Phase transitions and volunteering in spatial public goods games

Changes in political party systems arising from conflict and transfer among political parties

Conflict that arises between two groups of different paradigms is an inevitable phenomenon, and a representative example of the conflict among different groups is a conflict phenomenon caused by competition among political parties. In this paper, we study the dynamical behavior of a political party system. Considering three major political parties, we investigate how political party systems can be changed by employing a mathematical model. By considering the transfer mechanism of recruitment as well as conflict of competition between political parties, we found that all parties are likely to coexist when both the competition and transfer between the parties are weak, or if either mechanism can occur at a relatively low level. Otherwise, a political party system is changed to a single-party system. In addition, we found that when a party system was changed into a single-party system, it appeared to be either bistable or multistable, and has been elucidate by linear stability analysis. Our results may provide insights to understand mechanisms how political party systems can be changed by conflict and transfer.

Environmental Noise Could Promote Stochastic Local Stability of Behavioral Diversity Evolution

In this Letter, we investigate stochastic stability in a two-phenotype evolutionary game model for an infinite, well-mixed population undergoing discrete, nonoverlapping generations. We assume that the fitness of a phenotype is an exponential function of its expected payoff following random pairwise interactions whose outcomes randomly fluctuate with time. We show that the stochastic local stability of a constant interior equilibrium can be promoted by the random environmental noise even if the system may display a complicated nonlinear dynamics. This result provides a new perspective for a better understanding of how environmental fluctuations may contribute to the evolution of behavioral diversity.

Evolution of joint cooperation under phenotypic variations

Effects of phenotypic variation on the species-environment systems and the evolution of cooperation under prescribed phenotypic diversity have been well addressed respectively. Interspecies interactions in the context of evolvable phenotypic diversity remain largely unconsidered. We address the evolutionary dynamics by considering evolvable phenotypic variations under group interactions. Each individual carries a capacitor of phenotypes and pays a cost proportional to its volume. A random phenotype from the capacitor is expressed and the population is thus divided into subpopulations. Group interactions happen in each of these subpopulations, respectively. Competition is global. Results show that phenotypic diversity coevolves with cooperation under a wide range of conditions and that tradeoff between expanding capacitor and rising cost leads to an optimal level of phenotypic diversity best promoting cooperation. We also find that evolved high levels of phenotypic diversity can occasionally collapse due to the invasion of defector mutants, suggesting that cooperation and phenotypic diversity can mutually reinforce each other.

Conditional punishment is a double-edged sword in promoting cooperation

Punishment is widely recognized as an effective approach for averting from exploitation by free-riders in human society. However, punishment is costly, and thus rational individuals are unwilling to take the punishing action, resulting in the second-order free-rider problem. Recent experimental study evidences that individuals prefer conditional punishment, and their punishing decision depends on other members' punishing decisions. In this work, we thus propose a theoretical model for conditional punishment and investigate how such conditional punishment influences cooperation in the public goods game. Considering conditional punishers only take the punishing action when the number of unconditional punishers exceeds a threshold number, we demonstrate that such conditional punishment induces the effect of a double-edged sword on the evolution of cooperation both in well-mixed and structured populations. Specifically, when it is relatively easy for conditional punishers to engage in the punishment activity corresponding to a low threshold value, cooperation can be promoted in comparison with the case without conditional punishment. Whereas when it is relatively difficult for conditional punishers to engage in the punishment activity corresponding to a high threshold value, cooperation is inhibited in comparison with the case without conditional punishment. Moreover, we verify that such double-edged sword effect exists in a wide range of model parameters and can be still observed in other different punishment regimes.

Critical dynamics in population vaccinating behavior

Complex adaptive systems exhibit characteristic dynamics near tipping points such as critical slowing down (declining resilience to perturbations). We studied Twitter and Google search data about measles from California and the United States before and after the 2014–2015 Disneyland, California measles outbreak. We find critical slowing down starting a few years before the outbreak. However, population response to the outbreak causes resilience to increase afterward. A mathematical model of measles transmission and population vaccine sentiment predicts the same patterns. Crucially, critical slowing down begins long before a system actually reaches a tipping point. Thus, it may be possible to develop analytical tools to detect populations at heightened risk of a future episode of widespread vaccine refusal.

Vaccine refusal can lead to renewed outbreaks of previously eliminated diseases and even delay global eradication. Vaccinating decisions exemplify a complex, coupled system where vaccinating behavior and disease dynamics influence one another. Such systems often exhibit critical phenomena—special dynamics close to a tipping point leading to a new dynamical regime. For instance, critical slowing down (declining rate of recovery from small perturbations) may emerge as a tipping point is approached. Here, we collected and geocoded tweets about measles–mumps–rubella vaccine and classified their sentiment using machine-learning algorithms. We also extracted data on measles-related Google searches. We find critical slowing down in the data at the level of California and the United States in the years before and after the 2014–2015 Disneyland, California measles outbreak. Critical slowing down starts growing appreciably several years before the Disneyland outbreak as vaccine uptake declines and the population approaches the tipping point. However, due to the adaptive nature of coupled behavior–disease systems, the population responds to the outbreak by moving away from the tipping point, causing “critical speeding up” whereby resilience to perturbations increases. A mathematical model of measles transmission and vaccine sentiment predicts the same qualitative patterns in the neighborhood of a tipping point to greatly reduced vaccine uptake and large epidemics. These results support the hypothesis that population vaccinating behavior near the disease elimination threshold is a critical phenomenon. Developing new analytical tools to detect these patterns in digital social data might help us identify populations at heightened risk of widespread vaccine refusal.

Antisocial rewarding in structured populations

Cooperation in collective action dilemmas usually breaks down in the absence of additional incentive mechanisms. This tragedy can be escaped if cooperators have the possibility to invest in reward funds that are shared exclusively among cooperators (prosocial rewarding). Yet, the presence of defectors who do not contribute to the public good but do reward themselves (antisocial rewarding) deters cooperation in the absence of additional countermeasures. A recent simulation study suggests that spatial structure is sufficient to prevent antisocial rewarding from deterring cooperation. Here we reinvestigate this issue assuming mixed strategies and weak selection on a game-theoretic model of social interactions, which we also validate using individual-based simulations. We show that increasing reward funds facilitates the maintenance of prosocial rewarding but prevents its invasion, and that spatial structure can sometimes select against the evolution of prosocial rewarding. Our results suggest that, even in spatially structured populations, additional mechanisms are required to prevent antisocial rewarding from deterring cooperation in public goods dilemmas.

Win-stay-lose-learn promotes cooperation in the prisoner's dilemma game with voluntary participation

Voluntary participation, demonstrated to be a simple yet effective mechanism to promote persistent cooperative behavior, has been extensively studied. It has also been verified that the aspiration-based win-stay-lose-learn strategy updating rule promotes the evolution of cooperation. Inspired by this well-known fact, we combine the Win-Stay-Lose-Learn updating rule with voluntary participation: Players maintain their strategies when they are satisfied, or players attempt to imitate the strategy of one randomly chosen neighbor. We find that this mechanism maintains persistent cooperative behavior, even further promotes the evolution of cooperation under certain conditions.

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.

From degree-correlated to payoff-correlated activity for an optimal resolution of social dilemmas

An active participation of players in evolutionary games depends on several factors, ranging from personal stakes to the properties of the interaction network. Diverse activity patterns thus have to be taken into account when studying the evolution of cooperation in social dilemmas. Here we study the weak prisoner's dilemma game, where the activity of each player is determined in a probabilistic manner either by its degree or by its payoff. While degree-correlated activity introduces cascading failures of cooperation that are particularly severe on scale-free networks with frequently inactive hubs, payoff-correlated activity provides a more nuanced activity profile, which ultimately hinders systemic breakdowns of cooperation. To determine optimal conditions for the evolution of cooperation, we introduce an exponential decay to payoff-correlated activity that determines how fast the activity of a player returns to its default state. We show that there exists an intermediate decay rate at which the resolution of the social dilemma is optimal. This can be explained by the emerging activity patterns of players, where the inactivity of hubs is compensated effectively by the increased activity of average-degree players, who through their collective influence in the network sustain a higher level of cooperation. The sudden drops in the fraction of cooperators observed with degree-correlated activity therefore vanish, and so does the need for the lengthy spatiotemporal reorganization of compact cooperative clusters. The absence of such asymmetric dynamic instabilities thus leads to an optimal resolution of social dilemmas, especially when the conditions for the evolution of cooperation are strongly adverse.

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

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

Evolutionary game theory using agent-based methods

Evolutionary game theory is a successful mathematical framework geared towards understanding the selective pressures that affect the evolution of the strategies of agents engaged in interactions with potential conflicts. While a mathematical treatment of the costs and benefits of decisions can predict the optimal strategy in simple settings, more realistic settings such as finite populations, non-vanishing mutations rates, stochastic decisions, communication between agents, and spatial interactions, require agent-based methods where each agent is modeled as an individual, carries its own genes that determine its decisions, and where the evolutionary outcome can only be ascertained by evolving the population of agents forward in time. While highlighting standard mathematical results, we compare those to agent-based methods that can go beyond the limitations of equations and simulate the complexity of heterogeneous populations and an ever-changing set of interactors. We conclude that agent-based methods can predict evolutionary outcomes where purely mathematical treatments cannot tread (for example in the weak selection-strong mutation limit), but that mathematics is crucial to validate the computational simulations.

Reciprocity in spatial evolutionary public goods game on double-layered network

Spatial evolutionary games have mainly been studied on a single, isolated network. However, in real world systems, many interaction topologies are not isolated but many different types of networks are inter-connected to each other. In this study, we investigate the spatial evolutionary public goods game (SEPGG) on double-layered random networks (DRN). Based on the mean-field type arguments and numerical simulations, we find that SEPGG on DRN shows very rich interesting phenomena, especially, depending on the size of each layer, intra-connectivity, and inter-connected couplings, the network reciprocity of SEPGG on DRN can be drastically enhanced through the inter-connected coupling. Furthermore, SEPGG on DRN can provide a more general framework which includes the evolutionary dynamics on multiplex networks and inter-connected networks at the same time.

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.

Effect of spatial constraints on Hardy-Weinberg equilibrium

Panmixia is a key issue in maintaining genetic diversity, which facilitates evolutionary potential during environmental changes. Additionally, conservation biologists suggest the importance of avoiding small or subdivided populations, which are prone to losing genetic diversity. In this paper, computer simulations were performed to the genetic drift of neutral alleles in random mating populations with or without spatial constraints by randomly choosing a mate among the closest neighbours. The results demonstrated that the number of generations required for the neutral allele to become homozygous (T_h) varied proportionally to the population size and also strongly correlated with spatial constraints. The average T_h for populations of the same size with spatial constraints was approximately one-and-a-half times longer than without constraints. With spatial constraints, homozygous population clusters formed, which reduced local diversity but preserved global diversity. Therefore, panmixia might be harmful in preserving the genetic diversity of an entire population. The results also suggested that the gene flow or gene exchange among the subdivided populations must be carefully processed to restrict diseases transmission or death during transportation and to monitor the genetic diversity. The application of this concept to similar systems, such as information transfer among peers, is also discussed.

Computational complexity of ecological and evolutionary spatial dynamics

There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).

Cooperation in group-structured populations with two layers of interactions

Recently there has been a growing interest in studying multiplex networks where individuals are structured in multiple network layers. Previous agent-based simulations of games on multiplex networks reveal rich dynamics arising from interdependency of interactions along each network layer, yet there is little known about analytical conditions for cooperation to evolve thereof. Here we aim to tackle this issue by calculating the evolutionary dynamics of cooperation in group-structured populations with two layers of interactions. In our model, an individual is engaged in two layers of group interactions simultaneously and uses unrelated strategies across layers. Evolutionary competition of individuals is determined by the total payoffs accrued from two layers of interactions. We also consider migration which allows individuals to move to a new group within each layer. An approach combining the coalescence theory with the theory of random walks is established to overcome the analytical difficulty upon local migration. We obtain the exact results for all “isotropic” migration patterns, particularly for migration tuned with varying ranges. When the two layers use one game, the optimal migration ranges are proved identical across layers and become smaller as the migration probability grows.

Benefits of tolerance in public goods games

Leaving the joint enterprise when defection is unveiled is always a viable option to avoid being exploited. Although loner strategy helps the population not to be trapped into the tragedy of the commons state, it could offer only a modest income for nonparticipants. In this paper we demonstrate that showing some tolerance toward defectors could not only save cooperation in harsh environments but in fact results in a surprisingly high average payoff for group members in public goods games. Phase diagrams and the underlying spatial patterns reveal the high complexity of evolving states where cyclic dominant strategies or two-strategy alliances can characterize the final state of evolution. We identify microscopic mechanisms which are responsible for the superiority of global solutions containing tolerant players. This phenomenon is robust and can be observed both in well-mixed and in structured populations highlighting the importance of tolerance in our everyday life.

Zero-Determinant Strategies in Iterated Public Goods Game

Recently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner’s Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents’ payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

A double-edged sword: Benefits and pitfalls of heterogeneous punishment in evolutionary inspection games

As a simple model for criminal behavior, the traditional two-strategy inspection game yields counterintuitive results that fail to describe empirical data. The latter shows that crime is often recurrent, and that crime rates do not respond linearly to mitigation attempts. A more apt model entails ordinary people who neither commit nor sanction crime as the third strategy besides the criminals and punishers. Since ordinary people free-ride on the sanctioning efforts of punishers, they may introduce cyclic dominance that enables the coexistence of all three competing strategies. In this setup ordinary individuals become the biggest impediment to crime abatement. We therefore also consider heterogeneous punisher strategies, which seek to reduce their investment into fighting crime in order to attain a more competitive payoff. We show that this diversity of punishment leads to an explosion of complexity in the system, where the benefits and pitfalls of criminal behavior are revealed in the most unexpected ways. Due to the raise and fall of different alliances no less than six consecutive phase transitions occur in dependence on solely the temptation to succumb to criminal behavior, leading the population from ordinary people-dominated across punisher-dominated to crime-dominated phases, yet always failing to abolish crime completely.

Evolutionary dynamics for persistent cooperation in structured populations

The emergence and maintenance of cooperative behavior is a fascinating topic in evolutionary biology and social science. The public goods game (PGG) is a paradigm for exploring cooperative behavior. In PGG, the total resulting payoff is divided equally among all participants. This feature still leads to the dominance of defection without substantially magnifying the public good by a multiplying factor. Much effort has been made to explain the evolution of cooperative strategies, including a recent model in which only a portion of the total benefit is shared by all the players through introducing a new strategy named persistent cooperation. A persistent cooperator is a contributor who is willing to pay a second cost to retrieve the remaining portion of the payoff contributed by themselves. In a previous study, this model was analyzed in the framework of well-mixed populations. This paper focuses on discussing the persistent cooperation in lattice-structured populations. The evolutionary dynamics of the structured populations consisting of three types of competing players (pure cooperators, defectors, and persistent cooperators) are revealed by theoretical analysis and numerical simulations. In particular, the approximate expressions of fixation probabilities for strategies are derived on one-dimensional lattices. The phase diagrams of stationary states, and the evolution of frequencies and spatial patterns for strategies are illustrated on both one-dimensional and square lattices by simulations. Our results are consistent with the general observation that, at least in most situations, a structured population facilitates the evolution of cooperation. Specifically, here we find that the existence of persistent cooperators greatly suppresses the spreading of defectors under more relaxed conditions in structured populations compared to that obtained in well-mixed populations.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

Synchronization and extinction in cyclic games with mixed strategies

We consider cyclic Lotka-Volterra models with three and four strategies where at every interaction agents play a strategy using a time-dependent probability distribution. Agents learn from a loss by reducing the probability to play a losing strategy at the next interaction. For that, an agent is described as an urn containing β balls of three and four types, respectively, where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy. Using both mean-field rate equations and numerical simulations, we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time-dependent probability distributions. For the three-strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution. For large values of β, yielding a good approximation to a continuous distribution, spatially synchronized temporal oscillations dominate the system. For the four-strategy game the system is always neutrally stable, but different regimes emerge, depending on the size of the system and the level of discretization.

Social exclusion in finite populations

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

Spatial evolutionary public goods game on complete graph and dense complex networks

We study the spatial evolutionary public goods game (SEPGG) with voluntary or optional participation on a complete graph (CG) and on dense networks. Based on analyses of the SEPGG rate equation on finite CG, we find that SEPGG has two stable states depending on the value of multiplication factor r, illustrating how the "tragedy of the commons" and "an anomalous state without any active participants" occurs in real-life situations. When r is low (<<), the state with only loners is stable, and the state with only defectors is stable when r is high (>>). We also derive the exact scaling relation for r*. All of the results are confirmed by numerical simulation. Furthermore, we find that a cooperator-dominant state emerges when the number of participants or the mean degree, 〈k〉, decreases. We also investigate the scaling dependence of the emergence of cooperation on r and 〈k〉. These results show how "tragedy of the commons" disappears when cooperation between egoistic individuals without any additional socioeconomic punishment increases.

Dependency links can hinder the evolution of cooperation in the prisoner's dilemma game on lattices and networks

Networks with dependency links are more vulnerable when facing the attacks. Recent research also has demonstrated that the interdependent groups support the spreading of cooperation. We study the prisoner's dilemma games on spatial networks with dependency links, in which a fraction of individual pairs is selected to depend on each other. The dependency individuals can gain an extra payoff whose value is between the payoff of mutual cooperation and the value of temptation to defect. Thus, this mechanism reflects that the dependency relation is stronger than the relation of ordinary mutual cooperation, but it is not large enough to cause the defection of the dependency pair. We show that the dependence of individuals hinders, promotes and never affects the cooperation on regular ring networks, square lattice, random and scale-free networks, respectively. The results for the square lattice and regular ring networks are demonstrated by the pair approximation.

Impact of small groups with heterogeneous preference on behavioral evolution in population evacuation

Up to now, there have been a great number of mechanisms to explain the individual behavior and population traits, which seem of particular significance in evolutionary biology and social behavior analysis. Among them, small groups and heterogeneity are two useful frameworks to the above issue. However, vast majority of existing works separately consider both scenarios, which is inconsistent with realistic cases in our life. Here we propose the evolutionary games of heterogeneous small groups (namely, different small groups possess different preferences to dilemma) to study the collective behavior in population evacuation. Importantly, players usually face completely different dilemmas inside and outside the small groups. By means of numerous computation simulations, it is unveiled that the ratio of players in one certain small group directly decides the final behavior of the whole population. Moreover, it can also be concluded that heterogeneous degree of preference for different small groups plays a key role in the behavior traits of the system, which may validate some realistic social observations. The proposed framework is thus universally applicable and may shed new light into the solution of social dilemmas.

Effect of heterogeneous investments on the evolution of cooperation in spatial public goods game

Understanding the emergence of cooperation in spatial public goods game remains a grand challenge across disciplines. In most previous studies, it is assumed that the investments of all the cooperators are identical, and often equal to 1. However, it is worth mentioning that players are diverse and heterogeneous when choosing actions in the rapidly developing modern society and researchers have shown more interest to the heterogeneity of players recently. For modeling the heterogeneous players without loss of generality, it is assumed in this work that the investment of a cooperator is a random variable with uniform distribution, the mean value of which is equal to 1. The results of extensive numerical simulations convincingly indicate that heterogeneous investments can promote cooperation. Specifically, a large value of the variance of the random variable can decrease the two critical values for the result of behavioral evolution effectively. Moreover, the larger the variance is, the better the promotion effect will be. In addition, this article has discussed the impact of heterogeneous investments when the coevolution of both strategy and investment is taken into account. Comparing the promotion effect of coevolution of strategy and investment with that of strategy imitation only, we can conclude that the coevolution of strategy and investment decreases the asymptotic fraction of cooperators by weakening the heterogeneity of investments, which further demonstrates that heterogeneous investments can promote cooperation in spatial public goods game.

Cyclic dominance in evolutionary games: a review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

Stochastic evolution of staying together

Staying together means that replicating units do not separate after reproduction, but remain attached to each other or in close proximity. Staying together is a driving force for evolution of complexity, including the evolution of multi-cellularity and eusociality. We analyze the fixation probability of a mutant that has the ability to stay together. We assume that the size of the complex affects the reproductive rate of its units and the probability of staying together. We examine the combined effect of natural selection and random drift on the emergence of staying together in a finite sized population. The number of states in the underlying stochastic process is an exponential function of population size. We develop a framework for any intensity of selection and give closed form solutions for special cases. We derive general results for the limit of weak selection.

Evolution of staying together in the context of diffusible public goods

We study the coevolution of staying together and cooperation. Staying together means that replicating units do not separate after reproduction, but remain in proximity. For example, following cell division the two daughter cells may not fully separate but stay attached to each other. Repeated cell division thereby can lead to a simple multi-cellular complex. We assume that cooperators generate a diffusible public good, which can be absorbed by any cell in the system. The production of the public good entails a cost, while the absorption leads to a benefit. Defectors produce no public good. Defectors have a selective advantage unless a mechanism for evolution of cooperation is at work. Here we explore the idea that the public good produced by a cooperating cell is absorbed by cells of the same complex with a probability that depends on the size of the complex. Larger complexes are better at absorbing the public goods produced by their own individuals. We derive analytical conditions for the evolution of staying together, thereby studying the coevolution of clustering and cooperation. If cooperators and defectors differ in their intrinsic efficiency to absorb the public good, then we find multiple stable equilibria and the possibility for coexistence between cooperators and defectors. Finally we study the implications of disadvantages that might arise if complexes become too large.

Bluffing promotes overconfidence on social networks

The overconfidence, a well-established bias, in fact leads to unrealistic expectations or faulty assessment. So it remains puzzling why such psychology of self-deception is stabilized in human society. To investigate this problem, we draw lessons from evolutionary game theory which provides a theoretical framework to address the subtleties of cooperation among selfish individuals. Here we propose a spatial resource competition model showing that, counter-intuitively, moderate values rather than large values of resource-to-cost ratio boost overconfidence level most effectively. In contrast to theoretical results in infinite well-mixed populations, network plays a role both as a “catalyst” and a “depressant” in the spreading of overconfidence, especially when resource-to-cost ratio is in a certain range. Moreover, when bluffing is taken into consideration, overconfidence evolves to a higher level to counteract its detrimental effect, which may well explain the prosperity of this “erroneous” psychology.

Optional games on cycles and complete graphs

We study stochastic evolution of optional games on simple graphs. There are two strategies, A and B, whose interaction is described by a general payoff matrix. In addition, there are one or several possibilities to opt out from the game by adopting loner strategies. Optional games lead to relaxed social dilemmas. Here we explore the interaction between spatial structure and optional games. We find that increasing the number of loner strategies (or equivalently increasing mutational bias toward loner strategies) facilitates evolution of cooperation both in well-mixed and in structured populations. We derive various limits for weak selection and large population size. For some cases we derive analytic results for strong selection. We also analyze strategy selection numerically for finite selection intensity and discuss combined effects of optionality and spatial structure.

Evolutionary prisoner's dilemma on evolving random networks

In this paper, we investigate the evolution of cooperation in the spatial prisoner's dilemma game by incorporating partner choice into the framework of evolutionary game theory. Our research shows that the introduction of partner choice can notably promote the cooperative behavior in the prisoner's dilemma game. All the players are more likely to play the game with cooperators, which makes it easier for cooperators to form alliances. In particular, the system will be dominated completely by cooperators in a comfortable environment (i.e., with lower survival cost) because the cooperators can get more benefits than the defectors in this case due to their good reputation. In addition, we have found that the sustenance of cooperators improves notably as well in the snowdrift game and the stag-hunt game due to this introduction.

Role of investment heterogeneity in the cooperation on spatial public goods game

Public cooperation plays a significant role in the survival and maintenance of biological species, to elucidate its origin thus becomes an interesting question from various disciplines. Through long-term development, the public goods game has proven to be a useful tool, where cooperator making contribution can beat again the free-rides. Differentiating from the traditional homogeneous investment, individual trend of making contribution is more likely affected by the investment level of his neighborhood. Based on this fact, we here investigate the impact of heterogeneous investment on public cooperation, where the investment sum is mapped to the proportion of cooperators determined by parameter α. Interestingly, we find, irrespective of interaction networks, that the increment of α (increment of heterogeneous investment) is beneficial for promoting cooperation and even guarantees the complete cooperation dominance under weak replication factor. While this promotion effect can be attributed to the formation of more robust cooperator clusters and shortening END period. Moreover, we find that this simple mechanism can change the potential interaction network, which results in the change of phase diagrams. We hope that our work may shed light on the understanding of the cooperative behavior in other social dilemmas.

Social dilemma alleviated by sharing the gains with immediate neighbors

We study the evolution of cooperation in the evolutionary spatial prisoner's dilemma game (PDG) and snowdrift game (SG), within which a fraction α of the payoffs of each player gained from direct game interactions is shared equally by the immediate neighbors. The magnitude of the parameter α therefore characterizes the degree of the relatedness among the neighboring players. By means of extensive Monte Carlo simulations as well as an extended mean-field approximation method, we trace the frequency of cooperation in the stationary state. We find that plugging into relatedness can significantly promote the evolution of cooperation in the context of both studied games. Unexpectedly, cooperation can be more readily established in the spatial PDG than that in the spatial SG, given that the degree of relatedness and the cost-to-benefit ratio of mutual cooperation are properly formulated. The relevance of our model with the stakeholder theory is also briefly discussed.

Understanding recurrent crime as system-immanent collective behavior

Containing the spreading of crime is a major challenge for society. Yet, since thousands of years, no effective strategy has been found to overcome crime. To the contrary, empirical evidence shows that crime is recurrent, a fact that is not captured well by rational choice theories of crime. According to these, strong enough punishment should prevent crime from happening. To gain a better understanding of the relationship between crime and punishment, we consider that the latter requires prior discovery of illicit behavior and study a spatial version of the inspection game. Simulations reveal the spontaneous emergence of cyclic dominance between "criminals", "inspectors", and "ordinary people" as a consequence of spatial interactions. Such cycles dominate the evolutionary process, in particular when the temptation to commit crime or the cost of inspection are low or moderate. Yet, there are also critical parameter values beyond which cycles cease to exist and the population is dominated either by a stable mixture of criminals and inspectors or one of these two strategies alone. Both continuous and discontinuous phase transitions to different final states are possible, indicating that successful strategies to contain crime can be very much counter-intuitive and complex. Our results demonstrate that spatial interactions are crucial for the evolutionary outcome of the inspection game, and they also reveal why criminal behavior is likely to be recurrent rather than evolving towards an equilibrium with monotonous parameter dependencies.

Evolution of collective action in adaptive social structures

Many problems in nature can be conveniently framed as a problem of evolution of collective cooperative behaviour, often modelled resorting to the tools of evolutionary game theory in well-mixed populations, combined with an appropriate N-person dilemma. Yet, the well-mixed assumption fails to describe the population dynamics whenever individuals have a say in deciding which groups they will participate. Here we propose a simple model in which dynamical group formation is described as a result of a topological evolution of a social network of interactions. We show analytically how evolutionary dynamics under public goods games in finite adaptive networks can be effectively transformed into a N-Person dilemma involving both coordination and co-existence. Such dynamics would be impossible to foresee from more conventional 2-person interactions as well as from descriptions based on infinite, well-mixed populations. Finally, we show how stochastic effects help rendering cooperation viable, promoting polymorphic configurations in which cooperators prevail.

Costly advertising and the evolution of cooperation

In this paper, I investigate the co-evolution of fast and slow strategy spread and game strategies in populations of spatially distributed agents engaged in a one off evolutionary dilemma game. Agents are characterized by a pair of traits, a game strategy (cooperate or defect) and a binary ‘advertising’ strategy (advertise or don’t advertise). Advertising, which comes at a cost , allows investment into faster propagation of the agents’ traits to adjacent individuals. Importantly, game strategy and advertising strategy are subject to the same evolutionary mechanism. Via analytical reasoning and numerical simulations I demonstrate that a range of advertising costs exists, such that the prevalence of cooperation is significantly enhanced through co-evolution. Linking costly replication to the success of cooperators exposes a novel co-evolutionary mechanism that might contribute towards a better understanding of the origins of cooperation-supporting heterogeneity in agent populations.

Evolution of cooperation in a heterogeneous graph: fixation probabilities under weak selection

It has been shown that natural selection favors cooperation in a homogenous graph if the benefit-to-cost ratio exceeds the degree of the graph. However, most graphs related to interactions in real populations are heterogeneous, in which some individuals have many more neighbors than others. In this paper, we introduce a new state variable to measure the time evolution of cooperation in a heterogeneous graph. Based on the diffusion approximation, we find that the fixation probability of a single cooperator depends crucially on the number of its neighbors. Under weak selection, a cooperator with more neighbors has a larger probability of fixation in the population. We then investigate the average fixation probability of a randomly chosen cooperator. If a cooperator pays a cost for each of its neighbors (the so called fixed cost per game case), natural selection favors cooperation if the benefit-to-cost ratio is larger than the average degree. In contrast, if a cooperator pays a fixed cost and all its neighbors share the benefit (the fixed cost per individual case), cooperation is favored if the benefit-to-cost ratio is larger than the harmonic mean of the degree distribution. Moreover, increasing the graph heterogeneity will reduce the effect of natural selection.

The increased risk of joint venture promotes social cooperation

The joint venture of many members is common both in animal world and human society. In these public enterprizes, highly cooperative groups are more likely to while low cooperative groups are still possible but not probable to succeed. Existent literature mostly focuses on the traditional public goods game, in which cooperators create public wealth unconditionally and benefit all group members unbiasedly. We here institute a model addressing this public goods dilemma with incorporating the public resource foraging failure risk. Risk-averse individuals tend to lead a autarkic life, while risk-preferential ones tend to participate in the risky public goods game. For participants, group's success relies on its cooperativeness, with increasing contribution leading to increasing success likelihood. We introduce a function with one tunable parameter to describe the risk removal pattern and study in detail three representative classes. Analytical results show that the widely replicated population dynamics of cyclical dominance of loner, cooperator and defector disappear, while most of the time loners act as savors while eventually they also disappear. Depending on the way that group's success relies on its cooperativeness, either cooperators pervade the entire population or they coexist with defectors. Even in the later case, cooperators still hold salient superiority in number as some defectors also survive by parasitizing. The harder the joint venture succeeds, the higher level of cooperation once cooperators can win the evolutionary race. Our work may enrich the literature concerning the risky public goods games.

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.

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.

Evolutionary dynamics of group interactions on structured populations: a review

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.