Phase transitions and volunteering in spatial public goods games

Indirect reciprocity in the public goods game with collective reputations

Indirect reciprocity unveils how social cooperation is founded upon moral systems. Within the frame of dyadic games based on individual reputations, the 'leading-eight' strategies distinguish themselves in promoting and sustaining cooperation. However, in real-world societies, there are widespread interactions at the group level, where individuals need to make a singular action choice when facing multiple individuals with different reputations. Here, through introducing the assessment of collective reputations, we develop a framework that embeds group-level reputation structure into public goods games to study the evolution of group-level indirect reciprocity. We show that changing the criteria of group assessment destabilizes the reputation dynamics of leading-eight strategies. In a particular range of social assessment criteria, all leading-eight strategies can break the social dilemma in public goods games and sustain cooperation. Specifically, there exists an optimal, moderately set assessment criterion that is most conducive to promoting cooperation. Moreover, in the evolution of assessment criteria, the preference of the leading-eight strategies for social strictness is inversely correlated with the payoff level. Our work reveals the impact of social strictness on prosocial behaviour, highlighting the importance of group-level interactions in the analysis of evolutionary games and complex social dynamics.

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.

Evolutionary dynamics of cooperation coupled with ecological feedback compensation

A simple theoretical model (or a demonstrative example) was developed to illustrate how the evolution of cooperation can be affected by the density-dependent survival competition, in which we assume that the fertility of an individual depends only on the pairwise interaction between him and other individuals based on Prisoner's Dilemma game, while its viability is only related to the density-dependent survival competitiveness. Our results show that not only cooperation could be evolutionarily stable if the advantage of cooperators in viability can compensate for the cost they pay for their fertility, but also the long-term stable coexistence of cooperation and defection is possible if none of cooperation and defection is evolutionarily stable. Moreover, for the stochastic evolutionary dynamics in a finite population, our analysis shows that the increase (or decrease) of the survival competitiveness of cooperators (or defectors) should be conductive to the evolutionary emergence of cooperation.

Behavioural strategies in simultaneous and alternating prisoner's dilemma games with/without voluntary participation

The Prisoner's Dilemma is one of the most classic formats for exploring the principle of direct reciprocity. Although numerous theoretical and experimental studies have been conducted, little attention has been paid to the divergence between theoretical predictions and actual human behaviour. In addition, there are two additional essential challenges of experimental research. First, most experimental approaches have focused on games in which two players decide their actions simultaneously, but little is known about alternating games. Another is that there are few experiments on voluntary participation. Here, we conducted experiments on simultaneous games, alternating games, and games with and without voluntary participation for a total of four game patterns and examined the deviation from theoretical predictions for each. The results showed that, contrary to theoretical predictions, humans chose cooperation even after being exploited. We also observed that, with or without voluntary participation, people tended to take the same action they had taken in the previous round. Our results indicate that to understand the mechanisms of human behaviour, we need to integrate findings from behavioural science, psychology, and game theory.

Stability of pairwise social dilemma games: Destructive agents, constructive agents, and their joint effects

Destructive agents, who opt out of the game and indiscriminately harm others, paradoxically foster cooperation, representing an intriguing variant of the voluntary participation strategy. Yet, their impact on cooperation remains inadequately understood, particularly in the context of pairwise social dilemma games and in comparison to their counterparts, constructive agents, who opt out of the game but indiscriminately benefit others. Furthermore, little is known about the combined effects of both agent types on cooperation dynamics. Using replicator dynamics in infinite and well-mixed populations, we find that contrary to their role in facilitating cooperation in multiplayer games, destructive agents fail to encourage cooperation in pairwise social dilemmas. Instead, they replace defection in the prisoners' dilemma and stag-hunt games. Similarly, in the chicken game, they can destabilize or replace the mixed equilibrium of cooperation and defection, undermining cooperation in the harmony (trivial) game. Conversely, constructive agents, when their payoffs exceed their contributions to opponents, can exhibit effects similar to destructive agents. However, if their payoffs are lower, while they destabilize defection in prisoners' dilemma and stag-hunt games, they do not disrupt the cooperation equilibrium in harmony games and have a negligible impact on the coexistence of cooperation in chicken games. The combination of destructive and constructive agents does not facilitate cooperation, but instead generates complex evolutionary dynamics, including bistable, tristable, and quadstable states, with outcomes contingent on their relative payoffs and game types. These results, taken together, enhance our understanding of the impact of the voluntary participation mechanism on cooperation, contributing to a more comprehensive understanding of its influence.

Spontaneous symmetry breaking of cooperation between species

In mutualistic associations, two species cooperate by exchanging goods or services with members of another species for their mutual benefit. At the same time, competition for reproduction primarily continues with members of their own species. In intra-species interactions, the prisoner's dilemma is the leading mathematical metaphor to study the evolution of cooperation. Here we consider inter-species interactions in the spatial prisoner's dilemma, where members of each species reside on one lattice layer. Cooperators provide benefits to neighbouring members of the other species at a cost to themselves. Hence, interactions occur across layers but competition remains within layers. We show that rich and complex dynamics unfold when varying the cost-to-benefit ratio of cooperation, r. Four distinct dynamical domains emerge that are separated by critical phase transitions, each characterized by diverging fluctuations in the frequency of cooperation: (i) for large r cooperation is too costly and defection dominates; (ii) for lower r cooperators survive at equal frequencies in both species; (iii) lowering r further results in an intriguing, spontaneous symmetry breaking of cooperation between species with increasing asymmetry for decreasing r; (iv) finally, for small r, bursts of mutual defection appear that increase in size with decreasing r and eventually drive the populations into absorbing states. Typically, one species is cooperating and the other defecting and hence establish perfect asymmetry. Intriguingly and despite the symmetrical model set-up, natural selection can nevertheless favour the spontaneous emergence of asymmetric evolutionary outcomes where, on average, one species exploits the other in a dynamical equilibrium.

Temporal interaction and its role in the evolution of cooperation

This research investigates the impact of dynamic, time-varying interactions on cooperative behavior in social dilemmas. Traditional research has focused on deterministic rules governing pairwise interactions, yet the impact of interaction frequency and synchronization in groups on cooperation remains underexplored. Addressing this gap, our work introduces two temporal interaction mechanisms to model the stochastic or periodic participation of individuals in public goods games, acknowledging real-life variances due to exogenous temporal factors and geographical time differences. We consider that the interaction state significantly influences both game payoff calculations and the strategy updating process, offering new insights into the emergence and sustainability of cooperation. Our results indicate that maximum game participation frequency is suboptimal under a stochastic interaction mechanism. Instead, an intermediate activation probability maximizes cooperation, suggesting a vital balance between interaction frequency and inactivity security. Furthermore, local synchronization of interactions within specific areas is shown to be beneficial, as time differences hinder the spread of cross-structures but promote the formation of dense cooperative clusters with smoother boundaries. We also note that stronger clustering in networks, larger group sizes, and lower noise increase cooperation. This research contributes to understanding the role of node-based temporality and probabilistic interactions in social dilemmas, offering insights into fostering cooperation.

Evolutionary dynamics of any multiplayer game on regular graphs

Multiplayer games on graphs are at the heart of theoretical descriptions of key evolutionary processes that govern vital social and natural systems. However, a comprehensive theoretical framework for solving multiplayer games with an arbitrary number of strategies on graphs is still missing. Here, we solve this by drawing an analogy with the Balls-and-Boxes problem, based on which we show that the local configuration of multiplayer games on graphs is equivalent to distributing k identical co-players among n distinct strategies. We use this to derive the replicator equation for any n-strategy multiplayer game under weak selection, which can be solved in polynomial time. As an example, we revisit the second-order free-riding problem, where costly punishment cannot truly resolve social dilemmas in a well-mixed population. Yet, in structured populations, we derive an accurate threshold for the punishment strength, beyond which punishment can either lead to the extinction of defection or transform the system into a rock-paper-scissors-like cycle. The analytical solution also qualitatively agrees with the phase diagrams that were previously obtained for non-marginal selection strengths. Our framework thus allows an exploration of any multi-strategy multiplayer game on regular graphs.

The impact of mobility costs on cooperation and welfare in spatial social dilemmas

From over-exploitation of resources to urban pollution, sustaining well-being requires solving social dilemmas of cooperation. Often such dilemmas are studied assuming that individuals occupy fixed positions in a network or lattice. In spatial settings, however, agents can move, and such movements involve costs. Here we investigate how mobility costs impact cooperation dynamics. To this end, we study cooperation dilemmas where individuals are located in a two-dimensional space and can be of two types: cooperators-or cleaners, who pay an individual cost to have a positive impact on their neighbours-and defectors-or polluters, free-riding on others' effort to sustain a clean environment. Importantly, agents can pay a cost to move to a cleaner site. Both analytically and through agent-based simulations we find that, in general, introducing mobility costs increases pollution felt in the limit of fast movement (equivalently slow strategy revision). The effect on cooperation of increasing mobility costs is non-monotonic when mobility co-occurs with strategy revision. In such scenarios, low (yet non-zero) mobility costs minimise cooperation in low density environments; whereas high costs can promote cooperation even when a minority of agents initially defect. Finally, we find that heterogeneity in mobility cost affects the final distribution of strategies, leading to differences in who supports the burden of having a clean environment.

Strategy evolution on higher-order networks

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

Prosocial punishment bots breed social punishment in human players

Prosocial punishment, an important factor to stabilize cooperation in social dilemma games, often faces challenges like second-order free-riders-who cooperate but avoid punishing to save costs-and antisocial punishers, who defect and retaliate against cooperators. Addressing these challenges, our study introduces prosocial punishment bots that consistently cooperate and punish free-riders. Our findings reveal that these bots significantly promote the emergence of prosocial punishment among normal players due to their 'sticky effect'-an unwavering commitment to cooperation and punishment that magnetically attracts their opponents to emulate this strategy. Additionally, we observe that the prevalence of prosocial punishment is greatly enhanced when normal players exhibit a tendency to follow a 'copying the majority' strategy, or when bots are strategically placed in high-degree nodes within scale-free networks. Conversely, bots designed for defection or antisocial punishment diminish overall cooperation levels. This stark contrast underscores the critical role of strategic bot design in enhancing cooperative behaviours in human/AI interactions. Our findings open new avenues in evolutionary game theory, demonstrating the potential of human-machine collaboration in solving the conundrum of punishment.

Role polarization and its effects in the spatial ultimatum game

Human society is believed to be becoming increasingly polarized, yet it remains unclear how role polarization influences the evolution of fairness. In addition, little is known about role adaptation, despite the fact that altering the roles of players can often change the outcome of the ultimatum game. Unlike earlier static, random, symmetric role assignment, here I suggest a succeed-reinforce-fail-slacken role adaptation rule that encourages successful proposers in the present round to propose again in the next round and vice versa. The results demonstrate that this simple rule can tip the scales in favor of fair strategies when it comes to the proposer advantage, and therein lies the key to promoting fairness. Depending on its pace, notably, role adaptation can direct the system to equilibrium states that bear variable degrees of role polarization, with two consequences incidentally. Not only does it favor fairness, it also fosters empathy. Noise associated with role adaptation often reduces role polarization and thus has a negative impact on fairness and empathy. The comparison of experiments with various networks validates the substantial resilience of role polarization to structural changes. These findings add to the evidence for role polarization and highlight the centrality of role adaptation in the evolution of fairness.

Dual effects of conformity on the evolution of cooperation in social dilemmas

Human beings are easily impacted by social influences, due to their social nature. As an essential manifestation of social influences, conformity is associated with the frequency witnessed in others' behavior, involving normative conformity and informational conformity according to the reaction of individuals. The former comes from the fear of a normative environment, while the latter means most behaviors are followed due to information asymmetry. Normative conformity significantly enhances network reciprocity, producing optimal cooperation at a moderate proportion, which induces within-cluster behavioral homogeneity and between-cluster behavioral diversity. On the contrary, informational conformity has an inhibitory effect on the evolution of cooperation for a low proportion of the conformity population, which contributes to the formation of defectors' clusters. The symmetry and duality of the two types of conformity on cooperation evolution provide an interesting and unexplored approach for future research, revealing the mechanism of conformity in evolutionary games.

Small bots, big impact: solving the conundrum of cooperation in optional Prisoner's Dilemma game through simple strategies

Cooperation plays a crucial role in both nature and human society, and the conundrum of cooperation attracts the attention from interdisciplinary research. In this study, we investigated the evolution of cooperation in optional Prisoner's Dilemma games by introducing simple bots. We focused on one-shot and anonymous games, where the bots could be programmed to always cooperate, always defect, never participate or choose each action with equal probability. Our results show that cooperative bots facilitate the emergence of cooperation among ordinary players in both well-mixed populations and a regular lattice under weak imitation scenarios. Introducing loner bots has no impact on the emergence of cooperation in well-mixed populations, but it facilitates the dominance of cooperation in regular lattices under strong imitation scenarios. However, too many loner bots on a regular lattice inhibit the spread of cooperation and can eventually result in a breakdown of cooperation. Our findings emphasize the significance of bot design in promoting cooperation and offer useful insights for encouraging cooperation in real-world scenarios.

Super-rational aspiration promotes cooperation in the asymmetric game with peer exit punishment and reward

Super-rational aspiration induced strategy updating with exit rights has been considered in some previous studies, in which the players adjust strategies in line with their payoffs and aspirations, and they have access to exit the game. However, exit payoffs for exiting players are automatically allocated, which is clearly contrary to reality. In this study, evolutionary cooperation dynamics with super-rational aspiration and asymmetry in the Prisoner's Dilemma game is investigated, where exit payoffs are implemented by local peers. The results show that for different population structures, the asymmetry of the system is always contributive to the participation of the players. Furthermore, we show that under different exit payoffs, super-rationality and asymmetry are conductive to the evolution of cooperation.

Zealous cooperation does not always promote cooperation in public goods games

There is a conventional belief that prosocial behaviors cannot arise through selfish human nature, because defection always exploits cooperation to achieve a higher payoff at an individual level. Unyieldingly, some people hope to move society to cooperation through their zealous cooperation, regardless of payoffs. From the perspective of spatial evolutionary games, however, such zealous behavior is unnecessary because cooperation can emerge from selfish human nature by aggregating in evolution. Yet, to what extent can zealous cooperation induce others to cooperate? We assume a fraction of zealous agents in spatial public goods games who always cooperate. The results show that a moderate proportion of these zealous cooperators can diminish the cooperation level in the system, and cooperation is only promoted when zealots are many. Regarding spatial behaviors, the areas of zealous cooperation in a medium density can prevent evolutionary cooperation from passing through and aggregating. The phenomenon of zealous cooperation impeding cooperation becomes more pronounced when agents become less random and more selfish. This is because dotted zealous cooperation provides significant payoffs to neighboring defection, making them more solid in fitness. In this way, we also find that when zealous cooperators have low productivity, the neighbors receive fewer benefits by exploitation, thus allowing cooperation to spread. We also study replicator dynamics in unstructured populations where zealous cooperation always promotes cooperation, agreeing that zealous cooperation hindering cooperation is a spatial effect.

A social system to disperse the irrigation start date based on the spatial public goods game

In paddy rice cultivation, the amount of water used during the beginning of the irrigation season is the highest. However, there is a possibility of a water shortage at this season as climate change decreases snowfall. In this study, we propose new schemes based on the public goods game to reduce peak water volume during this season by dispersing the irrigation start dates. In our agent-based model, agents determine the irrigation start date based on the evolutionary game theory. This model considers the economic variables of individual farmers (e.g., gross cultivation profit and cultivation cost), the cost and subsidy for cooperation for the dispersion of the irrigation start dates, and the information-sharing network between farmers. Individual farmers update the cooperation/defection strategy at each time step based on their payoffs. Using this agent-based model simulation, we investigate a scheme that maximizes the dispersion of irrigation start dates among multiple scheme candidates. The results of the simulation show that, under the schemes in which one farmer belongs to a group and the groups do not overlap, the number of cooperating farmers did not increase, and the dispersion of irrigation start dates barely increased. By adopting a scheme in which one farmer belongs to multiple groups and the groups overlap, the number of cooperating farmers increased, while the dispersion of irrigation start dates maximized. Furthermore, the proposed schemes require the government to obtain information about the number of cooperators in each group to determine the subsidy amount. Therefore, we also proposed the method which allows estimating the number of cooperators in each group through the dispersion of irrigation start dates. This significantly reduces the cost of running the schemes and provides subsidization and policy evaluations unaffected by false declarations of farmers.

Towards preferential selection in the prisoner's dilemma game

In previous works, the choice of learning neighbor for an individual has generally obeyed pure random selection or preferential selection rules. In this paper, we introduce a tunable parameter ε to characterize the strength of preferential selection and focus on the transition towards preferential selection in the spatial evolutionary game by controlling ε to guide the system from pure random selection to preferential selection. Our simulation results reveal that the introduction of preferential selection can hugely alleviate social dilemmas and enhance network reciprocity. A larger ε leads to a higher critical threshold of the temptation b for the extinction of cooperators. Moreover, we provide some intuitive explanations for the above results from the perspective of strategy transition and cooperative clusters. Finally, we examine the robustness of the results for noise K and different topologies, find that qualitative features of the results are unchanged.

Evolution of cooperation under a generalized death-birth process

According to the evolutionary death-birth protocol, a player is chosen randomly to die and neighbors compete for the available position proportional to their fitness. Hence, the status of the focal player is completely ignored and has no impact on the strategy update. In this paper, we revisit and generalize this rule by introducing a weight factor to compare the payoff values of the focal and invading neighbors. By means of evolutionary graph theory, we analyze the model on joint transitive graphs to explore the possible consequences of the presence of a weight factor. We find that focal weight always hinders cooperation under weak selection strength. Surprisingly, the results show a nontrivial tipping point of the weight factor where the threshold of cooperation success shifts from positive to negative infinity. Once focal weight exceeds this tipping point, cooperation becomes unreachable. Our theoretical predictions are confirmed by Monte Carlo simulations on a square lattice of different sizes. We also verify the robustness of the conclusions to arbitrary two-player prisoner's dilemmas, to dispersal graphs with arbitrary edge weights, and to interaction and dispersal graphs overlapping arbitrarily.

Impact of social reward on the evolution of cooperation in voluntary prisoner's dilemma

The existence and sustainability of cooperation is a critical issue in nature and social systems. Reward is an essential mechanism to enhance cooperation. Meanwhile, some individuals loathe competition and then choose to escape and become a loner in competition. In this scenario, we propose a four-strategy networked evolutionary game model involving rewarders, loners, cooperators, and defectors. The classical square lattice and the Erdös-Rényi random network are adopted to describe the interaction between individuals. The four-strategy model is an extension of the classic prisoner's dilemma game model. The simulation results show that the introduction of new strategic choices can significantly improve cooperation in the population. The promotion level of cooperation is directly correlated with reward intensity and negatively correlated with reward cost. With regard to the evolution of altruistic behaviors, the fixed income from interactions with loners has an impact that is connected to the temptation to defect. Furthermore, by analyzing characteristic snapshots of four strategies, we further dissect the essence of the evolution of cooperation. As the temptation value increases, cooperators and rewarders first form compact clusters, then more and more loners join to resist the intrusion of defectors. Eventually, the three strategies coexist stably in a spatially structured population. Our research may shed some light on exploring the nature of cooperation and solving social dilemmas in the future.

Mean-field models of dynamics on networks via moment closure: An automated procedure

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equation is a function of higher-order moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

Social relationship adjustments within the same sex promote marital bliss

Parental care is essential for biological systems. Marital bliss is one of the ideal paradigms for parental care, in which males contribute in raising offspring and females require a courtship time. Yet marital bliss state is neither Nash equilibrium nor Pareto optimum for the classic Battle of the Sexes. It thus leads to a gap between evolutionary theory and marital bliss. Previous works concentrate on the pairwise interactions between the two sexes to fill this gap, such as the courtship time and encounter rate. The social relationships within the same sex, however, receives much less attention. Here we investigate how social relationships within the same sex change marital bliss by introducing the coevolution of strategy and social network. Based on the time scale separation, it is found that a symmetric game is emergent via social adjustments within each sex, and the evolutionary outcome is determined by the interplay between the emergent symmetric game and the Battle of the Sexes. We find that marital bliss can be promoted when males are rational (strong selection limit) and females are irrational (weak selection limit); the stable Coy-Coy social relationships both stabilize and speed up marital bliss; the general criterion of stabilising marital bliss for arbitrary imitation function are found, which are verified by simulations. Furthermore, the emergent symmetric games are insightful for determining whether the stable marital bliss is global stable. Our work provides an alternative avenue to facilitate marital bliss, which can be applied for general asymmetric games on dynamical networks.

Defectors in bad circumstances possessing higher reputation can promote cooperation

In nature and human society, social relationships and behavior patterns are usually unpredictable. In any interaction, individuals will constantly have to deal with prior uncertainty. The concept of "reputation" can provide some information to mitigate such uncertainty. In previous studies, researchers have considered that only cooperators are able to maintain a high reputation; no matter the circumstances of a defector, they are classified as a faithless individual. In reality, however, some individuals will be forced to defect to protect themselves against exploitation. Therefore, it makes sense that defectors in bad circumstances could also obtain higher reputations, and cooperators can maintain higher reputations in comfortable circumstances. In this work, the reputations of individuals are calculated using the fraction of their neighbors who have the same strategy. In this way, some defectors in a population may obtain higher reputations than some cooperators. We introduce this reputation rule using heterogeneous investments in public goods games. Our numerical simulation results indicate that this reputation rule and heterogeneous investments can better stimulate cooperation. Additionally, stronger investment heterogeneity can further increase the level of cooperation. To explain this phenomenon, dynamical evolution is observed in Monte Carlo simulations. We also investigated the effects of the noise intensity of the irrational population and the original proportion of cooperation in the population. The robustness of this cooperation model was also considered with respect to the network structure and total investment, and we found that the conclusions remained the same.

Weak selection helps cheap but harms expensive cooperation in spatial threshold dilemmas

Public Goods Games (PGGs) are n-person games with dependence of individual fitness benefits on the collective investment by the players. We have studied a simple PGG scenario played out by cooperating (C) and defecting (D) agents, applying the highly nonlinear threshold benefit function in an individual-based lattice model. A semi-analytical approximation of the lattice model has been developed and shown to describe the dynamics fairly well in the vicinity of the steady state. Besides the expected outcomes (i.e., the negative effect on cooperator persistence of higher cooperation costs and/or more intensive mixing of the population) we have found a surprising, counter-intuitive effect of the strength of selection on the steady state of the model. The effect is different at low and high cooperation costs, and it shows up only in the lattice model, suggesting that stochastic effects and higher order spatial correlations due to the emergent spatial clustering of cooperators (not taken into account in the semi-analytical approximation) must be responsible for the unexpected results for which we propose an intuitive explanation, present a tentative demonstration, and shortly discuss their biological relevance.

Evolution of cooperation and consistent personalities in public goods games

The evolution of cooperation has remained an important problem in evolutionary theory and social sciences. In this regard, a curious question is why consistent cooperative and defective personalities exist and if they serve a role in the evolution of cooperation? To shed light on these questions, here, I consider a population of individuals who possibly play two consecutive rounds of public goods game, with different enhancement factors. Importantly, individuals have independent strategies in the two rounds. However, their strategy in the first round affects the game they play in the second round. I consider two different scenarios where either only first-round cooperators play a second public goods game, or both first-round cooperators and first-round defectors play a second public goods game, but in different groups. The first scenario can be considered a reward dilemma, and the second can be considered an assortative public goods game but with independent strategies of the individuals in the two rounds. Both models show cooperators can survive either in a fixed point or through different periodic orbits. Interestingly, due to the emergence of a correlation between the strategies of the individuals in the two rounds, individuals develop consistent personalities during the evolution. This, in turn, helps cooperation to flourish. These findings shed new light on the evolution of cooperation and show how consistent cooperative and defective personalities can evolve and play a positive role in solving social dilemmas.

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

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

Spatial social dilemmas promote diversity

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

The study on the role of dedicators on promoting cooperation in public goods game

In daily life, some people are always seen dedicating available resources to support collective activities. In this paper, we call these people who care group goals more than individual goals dedicators. Inspired by this phenomenon, we studied the role of dedicators on the evolution of cooperation in public goods game (PGG) based on a Chinese Folk Spring Festival Gala. Three types of agents were introduced into our PGG model including cooperators, defectors and dedicators. Dedicators tried to donate when the Gala was short of funds. Cooperators and defectors imitated the strategy of the highest-payoff neighbor based on the rational mechanism. And their imitating probability was modified on account of the emotional mechanism for positive effect of the dedicator's donating behavior and negative effect of continuous poor performance. Through numerical simulations, we found that the existence of dedicators can indeed promote cooperation in PGG. It should be noted that dedicators' willingness to donate was more important than their donation quantity in facilitating cooperation. And the stronger the emotional effect intensity of dedicators' donating behavior was, the better. So, the selfless dedication of participants should be praised to promote cooperation by improving their emotional effect intensity. Last but not least, a reasonable activity budget was needed to sustain the highest level of cooperation.

Density-dependent private benefit leads to bacterial mutualism

Microorganisms produce and secrete materials that are beneficial for themselves and their neighbors. We modeled the situation when cells can produce different costly secretions which increase the carrying capacity of the population. Strains that lose the function of producing one or more secretions avoid the cost of production and can exhaust the producers. However, secreting substances provides a private benefit for the producers in a density-dependent way. We developed a model to examine the outcome of the selection among different types of producer strains from the nonproducer strain to the partial producers, to the full producer strain. We were interested in circumstances under which selection maintains partners that produce complementary secreted materials thus forming an interdependent mutualistic interaction. We show that interdependent mutualism is selected under a broad range of conditions if private benefit decreases with density. Selection frequently causes the coexistence of more and less generalist cooperative strains, thus cooperation and exploitation co-occur. Interdependent mutualism is evolved under more specific circumstances if private benefit increases with density and these general observations are valid in a well-mixed and a structured deme model. We show that the applied population structure allows the invasion of rare cooperators and supports cooperation in general.

Assortative clustering in a one-dimensional population with replication strategies

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

Low-carbon technology collaborative innovation in industrial cluster with social exclusion: An evolutionary game theory perspective

As governments implement low-carbon economy widely, boosting low-carbon transformation in industrial clusters has become a challenge. This study establishes an evolutionary game model of low-carbon technology collaborative innovation based on spatial public goods game to solve the free-riding problem effectively in research and development. By introducing a social exclusion mechanism, we explore the requirements for the emergence of cooperation between enterprises, and we consider the heterogeneity and scale-free characteristics of industrial clusters comprehensively. Simulation results confirm that social exclusion can significantly promote cooperation as a form of cooperation with additional cost. When exclusion cost decreases and probability increases, an excluder can survive in a lower enhancement factor, which guarantees a stable exclusion mechanism. Furthermore, this mechanism is key to forming and maintaining cooperative behavior. When a cluster follows a scale-free distribution, the sparse network structure can avoid cooperation collapse. Moreover, heterogeneous investment is a robust alternative in the face of invading defectors. This study provides a new understanding to promote the collaborative innovation of enterprises in industrial clusters.

Freedom to choose between public resources promotes cooperation

As cooperation incurs a cost to the cooperator for others to benefit, its evolution seems to contradict natural selection. How evolution has resolved this obstacle has been among the most intensely studied questions in evolutionary theory in recent decades. Here, we show that having a choice between different public resources provides a simple mechanism for cooperation to flourish. Such a mechanism can be at work in many biological or social contexts where individuals can form different groups or join different institutions to perform a collective action task, or when they can choose between collective actions with different profitability. As a simple evolutionary model suggests, defectors tend to join the highest quality resource in such a context. This allows cooperators to survive and out-compete defectors by sheltering in a lower quality resource. Cooperation is maximized, however, when the qualities of the two highest quality resources are similar, and thus, they are almost interchangeable.

Exit rights open complex pathways to cooperation

We study the evolutionary dynamics of the Prisoner's Dilemma game in which cooperators and defectors interact with another actor type called exiters. Rather than being exploited by defectors, exiters exit the game in favour of a small pay-off. We find that this simple extension of the game allows cooperation to flourish in well-mixed populations when iterations or reputation are added. In networked populations, however, the exit option is less conducive to cooperation. Instead, it enables the coexistence of cooperators, defectors, and exiters through cyclic dominance. Other outcomes are also possible as the exit pay-off increases or the network structure changes, including network-wide oscillations in actor abundances that may cause the extinction of exiters and the domination of defectors, although game parameters should favour exiting. The complex dynamics that emerges in the wake of a simple option to exit the game implies that nuances matter even if our analyses are restricted to incentives for rational behaviour.

Best response dynamics improve sustainability and equity outcomes in common-pool resources problems, compared to imitation dynamics

Shared resource extraction among profit-seeking individuals involves a tension between individual benefit and the collective well-being represented by the persistence of the resource. Many game theoretic models explore this scenario, but these models tend to assume either best response dynamics (where individuals instantly switch to better paying strategies) or imitation dynamics (where individuals copy successful strategies from neighbours), and do not systematically compare predictions under the two assumptions. Here we propose an iterated game on a social network with payoff functions that depend on the state of the resource. Agents harvest the resource, and the strategy composition of the population evolves until an equilibrium is reached. The system is then repeatedly perturbed and allowed to re-equilibrate. We compare model predictions under best response and imitation dynamics. Compared to imitation dynamics, best response dynamics increase sustainability of the system, the persistence of cooperation while decreasing inequality and debt corresponding to the Gini index in the agents' cumulative payoffs. Additionally, for best response dynamics, the number of strategy switches before equilibrium fits a power-law distribution under a subset of the parameter space, suggesting the system is in a state of self-organized criticality. We find little variation in most mean results over different network topologies; however, there is significant variation in the distributions of the raw data, equality of payoff, clustering of like strategies and power-law fit. We suggest the primary mechanisms driving the difference in sustainability between the two strategy update rules to be the clustering of like strategies as well as the time delay imposed by an imitation processes. Given the strikingly different outcomes for best response versus imitation dynamics for common-pool resource systems, our results suggest that modellers should choose strategy update rules that best represent decision-making in their study systems.

Coevolution of cooperation and language

As a cooperative act decreases an individual's fitness for others to benefit, it is expected to be selected against by natural selection. That, how contrary to this naive expectation cooperation has evolved, is a fundamental problem in evolutionary biology and social sciences. Here, by introducing a mathematical model, we show that coevolution of cooperation and language can provide an avenue through which both cooperation and language evolve. In this model, individuals in a population play a prisoner's dilemma game and at the same time try to communicate a set of representations by producing signals. For this purpose, individuals try to build a common language, which is composed of a set of signal-representation associations. Individuals decide in language learning based on their payoff from the prisoner's dilemma game and decide about their strategy in the prisoner's dilemma game based on their success in conveying symbolic information. The model shows cooperators are able to build a common language and protect it against defectors' attempt to decode it. The language channels the benefit of cooperation toward cooperators, and defectors, being banished from the language, are unable to exploit cooperators, and are doomed to extinction.

Evolutionary dynamics drives role specialization in a community of players

The progression of game theory from classical to evolutionary and spatial games provided a powerful means to study cooperation, and enabled a better understanding of general cooperation-promoting mechanisms. However, current standard models assume that at any given point players must choose either cooperation or defection, meaning that regardless of the spatial structure in which they exist, they cannot differentiate between their neighbours and adjust their behaviour accordingly. This is at odds with interactions among organisms in nature who are well capable of behaving differently towards different members of their communities. We account for this natural fact by introducing a new type of player-dubbed link players-who can adjust their behaviour to each individual neighbour. This is in contrast to more common node players whose behaviour affects all neighbours in the same way. We proceed to study cooperation in pure and mixed populations, showing that cooperation peaks at moderately low densities of link players. In such conditions, players naturally specialize in different roles. Node players tend to be either cooperators or defectors, while link players form social insulation between cooperative and defecting clusters by acting both as cooperators and defectors. Such fairly complex processes emerging from a simple model reflect some of the complexities observed in experimental studies on social behaviour in microbes and pave a way for the development of richer game models.

Coalition-structured governance improves cooperation to provide public goods

While the benefits of common and public goods are shared, they tend to be scarce when contributions are provided voluntarily. Failure to cooperate in the provision or preservation of these goods is fundamental to sustainability challenges, ranging from local fisheries to global climate change. In the real world, such cooperative dilemmas occur in multiple interactions with complex strategic interests and frequently without full information. We argue that voluntary cooperation enabled across overlapping coalitions (akin to polycentricity) not only facilitates a higher generation of non-excludable public goods, but it may also allow evolution toward a more cooperative, stable, and inclusive approach to governance. Contrary to any previous study, we show that these merits of multi-coalition governance are far more general than the singular examples occurring in the literature, and they are robust under diverse conditions of excludability, congestion of the non-excludable public good, and arbitrary shapes of the return-to-contribution function. We first confirm the intuition that a single coalition without enforcement and with players pursuing their self-interest without knowledge of returns to contribution is prone to cooperative failure. Next, we demonstrate that the same pessimistic model but with a multi-coalition structure of governance experiences relatively higher cooperation by enabling recognition of marginal gains of cooperation in the game at stake. In the absence of enforcement, public-goods regimes that evolve through a proliferation of voluntary cooperative forums can maintain and increase cooperation more successfully than singular, inclusive regimes.

Evolutionary public goods game on the birandom geometric graph

To investigate the evolution of cooperation in spatial public goods games, this paper establishes a birandom geometric graph, in which two types of nodes, representing players and public goods respectively, are placed at random locations in the unit square. Each public good has a limit influence range and the individuals that fall into the same range engage in a public good game. In contrast to the classical network models consisting of only one type of nodes, the birandom geometric graph provides a natural way to describe the scenarios where individuals and public resources are independent of each other. Numerical simulations reveal that cooperation can be significantly promoted when the group size and the average number of groups that each player participates in are relatively small, which is at odds with the results on the square lattice, but is consistent with a body of empirical evidence reported by Ostrom and Olson et al. Analysis of the evolutionary process suggests that the facilitation of cooperation is due primarily to the formation of the cooperative clusters, which can effectively resist the invasion of the defectors.

Understanding collective behaviors in reinforcement learning evolutionary games via a belief-based formalization

Collective behaviors by self-organization are ubiquitous in nature and human society and extensive efforts have been made to explore the mechanisms behind them. Artificial intelligence (AI) as a rapidly developing field is of great potential for these tasks. By combining reinforcement learning with evolutionary game (RLEG), we numerically discover a rich spectrum of collective behaviors-explosive events, oscillation, and stable states, etc., that are also often observed in the human society. In this work, we aim to provide a theoretical framework to investigate the RLEGs systematically. Specifically, we formalize AI-agents' learning processes in terms of belief switches and behavior modes defined as a series of actions following beliefs. Based on the preliminary results in the time-independent environment, we investigate the stability at the mixed equilibrium points in RLEGs generally, in which agents reside in one of the optimal behavior modes. Moreover, we adopt the maximum entropy principle to infer the composition of agents residing in each mode at a strictly stable point. When the theoretical analysis is applied to the 2×2 game setting, we can explain the uncovered collective behaviors and are able to construct equivalent systems intuitively. Also, the inferred composition of different modes is consistent with simulations. Our work may be helpful to understand the related collective emergence in human society as well as behavioral patterns at the individual level and potentially facilitate human-computer interactions in the future.

Co-Evolution of Complex Network Public Goods Game under the Edges Rules

The reconnection of broken edges is an effective way to avoid drawback for the commons in past studies. Inspired by this, we proposed a public goods game model under the edges rules, where we evaluate the weight of edges by their nodes' payoff. The results proved that the game obtains a larger range of cooperation with a small gain factor by this proposed model by consulting Monte Carlo simulations (MCS) and real experiments. Furthermore, as the following the course of game and discussing the reason of cooperation, in the research, we found that the distribution entropy of the excess average degree is able to embody and predict the presence of cooperation.

Research on an evolutionary game model and simulation of a cluster innovation network based on fairness preference

The cluster innovation network is an important part of regional economic development. In addition, the fairness preference of internal innovators in the processes of investment and benefit distribution are particularly important for curbing "free riding" and other speculative behaviors and for creating a good cooperation environment. Therefore, taking a cluster innovation network constructed by the weighted evolutionary BBV model as the research subject, this paper constructs an evolutionary game model of a cluster innovation network based on a spatial public goods game and the theory of fairness preferences, which involves the processes of investment and payoff allocation. Using simulation analysis, this paper studies the evolution of innovators’ cooperative behaviors and benefits in cluster innovation network under the conditions of a fairness preference and a return intensity. The results show that an increase in the weight coefficient, gain coefficient and degree of differentiation between the previous income and current investment can effectively promote improvements in the level of enterprise cooperation. Indeed, the greater the weight coefficient, the gain coefficient and the degree of differentiation are, the more substantial the improvement in the level of enterprise cooperation will be. Moreover, an improvement in the differentiation of the breadth and depth of enterprise cooperation has an inhibitory effect on enterprise cooperation. Furthermore, whereas increases in regulation and gain coefficients can effectively promote enterprise cooperation. However, the increase in the weight coefficient has a different effect on enterprise benefit in terms of the breadth and depth of cooperation. Finally, we hope to improve the overall cooperation level and cooperation income of the network by deeply understanding the fair preferences of innovators in the processes of investment and benefit distribution, which is helpful for promoting the evolution and development of cluster innovation networks.

Probabilistic punishment and reward under rule of trust-based decision-making in continuous public goods game

Altruistic punishment and reward have been proved to promote the evolution of cooperation in the public goods game(PGG), but the punishers and the rewarders have to pay a price for these behaviors and that results in overall loss of interest. In present work, probabilistic punishment and reward are introduced to PGG. Probabilistic punishment and reward mean that punishment and reward are executed with a certain probability. Although that will reduce unnecessary costs, occasional absence of execution can lead to distrust. Thus we focus on how to implement punishment and reward efficiently within the structured populations. Numerical simulations are performed and prove that probabilistic punishment and reward can promote the evolution of cooperation more effectively. Further researches indicate that there is an optimal executing probability to promote cooperation and maximize reduction of cost. In addition, when the unit cost is high, the PGG with probabilistic punishment and reward still helps the evolution of altruistic punishers and rewarders, thereby avoiding collapse of cooperation.

Effect of voluntary participation on an alternating and a simultaneous prisoner's dilemma

We studied the evolution of cooperation in the framework of evolutionary game theory, implementing voluntary participation in the prisoner's dilemma. Although previous studies have tried to overcome the dilemma by introducing voluntary participation called a "loner," the question of which strategies among various strategies including voluntary participation are adaptive under competitive circumstances is still an unsolved puzzle. Here we have developed a model that consists of all possible strategies using a one-period memory of past actions. This model enables us to analyze a "melting pot" of strategies, wherein several strategies interact and compete with each other. Our results revealed that one strategy, in which one escapes if a partner defects or cooperates if a partner becomes a loner, dominates and maintains cooperation in an alternating prisoner's dilemma game. However, the so-called "win-stay, lose-shift" strategy dominates in a simultaneous prisoner's dilemma game. Our simulations clearly show that voluntary participation in the prisoner's dilemma game works in the alternating situation rather than the simultaneous one.

Evolution of costly signaling and partial cooperation

Two seemingly unrelated, but fundamental challenges in evolutionary theory, are the evolution of costly signals and costly cooperative traits, both expected to reduce an individual’s fitness and diminish by natural selection. Here, by considering a well mixed population of individuals who produce signals and decide on their strategies in a game they play, based on the signals, we show that costly signals and costly cooperative strategies can co-evolve as a result of internal dynamics of the system. Costly signals evolve, despite their apparent cost, due to a favorable cooperative response they elicit. This favorable strategic response can be quantified in a fitness term which governs the distribution of costly signals better than their apparent cost. In the same way, cooperative strategies evolve as they can reach a high fitness due to the internal dynamics of the systems.

Evolutionary multiplayer games on graphs with edge diversity

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

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.

Investment preference promotes cooperation in spatial public goods game

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

Thermodynamics of evolutionary games

How cooperation can evolve between players is an unsolved problem of biology. Here we use Hamiltonian dynamics of models of the Ising type to describe populations of cooperating and defecting players to show that the equilibrium fraction of cooperators is given by the expectation value of a thermal observable akin to a magnetization. We apply the formalism to the public goods game with three players and show that a phase transition between cooperation and defection occurs that is equivalent to a transition in one-dimensional Ising crystals with long-range interactions. We then investigate the effect of punishment on cooperation and find that punishment plays the role of a magnetic field that leads to an "alignment" between players, thus encouraging cooperation. We suggest that a thermal Hamiltonian picture of the evolution of cooperation can generate other insights about the dynamics of evolving groups by mining the rich literature of critical dynamics in low-dimensional spin systems.