity of opponents. Here, we study the evolution of cooperation when behavior is conditioned by social context, but behaviors can spill over between contexts. Our mathematical analysis shows that contextualized behavior rescues cooperation across a broad range of population structures, even when the number of social contexts is small. Increasing the number of social contexts further promotes cooperation by orders of magnitude.

Evolution of cooperation with asymmetric social interactions

How cooperation emerges in human societies is both an evolutionary enigma and a practical problem with tangible implications for societal health. Population structure has long been recognized as a catalyst for cooperation because local interactions facilitate reciprocity. Analysis of population structure typically assumes bidirectional social interactions. But human social interactions are often unidirectional-where one individual has the opportunity to contribute altruistically to another, but not conversely-as the result of organizational hierarchies, social stratification, popularity effects, and endogenous mechanisms of network growth. Here we expand the theory of cooperation in structured populations to account for both uni- and bidirectional social interactions. Even though unidirectional interactions remove the opportunity for reciprocity, we find that cooperation can nonetheless be favored in directed social networks and that cooperation is provably maximized for networks with an intermediate proportion of unidirectional interactions, as observed in many empirical settings. We also identify two simple structural motifs that allow efficient modification of interaction directions to promote cooperation by orders of magnitude. We discuss how our results relate to the concepts of generalized and indirect reciprocity.

Evolution of prosocial behaviours in multilayer populations

Human societies include diverse social relationships. Friends, family, business colleagues and online contacts can all contribute to one's social life. Individuals may behave differently in different domains, but success in one domain may engender success in another. Here, we study this problem using multilayer networks to model multiple domains of social interactions, in which individuals experience different environments and may express different behaviours. We provide a mathematical analysis and find that coupling between layers tends to promote prosocial behaviour. Even if prosociality is disfavoured in each layer alone, multilayer coupling can promote its proliferation in all layers simultaneously. We apply this analysis to six real-world multilayer networks, ranging from the socio-emotional and professional relationships in a Zambian community, to the online and offline relationships within an academic university. We discuss the implications of our results, which suggest that small modifications to interactions in one domain may catalyse prosociality in a different domain.

Aspiration dynamics generate robust predictions in heterogeneous populations

Update rules, which describe how individuals adjust their behavior over time, affect the outcome of social interactions. Theoretical studies have shown that evolutionary outcomes are sensitive to model details when update rules are imitation-based but are robust when update rules are self-evaluation based. However, studies of self-evaluation based rules have focused on homogeneous population structures where each individual has the same number of neighbors. Here, we consider heterogeneous population structures represented by weighted networks. Under weak selection, we analytically derive the condition for strategy success, which coincides with the classical condition of risk-dominance. This condition holds for all weighted networks and distributions of aspiration levels, and for individualized ways of self-evaluation. Our findings recover previous results as special cases and demonstrate the universality of the robustness property under self-evaluation based rules. Our work thus sheds light on the intrinsic difference between evolutionary dynamics under self-evaluation based and imitation-based update rules.

Social interaction outcomes can depend on the type of information individuals possess and how it is used in decision-making. Here, Zhou et al. find that self-evaluation based decision-making rules lead to evolutionary outcomes that are robust to different population structures and ways of self-evaluation.

Role of synergy and antagonism in designing multidrug adaptive chemotherapy schedules

Chemotherapeutic resistance via the mechanism of competitive release of resistant tumor cell subpopulations is a major problem associated with cancer treatments and one of the main causes of tumor recurrence. Often, chemoresistance is mitigated by using multidrug schedules (two or more combination therapies) that can act synergistically, additively, or antagonistically on the heterogeneous population of cells as they evolve. In this paper, we develop a three-component evolutionary game theory model to design two-drug adaptive schedules that mitigate chemoresistance and delay tumor recurrence in an evolving collection of tumor cells with two resistant subpopulations and one chemosensitive population that has a higher baseline fitness but is not resistant to either drug. Using the nonlinear replicator dynamical system with a payoff matrix of Prisoner's Dilemma (PD) type (enforcing a cost to resistance), we investigate the nonlinear dynamics of this three-component system along with an additional tumor growth model whose growth rate is a function of the fitness landscape of the tumor cell populations. A key parameter determines whether the two drugs interact synergistically, additively, or antagonistically. We show that antagonistic drug interactions generally result in slower rates of adaptation of the resistant cells than synergistic ones, making them more effective in combating the evolution of resistance. We then design evolutionary cycles (closed loops) in the three-component phase space by shaping the fitness landscape of the cell populations (i.e., altering the evolutionary stable states of the game) using appropriately designed time-dependent schedules (adaptive therapy), altering the dosages and timing of the two drugs. We describe two key bifurcations associated with our drug interaction parameter which help explain why antagonistic interactions are more effective at controlling competitive release of the resistant population than synergistic interactions in the context of an evolving tumor.

Stochastic evolutionary dynamics of trust games with asymmetric parameters

Trusting in others and reciprocating that trust with trustworthy actions are crucial to successful and prosperous societies. The trust game has been widely used to quantitatively study trust and trustworthiness, involving a sequential exchange between an investor and a trustee. Deterministic evolutionary game theory predicts no trust and no trustworthiness, whereas the behavioral experiments with the one-shot anonymous trust game show that people substantially trust and respond trustworthily. To explain these discrepancies, previous works often turn to additional mechanisms, which are borrowed from other games such as the prisoner's dilemma. Although these mechanisms lead to the evolution of trust and trustworthiness to an extent, the optimal or the most common strategy often involves no trustworthiness. In this paper, we study the impact of asymmetric demographic parameters (e.g., different population sizes) on game dynamics of the trust game. We show that, in a weak-mutation limit, stochastic evolutionary dynamics with the asymmetric parameters can lead to the evolution of high trust and high trustworthiness without any additional mechanisms in well-mixed finite populations. Even full trust and near full trustworthiness can be the most common strategies. These results are qualitatively different from those of the previous works. Our results thereby demonstrate rich evolutionary dynamics of the asymmetric trust game.

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.

Evolutionary dynamics with game transitions

The environment has a strong influence on a population's evolutionary dynamics. Driven by both intrinsic and external factors, the environment is subject to continual change in nature. To capture an ever-changing environment, we consider a model of evolutionary dynamics with game transitions, where individuals' behaviors together with the games that they play in one time step influence the games to be played in the next time step. Within this model, we study the evolution of cooperation in structured populations and find a simple rule: Weak selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, b, to the corresponding cost, c, exceeds [Formula: see text], where k is the average number of neighbors of an individual and [Formula: see text] captures the effects of the game transitions. Even if cooperation cannot be favored in each individual game, allowing for a transition to a relatively valuable game after mutual cooperation and to a less valuable game after defection can result in a favorable outcome for cooperation. In particular, small variations in different games being played can promote cooperation markedly. Our results suggest that simple game transitions can serve as a mechanism for supporting prosocial behaviors in highly connected populations.

Evolutionary game dynamics of combining the imitation and aspiration-driven update rules

So far, most studies on evolutionary game dynamics in finite populations have concentrated on a single update rule. However, given the impacts of the environment and individual cognition, individuals may use different update rules to change their current strategies. In light of this, the current paper reports on a study that constructed a mixed stochastic evolutionary game dynamic by combining the imitation and aspiration-driven update processes. The target was to clarify the influences of the aspiration-driven process on the evolution of the level of cooperation by considering the behavior of a population in which individuals have two strategies available: cooperation and defection. Through a numerical analysis of unstructured populations and simulation analyses of structured populations and of the random-matching model, the following results were found. First, the mean fraction of cooperators varied alongside the probability with which the individual adopted the aspiration-driven update rule. In the Prisoner's Dilemma and coexistence games, the aspiration-driven update process promoted cooperation in the well-mixed population but inhibited it in structured ones and the random-matching model; however, in the coordination game, the aspiration-driven update process was seen to exert the opposite effect on cooperation by inhibiting the latter in a homogeneously mixed population but promoting it in structured ones and in the random-matching model. Second, the mean fraction of cooperators changed with the aspiration level in the differently structured populations and random-matching model, and there appeared a phase transition point. Third, the evolutionary characteristics of the mean fraction of cooperators maintained robustness in the differently structured populations and random-matching model. These results extend evolutionary game theory.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

Evolutionary multiplayer games on graphs with edge diversity

Evolutionary game dynamics in structured populations has been extensively explored in past decades. However, most previous studies assume that payoffs of individuals are fully determined by the strategic behaviors of interacting parties, and social ties between them only serve as the indicator of the existence of interactions. This assumption neglects important information carried by inter-personal social ties such as genetic similarity, geographic proximity, and social closeness, which may crucially affect the outcome of interactions. To model these situations, we present a framework of evolutionary multiplayer games on graphs with edge diversity, where different types of edges describe diverse social ties. Strategic behaviors together with social ties determine the resulting payoffs of interactants. Under weak selection, we provide a general formula to predict the success of one behavior over the other. We apply this formula to various examples which cannot be dealt with using previous models, including the division of labor and relationship- or edge-dependent games. We find that labor division can promote collective cooperation markedly. The evolutionary process based on relationship-dependent games can be approximated by interactions under a transformed and unified game. Our work stresses the importance of social ties and provides effective methods to reduce the calculating complexity in analyzing the evolution of realistic systems.

Evolutionary game dynamics of the Wright-Fisher process with different selection intensities

Evolutionary game dynamics in finite populations can be described by a frequency-dependent, stochastic Wright-Fisher process. The fitness of individuals in a population is not only linked to environmental conditions but also tightly coupled to the types and frequencies of competitors, leading to different types of individuals with different selection intensities. We studied a 2 × 2 symmetric game in a finite population and established a dynamic model of the Wright-Fisher process by introducing different selection intensities for different strategies. Thus, we provided another effective way to study the evolutionary dynamics of a finite population and obtained the analytical expressions of fixation probabilities under weak selection. The fixation probability of a strategy is not only related to a game matrix but also to different selection intensities. The conditions required for natural selection to favor one strategy and for that strategy to be an evolutionary stable strategy (ESS_N) are specified in our model. We compared our results with those of a Moran dynamic process with different selection intensities to explore these two processes better. In the two processes, the conditions conducive to the strategy's taking fixation are the same. By simulation analysis, the dynamic relationships between the fixation probabilities and selection intensities were intuitively observed in the prisoner's dilemma, coordination, and coexistence games. The fixation probability of the cooperative strategy in the prisoner's dilemma decreases with the increase of its own selection intensity. In the coexistence and coordination games, the fixation probability of the cooperative strategy increases with its own selection intensity. For the three types of games, the fixation probability of the cooperative strategy decreases with the increase of the selection intensity of the defection strategy.

Stochastic dynamics in the fitness-based process which can be on behalf of the standard Moran, local and Wright-Fisher processes

In this paper, we proposed a frequency dependent fitness-based process, which is an extension of both the standard Moran process and the Wright-Fisher process. Some individuals are selected into a parent's pool and reproduce. Then the offspring is selected to replace individuals in the entire parent generation. We explored the influence of the size of parent pool and the number of offspring on a single cooperator's fixation. The less offspring leads to higher fixation probability of s single cooperator. Meanwhile, the fixation probability decreases with the growth of the local level. In other words, the direction of the number of offspring's impact on fixation probability is in accordance with that of the local level's impact. The less offspring in one generation or the smaller parent's pool contributes to promoting cooperation with the fitness-based updating rule.

A two-player iterated survival game

We describe an iterated game between two players, in which the payoff is to survive a number of steps. Expected payoffs are probabilities of survival. A key feature of the game is that individuals have to survive on their own if their partner dies. We consider individuals with hardwired, unconditional behaviors or strategies. When both players are present, each step is a symmetric two-player game. The overall survival of the two individuals forms a Markov chain. As the number of iterations tends to infinity, all probabilities of survival decrease to zero. We obtain general, analytical results for n-step payoffs and use these to describe how the game changes as n increases. In order to predict changes in the frequency of a cooperative strategy over time, we embed the survival game in three different models of a large, well-mixed population. Two of these models are deterministic and one is stochastic. Offspring receive their parent's type without modification and fitnesses are determined by the game. Increasing the number of iterations changes the prospects for cooperation. All models become neutral in the limit (n→∞). Further, if pairs of cooperative individuals survive together with high probability, specifically higher than for any other pair and for either type when it is alone, then cooperation becomes favored if the number of iterations is large enough. This holds regardless of the structure of pairwise interactions in a single step. Even if the single-step interaction is a Prisoner's Dilemma, the cooperative type becomes favored. Enhanced survival is crucial in these iterated evolutionary games: if players in pairs start the game with a fitness deficit relative to lone individuals, the prospects for cooperation can become even worse than in the case of a single-step game.

Individualised aspiration dynamics: Calculation by proofs

Cooperation is key for the evolution of biological systems ranging from bacteria communities to human societies. Evolutionary processes can dramatically alter the cooperation level. Evolutionary processes are typically of two classes: comparison based and self-evaluation based. The fate of cooperation is extremely sensitive to the details of comparison based processes. For self-evaluation processes, however, it is still unclear whether the sensitivity remains. We concentrate on a class of self-evaluation processes based on aspiration, where all the individuals adjust behaviors based on their own aspirations. We prove that the evolutionary outcome with heterogeneous aspirations is the same as that of the homogeneous one for regular networks under weak selection limit. Simulation results further suggest that it is also valid for general networks across various distributions of personalised aspirations. Our result clearly indicates that self-evaluation processes are robust in contrast with comparison based rules. In addition, our result greatly simplifies the calculation of the aspiration dynamics, which is computationally expensive.

Cooperation is the cornerstone to understand how biological systems evolve. Previous studies have shown that cooperation is sensitive to the details of evolutionary processes, even if all the individuals update strategies in the same way. Here we propose a class of updating rules driven by self-evaluation, where each individual has its personal aspiration. The evolutionary outcome is the same as if all the individuals adopt the same aspiration for regular networks, provided the selection intensity is weak enough. In addition, we provide a simple numerical method to identify the favored strategy. Our result shows a very robust class of strategy updating rules. And it implies that complexity in updating rules does not necessarily lead to the sensitivity of evolutionary outcomes.

Extrapolating weak selection in evolutionary games

This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes to fixation before the next one occurs. It has been shown that for [Formula: see text] games the ranking of the strategies does not change as strength of selection is increased (Wu et al. in Phys Rev 82:046106, 2010). The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu et al. (2013) did numerical computations for a fixed population size N. Here, we will instead let the strength of selection [Formula: see text] where c is fixed and let [Formula: see text] to obtain formulas for the invadability probabilities [Formula: see text] that determine the rankings. These formulas, which are integrals on [0, 1], are intractable calculus problems, but can be easily evaluated numerically. Here, we use them to derive simple formulas for the ranking order when c is small or c is large.

Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics

In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk-Dove example.

Public goods games in populations with fluctuating size

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.

Promotion of cooperation in evolutionary game dynamics with local information

In this paper, we propose a strategy-updating rule driven by local information, which is called Local process. Unlike the standard Moran process, the Local process does not require global information about the strategic environment. By analyzing the dynamical behavior of the system, we explore how the local information influences the fixation of cooperation in two-player evolutionary games. Under weak selection, the decreasing local information leads to an increase of the fixation probability when natural selection does not favor cooperation replacing defection. In the limit of sufficiently large selection, the analytical results indicate that the fixation probability increases with the decrease of the local information, irrespective of the evolutionary games. Furthermore, for the dominance of defection games under weak selection and for coexistence games, the decreasing of local information will lead to a speedup of a single cooperator taking over the population. Overall, to some extent, the local information is conducive to promoting the cooperation.

Voluntary vaccination dilemma with evolving psychological perceptions

Voluntary vaccination is a universal control protocol for infectious diseases. Yet there exists a social dilemma between individual benefits and public health: non-vaccinators free ride via the herd immunity from adequate vaccinators who bear vaccination cost. This is due to the interplay between disease prevalence and individual vaccinating behavior. To complicate matters further, individual vaccinating behavior depends on the perceived vaccination cost rather than the actual one. The perception of vaccination cost is an individual trait, which varies from person to person, and evolves in response to the disease prevalence and vaccination coverage. To explore how evolving perception shapes individual vaccinating behavior and thus the vaccination dynamics, we provide a model combining epidemic dynamics with evolutionary game theory which captures the voluntary vaccination dilemma. In particular, individuals adjust their perception based on the inertia effect in psychology and then update their vaccinating behavior through imitating the behavior of a more successful peer. We find that i) vaccination is acceptable when the expected vaccination cost considering perception and actual vaccination cost is less than the maximum of the expected non-vaccination cost; ii) the evolution of perception is a "double-edged sword" for vaccination dynamics: it can improve vaccination coverage when most individuals perceive exaggerated vaccination cost, and it inhibits vaccination coverage in the other cases.

Evolving synergetic interactions

Cooperators forgo their own interests to benefit others. This reduces their fitness and thus cooperators are not likely to spread based on natural selection. Nonetheless, cooperation is widespread on every level of biological organization ranging from bacterial communities to human society. Mathematical models can help to explain under which circumstances cooperation evolves. Evolutionary game theory is a powerful mathematical tool to depict the interactions between cooperators and defectors. Classical models typically involve either pairwise interactions between individuals or a linear superposition of these interactions. For interactions within groups, however, synergetic effects may arise: their outcome is not just the sum of its parts. This is because the payoffs via a single group interaction can be different from the sum of any collection of two-player interactions. Assuming that all interactions start from pairs, how can such synergetic multiplayer games emerge from simpler pairwise interactions? Here, we present a mathematical model that captures the transition from pairwise interactions to synergetic multiplayer ones. We assume that different social groups have different breaking rates. We show that non-uniform breaking rates do foster the emergence of synergy, even though individuals always interact in pairs. Our work sheds new light on the mechanisms underlying such synergetic interactions.

Extreme Selection Unifies Evolutionary Game Dynamics in Finite and Infinite Populations

We show that when selection is extreme-the fittest strategy always reproduces or is imitated-the unequivalence between the possible evolutionary game scenarios in finite and infinite populations resolves, in the sense that the three generic outcomes-dominance, coexistence, and mutual exclusion-emerge in well-mixed populations of any size. We consider the simplest setting of a 2-player-2-strategy symmetric game and the two most common microscopic definitions of strategy spreading-the frequency-dependent Moran process and the imitation process by pairwise comparison-both in the case allowing any intensity of selection. We show that of the seven different invasion and fixation scenarios that are generically possible in finite populations-fixation being more or less likely to occur and rapid compared to the neutral game-the three that are possible in large populations are the same three that occur for sufficiently strong selection: (1) invasion and fast fixation of one strategy; (2) mutual invasion and slow fixation of one strategy; (3) no invasion and no fixation. Moreover (and interestingly), in the limit of extreme selection 2 becomes mutual invasion and no fixation, a case not possible for finite intensity of selection that better corresponds to the deterministic case of coexistence. In the extreme selection limit, we also derive the large population deterministic limit of the two considered stochastic processes.

Evolutionary games on cycles with strong selection

Evolutionary games on graphs describe how strategic interactions and population structure determine evolutionary success, quantified by the probability that a single mutant takes over a population. Graph structures, compared to the well-mixed case, can act as amplifiers or suppressors of selection by increasing or decreasing the fixation probability of a beneficial mutant. Properties of the associated mean fixation times can be more intricate, especially when selection is strong. The intuition is that fixation of a beneficial mutant happens fast in a dominance game, that fixation takes very long in a coexistence game, and that strong selection eliminates demographic noise. Here we show that these intuitions can be misleading in structured populations. We analyze mean fixation times on the cycle graph under strong frequency-dependent selection for two different microscopic evolutionary update rules (death-birth and birth-death). We establish exact analytical results for fixation times under strong selection and show that there are coexistence games in which fixation occurs in time polynomial in population size. Depending on the underlying game, we observe inherence of demographic noise even under strong selection if the process is driven by random death before selection for birth of an offspring (death-birth update). In contrast, if selection for an offspring occurs before random removal (birth-death update), then strong selection can remove demographic noise almost entirely.

Fixation Probabilities for Any Configuration of Two Strategies on Regular Graphs

Population structure and spatial heterogeneity are integral components of evolutionary dynamics, in general, and of evolution of cooperation, in particular. Structure can promote the emergence of cooperation in some populations and suppress it in others. Here, we provide results for weak selection to favor cooperation on regular graphs for any configuration, meaning any arrangement of cooperators and defectors. Our results extend previous work on fixation probabilities of rare mutants. We find that for any configuration cooperation is never favored for birth-death (BD) updating. In contrast, for death-birth (DB) updating, we derive a simple, computationally tractable formula for weak selection to favor cooperation when starting from any configuration containing any number of cooperators. This formula elucidates two important features: (i) the takeover of cooperation can be enhanced by the strategic placement of cooperators and (ii) adding more cooperators to a configuration can sometimes suppress the evolution of cooperation. These findings give a formal account for how selection acts on all transient states that appear in evolutionary trajectories. They also inform the strategic design of initial states in social networks to maximally promote cooperation. We also derive general results that characterize the interaction of any two strategies, not only cooperation and defection.

Impact of migration on the multi-strategy selection in finite group-structured populations

For large quantities of spatial models, the multi-strategy selection under weak selection is the sum of two competition terms: the pairwise competition and the competition of multiple strategies with equal frequency. Two parameters σ₁ and σ₂ quantify the dependence of the multi-strategy selection on these two terms, respectively. Unlike previous studies, we here do not require large populations for calculating σ₁ and σ₂, and perform the first quantitative analysis of the effect of migration on them in group-structured populations of any finite sizes. The Moran and the Wright-Fisher process have the following common findings. Compared with well-mixed populations, migration causes σ₁ to change with the mutation probability from a decreasing curve to an inverted U-shaped curve and maintains the increase of σ₂. Migration (probability and range) leads to a significant change of σ₁ but a negligible one of σ₂. The way that migration changes σ₁ is qualitatively similar to its influence on the single parameter characterizing the two-strategy selection. The Moran process is more effective in increasing σ₁ for most migration probabilities and the Wright-Fisher process is always more effective in increasing σ₂. Finally, our findings are used to study the evolution of cooperation under direct reciprocity.

Fixation probabilities of evolutionary coordination games on two coupled populations

Evolutionary forces resulted from competitions between different populations are common, which change the evolutionary behavior of a single population. In an isolated population of coordination games of two strategies (e.g., s_{1} and s_{2}), the previous studies focused on determining the fixation probability that the system is occupied by only one strategy (s_{1}) and their expectation times, given an initial mixture of two strategies. In this work, we propose a model of two interdependent populations, disclosing the effects of the interaction strength on fixation probabilities. In the well-mixing limit, a detailed linear stability analysis is performed, which allows us to find and to classify the different equilibria, yielding a clear picture of the bifurcation patterns in phase space. We demonstrate that the interactions between populations crucially alter the dynamic behavior. More specifically, if the coupling strength is larger than some threshold value, the critical initial density of one strategy (s_{1}) that corresponds to fixation is significantly delayed. Instead, the two populations evolve to the opposite state of all (s_{2}) strategy, which are in favor of the red queen hypothesis. We delineate the extinction time of strategy (s_{1}) explicitly, which is an exponential form. These results are validated by systematic numerical simulations.

Reputation-Based Investment Helps to Optimize Group Behaviors in Spatial Lattice Networks

Encouraging cooperation among selfish individuals is crucial in many real-world systems, where individuals’ collective behaviors can be analyzed using evolutionary public goods game. Along this line, extensive studies have shown that reputation is an effective mechanism to investigate the evolution of cooperation. In most existing studies, participating individuals in a public goods game are assumed to contribute unconditionally into the public pool, or they can choose partners based on a common reputation standard (e.g., preferences or characters). However, to assign one reputation standard for all individuals is impractical in many real-world deployment. In this paper, we introduce a reputation tolerance mechanism that allows an individual to select its potential partners and decide whether or not to contribute an investment to the public pool based on its tolerance to other individuals’ reputation. Specifically, an individual takes part in a public goods game only if the number of participants with higher reputation exceeds the value of its tolerance. Moreover, in this paper, an individual’s reputation can increase or decrease in a bounded interval based on its historical behaviors. We explore the principle that how the reputation tolerance and conditional investment mechanisms can affect the evolution of cooperation in spatial lattice networks. Our simulation results demonstrate that a larger tolerance value can achieve an environment that promote the cooperation of participants.

Aspiration promotes cooperation in the prisoner's dilemma game with the imitation rule

A model of stochastic evolutionary game dynamics with finite population of size N+M was built. Among these individuals, N individuals update strategies with aspiration updating, while the other M individuals update strategies with imitation updating. In the proposed model, we obtain the expression of the mean fraction of cooperators and analyze some concrete cases. Compared with the standard imitation dynamics, there is always a positive probability to support the formation of cooperation in the system with the aspiration and imitation rules. Moreover, the numerical results indicate that more aspiration-driven individuals lead to a higher mean fraction of imitation-driven cooperators, which means the invasion of the aspiration-driven individuals is conducive to promoting the cooperation of the imitation-driven individuals.

Speed of evolution on graphs

The likelihood that a mutant fixates in the wild population, i.e., fixation probability, has been intensively studied in evolutionary game theory, where individuals' fitness is frequency dependent. However, it is of limited interest when it takes long to take over. Thus the speed of evolution becomes an important issue. In general, it is still unclear how fixation times are affected by the population structure, although the fixation times have already been addressed in the well-mixed populations. Here we theoretically address this issue by pair approximation and diffusion approximation on regular graphs. It is shown (i) that under neutral selection, both unconditional and conditional fixation time are shortened by increasing the number of neighbors; (ii) that under weak selection, for the simplified prisoner's dilemma game, if benefit-to-cost ratio exceeds the degree of the graph, then the unconditional fixation time of a single cooperator is slower than that in the neutral case; and (iii) that under weak selection, for the conditional fixation time, limited neighbor size dilutes the counterintuitive stochastic slowdown which was found in well-mixed populations. Interestingly, we find that all of our results can be interpreted as that in the well-mixed population with a transformed payoff matrix. This interpretation is also valid for both death-birth and birth-death processes on graphs. This interpretation bridges the fixation time in the structured population and that in the well-mixed population. Thus it opens the avenue to investigate the challenging fixation time in structured populations by the known results in well-mixed populations.

Modes of migration and multilevel selection in evolutionary multiplayer games

The evolution of cooperation in group-structured populations has received much attention, but little is known about the effects of different modes of migration of individuals between groups. Here, we have incorporated four different modes of migration that differ in the degree of coordination among the individuals. For each mode of migration, we identify the set of multiplayer games in which the cooperative strategy has higher fixation probability than defection. The comparison shows that the set of games under which cooperation may evolve generally expands depending upon the degree of coordination among the migrating individuals. Weak altruism can evolve under all modes of individual migration, provided that the benefit to cost ratio is high enough. Strong altruism, however, evolves only if the mode of migration involves coordination of individual actions. Depending upon the migration frequency and degree of coordination among individuals, conditions that allow selection to work at the level of groups can be established.

Asymmetric Evolutionary Games

Evolutionary game theory is a powerful framework for studying evolution in populations of interacting individuals. A common assumption in evolutionary game theory is that interactions are symmetric, which means that the players are distinguished by only their strategies. In nature, however, the microscopic interactions between players are nearly always asymmetric due to environmental effects, differing baseline characteristics, and other possible sources of heterogeneity. To model these phenomena, we introduce into evolutionary game theory two broad classes of asymmetric interactions: ecological and genotypic. Ecological asymmetry results from variation in the environments of the players, while genotypic asymmetry is a consequence of the players having differing baseline genotypes. We develop a theory of these forms of asymmetry for games in structured populations and use the classical social dilemmas, the Prisoner’s Dilemma and the Snowdrift Game, for illustrations. Interestingly, asymmetric games reveal essential differences between models of genetic evolution based on reproduction and models of cultural evolution based on imitation that are not apparent in symmetric games.

Biological interactions, even between members of the same species, are almost always asymmetric due to differences in size, access to resources, or past interactions. However, classical game-theoretical models of evolution fail to account for sources of asymmetry in a comprehensive manner. Here, we extend the theory of evolutionary games to two general classes of asymmetry arising from environmental variation and individual differences, covering much of the heterogeneity observed in nature. If selection is weak, evolutionary processes based on asymmetric interactions behave macroscopically like symmetric games with payoffs that may depend on the resource distribution in the population or its structure. Asymmetry uncovers differences between genetic and cultural evolution that are not apparent when interactions are symmetric.

Social exclusion in finite populations

Social exclusion, keeping free riders from benefit sharing, plays an important role in sustaining cooperation in our world. Here we propose two different exclusion regimes, namely, peer exclusion and pool exclusion, to investigate the evolution of social exclusion in finite populations. In the peer exclusion regime, each excluder expels all the defectors independently, and thus bears the total cost on his own, while in the pool exclusion regime, excluders spontaneously form an institution to carry out rejection of the free riders, and each excluder shares the cost equally. In a public goods game containing only excluders and defectors, it is found that peer excluders outperform pool excluders if the exclusion costs are small, and the situation is converse once the exclusion costs exceed some critical points, which holds true for all the selection intensities and different update rules. Moreover, excluders can dominate the whole population under a suitable parameters range in the presence of second-order free riders (cooperators), showing that exclusion has prominent advantages over common costly punishment. More importantly, our finding indicates that the group exclusion mechanism helps the cooperative union to survive under unfavorable conditions. Our results may give some insights into better understanding the prevalence of such a strategy in the real world and its significance in sustaining cooperation.