How mutation affects evolutionary games on graphs

Value-behavior inconsistency is robust to promote cooperative behavior in structured populations

The evolution of cooperation is a theme commonly studied in biology, psychology, sociology, and economics. Mechanisms that promote cooperative behavior in structured populations have been intensively studied. However, individuals' values, specifically, their opinions have been rarely taken into account so far. Inspired by cognition dissonance theory, we assume that individuals pay the cost of guiltiness if the behavior is defection but the opinion deviates from defection, and pay the cost of regret if the behavior is cooperation but the opinion deviates from cooperation. For all general stochastic evolutionary dynamics on arbitrary static networks with multiple opinions, we prove in the weak selection limit that: (i) value-behavior inconsistency cost promotes cooperative behavior if and only if the average cost of regret is less than that of guiltiness; (ii) individuals with value-behavior consistency are more abundant than that with value-behavior inconsistency. This is in contrast with other mechanisms that are at work for cooperation for one population structure but not others. Furthermore, it is also validated on an empirical network and for non-weak selection intensity. The value-behavior inconsistency is thus a robust mechanism to promote cooperative behavior in structured populations. Our results shed light on the importance of the co-evolutionary dynamics of opinion and behavior, which opens an avenue for cooperation.

The coalescent in finite populations with arbitrary, fixed structure

The coalescent is a stochastic process representing ancestral lineages in a population undergoing neutral genetic drift. Originally defined for a well-mixed population, the coalescent has been adapted in various ways to accommodate spatial, age, and class structure, along with other features of real-world populations. To further extend the range of population structures to which coalescent theory applies, we formulate a coalescent process for a broad class of neutral drift models with arbitrary - but fixed - spatial, age, sex, and class structure, haploid or diploid genetics, and any fixed mating pattern. Here, the coalescent is represented as a random sequence of mappings [Formula: see text] from a finite set G to itself. The set G represents the "sites" (in individuals, in particular locations and/or classes) at which these alleles can live. The state of the coalescent, Ct:G→G, maps each site g∈G to the site containing g's ancestor, t time-steps into the past. Using this representation, we define and analyze coalescence time, coalescence branch length, mutations prior to coalescence, and stationary probabilities of identity-by-descent and identity-by-state. For low mutation, we provide a recipe for computing identity-by-descent and identity-by-state probabilities via the coalescent. Applying our results to a diploid population with arbitrary sex ratio r, we find that measures of genetic dissimilarity, among any set of sites, are scaled by 4r(1-r) relative to the even sex ratio case.

Symmetry in models of natural selection

Symmetry arguments are frequently used-often implicitly-in mathematical modelling of natural selection. Symmetry simplifies the analysis of models and reduces the number of distinct population states to be considered. Here, I introduce a formal definition of symmetry in mathematical models of natural selection. This definition applies to a broad class of models that satisfy a minimal set of assumptions, using a framework developed in previous works. In this framework, population structure is represented by a set of sites at which alleles can live, and transitions occur via replacement of some alleles by copies of others. A symmetry is defined as a permutation of sites that preserves probabilities of replacement and mutation. The symmetries of a given selection process form a group, which acts on population states in a way that preserves the Markov chain representing selection. Applying classical results on group actions, I formally characterize the use of symmetry to reduce the states of this Markov chain, and obtain bounds on the number of states in the reduced chain.

Behavioural, demographic and fitness consequences of social instability in cooperatively breeding dwarf mongoose groups

Social instability frequently arises in group-living species, but the potential costs have rarely been investigated in free-living cooperative breeders, especially across different timeframes. Using natural observations, body mass measurements and life-history data from dwarf mongooses (Helogale parvula), we determined the short- and long-term consequences of a change in one of the dominant breeding pairs. We found that a new breeder led to alterations in both collective and individual behaviours (i.e. increases in communal scent-marking, engagement in intergroup interactions, sentinel activity and within-group grooming), as well as reduced body mass gain, further demographic changes and decreased reproductive success (i.e. fewer pups surviving to adulthood). The effects were particularly apparent when it was the female breeder who changed; new female breeders were younger than more experienced counterparts. Our findings support the idea that stability and cooperation are strongly linked and provide potential reasons for previously documented health and fitness benefits of social stability.

Evolutionary Kuramoto dynamics

Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural dimensions, a phase and a communication strategy, along which evolution occurs. We measure success of mutations by comparing the benefit of synchronization balanced against the cost of connections between the oscillators. Despite the simple set-up, this model exhibits non-trivial behaviours mimicking several different classical games—the Prisoner’s Dilemma, snowdrift games, coordination games—as the landscape of the oscillators changes over time. Across many situations, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization is greater than twice the cost, the system will evolve towards complete communication and phase synchronization.

The Evolution of Cooperation in Two-Dimensional Mobile Populations with Random and Strategic Dispersal

We investigate the effect of the environment dimensionality and different dispersal strategies on the evolution of cooperation in a finite structured population of mobile individuals. We consider a population consisting of cooperators and free-riders residing on a two-dimensional lattice with periodic boundaries. Individuals explore the environment according to one of the four dispersal strategies and interact with each other via a public goods game. The population evolves according to a birth–death–birth process with the fitness of the individuals deriving from the game-induced payouts. We found that the outcomes of the strategic dispersal strategies in the two-dimensional setting are identical to the outcomes in the one-dimensional setting. The random dispersal strategy, not surprisingly, resulted in the worst outcome for cooperators.

Dynamical analysis of evolutionary public goods game on signed networks

In evolutionary dynamics, the population structure and multiplayer interactions significantly impact the evolution of cooperation levels. Previous works mainly focus on the theoretical analysis of multiplayer games on regular networks or pairwise games on complex networks. Combining these two factors, complex networks and multiplayer games, we obtain the fixation probability and fixation time of the evolutionary public goods game in a structured population represented by a signed network. We devise a stochastic framework for estimating fixation probability with weak mistrust or strong mistrust mechanisms and develop a deterministic replicator equation to predict the expected density of cooperators when the system evolves to the equilibrium on a signed network. Specifically, the most interesting result is that negative edges diversify the cooperation steady state, evolving in three different patterns of fixed probability in Erdös-Rényi signed and Watts-Strogatz signed networks with the new "strong mistrust" mechanism.

The characteristics of average abundance function with mutation of multi-player threshold public goods evolutionary game model under redistribution mechanism

BACKGROUND

In recent years, the average abundance function has attracted much attention as it reflects the degree of cooperation in the population. Then it is significant to analyse how average abundance functions can be increased to promote the proliferation of cooperative behaviour. However, further theoretical analysis for average abundance function with mutation under redistribution mechanism is still lacking. Furthermore, the theoretical basis for the corresponding numerical simulation is not sufficiently understood.

RESULTS

We have deduced the approximate expressions of average abundance function with mutation under redistribution mechanism on the basis of different levels of selection intensity [Formula: see text] (sufficiently small and large enough). In addition, we have analysed the influence of the size of group d, multiplication factor r, cost c, aspiration level [Formula: see text] on average abundance function from both quantitative and qualitative aspects.

CONCLUSIONS

(1) The approximate expression will become the linear equation related to selection intensity when [Formula: see text] is sufficiently small. (2) On one hand, approximation expression when [Formula: see text] is large enough is not available when r is small and m is large. On the other hand, this approximation expression will become more reliable when [Formula: see text] is larger. (3) On the basis of the expected payoff function [Formula: see text] and function [Formula: see text], the corresponding results for the effects of parameters (d,r,c,[Formula: see text]) on average abundance function [Formula: see text] have been explained.

Stable leaders pave the way for cooperation under time-dependent exploration rates

The exploration of different behaviours is part of the adaptation repertoire of individuals to new environments. Here, we explore how the evolution of cooperative behaviour is affected by the interplay between exploration dynamics and social learning, in particular when individuals engage on prisoner's dilemma along the edges of a social network. We show that when the population undergoes a transition from strong to weak exploration rates a decline in the overall levels of cooperation is observed. However, if the rate of decay is lower in highly connected individuals (Leaders) than for the less connected individuals (Followers) then the population is able to achieve higher levels of cooperation. Finally, we show that minor differences in selection intensities (the degree of determinism in social learning) and individual exploration rates, can translate into major differences in the observed collective dynamics.

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.

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.

Dynamics of multiplayer games on complex networks using territorial interactions

The modeling of evolution in structured populations has been significantly advanced by evolutionary graph theory, which incorporates pairwise relationships between individuals on a network. More recently, a new framework has been developed to allow for multiplayer interactions of variable size in more flexible and potentially changing population structures. While the theory within this framework has been developed and simple structures considered, there has been no systematic consideration of a large range of different population structures, which is the subject of this paper. We consider a large range of underlying graphical structures for the territorial raider model, the most commonly used model in the new structure, and consider a variety of important properties of our structures with the aim of finding factors that determine the fixation probability of mutants. We find that the graphical temperature and the average group size, as previously defined, are strong predictors of fixation probability, while all other properties considered are poor predictors, although the clustering coefficient is a useful secondary predictor when combined with either temperature or group size. The relationship between temperature or average group size and fixation probability is sometimes, however, nonmonotonic, with a directional reverse occurring around the temperature associated with what we term "completely mixed" populations in the case of the hawk-dove game, but not the public goods game.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Biodiversity in the cyclic competition system of three species according to the emergence of mutant species

Understanding mechanisms which promote or hinder existing ecosystems are important issues in ecological sciences. In addition to fundamental interactions such as competition and migration among native species, existing ecosystems can be easily disturbed by external factors, and the emergence of new species may be an example in such cases. The new species which does not exist in a current ecosystem can be regarded as either alien species entered from outside or mutant species born by mutation in existing normal species. Recently, as existing ecosystems are getting influenced by various physical/chemical external factors, mutation due to anthropogenic and environmental factors can occur more frequently and is thus attracting much attention for the maintenance of ecosystems. In this paper, we consider emergences of mutant species among self-competing three species in the cyclic dominance. By defining mutation as the birth of mutant species, we investigate how mutant species can affect biodiversity in the existing ecosystem. Through microscopic and macroscopic approaches, we have found that the society of existing normal species can be disturbed by mutant species either the society is maintained accompanying with the coexistence of all species or jeopardized by occupying of mutant species. Due to the birth of mutant species, the existing society may be more complex by constituting two different groups of normal and mutant species, and our results can be contributed to analyze complex ecosystems of many species. We hope our findings may propose a new insight on mutation in cyclic competition systems of many species.

Population Structure Promotes the Evolution of Intuitive Cooperation and Inhibits Deliberation

Spatial structure is one of the most studied mechanisms in evolutionary game theory. Here, we explore the consequences of spatial structure for a question which has received considerable empirical and theoretical attention in recent years, but has not yet been studied from a network perspective: whether cooperation relies on intuitive predispositions or deliberative self-control. We examine this question using a model which integrates the “dual-process” framework from cognitive science with evolutionary game theory, and considers the evolution of agents who are embedded within a social network and only interact with their neighbors. In line with past work in well-mixed populations, we find that selection favors either the intuitive defector strategy which never deliberates, or the dual-process cooperator strategy which intuitively cooperates but uses deliberation to switch to defection when doing so is payoff-maximizing. We find that sparser networks (i.e., smaller average degree) facilitate the success of dual-process cooperators over intuitive defectors, while also reducing the level of deliberation that dual-process cooperators engage in; and that these results generalize across different kinds of networks. These observations demonstrate the important role that spatial structure can have not just on the evolution of cooperation, but on the co-evolution of cooperation and cognition.

Evolution of cooperation with interactive identity and diversity

Interactive identity and interactive diversity are generally regarded as two typical interaction patterns in living systems. The former describes that in each generation every individual behaves identically to all of its opponents, and the latter allows each individual to behave diversely to its distinct opponents. Most traditional research on the evolution of cooperation, however, has been confined to populations with a uniform interaction pattern. Here we study the cooperation conundrum in a diverse population comprising players with interactive identity and with interactive diversity. We find that in homogeneous networks a small fraction of players taking interactive diversity are enough to stabilize cooperation for a wide range of payoff values even in a noisy environment. When assigned to heterogeneous networks, players in high-degree nodes taking interactive diversity significantly strengthen systems' resilience against the shifty environment and enlarge the survival region of cooperation. However, they fail to establish a homogeneous strategy 'cloud' in the neighborhood and thus can not coordinate players in low-degree nodes to reach a socially optimal cooperation level. The most favorable outcome emerges when players in high-degree nodes take interactive identity and meanwhile others adopt interactive diversity. Our findings reveal the significance of the two typical interaction patterns and could be a good heuristic in coordinating them to achieve the social optimum in cooperation.

Second-order cooperation: Cooperative offspring as a living public good arising from second-order selection on non-cooperative individuals

Switching rate between cooperating and non-cooperating genotypes is a crucial social evolution factor, often neglected by game theory-inspired theoretical and experimental frameworks. We show that the evolution of alleles increasing the mutation or phenotypic switching rates toward cooperation is in itself a social dilemma. Although cooperative offspring are often unlikely to reproduce, due to high cost of cooperation, they can be seen both as a living public good and a part of the extended parental phenotype. The competition between individuals that generate cooperators and ones that do not is often more relevant than the competition between cooperators and non-cooperators. The dilemma of second-order cooperation we describe relates directly to eusociality, but can be also interpreted as a division of labor or a soma-germline distinction. The results of our simulations shine a new light on what Darwin had already termed a "special difficulty" of evolutionary theory and describe a novel type of cooperation dynamics.

Evolutionary dynamics on any population structure

Evolution occurs in populations of reproducing individuals. The structure of a population can affect which traits evolve. Understanding evolutionary game dynamics in structured populations remains difficult. Mathematical results are known for special structures in which all individuals have the same number of neighbours. The general case, in which the number of neighbours can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class that suggests there is no efficient algorithm. Whether a simple solution for weak selection exists has remained unanswered. Here we provide a solution for weak selection that applies to any graph or network. Our method relies on calculating the coalescence times of random walks. We evaluate large numbers of diverse population structures for their propensity to favour cooperation. We study how small changes in population structure-graph surgery-affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.

Fidelity of parent-offspring transmission and the evolution of social behavior in structured populations

The theoretical investigation of how spatial structure affects the evolution of social behavior has mostly been done under the assumption that parent-offspring strategy transmission is perfect, i.e., for genetically transmitted traits, that mutation is very weak or absent. Here, we investigate the evolution of social behavior in structured populations under arbitrary mutation probabilities. We consider populations of fixed size N, structured such that in the absence of selection, all individuals have the same probability of reproducing or dying (neutral reproductive values are the all same). Two types of individuals, A and B, corresponding to two types of social behavior, are competing; the fidelity of strategy transmission from parent to offspring is tuned by a parameter μ. Social interactions have a direct effect on individual fecundities. Under the assumption of small phenotypic differences (implying weak selection), we provide a formula for the expected frequency of type A individuals in the population, and deduce conditions for the long-term success of one strategy against another. We then illustrate our results with three common life-cycles (Wright-Fisher, Moran Birth-Death and Moran Death-Birth), and specific population structures (graph-structured populations). Qualitatively, we find that some life-cycles (Moran Birth-Death, Wright-Fisher) prevent the evolution of altruistic behavior, confirming previous results obtained with perfect strategy transmission. We also show that computing the expected frequency of altruists on a regular graph may require knowing more than just the graph's size and degree.

Games of multicellularity

Evolutionary game dynamics are often studied in the context of different population structures. Here we propose a new population structure that is inspired by simple multicellular life forms. In our model, cells reproduce but can stay together after reproduction. They reach complexes of a certain size, n, before producing single cells again. The cells within a complex derive payoff from an evolutionary game by interacting with each other. The reproductive rate of cells is proportional to their payoff. We consider all two-strategy games. We study deterministic evolutionary dynamics with mutations, and derive exact conditions for selection to favor one strategy over another. Our main result has the same symmetry as the well-known sigma condition, which has been proven for stochastic game dynamics and weak selection. For a maximum complex size of n=2 our result holds for any intensity of selection. For n≥3 it holds for weak selection. As specific examples we study the prisoner's dilemma and hawk-dove games. Our model advances theoretical work on multicellularity by allowing for frequency-dependent interactions within groups.

The Inevitability of Ethnocentrism Revisited: Ethnocentrism Diminishes As Mobility Increases

Nearly all major conflicts across the globe, both current and historical, are characterized by individuals defining themselves and others by group membership. This existence of group-biased behavior (in-group favoring and out-group hostile) has been well established empirically, and has been shown to be an inevitable outcome in many evolutionary studies. Thus it is puzzling that statistics show violence and out-group conflict declining dramatically over the past few centuries of human civilization. Using evolutionary game-theoretic models, we solve this puzzle by showing for the first time that out-group hostility is dramatically reduced by mobility. Technological and societal advances over the past centuries have greatly increased the degree to which humans change physical locations, and our results show that in highly mobile societies, one's choice of action is more likely to depend on what individual one is interacting with, rather than the group to which the individual belongs. Our empirical analysis of archival data verifies that contexts with high residential mobility indeed have less out-group hostility than those with low mobility. This work suggests that, in fact, group-biased behavior that discriminates against out-groups is not inevitable after all.

Cellular cooperation with shift updating and repulsion

Population structure can facilitate evolution of cooperation. In a structured population, cooperators can form clusters which resist exploitation by defectors. Recently, it was observed that a shift update rule is an extremely strong amplifier of cooperation in a one dimensional spatial model. For the shift update rule, an individual is chosen for reproduction proportional to fecundity; the offspring is placed next to the parent; a random individual dies. Subsequently, the population is rearranged (shifted) until all individual cells are again evenly spaced out. For large population size and a one dimensional population structure, the shift update rule favors cooperation for any benefit-to-cost ratio greater than one. But every attempt to generalize shift updating to higher dimensions while maintaining its strong effect has failed. The reason is that in two dimensions the clusters are fragmented by the movements caused by rearranging the cells. Here we introduce the natural phenomenon of a repulsive force between cells of different types. After a birth and death event, the cells are being rearranged minimizing the overall energy expenditure. If the repulsive force is sufficiently high, shift becomes a strong promoter of cooperation in two dimensions.

Random and non-random mating populations: Evolutionary dynamics in meiotic drive

Game theoretic tools are utilized to analyze a one-locus continuous selection model of sex-specific meiotic drive by considering nonequivalence of the viabilities of reciprocal heterozygotes that might be noticed at an imprinted locus. The model draws attention to the role of viability selections of different types to examine the stable nature of polymorphic equilibrium. A bridge between population genetics and evolutionary game theory has been built up by applying the concept of the Fundamental Theorem of Natural Selection. In addition to pointing out the influences of male and female segregation ratios on selection, configuration structure reveals some noted results, e.g., Hardy-Weinberg frequencies hold in replicator dynamics, occurrence of faster evolution at the maximized variance fitness, existence of mixed Evolutionarily Stable Strategy (ESS) in asymmetric games, the tending evolution to follow not only a 1:1 sex ratio but also a 1:1 different alleles ratio at particular gene locus. Through construction of replicator dynamics in the group selection framework, our selection model introduces a redefining bases of game theory to incorporate non-random mating where a mating parameter associated with population structure is dependent on the social structure. Also, the model exposes the fact that the number of polymorphic equilibria will depend on the algebraic expression of population structure.

Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

Static network structure can stabilize human cooperation

The evolution of cooperation in network-structured populations has been a major focus of theoretical work in recent years. When players are embedded in fixed networks, cooperators are more likely to interact with, and benefit from, other cooperators. In theory, this clustering can foster cooperation on fixed networks under certain circumstances. Laboratory experiments with humans, however, have thus far found no evidence that fixed network structure actually promotes cooperation. Here, we provide such evidence and help to explain why others failed to find it. First, we show that static networks can lead to a stable high level of cooperation, outperforming well-mixed populations. We then systematically vary the benefit that cooperating provides to one's neighbors relative to the cost required to cooperate (b/c), as well as the average number of neighbors in the network (k). When b/c > k, we observe high and stable levels of cooperation. Conversely, when b/c ≤ k or players are randomly shuffled, cooperation decays. Our results are consistent with a quantitative evolutionary game theoretic prediction for when cooperation should succeed on networks and, for the first time to our knowledge, provide an experimental demonstration of the power of static network structure for stabilizing human cooperation.

Evolutionary prisoner's dilemma game on graphs and social networks with external constraint

A game-theoretical model is constructed to capture the effect of external constraint on the evolution of cooperation. External constraint describes the case where individuals are forced to cooperate with a given probability in a society. Mathematical analyses are conducted via pair approximation and diffusion approximation methods. The results show that the condition for cooperation to be favored on graphs with constraint is b¯/c¯>k/A¯ (A¯=1+kp/(1-p)), where b¯ and c¯ represent the altruistic benefit and cost, respectively, k is the average degree of the graph and p is the probability of compulsory cooperation by external enforcement. Moreover, numerical simulations are also performed on a repeated game with three strategies, always defect (ALLD), tit-for-tat (TFT) and always cooperate (ALLC). These simulations demonstrate that a slight enforcement of ALLC can only promote cooperation when there is weak network reciprocity, while the catalyst effect of TFT on cooperation is verified. In addition, the interesting phenomenon of stable coexistence of the three strategies can be observed. Our model can represent evolutionary dynamics on a network structure which is disturbed by a specified external constraint.

Spatial dilemmas of diffusible public goods

The emergence of cooperation is a central question in evolutionary biology. Microorganisms often cooperate by producing a chemical resource (a public good) that benefits other cells. The sharing of public goods depends on their diffusion through space. Previous theory suggests that spatial structure can promote evolution of cooperation, but the diffusion of public goods introduces new phenomena that must be modeled explicitly. We develop an approach where colony geometry and public good diffusion are described by graphs. We find that the success of cooperation depends on a simple relation between the benefits and costs of the public good, the amount retained by a producer, and the average amount retained by each of the producer's neighbors. These quantities are derived as analytic functions of the graph topology and diffusion rate. In general, cooperation is favored for small diffusion rates, low colony dimensionality, and small rates of decay of the public good. DOI: http://dx.doi.org/10.7554/eLife.01169.001.

Genetic architecture promotes the evolution and maintenance of cooperation

When cooperation has a direct cost and an indirect benefit, a selfish behavior is more likely to be selected for than an altruistic one. Kin and group selection do provide evolutionary explanations for the stability of cooperation in nature, but we still lack the full understanding of the genomic mechanisms that can prevent cheater invasion. In our study we used Aevol, an agent-based, in silico genomic platform to evolve populations of digital organisms that compete, reproduce, and cooperate by secreting a public good for tens of thousands of generations. We found that cooperating individuals may share a phenotype, defined as the amount of public good produced, but have very different abilities to resist cheater invasion. To understand the underlying genetic differences between cooperator types, we performed bio-inspired genomics analyses of our digital organisms by recording and comparing the locations of metabolic and secretion genes, as well as the relevant promoters and terminators. Association between metabolic and secretion genes (promoter sharing, overlap via frame shift or sense-antisense encoding) was characteristic for populations with robust cooperation and was more likely to evolve when secretion was costly. In mutational analysis experiments, we demonstrated the potential evolutionary consequences of the genetic association by performing a large number of mutations and measuring their phenotypic and fitness effects. The non-cooperating mutants arising from the individuals with genetic association were more likely to have metabolic deleterious mutations that eventually lead to selection eliminating such mutants from the population due to the accompanying fitness decrease. Effectively, cooperation evolved to be protected and robust to mutations through entangled genetic architecture. Our results confirm the importance of second-order selection on evolutionary outcomes, uncover an important genetic mechanism for the evolution and maintenance of cooperation, and suggest promising methods for preventing gene loss in synthetically engineered organisms.

Adaptive dynamics with interaction structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

Global migration can lead to stronger spatial selection than local migration

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.

Evolution of emotions on networks leads to the evolution of cooperation in social dilemmas

We show that the resolution of social dilemmas in random graphs and scale-free networks is facilitated by imitating not the strategy of better-performing players but, rather, their emotions. We assume sympathy and envy to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network.

Evolution of cooperation in spatial traveler's dilemma game

Traveler's dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler's dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth-death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

Consistent strategy updating in spatial and non-spatial behavioral experiments does not promote cooperation in social networks

The presence of costly cooperation between otherwise selfish actors is not trivial. A prominent mechanism that promotes cooperation is spatial population structure. However, recent experiments with human subjects report substantially lower level of cooperation then predicted by theoretical models. We analyze the data of such an experiment in which a total of 400 players play a Prisoner's Dilemma on a square lattice in two treatments, either interacting via a fixed square lattice (15 independent groups) or with a population structure changing after each interaction (10 independent groups). We analyze the statistics of individual decisions and infer in which way they can be matched with the typical models of evolutionary game theorists. We find no difference in the strategy updating between the two treatments. However, the strategy updates are distinct from the most popular models which lead to the promotion of cooperation as shown by computer simulations of the strategy updating. This suggests that the promotion of cooperation by population structure is not as straightforward in humans as often envisioned in theoretical models.

Wisdom of groups promotes cooperation in evolutionary social dilemmas

Whether or not to change strategy depends not only on the personal success of each individual, but also on the success of others. Using this as motivation, we study the evolution of cooperation in games that describe social dilemmas, where the propensity to adopt a different strategy depends both on individual fitness as well as on the strategies of neighbors. Regardless of whether the evolutionary process is governed by pairwise or group interactions, we show that plugging into the “wisdom of groups” strongly promotes cooperative behavior. The more the wider knowledge is taken into account the more the evolution of defectors is impaired. We explain this by revealing a dynamically decelerated invasion process, by means of which interfaces separating different domains remain smooth and defectors therefore become unable to efficiently invade cooperators. This in turn invigorates spatial reciprocity and establishes decentralized decision making as very beneficial for resolving social dilemmas.

Evolutionary shift dynamics on a cycle

We present a new model of evolutionary dynamics in one-dimensional space. Individuals are arranged on a cycle. When a new offspring is born, another individual dies and the rest shift around the cycle to make room. This rule, which is inspired by spatial evolution in somatic tissue and microbial colonies, has the remarkable property that, in the limit of large population size, evolution acts to maximize the payoff of the whole population. Therefore, social dilemmas, in which some individuals benefit at the expense of others, are resolved. We demonstrate this principle for both discrete and continuous games. We also discuss extensions of our model to other one-dimensional spatial configurations. We conclude that shift dynamics in one dimension is an unusually strong promoter of cooperative behavior.