Invasion and expansion of cooperators in lattice populations: prisoner's dilemma vs. snowdrift games

Enhancing social cohesion with cooperative bots in societies of greedy, mobile individuals

Addressing collective issues in social development requires a high level of social cohesion, characterized by cooperation and close social connections. However, social cohesion is challenged by selfish, greedy individuals. With the advancement of artificial intelligence (AI), the dynamics of human-machine hybrid interactions introduce new complexities in fostering social cohesion. This study explores the impact of simple bots on social cohesion from the perspective of human-machine hybrid populations within network. By investigating collective self-organizing movement during migration, results indicate that cooperative bots can promote cooperation, facilitate individual aggregation, and thereby enhance social cohesion. The random exploration movement of bots can break the frozen state of greedy population, help to separate defectors in cooperative clusters, and promote the establishment of cooperative clusters. However, the presence of defective bots can weaken social cohesion, underscoring the importance of carefully designing bot behavior. Our research reveals the potential of bots in guiding social self-organization and provides insights for enhancing social cohesion in the era of human-machine interaction within social networks.

A geometric process of evolutionary game dynamics

Many evolutionary processes occur in phenotype spaces which are continuous. It is therefore of interest to explore how selection operates in continuous spaces. One approach is adaptive dynamics, which assumes that mutants are local. Here we study a different process which also allows non-local mutants. We assume that a resident population is challenged by an invader who uses a strategy chosen from a random distribution on the space of all strategies. We study the repeated donation game of direct reciprocity. We consider reactive strategies given by two probabilities, denoting respectively the probability to cooperate after the co-player has cooperated or defected. The strategy space is the unit square. We derive analytic formulae for the stationary distribution of evolutionary dynamics and for the average cooperation rate as function of the cost-to-benefit ratio. For positive reactive strategies, we prove that cooperation is more abundant than defection if the area of the cooperative region is greater than 1/2 which is equivalent to benefit, b, divided by cost, c, exceeding [Formula: see text]. We introduce the concept of strategies that are stable with probability one. We also study an extended process and discuss other games.

A game-based approach for designing a collaborative evolution mechanism for unmanned swarms on community networks

Intelligent and coordinated unmanned aerial vehicle (UAV) swarm combat will be the main mode of warfare in the future, and mechanistic design of autonomous cooperation within swarms is the key to enhancing combat effectiveness. Exploration of the essential features and patterns of autonomous collaboration in unmanned swarms has become the focus of scientific research and technological applications, in keeping with the evolving conceptions of the military theatre. However, given the unique attributes of the military and the novelty of the warfare mode of unmanned swarms, few achievements have been reported in the existing research. In this study, we analysed the military requirements of unmanned swarm operations and proposed an analytic framework for autonomous collaboration. Then, a literature review addressing swarm evolution dynamics, game-based swarm collaboration, and collaborative evolution on complex networks was conducted. Next, on the basis of the above work, we designed a community network for unmanned swarm cooperation and constructed a collaborative evolution model based on the multiplayer public goods game (PGG). Furthermore, according to the “network” and “model”, the dynamic evolution process of swarm collaboration was formally deduced. Finally, a simulation was conducted to analyse the influence of relevant parameters (i.e., swarm size, degree distribution, cost, multiplication factor) on the collaborative behaviour of unmanned swarms. According to the simulation results, some reasonable suggestions for collaborative management and control in swarm operation are given, which can provide theoretical reference and decision-making support for the design of coordination mechanisms and improved combat effectiveness in unmanned swarm operation.

Global dynamics of microbial communities emerge from local interaction rules

Most microbes live in spatially structured communities (e.g., biofilms) in which they interact with their neighbors through the local exchange of diffusible molecules. To understand the functioning of these communities, it is essential to uncover how these local interactions shape community-level properties, such as the community composition, spatial arrangement, and growth rate. Here, we present a mathematical framework to derive community-level properties from the molecular mechanisms underlying the cell-cell interactions for systems consisting of two cell types. Our framework consists of two parts: a biophysical model to derive the local interaction rules (i.e. interaction range and strength) from the molecular parameters underlying the cell-cell interactions and a graph based model to derive the equilibrium properties of the community (i.e. composition, spatial arrangement, and growth rate) from these local interaction rules. Our framework shows that key molecular parameters underlying the cell-cell interactions (e.g., the uptake and leakage rates of molecules) determine community-level properties. We apply our model to mutualistic cross-feeding communities and show that spatial structure can be detrimental for these communities. Moreover, our model can qualitatively recapitulate the properties of an experimental microbial community. Our framework can be extended to a variety of systems of two interacting cell types, within and beyond the microbial world, and contributes to our understanding of how community-level properties emerge from microscopic interactions between cells.

Microorganisms perform essential processes on our planet. Many of these processes result from interactions between different species growing in spatially structured communities. A central goal is to understand how community processes emerge from such interactions between cells. Here we develop a mathematical framework to derive community-level properties, such as the community composition, growth rate, and spatial organization, from the molecular mechanisms underlying these cell-cell interactions. We focus on mutualistic communities consisting of two cell types that need to interact with each other in order to grow. We derive equations that describe how changes in the molecular parameters of cellular interactions affect individuals’ and community properties. We find that spatial structure has a negative impact on these mutualistic communities: as cells become surrounded by their own type, they have less access to the other cell type with which they need to interact to grow well. We show that our framework can also be applied to other types of microbial communities and potentially to non-microbial systems such as tissues. More generally, this work advances our understanding of how scales are connected in biological systems, both in the microbial world and beyond.

Snowdrift game induces pattern formation in systems of self-propelled particles

Evolutionary games between species are known to lead to intriguing spatiotemporal patterns in systems of diffusing agents. However, the role of interspecies interactions is hardly studied when agents are (self-)propelled, as is the case in many biological systems. Here, we combine aspects from active matter and evolutionary game theory and study a system of two species whose individuals are (self-)propelled and interact through a snowdrift game. We derive hydrodynamic equations for the density and velocity fields of both species from which we identify parameter regimes in which one or both species form macroscopic orientational order as well as regimes of propagating wave patterns. Interestingly, we find simultaneous wave patterns in both species that result from the interplay between alignment and snowdrift interactions-a feedback mechanism that we call game-induced pattern formation. We test these results in agent-based simulations and confirm the different regimes of order and spatiotemporal patterns as well as game-induced pattern formation.

Spatial social dilemmas promote diversity

Cooperative investments in social dilemmas can spontaneously diversify into stably coexisting high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation, we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual-based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavorable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

Assortative clustering in a one-dimensional population with replication strategies

In a geographically distributed population, assortative clustering plays an important role in evolution by modifying local environments. To examine its effects in a linear habitat, we consider a one-dimensional grid of cells, where each cell is either empty or occupied by an organism whose replication strategy is genetically inherited to offspring. The strategy determines whether to have offspring in surrounding cells, as a function of the neighborhood configuration. If more than one offspring compete for a cell, then they can be all exterminated due to the cost of conflict depending on environmental conditions. We find that the system is more densely populated in an unfavorable environment than in a favorable one because only the latter has to pay the cost of conflict. This observation agrees reasonably well with a mean-field analysis which takes assortative clustering of strategies into consideration. Our finding suggests a possibility of intrinsic nonlinearity between environmental conditions and population density when an evolutionary process is involved.

Pair approximation model for the vaccination game: predicting the dynamic process of epidemic spread and individual actions against contagion

We successfully establish a theoretical framework of pairwise approximation for the vaccination game in which both the dynamic process of epidemic spread and individual actions in helping prevent social behaviours are quantitatively evaluated. In contrast with mean-field approximation, our model captures higher-order effects from neighbours by using an underlying network that shows how the disease spreads and how individual decisions evolve over time. This model considers not only imperfect vaccination but also intermediate protective measures other than vaccines. Our analytical predictions are validated by multi-agent simulation results that estimate random regular graphs at varying degrees.

Cooperation in Microbial Populations: Theory and Experimental Model Systems

Cooperative behavior, the costly provision of benefits to others, is common across all domains of life. This review article discusses cooperative behavior in the microbial world, mediated by the exchange of extracellular products called public goods. We focus on model species for which the production of a public good and the related growth disadvantage for the producing cells are well described. To unveil the biological and ecological factors promoting the emergence and stability of cooperative traits we take an interdisciplinary perspective and review insights gained from both mathematical models and well-controlled experimental model systems. Ecologically, we include crucial aspects of the microbial life cycle into our analysis and particularly consider population structures where ensembles of local communities (subpopulations) continuously emerge, grow, and disappear again. Biologically, we explicitly consider the synthesis and regulation of public good production. The discussion of the theoretical approaches includes general evolutionary concepts, population dynamics, and evolutionary game theory. As a specific but generic biological example, we consider populations of Pseudomonas putida and its regulation and use of pyoverdines, iron scavenging molecules, as public goods. The review closes with an overview on cooperation in spatially extended systems and also provides a critical assessment of the insights gained from the experimental and theoretical studies discussed. Current challenges and important new research opportunities are discussed, including the biochemical regulation of public goods, more realistic ecological scenarios resembling native environments, cell-to-cell signaling, and multispecies communities.

Close spatial arrangement of mutants favors and disfavors fixation

Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in the expansion of mutants. However, the clustering of mutants and the game dynamics are typically coupled. It is still unclear how the clustering of mutants alone alters the evolutionary dynamics. To this end, we employ a minimal model with frequency independent fitness on a circle. It disentangles the clustering of mutants from game dynamics. The distance between two mutants on the circle is adopted as a natural indicator for the clustering of mutants or assortment. We find that the assortment is an amplifier of the selection for the connected mutants compared with the separated ones. Nevertheless, as mutants are separated, the more dispersed mutants are, the greater the chance of invasion is. It gives rise to the non-monotonic effect of clustering, which is counterintuitive. On the other hand, we find that less assortative mutants speed up fixation. Our model shows that the clustering of mutants plays a non-trivial role in fixation, which has emerged even if the game interaction is absent.

Evolutionary dynamics on networks are key for biological and social evolution. Typically, the clustering mutants on networks can dramatically alter the direction of selection. Previous studies on the assortment of mutants assume that individuals interact in a frequency-dependent way. It is hard to tell how assortment alone alters the evolutionary fate. We establish a minimal network model to disentangle the assortment from the game interaction. We find that for weak selection limit, the assortment of mutants plays little role in fixation probability. For strong selection limit, connected mutants, i.e., the maximum assortment, are best for fixation. When the mutants are separated by only one wild-type individual, it is worse off than that separated by more than one wild-type individual in fixation probability. Our results show the nontrivial yet fundamental effect of the clustering on fixation. Noteworthily, it has already arisen, even if the game interaction is absent.

Investment preference promotes cooperation in spatial public goods game

It is usually assumed that each cooperator contributes equally to different public pools in spatial public goods game. However, it is more reasonable to invest differently according to individual investment preference. In this paper, an extended public goods game, in which cooperators contribute to the groups according to the investment preference, is developed. The investment preference of a cooperator is characterized by the fraction of the cooperator from his/her own memory about a group and the intensity of investment preference is represented by a tunable parameter α. The well-mixed population and the structured population are analyzed under this mechanism. It is shown that the investment preference can give rise to coordination. Moreover, the extensive numerical simulation results show that with the increasing of investment preference density or memory length, the proportion of cooperation can increase monotonously. This is because the investment preference could help cooperators resist the invasion from defectors. Compared with the basic version, the new mechanism is able to promote cooperation effectively. Our research may provide a valuable insight for further exploring the nature of cooperation in the real world.

Sequential interactions-in which one player plays first and another responds-promote cooperation in evolutionary-dynamical simulations of single-shot Prisoner's Dilemma and Snowdrift games

Cooperation is a central topic in evolutionary biology because (a) it is difficult to reconcile why individuals would act in a way that benefits others if such action is costly to themselves, and (b) it underpins many of the 'major transitions of evolution', making it essential for explaining the origins of successively higher levels of biological organization. Within evolutionary game theory, the Prisoner's Dilemma and Snowdrift games are the main theoretical constructs used to study the evolution of cooperation in dyadic interactions. In single-shot versions of these games, wherein individuals play each other only once, players typically act simultaneously rather than sequentially. Allowing one player to respond to the actions of its co-player-in the absence of any possibility of the responder being rewarded for cooperation or punished for defection, as in simultaneous or sequential iterated games-may seem to invite more incentive for exploitation and retaliation in single-shot games, compared to when interactions occur simultaneously, thereby reducing the likelihood that cooperative strategies can thrive. To the contrary, I use lattice-based, evolutionary-dynamical simulation models of single-shot games to demonstrate that under many conditions, sequential interactions have the potential to enhance unilaterally or mutually cooperative outcomes and increase the average payoff of populations, relative to simultaneous interactions-benefits that are especially prevalent in a spatially explicit context. This surprising result is attributable to the presence of conditional strategies that emerge in sequential games that can't occur in the corresponding simultaneous versions.

Invasion of Cooperation in Scale-Free Networks: Accumulated versus Average Payoffs

It is well known that cooperation cannot be an evolutionarily stable strategy for a non-iterative game in a well-mixed population. In contrast, structured populations favor cooperation, since cooperators can benefit each other by forming local clusters. Previous studies have shown that scale-free networks strongly promote cooperation. However, little is known about the invasion mechanism of cooperation in scale-free networks. To study microscopic and macroscopic behaviors of cooperators' invasion, we conducted computational experiments on the evolution of cooperation in scale-free networks where, starting from all defectors, cooperators can spontaneously emerge by mutation. Since the evolutionary dynamics are influenced by the definition of fitness, we tested two commonly adopted fitness functions: accumulated payoff and average payoff. Simulation results show that cooperation is strongly enhanced with the accumulated payoff fitness compared to the average payoff fitness. However, the difference between the two functions decreases as the average degree increases. As the average degree increases, cooperation decreases for the accumulated payoff fitness, while it increases for the average payoff fitness. Moreover, for the average payoff fitness, low-degree nodes play a more important role in spreading cooperative strategies than for the accumulated payoff fitness.

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.

Coevolution of Cooperation and Partner Rewiring Range in Spatial Social Networks

In recent years, there has been growing interest in the study of coevolutionary games on networks. Despite much progress, little attention has been paid to spatially embedded networks, where the underlying geographic distance, rather than the graph distance, is an important and relevant aspect of the partner rewiring process. It thus remains largely unclear how individual partner rewiring range preference, local vs. global, emerges and affects cooperation. Here we explicitly address this issue using a coevolutionary model of cooperation and partner rewiring range preference in spatially embedded social networks. In contrast to local rewiring, global rewiring has no distance restriction but incurs a one-time cost upon establishing any long range link. We find that under a wide range of model parameters, global partner switching preference can coevolve with cooperation. Moreover, the resulting partner network is highly degree-heterogeneous with small average shortest path length while maintaining high clustering, thereby possessing small-world properties. We also discover an optimum availability of reputation information for the emergence of global cooperators, who form distant partnerships at a cost to themselves. From the coevolutionary perspective, our work may help explain the ubiquity of small-world topologies arising alongside cooperation in the real world.

Extension of a spatial evolutionary coordination game with neutral options

The multiagent evolutionary games on a lattice are equivalent to a kinetic Ising model if the uniform pair interactions are defined by a two-strategy coordination game and the logit rule controls the strategy updates. Now we extend this model by allowing the players to use additional neutral strategies that provide zero payoffs for both players if one of them selects one of the neutral strategies. In the resulting n-strategy evolutionary games the analytical methods and numerical simulations indicate continuous order-disorder phase transitions when increasing the noise level if n does not exceed a threshold value. For larger n the system exhibits a first order phase transition at a critical noise level decreasing asymptotically as 2/ln(n).

Intermediate-Range Migration Furnishes a Narrow Margin of Efficiency in the Two-Strategy Competition

It is well-known that the effects of spatial selection on the two-strategy competition can be quantified by the structural coefficient σ under weak selection. We here calculate the accurate value of σ in group-structured populations of any finite size. In previous similar models, the large population size has been explicitly required for obtaining σ, and here we analyze quantitatively how large the population should be. Unlike previous models which have only involved the influences of the longest and the shortest migration rang on σ, we consider all migration ranges together. The new phenomena are that an intermediate range maximizes σ for medium migration probabilities which are of the tiny minority and the maximum value is slightly larger than those for other ranges. Furthermore, we find the ways that migration or mutation changes σ can vary significantly through determining analytically how the high-frequency steady states (distributions of either strategy over all groups) impact the expression of σ obtained before. Our findings can be directly used to resolve the dilemma of cooperation and provide a more intuitive understanding of spatial selection.

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.

First carrot, then stick: how the adaptive hybridization of incentives promotes cooperation

Social institutions often use rewards and penalties to promote cooperation. Providing incentives tends to be costly, so it is important to find effective and efficient policies for the combined use of rewards and penalties. Most studies of cooperation, however, have addressed rewarding and punishing in isolation and have focused on peer-to-peer sanctioning as opposed to institutional sanctioning. Here, we demonstrate that an institutional sanctioning policy we call ‘first carrot, then stick’ is unexpectedly successful in promoting cooperation. The policy switches the incentive from rewarding to punishing when the frequency of cooperators exceeds a threshold. We find that this policy establishes and recovers full cooperation at lower cost and under a wider range of conditions than either rewards or penalties alone, in both well-mixed and spatial populations. In particular, the spatial dynamics of cooperation make it evident how punishment acts as a ‘booster stage’ that capitalizes on and amplifies the pro-social effects of rewarding. Together, our results show that the adaptive hybridization of incentives offers the ‘best of both worlds’ by combining the effectiveness of rewarding in establishing cooperation with the effectiveness of punishing in recovering it, thereby providing a surprisingly inexpensive and widely applicable method of promoting cooperation.

Evolutionary dynamics of fairness on graphs with migration

Individual migration plays a crucial role in evolutionary dynamics of population on networks. In this paper, we generalize the networked ultimatum game by diluting population structures as well as endowing individuals with migration ability, and investigate evolutionary dynamics of fairness on graphs with migration in the ultimatum game. We first revisit the impact of node degree on the evolution of fairness. Interestingly, numerical simulations reveal that there exists an optimal value of node degree resulting in the maximal offer level of populations. Then we explore the effects of dilution and migration on the evolution of fairness, and find that both the dilution of population structures and the endowment of migration ability to individuals would lead to the drop of offer level, while the rise of acceptance level of populations. Notably, natural selection even favors the evolution of self-incompatible strategies, when either vacancy rate or migration rate exceeds a critical threshold. To confirm our simulation results, we also propose an analytical method to study the evolutionary dynamics of fairness on graphs with migration. This method can be applied to explore any games governed by pairwise interactions in finite populations.

The expansion of neighborhood and pattern formation on spatial prisoner's dilemma

The prisoner's dilemma (PD), in which players can either cooperate or defect, is considered a paradigm for studying the evolution of cooperation in spatially structured populations. There the compact cooperator cluster is identified as a characteristic pattern and the probability of forming such pattern in turn depends on the features of the networks. In this paper, we investigate the influence of expansion of neighborhood on pattern formation by taking a weak PD game with one free parameter T, the temptation to defect. Two different expansion methods of neighborhood are considered. One is based on a square lattice and expanses along four directions generating networks with degree increasing with K=4m. The other is based on a lattice with Moore neighborhood and expanses along eight directions, generating networks with degree of K=8m. Individuals are placed on the nodes of the networks, interact with their neighbors and learn from the better one. We find that cooperator can survive for a broad degree 4≤K≤70 by taking a loose type of cooperator clusters. The former simple corresponding relationship between macroscopic patterns and the microscopic PD interactions is broken. Under a condition that is unfavorable for cooperators such as large T and K, systems prefer to evolve to a loose type of cooperator clusters to support cooperation. However, compared to the well-known compact pattern, it is a suboptimal strategy because it cannot help cooperators dominating the population and always corresponding to a low cooperation level.

Aspiration dynamics in structured population acts as if in a well-mixed one

Understanding the evolution of human interactive behaviors is important. Recent experimental results suggest that human cooperation in spatial structured population is not enhanced as predicted in previous works, when payoff-dependent imitation updating rules are used. This constraint opens up an avenue to shed light on how humans update their strategies in real life. Studies via simulations show that, instead of comparison rules, self-evaluation driven updating rules may explain why spatial structure does not alter the evolutionary outcome. Though inspiring, there is a lack of theoretical result to show the existence of such evolutionary updating rule. Here we study the aspiration dynamics, and show that it does not alter the evolutionary outcome in various population structures. Under weak selection, by analytical approximation, we find that the favored strategy in regular graphs is invariant. Further, we show that this is because the criterion under which a strategy is favored is the same as that of a well-mixed population. By simulation, we show that this holds for random networks. Although how humans update their strategies is an open question to be studied, our results provide a theoretical foundation of the updating rules that may capture the real human updating rules.

Universality of fixation probabilities in randomly structured populations

The stage of evolution is the population of reproducing individuals. The structure of the population is known to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. The most general result so far, the isothermal theorem, assumes the propensity for change in each position is exactly the same, but realistic biological structures are always subject to variation and noise. We consider a finite population under constant selection whose structure is given by a variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result for the isothermal theorem and using large deviation estimates to understand the typical structure of random graphs, we prove that for a generalization of the Erdős-Rényi model, the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. Simulations of perturbed lattices, small-world networks, and scale-free networks behave similarly. We conjecture that the fixation probability in a well-mixed population, (1 − r⁻¹)/(1 − r⁻ⁿ), is universal: for many random graph models, the fixation probability approaches the above function uniformly as the graphs become large.

Different perceptions of social dilemmas: evolutionary multigames in structured populations

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

Inequity aversion and the evolution of cooperation

Evolution of cooperation is a widely studied problem in biology, social science, economics, and artificial intelligence. Most of the existing approaches that explain cooperation rely on some notion of direct or indirect reciprocity. These reciprocity based models assume agents recognize their partner and know their previous interactions, which requires advanced cognitive abilities. In this paper we are interested in developing a model that produces cooperation without requiring any explicit memory of previous game plays. Our model is based on the notion of inequity aversion, a concept introduced within behavioral economics, whereby individuals care about payoff equality in outcomes. Here we explore the effect of using income inequality to guide partner selection and interaction. We study our model by considering both the well-mixed and the spatially structured population and present the conditions under which cooperation becomes dominant. Our results support the hypothesis that inequity aversion promotes cooperative relationship among nonkin.

Direct sum matrix game with prisoner's dilemma and snowdrift game

A direct sum form is proposed for constructing a composite game from two 2 x 2 games, prisoner's dilemma and snowdrift game. This kind of direct sum form game is called a multiple roles game. The replicator dynamics of the multiple roles game with will-mixed populations is explored. The dynamical behaviors on square lattice are investigated by numerical simulation. It is found that the dynamical behaviors of population on square lattice depend on the mixing proportion of the two simple games. Mixing SD activities to pure PD population inhibits the proportion of cooperators in PD, and mixing PD activities to pure SD population stimulates the proportion of cooperators in SD. Besides spatial reciprocity, our results show that there are roles reciprocities between different types of individuals.

Effect of initial fraction of cooperators on cooperative behavior in evolutionary prisoner's dilemma game

We investigate the influence of initial fraction of cooperators on the evolution of cooperation in spatial prisoner's dilemma games. Compared with the results of heterogeneous networks, we find that there is a relatively low initial fraction of cooperators to guarantee higher equilibrium cooperative level. While this interesting phenomenon is contrary to the commonly shared knowledge that higher initial fraction of cooperators can provide better environment for the evolution of cooperation. To support our outcome, we explore the time courses of cooperation and find that the whole course can be divided into two sequent stages: enduring (END) and expanding (EXP) periods. At the end of END period, thought there is a limited number of cooperator clusters left for the case of low initial setup, these clusters can smoothly expand to hold the whole system in the EXP period. However, for high initial fraction of cooperators, superfluous cooperator clusters hinder their effective expansion, which induces many remaining defectors surrounding the cooperator clusters. Moreover, through intensive analysis, we also demonstrate that when the tendency of three cooperation cluster characteristics (cluster size, cluster number and cluster shape) are consistent within END and EXP periods, the state that maximizes cooperation can be favored.

Rapid evolution of cooperation in group-living animals

BACKGROUND

It is often assumed that evolution takes place on very large timescales. Countering this assumption, rapid evolutionary dynamics are increasingly documented in biological systems, e.g. in the context of predator-prey interactions, species coexistence and invasion. It has also been shown that rapid evolution can facilitate the evolution of cooperation. In this context often evolutionary dynamics influence population dynamics, but in spatial models rapid evolutionary dynamics also emerge with constant population sizes. Currently it is not clear how well these spatial models apply to species in which individuals are not embedded in fixed spatial structures. To address this issue we employ an agent-based model of group living individuals. We investigate how positive assortment between cooperators and defectors and pay-off differences between cooperators and defectors depend on the occurrence of evolutionary dynamics.

RESULTS

We find that positive assortment and pay-off differences between cooperators and defectors differ when comparing scenarios with and without selection, which indicates that rapid evolutionary dynamics are occurring in the selection scenarios. Specifically, rapid evolution occurs because changes in positive assortment feed back on evolutionary dynamics, which crucially impacts the evolution of cooperation. At high frequencies of cooperators these feedback dynamics increase positive assortment facilitating the evolution of cooperation. In contrast, at low frequencies of cooperators rapid evolutionary dynamics lead to a decrease in assortment, which acts against the evolution of cooperation. The contrasting dynamics at low and high frequencies of cooperators create positive frequency-dependent selection.

CONCLUSIONS

Rapid evolutionary dynamics can influence the evolution of cooperation in group-living species and lead to positive frequency-dependent selection even if population size and maximum group-size are not affected by evolutionary dynamics. Rapid evolutionary dynamics can emerge in this case because sufficiently strong selective pressures allow evolutionary and demographic dynamics, and consequently also feedback between assortment and evolution, to occur on the same timescale. In particular, emerging positive frequency-dependent selection could be an important explanation for differences in cooperative behaviors among different species with similar population structures such as humans and chimpanzees.

Static cooperator-defector patterns in models of the snowdrift game played on cycle graphs

Evolutionary graph theory is an extension of evolutionary game theory in which each individual agent, represented by a node, interacts only with a subset of the entire population to which it belongs (i.e., those to which it is connected by edges). In the context of the evolution of cooperation, in which individuals playing the cooperator strategy interact with individuals playing the defector strategy and game payoffs are equated with fitness, evolutionary games on graphs lead to global standoffs (i.e., static patterns) when all cooperators in a population have the same payoff as any defectors with which they share an edge. I consider the simplest type of regular-connected graph, the cycle graph, in which every node has exactly two edges (k = 2), for the prisoner's dilemma game and the snowdrift game, the two most important pairwise games in cooperation theory. I show that for simplified payoff structures associated with these games, standoffs are only possible for two valid cost-benefit ratios in the snowdrift game. I further show that only the greater of these two cost-benefit ratios is likely to be attracting in most situations (i.e., likely to spontaneously result in a global standoff when starting from nonstandoff conditions). Numerical simulations confirm this prediction. This work contributes to our understanding of the evolution of pattern formation in games played in finite, sparsely connected populations.

Mixed strategy under generalized public goods games

The relationship between group's contribution and public goods produced often exhibits nonlinearity, which constitutes the generalized public goods game. Far less attention has been paid to how the mixed strategy evolves in such generalized games. Here, we study the effects of nonlinear production functions on the evolution of the mixed strategy in finite populations for the first time. When the group size and the population size are comparable, cooperation is doomed irrespective of the production function. Otherwise, nonlinear production functions may induce a convergent evolutionary stable strategy (CESS) or a repeller, but cannot yield the evolutionary branching. Moreover, we particularly consider three representative families of production functions, intriguingly which all display the hysteresis effect. For two families of production functions including concave and convex curves, a unique CESS or a unique repeller may occur even if the group size is two. Whereas for the third class encompassing symmetrically sigmoidal and inverse sigmoidal curves, the coexistence of a CESS and a repeller only occurs if group size is above two, and two saddle-node bifurcations appear. Our work includes some evidently different results by comparing with the evolution of continuous investment or binary strategy.

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.

Adaptive tag switching reinforces the coevolution of contingent cooperation and tag diversity

Most of the previous studies concerning the similarity-based interaction have assumed that the change of tags just happens in the imitation stage. Individuals actually can adjust their tags whenever the environments related to these tags grow nasty. We institute a spatial model to investigate the effect of the coevolution of tag and strategy on the evolution of cooperation in the context of the Prisoner's Dilemma game. Interactions just happen between tag-identical neighbors. Individuals exploited by defectors change their current tags at a certain cost. The time-scale ratio controls how fast interaction happens relatively to selection. Results show that whenever individuals have enough chance to adapt to the environment, cooperation is greatly improved even for quite large temptation to defect. Intensive exploration reveals that both little and large costs of tag switching can further favor the establishment of cooperation. Our work may add more into the literature concerning games on adaptive networks.

Mobility, fitness collection, and the breakdown of cooperation

The spatial arrangement of individuals is thought to overcome the dilemma of cooperation: When cooperators engage in clusters, they might share the benefit of cooperation while being more protected against noncooperating individuals, who benefit from cooperation but save the cost of cooperation. This is paradigmatically shown by the spatial prisoner's dilemma model. Here, we study this model in one and two spatial dimensions, but explicitly take into account that in biological setups, fitness collection and selection are separated processes occurring mostly on vastly different time scales. This separation is particularly important to understand the impact of mobility on the evolution of cooperation. We find that even small diffusive mobility strongly restricts cooperation since it enables noncooperative individuals to invade cooperative clusters. Thus, in most biological scenarios, where the mobility of competing individuals is an irrefutable fact, the spatial prisoner's dilemma alone cannot explain stable cooperation, but additional mechanisms are necessary for spatial structure to promote the evolution of cooperation. The breakdown of cooperation is analyzed in detail. We confirm the existence of a phase transition, here controlled by mobility and costs, which distinguishes between purely cooperative and noncooperative absorbing states. While in one dimension the model is in the class of the voter model, it belongs to the directed percolation universality class in two dimensions.

Consolidating birth-death and death-birth processes in structured populations

Network models extend evolutionary game theory to settings with spatial or social structure and have provided key insights on the mechanisms underlying the evolution of cooperation. However, network models have also proven sensitive to seemingly small details of the model architecture. Here we investigate two popular biologically motivated models of evolution in finite populations: Death-Birth (DB) and Birth-Death (BD) processes. In both cases reproduction is proportional to fitness and death is random; the only difference is the order of the two events at each time step. Although superficially similar, under DB cooperation may be favoured in structured populations, while under BD it never is. This is especially troubling as natural populations do not follow a strict one birth then one death regimen (or vice versa); such constraints are introduced to make models more tractable. Whether structure can promote the evolution of cooperation should not hinge on a simplifying assumption. Here, we propose a mixed rule where in each time step DB is used with probability and BD is used with probability . We derive the conditions for selection favouring cooperation under the mixed rule for all social dilemmas. We find that the only qualitatively different outcome occurs when using just BD (). This case admits a natural interpretation in terms of kin competition counterbalancing the effect of kin selection. Finally we show that, for any mixed BD-DB update and under weak selection, cooperation is never inhibited by population structure for any social dilemma, including the Snowdrift Game.

Beyond pairwise strategy updating in the prisoner's dilemma game

In spatial games players typically alter their strategy by imitating the most successful or one randomly selected neighbor. Since a single neighbor is taken as reference, the information stemming from other neighbors is neglected, which begets the consideration of alternative, possibly more realistic approaches. Here we show that strategy changes inspired not only by the performance of individual neighbors but rather by entire neighborhoods introduce a qualitatively different evolutionary dynamics that is able to support the stable existence of very small cooperative clusters. This leads to phase diagrams that differ significantly from those obtained by means of pairwise strategy updating. In particular, the survivability of cooperators is possible even by high temptations to defect and over a much wider uncertainty range. We support the simulation results by means of pair approximations and analysis of spatial patterns, which jointly highlight the importance of local information for the resolution of social dilemmas.

Approximating evolutionary dynamics on networks using a Neighbourhood Configuration model

Evolutionary dynamics have been traditionally studied on homogeneously mixed and infinitely large populations. However, real populations are finite and characterised by complex interactions among individuals. Recent studies have shown that the outcome of the evolutionary process might be significantly affected by the population structure. Although an analytic investigation of the process is possible when the contact structure of the population has a simple form, this is usually infeasible on complex structures and the use of various assumptions and approximations is necessary. In this paper, we adopt an approximation method which has been recently used for the modelling of infectious disease transmission to model evolutionary game dynamics on complex networks. Comparisons of the predictions of the model constructed with the results of computer simulations reveal the effectiveness of the method and the improved accuracy that it provides when, for example, compared to well-known pair approximation methods. This modelling framework offers a flexible way to carry out a systematic analysis of evolutionary game dynamics on graphs and to establish the link between network topology and potential system behaviours. As an example, we investigate how the Hawk and Dove strategies in a Hawk-Dove game spread in a population represented by a random regular graph, a random graph and a scale-free network, and we examine the features of the graph which affect the evolution of the population in this particular game.

If players are sparse social dilemmas are too: Importance of percolation for evolution of cooperation

Spatial reciprocity is a well known tour de force of cooperation promotion. A thorough understanding of the effects of different population densities is therefore crucial. Here we study the evolution of cooperation in social dilemmas on different interaction graphs with a certain fraction of vacant nodes. We find that sparsity may favor the resolution of social dilemmas, especially if the population density is close to the percolation threshold of the underlying graph. Regardless of the type of the governing social dilemma as well as particularities of the interaction graph, we show that under pairwise imitation the percolation threshold is a universal indicator of how dense the occupancy ought to be for cooperation to be optimally promoted. We also demonstrate that myopic updating, due to the lack of efficient spread of information via imitation, renders the reported mechanism dysfunctional, which in turn further strengthens its foundations.

Inertia in strategy switching transforms the strategy evolution

A recent experimental study [Traulsen et al., Proc. Natl. Acad. Sci. 107, 2962 (2010)] shows that human strategy updating involves both direct payoff comparison and the cost of switching strategy, which is equivalent to inertia. However, it remains largely unclear how such a predisposed inertia affects 2 × 2 games in a well-mixed population of finite size. To address this issue, the "inertia bonus" (strategy switching cost) is added to the learner payoff in the Fermi process. We find how inertia quantitatively shapes the stationary distribution and that stochastic stability under inertia exhibits three regimes, with each covering seven regions in the plane spanned by two inertia parameters. We also obtain the extended "1/3" rule with inertia and the speed criterion with inertia; these two findings hold for a population above two. We illustrate the above results in the framework of the Prisoner's Dilemma game. As inertia varies, two intriguing stationary distributions emerge: the probability of coexistence state is maximized, or those of two full states are simultaneously peaked. Our results may provide useful insights into how the inertia of changing status quo acts on the strategy evolution and, in particular, the evolution of cooperation.

Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics

Agent-based stochastic models for finite populations have recently received much attention in the game theory of evolutionary dynamics. Both the ultimate fixation and the pre-fixation transient behavior are important to a full understanding of the dynamics. In this paper, we study the transient dynamics of the well-mixed Moran process through constructing a landscape function. It is shown that the landscape playing a central theoretical "device" that integrates several lines of inquiries: the stable behavior of the replicator dynamics, the long-time fixation, and continuous diffusion approximation associated with asymptotically large population. Several issues relating to the transient dynamics are discussed: (i) multiple time scales phenomenon associated with intra- and inter-attractoral dynamics; (ii) discontinuous transition in stochastically stationary process akin to Maxwell construction in equilibrium statistical physics; and (iii) the dilemma diffusion approximation facing as a continuous approximation of the discrete evolutionary dynamics. It is found that rare events with exponentially small probabilities, corresponding to the uphill movements and barrier crossing in the landscape with multiple wells that are made possible by strong nonlinear dynamics, plays an important role in understanding the origin of the complexity in evolutionary, nonlinear biological systems.

Resolution of the stochastic strategy spatial prisoner's dilemma by means of particle swarm optimization

We study the evolution of cooperation among selfish individuals in the stochastic strategy spatial prisoner's dilemma game. We equip players with the particle swarm optimization technique, and find that it may lead to highly cooperative states even if the temptations to defect are strong. The concept of particle swarm optimization was originally introduced within a simple model of social dynamics that can describe the formation of a swarm, i.e., analogous to a swarm of bees searching for a food source. Essentially, particle swarm optimization foresees changes in the velocity profile of each player, such that the best locations are targeted and eventually occupied. In our case, each player keeps track of the highest payoff attained within a local topological neighborhood and its individual highest payoff. Thus, players make use of their own memory that keeps score of the most profitable strategy in previous actions, as well as use of the knowledge gained by the swarm as a whole, to find the best available strategy for themselves and the society. Following extensive simulations of this setup, we find a significant increase in the level of cooperation for a wide range of parameters, and also a full resolution of the prisoner's dilemma. We also demonstrate extreme efficiency of the optimization algorithm when dealing with environments that strongly favor the proliferation of defection, which in turn suggests that swarming could be an important phenomenon by means of which cooperation can be sustained even under highly unfavorable conditions. We thus present an alternative way of understanding the evolution of cooperative behavior and its ubiquitous presence in nature, and we hope that this study will be inspirational for future efforts aimed in this direction.

Feedback reciprocity mechanism promotes the cooperation of highly clustered scale-free networks

We study how the clustering coefficient influences the evolution of cooperation in scale-free public goods games. In games played by groups of individuals, triangle loops provide stronger support for mutual cooperation to resist invasion of selfish behavior than that in the absence of such loops, so that diffusion of cooperative behavior is relatively promoted. The feedback reciprocity mechanism of triangle plays a key role in facilitating cooperation in high clustered networks.