Impact of critical mass on the evolution of cooperation in spatial public goods games

Evolutionary dynamics of continuous public goods games in structured populations

Over the past few decades, many works have studied the evolutionary dynamics of continuous games. However, previous works have primarily focused on two-player games with pairwise interactions. Indeed, group interactions rather than pairwise interactions are usually found in real situations. The public goods game serves as a paradigm of multi-player interactions. Notably, various types of benefit functions are typically considered in public goods games, including linear, saturating, and sigmoid functions. Thus far, the evolutionary dynamics of cooperation in continuous public goods games with these benefit functions remain unknown in structured populations. In this paper, we consider the continuous public goods game in structured populations. By employing the pair approximation approach, we derive the analytical expressions for invasion fitness. Furthermore, we explore the adaptive dynamics of cooperative investments in the game with various benefit functions. First, for the linear public goods game, we find that there is no singular strategy, and the cooperative investments evolve to either the maximum or minimum depending on the benefit-to-cost ratio. Subsequently, we examine the game with saturating benefit functions and demonstrate the potential existence of an evolutionarily stable strategy (ESS). Additionally, for the game with the sigmoid benefit function, we observe that the evolutionary outcomes are closely related to the threshold value. When the threshold is small, a unique ESS emerges. For intermediate threshold values, both the ESS and repellor singular strategies can coexist. When the threshold value is large, a unique repellor displays. Finally, we perform individual-based simulations to validate our theoretical results.

Evolutionary dynamics in state-feedback public goods games with peer punishment

Public goods game serves as a valuable paradigm for studying the challenges of collective cooperation in human and natural societies. Peer punishment is often considered an effective incentive for promoting cooperation in such contexts. However, previous related studies have mostly ignored the positive feedback effect of collective contributions on individual payoffs. In this work, we explore global and local state-feedback, where the multiplication factor is positively correlated with the frequency of contributors in the entire population or within the game group, respectively. By using replicator dynamics in an infinite well-mixed population, we reveal that state-based feedback plays a crucial role in alleviating the cooperative dilemma by enhancing and sustaining cooperation compared to the feedback-free case. Moreover, when the feedback strength is sufficiently strong or the baseline multiplication factor is sufficiently high, the system with local state-feedback provides full cooperation, hence supporting the "think globally, act locally" principle. Besides, we show that the second-order free-rider problem can be partially mitigated under certain conditions when the state-feedback is employed. Importantly, these results remain robust with respect to variations in punishment cost and fine.

Evolution of cooperation under threshold public goods game

When the number of cooperators does not reach the collective target, resulting in the collective risk social dilemma, the self-organizing behavior of the group leads to the loss of collective interest and the government intervention leads to the increase of collective interest. For these two situations, we study the evolution of cooperation under threshold public goods game in well-mixed population. The results show that the introduction of the threshold makes it possible to generate complex dynamics with two interior equilibria in the replication equation. For self-organizing behavior, increasing the threshold is a double-edged sword. Although raising the threshold may increase the level of cooperation, the risk of dominance of defection also increases. For government intervention behavior, increasing the threshold significantly promotes cooperation. The common result is that individuals tend to free-ride when the group size increases. In addition, the greater the impact of the two behaviors on collective interest, the more conducive to promoting cooperation. When there are fewer initial cooperators, the government should intervene in time to generate cooperation. When there are more initial cooperators, self-organizing behavior is more conducive to attracting cooperators than government intervention. We hope that the model and results proposed in this paper can contribute to addressing the challenges posed by global issues such as climate change and the degradation of the ecological environment.

Enhanced cooperation in multiplayer snowdrift games with random and dynamic groupings

An analytically tractable generalization of the N-person snowdrift (NSG) game that illustrates how cooperation can be enhanced is proposed and studied. The number of players competing within a NSG varies from one time step to another. Exact equations governing the frequency of cooperation f_{c}(r) as a function of the cost-to-benefit ratio r within an imitation strategy updating scheme are presented. For group sizes g uniformly distributed within the range g∈[1,g_{m}], an analytic formula for the critical value r_{c}(g_{m}), below which the system evolves into a totally cooperative (AllC) state, is derived. In contrast, a fixed group size NSG does not support an AllC state. The result r_{c}(g_{m}) requires the presence of sole-player groups and involves the inverse of the harmonic numbers and, more generally, the inverse first moment of the group size distribution. For r>r_{c}(g_{m}), the equation that determines the dynamical mixed states f_{c}(r) is given, with exact solutions existing for g_{m}≤5. The exact treatment allows the study of the phase boundary between the AllC state and the mixed states. The analytic results are checked against simulation results and exact agreements are demonstrated. The analytic form of the critical r_{c}(g_{m}) illustrates the necessity of having groups of a sole player in the evolutionary process. This result is supported by simulations with group sizes excluding the sole groups for which no AllC state emerges. A physically transparent picture of the importance of the sole players in inducing an AllC state is further presented based on the last surviving pattern before the AllC state is attained. The exact expression r_{c}(g_{m}) turns out to remain valid for nonuniform group-size distributions. Our analytical tractable generalization, therefore, sheds light on how a competing environment with variable group sizes could enhance cooperation and induce an AllC state.

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

BACKGROUND

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

RESULTS

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

CONCLUSIONS

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

Two-dimensional adaptive dynamics of evolutionary public goods games: finite-size effects on fixation probability and branching time

Public goods games (PGGs) describe situations in which individuals contribute to a good at a private cost, but others can free-ride by receiving a share of the public benefit at no cost. The game occurs within local neighbourhoods, which are subsets of the whole population. Free-riding and maximal production are two extremes of a continuous spectrum of traits. We study the adaptive dynamics of production and neighbourhood size. We allow the public good production and the neighbourhood size to coevolve and observe evolutionary branching. We explain how an initially monomorphic population undergoes evolutionary branching in two dimensions to become a dimorphic population characterized by extremes of the spectrum of trait values. We find that population size plays a crucial role in determining the final state of the population. Small populations may not branch or may be subject to extinction of a subpopulation after branching. In small populations, stochastic effects become important and we calculate the probability of subpopulation extinction. Our work elucidates the evolutionary origins of heterogeneity in local PGGs among individuals of two traits (production and neighbourhood size), and the effects of stochasticity in two-dimensional trait space, where novel effects emerge.

High thresholds encouraging the evolution of cooperation in threshold public-good games

For a well-mixed population, we consider a threshold public good game where group members only obtain benefits from a public good if a sufficiently large number of them cooperates. We investigate the effect of an increase in the threshold on the level of cooperation that evolves. It is shown that for sufficiently large participation costs, the level of cooperation is higher for low and for high thresholds, than it is for intermediate thresholds – where in the latter case cooperation may not evolve at all. The counterintuitive effect where an increase in the threshold from an intermediate to a high one decreases the probability of cooperation, is related to the so-called common-enemy hypothesis of the evolution of cooperation. We further apply our analysis to assess the relative weight of different game types across the parameter space, and show that game types where either a small, or a large fraction of the population evolves as cooperators, receive more weight compared to game types where an intermediate fraction of cooperators evolves.

Synergistic third-party rewarding and punishment in the public goods game

We study the evolution of cooperation in the spatial public goods game in the presence of third-party rewarding and punishment. The third party executes public intervention, punishing groups where cooperation is weak and rewarding groups where cooperation is strong. We consider four different scenarios to determine what works best for cooperation, in particular, neither rewarding nor punishment, only rewarding, only punishment or both rewarding and punishment. We observe strong synergistic effects when rewarding and punishment are simultaneously applied, which are absent if neither of the two incentives or just each individual incentive is applied by the third party. We find that public cooperation can be sustained at comparatively low third-party costs under adverse conditions, which is impossible if just positive or negative incentives are applied. We also examine the impact of defection tolerance and application frequency, showing that the higher the tolerance and the frequency of rewarding and punishment, the more cooperation thrives. Phase diagrams and characteristic spatial distributions of strategies are presented to corroborate these results, which will hopefully prove useful for more efficient public policies in support of cooperation in social dilemmas.

The role of memory in human strategy updating in optional public goods game

Most research suggests that humans can optimize their behavior by imitating other humans. However, it remains unclear whether humans actually imitate others in real-life situations. To address this question, we conducted spatial public goods experiments with voluntary participation. In direct contrast to the prevailing view, the results of our experiments show that imitation plays an insignificant role in the decision making process. Furthermore, we found that the nature of human decision making relied more on their performances in the game's earlier history rather than the performance they observed in others. The action that gained better results in one's own history had a higher chance to be adopted even if this action did not work for the others.

Dynamics of multiplayer games on complex networks using territorial interactions

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

Reciprocity-based cooperative phalanx maintained by overconfident players

According to the evolutionary game theory principle, a strategy representing a higher payoff can spread among competitors. But there are cases when a player consistently overestimates or underestimates her own payoff, which undermines proper comparison. Interestingly, both underconfident and overconfident individuals are capable of elevating the cooperation level significantly. While former players stimulate a local coordination of strategies, the presence of overconfident individuals enhances the spatial reciprocity mechanism. In both cases the propagations of competing strategies are influenced in a biased way resulting in a cooperation supporting environment. These effects are strongly related to the nonlinear character of invasion probabilities which is a common and frequently observed feature of microscopic dynamics.

Divide-and-conquer Tournament on Social Networks

In social gaming networks, previous studies extensively investigated the influence of a variety of strategies on reciprocal behaviors in the prisoner's dilemma game. The studied frameworks range from the case that an individual uniformly cooperates or defects with all social contacts, to the recently reported divide-and-conquer games, where an individual can choose a particular move to play with each neighbor. In this paper, we investigate a divide-and-conquer tournament among 14 well-known strategies on social gaming networks. In the tournament, an individual's fitness is measured by accumulated and average payoff aggregated for a certain number of rounds. On the base of their fitness, the evolution of the population follows a local learning mechanism. Our observation indicates that the distribution of individuals adopting a strategy in degree ranking fundamentally changes the frequency of the strategy. In the divide-and-conquer gaming networks, our result suggests that the connectivity in social networks and strategy are two key factors that govern the evolution of the population.

Evolutionary dynamics and the evolution of multiplayer cooperation in a subdivided population

The classical models of evolution have been developed to incorporate structured populations using evolutionary graph theory and, more recently, a new framework has been developed to allow for more flexible population structures which potentially change through time and can accommodate multiplayer games with variable group sizes. In this paper we extend this work in three key ways. Firstly by developing a complete set of evolutionary dynamics so that the range of dynamic processes used in classical evolutionary graph theory can be applied. Secondly, by building upon previous models to allow for a general subpopulation structure, where all subpopulation members have a common movement distribution. Subpopulations can have varying levels of stability, represented by the proportion of interactions occurring between subpopulation members; in our representation of the population all subpopulation members are represented by a single vertex. In conjunction with this we extend the important concept of temperature (the temperature of a vertex is the sum of all the weights coming into that vertex; generally, the higher the temperature, the higher the rate of turnover of individuals at a vertex). Finally, we have used these new developments to consider the evolution of cooperation in a class of populations which possess this subpopulation structure using a multiplayer public goods game. We show that cooperation can evolve providing that subpopulations are sufficiently stable, with the smaller the subpopulations the easier it is for cooperation to evolve. We introduce a new concept of temperature, namely "subgroup temperature", which can be used to explain our results.

When agreement-accepting free-riders are a necessary evil for the evolution of cooperation

Agreements and commitments have provided a novel mechanism to promote cooperation in social dilemmas in both one-shot and repeated games. Individuals requesting others to commit to cooperate (proposers) incur a cost, while their co-players are not necessarily required to pay any, allowing them to free-ride on the proposal investment cost (acceptors). Although there is a clear complementarity in these behaviours, no dynamic evidence is currently available that proves that they coexist in different forms of commitment creation. Using a stochastic evolutionary model allowing for mixed population states, we identify non-trivial roles of acceptors as well as the importance of intention recognition in commitments. In the one-shot prisoner's dilemma, alliances between proposers and acceptors are necessary to isolate defectors when proposers do not know the acceptance intentions of the others. However, when the intentions are clear beforehand, the proposers can emerge by themselves. In repeated games with noise, the incapacity of proposers and acceptors to set up alliances makes the emergence of the first harder whenever the latter are present. As a result, acceptors will exploit proposers and take over the population when an apology-forgiveness mechanism with too low apology cost is introduced, and hence reduce the overall cooperation level.

Cooperation Survives and Cheating Pays in a Dynamic Network Structure with Unreliable Reputation

In a networked society like ours, reputation is an indispensable tool to guide decisions about social or economic interactions with individuals otherwise unknown. Usually, information about prospective counterparts is incomplete, often being limited to an average success rate. Uncertainty on reputation is further increased by fraud, which is increasingly becoming a cause of concern. To address these issues, we have designed an experiment based on the Prisoner’s Dilemma as a model for social interactions. Participants could spend money to have their observable cooperativeness increased. We find that the aggregate cooperation level is practically unchanged, i.e., global behavior does not seem to be affected by unreliable reputations. However, at the individual level we find two distinct types of behavior, one of reliable subjects and one of cheaters, where the latter artificially fake their reputation in almost every interaction. Cheaters end up being better off than honest individuals, who not only keep their true reputation but are also more cooperative. In practice, this results in honest subjects paying the costs of fraud as cheaters earn the same as in a truthful environment. These findings point to the importance of ensuring the truthfulness of reputation for a more equitable and fair society.

Unfavorable Individuals in Social Gaming Networks

In social gaming networks, the current research focus has been on the origin of widespread reciprocal behaviors when individuals play non-cooperative games. In this paper, we investigate the topological properties of unfavorable individuals in evolutionary games. The unfavorable individuals are defined as the individuals gaining the lowest average payoff in a round of game. Since the average payoff is normally considered as a measure of fitness, the unfavorable individuals are very likely to be eliminated or change their strategy updating rules from a Darwinian perspective. Considering that humans can hardly adopt a unified strategy to play with their neighbors, we propose a divide-and-conquer game model, where individuals can interact with their neighbors in the network with appropriate strategies. We test and compare a series of highly rational strategy updating rules. In the tested scenarios, our analytical and simulation results surprisingly reveal that the less-connected individuals in degree-heterogeneous networks are more likely to become the unfavorable individuals. Our finding suggests that the connectivity of individuals as a social capital fundamentally changes the gaming environment. Our model, therefore, provides a theoretical framework for further understanding the social gaming networks.

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.

Group Size Effect on Cooperation in One-Shot Social Dilemmas II: Curvilinear Effect

In a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.

How insurance affects altruistic provision in threshold public goods games

The occurrence and maintenance of cooperative behaviors in public goods systems have attracted great research attention across multiple disciplines. A threshold public goods game requires a minimum amount of contributions to be collected from a group of individuals for provision to occur. Here we extend the common binary-strategy combination of cooperation and defection by adding a third strategy, called insured cooperation, which corresponds to buying an insurance covering the potential loss resulted from the unsuccessful public goods game. Particularly, only the contributing agents can opt to be insured, which is an effort decreasing the amount of the potential loss occurring. Theoretical computations suggest that when agents face the potential aggregate risk in threshold public goods games, more contributions occur with increasing compensation from insurance. Moreover, permitting the adoption of insurance significantly enhances individual contributions and facilitates provision, especially when the required threshold is high. This work also relates the strategy competition outcomes to different allocation rules once the resulted contributions exceed the threshold point in populations nested within a dilemma.

Group size effect on cooperation in one-shot social dilemmas

Social dilemmas are central to human society. Depletion of natural resources, climate protection, security of energy supply, and workplace collaborations are all examples of social dilemmas. Since cooperative behaviour in a social dilemma is individually costly, Nash equilibrium predicts that humans should not cooperate. Yet experimental studies show that people do cooperate even in anonymous one-shot interactions. In spite of the large number of participants in many modern social dilemmas, little is known about the effect of group size on cooperation. Does larger group size favour or prevent cooperation? We address this problem both experimentally and theoretically. Experimentally, we find that there is no general answer: it depends on the strategic situation. Specifically, we find that larger groups are more cooperative in the Public Goods game, but less cooperative in the N-person Prisoner's dilemma. Theoretically, we show that this behaviour is not consistent with either the Fehr & Schmidt model or (a one-parameter version of) the Charness & Rabin model, but it is consistent with the cooperative equilibrium model introduced by the second author.

Asymmetric interaction paired with a super-rational strategy might resolve the tragedy of the commons without requiring recognition or negotiation

Avoiding the tragedy of the commons requires that one or more individuals in a group or partnership “volunteer”, benefiting the group at a cost to themselves. Recognition and negotiation with social partners can maintain cooperation, but are often not possible. If recognition and negotiation are not always the mechanism by which cooperative partnerships avoid collective tragedies, what might explain the diverse social cooperation observed in nature? Assuming that individuals interact asymmetrically and that both “weak” and “strong” players employ a super-rational strategy, we find that tragedy of the commons can be avoided without requiring either recognition or negotiation. Whereas in the volunteer's dilemma game a rational “strong” player is less likely to volunteer to provide a common good in larger groups, we show that under a wide range of conditions a super-rational “strong” player is more likely to provide a common good. These results imply that the integration of super-rationality and asymmetric interaction might have the potential to resolve the tragedy of the commons. By illuminating the conditions under which players are likely to volunteer, we shed light on the patterns of volunteerism observed in variety of well-studied cooperative social systems, and explore how societies might avert social tragedies.

Invasion of cooperators in lattice populations: linear and non-linear public good games

A generalized version of the N-person volunteer's dilemma (NVD) Game has been suggested recently for illustrating the problem of N-person social dilemmas. Using standard replicator dynamics it can be shown that coexistence of cooperators and defectors is typical in this model. However, the question of how a rare mutant cooperator could invade a population of defectors is still open. Here we examined the dynamics of individual based stochastic models of the NVD. We analyze the dynamics in well-mixed and viscous populations. We show in both cases that coexistence between cooperators and defectors is possible; moreover, spatial aggregation of types in viscous populations can easily lead to pure cooperation. Furthermore we analyze the invasion of cooperators in populations consisting predominantly of defectors. In accordance with analytical results, in deterministic systems, we found the invasion of cooperators successful in the well-mixed case only if their initial concentration was higher than a critical threshold, defined by the replicator dynamics of the NVD. In the viscous case, however, not the initial concentration but the initial number determines the success of invasion. We show that even a single mutant cooperator can invade with a high probability, because the local density of aggregated cooperators exceeds the threshold defined by the game. Comparing the results to models using different benefit functions (linear or sigmoid), we show that the role of the benefit function is much more important in the well-mixed than in the viscous case.

Evolution of cooperation in spatial traveler's dilemma game

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

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.

Risk-driven migration and the collective-risk social dilemma

A collective-risk social dilemma implies that personal endowments will be lost if contributions to the common pool within a group are too small. Failure to reach the collective target thus has dire consequences for all group members, independently of their strategies. Wanting to move away from unfavorable locations is therefore anything but surprising. Inspired by these observations, we here propose and study a collective-risk social dilemma where players are allowed to move if the collective failure becomes too probable. More precisely, this so-called risk-driven migration is launched depending on the difference between the actual contributions and the declared target. Mobility therefore becomes an inherent property that is utilized in an entirely self-organizing manner. We show that under these assumptions cooperation is promoted much more effectively than under the action of manually determined migration rates. For the latter, we in fact identify parameter regions where the evolution of cooperation is greatly inhibited. Moreover, we find unexpected spatial patterns where cooperators that do not form compact clusters outperform those that do, and where defectors are able to utilize strikingly different ways of invasion. The presented results support the recently revealed importance of percolation for the successful evolution of public cooperation, while at the same time revealing surprisingly simple methods of self-organization towards socially desirable states.

Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies

Cooperation and defection may be considered to be two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases, the characteristic threshold decreases while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation.

Adaptive and bounded investment returns promote cooperation in spatial public goods games

The public goods game is one of the most famous models for studying the evolution of cooperation in sizable groups. The multiplication factor in this game can characterize the investment return from the public good, which may be variable depending on the interactive environment in realistic situations. Instead of using the same universal value, here we consider that the multiplication factor in each group is updated based on the differences between the local and global interactive environments in the spatial public goods game, but meanwhile limited to within a certain range. We find that the adaptive and bounded investment returns can significantly promote cooperation. In particular, full cooperation can be achieved for high feedback strength when appropriate limitation is set for the investment return. Also, we show that the fraction of cooperators in the whole population can become larger if the lower and upper limits of the multiplication factor are increased. Furthermore, in comparison to the traditionally spatial public goods game where the multiplication factor in each group is identical and fixed, we find that cooperation can be better promoted if the multiplication factor is constrained to adjust between one and the group size in our model. Our results highlight the importance of the locally adaptive and bounded investment returns for the emergence and dominance of cooperative behavior in structured populations.

Conditional strategies and the evolution of cooperation in spatial public goods games

The fact that individuals will most likely behave differently in different situations begets the introduction of conditional strategies. Inspired by this, we study the evolution of cooperation in the spatial public goods game, where, besides unconditional cooperators and defectors, also different types of conditional cooperators compete for space. Conditional cooperators will contribute to the public good only if other players within the group are likely to cooperate as well but will withhold their contribution otherwise. Depending on the number of other cooperators that are required to elicit cooperation of a conditional cooperator, the latter can be classified in as many types as there are players within each group. We find that the most cautious cooperators, who require all other players within a group to be conditional cooperators, are the undisputed victors of the evolutionary process, even at very low synergy factors. We show that the remarkable promotion of cooperation is due primarily to the spontaneous emergence of quarantining of defectors, who become surrounded by conditional cooperators and are forced into isolated convex "bubbles" from which they are unable to exploit the public good. This phenomenon can be observed only in structured populations, thus adding to the relevance of pattern formation for the successful evolution of cooperation.

Adaptive evolution of cooperation through Darwinian dynamics in Public Goods games

The linear or threshold Public Goods game (PGG) is extensively accepted as a paradigmatic model to approach the evolution of cooperation in social dilemmas. Here we explore the significant effect of nonlinearity of the structures of public goods on the evolution of cooperation within the well-mixed population by adopting Darwinian dynamics, which simultaneously consider the evolution of populations and strategies on a continuous adaptive landscape, and extend the concept of evolutionarily stable strategy (ESS) as a coalition of strategies that is both convergent-stable and resistant to invasion. Results show (i) that in the linear PGG contributing nothing is an ESS, which contradicts experimental data, (ii) that in the threshold PGG contributing the threshold value is a fragile ESS, which cannot resist the invasion of contributing nothing, and (iii) that there exists a robust ESS of contributing more than half in the sigmoid PGG if the return rate is relatively high. This work reveals the significant effect of the nonlinearity of the structures of public goods on the evolution of cooperation, and suggests that, compared with the linear or threshold PGG, the sigmoid PGG might be a more proper model for the evolution of cooperation within the well-mixed population.

Geographic constraints on social network groups

Social groups are fundamental building blocks of human societies. While our social interactions have always been constrained by geography, it has been impossible, due to practical difficulties, to evaluate the nature of this restriction on social group structure. We construct a social network of individuals whose most frequent geographical locations are also known. We also classify the individuals into groups according to a community detection algorithm. We study the variation of geographical span for social groups of varying sizes, and explore the relationship between topological positions and geographic positions of their members. We find that small social groups are geographically very tight, but become much more clumped when the group size exceeds about 30 members. Also, we find no correlation between the topological positions and geographic positions of individuals within network communities. These results suggest that spreading processes face distinct structural and spatial constraints.

Different gain/loss sensitivity and social adaptation ability in gifted adolescents during a public goods game

Gifted adolescents are considered to have high IQs with advanced mathematical and logical performances, but are often thought to suffer from social isolation or emotional mal-adaptation to the social group. The underlying mechanisms that cause stereotypic portrayals of gifted adolescents are not well known. We aimed to investigate behavioral performance of gifted adolescents during social decision-making tasks to assess their affective and social/non-social cognitive abilities. We examined cooperation behaviors of 22 gifted and 26 average adolescents during an iterative binary public goods (PG) game, a multi-player social interaction game, and analyzed strategic decision processes that include cooperation and free-riding. We found that the gifted adolescents were more cooperative than average adolescents. Particularly, comparing the strategies for the PG game between the two groups, gifted adolescents were less sensitive to loss, yet were more sensitive to gain. Additionally, the behavioral characteristics of average adolescents, such as low trust of the group and herding behavior, were not found in gifted adolescents. These results imply that gifted adolescents have a high cognitive ability but a low ability to process affective information or to adapt in social groups compared with average adolescents. We conclude that gain/loss sensitivity and the ability to adapt in social groups develop to different degrees in average and gifted adolescents.

Local interaction scale controls the existence of a nontrivial optimal critical mass in opinion spreading

We study a model of opinion formation where the collective decision of a group is said to happen if the fraction of agents having the most common opinion exceeds a threshold value, a critical mass. We find that there exists a unique nontrivial critical mass giving the most efficient convergence to consensus. In addition, we observe that for small critical masses, the characteristic time scale for the relaxation to consensus splits into two. The shorter time scale corresponds to a direct relaxation and the longer one can be explained by the existence of intermediate metastable states similar to those found in [P. Chen and S. Redner, Phys. Rev. E 71, 036101 (2005)]. This longer time scale is dependent on the precise condition for consensus-with a modification of the condition it can go away.