Optional contributions have positive effects for volunteering public goods games

Public goods (PG) games with the volunteering mechanism are referred to as volunteering public goods (VPG) games, in which loners are introduced to the PG games, and a loner obtains a constant payoff but not participating the game. Considering that small contributions may have positive effects to encourage more players with bounded rationality to contribute, this paper introduces optional contributions (high value or low value) to these typical VPG games—a cooperator can contribute a high or low payoff to the public pools. With the low contribution, the logit dynamics show that cooperation can be promoted in a well mixed population comparing to the typical VPG games, furthermore, as the multiplication factor is greater than a threshold, the average payoff of the population is also enhanced. In spatial VPG games, we introduce a new adjusting mechanism that is an approximation to best response. Some results in agreement with the prediction of the logit dynamics are found. These simulation results reveal that for VPG games the option of low contributions may be a better method to stimulate the growth of cooperation frequency and the average payoff of the population.     

Role of recommendation in spatial public goods games

We study the role of recommendation in a co-evolutionary public goods game in which groups can recommend their members for establishment of new relationships with individuals outside the current group according to group quality. Intriguingly, for square lattices and ER graphs there exists optimal group quality for recommendation that induces positive feedback between cooperation and recommendation. Snapshots of spatial patterns of cooperators, defectors, recommended cooperators and recommended defectors show that if group quality is appropriate for recommendation, cooperation and recommendation can simultaneously emerge. Moreover, we find that local recommendation improves cooperation more than global recommendation. As an extension, we also present results for Barabási–Albert networks. The positive effect of recommendation on cooperation for Barabási–Albert networks is independent of group quality. Our results provide an insight into the evolution of cooperation in real social systems.     

Heritability promotes cooperation in spatial public goods games

Heritability is ubiquitous within most real biological or social systems. A heritable trait is most simply an offspring’s trait that resembles the parent’s corresponding trait, which can be fitness, strategy, or the way of strategy adoption for evolutionary games. Here we study the effects of heritability on the evolution of spatial public goods games. In our model, the fitness of players is determined by the payoffs from the current interactions and their history. Based on extensive simulations, we find that the density of cooperators is enhanced by increasing the heritability of players over a wide range of the multiplication factor. We attribute the enhancement of cooperation to the inherited fitness that stabilizes the fitness of players, and thus prevents the expansion of defectors effectively.     

Benefits of intervention in spatial public goods games

The anarchy driven by private punishment is inopportune and inappropriate in modern human societies. It is necessary for a person to appeal to a higher authority such as the police so that a society can maintain more stable. We introduce strategy intervention instead of punishment in spatial public goods game. Some defectors are forced to contribute to the common pool. We show how strategy intervention affects cooperation of a population. Interestingly, weak intervention restrains the impact of spatial reciprocity leading to a lower level of cooperation or even a full defection state. Such phenomena are in contrast to ordinary intuitions. Intervention is enforced by a higher authority which avoids the second-order problems. Furthermore, high synergy factor and proper intervention has a mutual impact on increasing group incomes. We highlight the importance of institutional intervention in a stable society.     

Punishment in optional public goods games

In this work,the optional public goods games with punishment are studied.By adopting the approximate best response dynamics,a micro model is given to explain the evolutionary process.Simultaneously,the magnitude of rationality is also considered.Under the condition of bounded rationality which provides a light to interpret phenomena in human society,the model leads to two types of equilibriums.One is the equilibrium without punishers and the other is the equilibrium including only punishers and cooperators.In addition,the effects of rationality on equilibriums are briefly investigated.     

Static and evolutionary quantum public goods games

We apply the continuous-variable quantization scheme to quantize public goods game and find that new pure strategy Nash equilibria emerge in the static case. Furthermore, in the evolutionary public goods game, entanglement can also contribute to the persistence of cooperation under various population structures without altruism, voluntary participation, and punishment.     

自我质疑机制下公共物品博弈模型的相变特性

公共物品博弈是研究群体相互作用的经典模型,广泛用于解释自私个体间合作的涌现和保持.本文从理论分析和蒙特卡罗模拟两个方面研究了二维正方格子上一个有偿惩罚机制下随自我质疑更新规则演化的公共物品博弈模型的相变特性.理论分析方面,将公共物品博弈模型转化为一个外场不为零的铁磁Ising模型.通过有效能量发现:不存在惩罚时,个体间的耦合强度为零,体系只有外场作用;存在惩罚时,个体间包含最近邻、次近邻和第三近邻相互作用且外场不为零.蒙特卡罗模拟方面,首先验证了理论分析的正确性,然后对公共物品博弈模型相关的一级相变和二级相变进行了有限尺度标度分析.研究发现:1)蒙特卡罗模拟所得结果与类Ising模型分析结果完全吻合;2)相比二维Ising模型,公共物品博弈的二级相变临界指数发生了变化;3)公共物品博弈的一级相变与二维Ising模型相同.     

Diversity of contribution promotes cooperation in public goods games

In most previous studies of public goods game, individuals conventionally donate their contributions equally to the games they participate in. We develop an extended public goods game model, in which individuals distribute their contributions based on the groups’ qualities. Namely, the individuals are allowed to increase their investment to the superior groups at the expense of the nasty ones. The quality of a group is positively correlated with its cooperation level. In numerical simulations, synchronized stochastic strategy updating rule based on pairwise comparison for a fixed noise level is adopted. The results show that the high-quality group preference mechanism can greatly improve cooperation, compared with conventional models. Besides, the system with stronger preference toward high-quality groups performs better. Investigation of wealth distribution at equilibrium reveals that cooperators’ wealth appreciates with the increase of preference degree when cooperators take up the same fraction of the population.     

Heterogeneity of allocation promotes cooperation in public goods games

We investigate the effects of heterogeneous investment and distribution on the evolution of cooperation in the context of the public goods games. To do this, we develop a simple model in which each individual allocates differing funds to his direct neighbors based upon their difference in connectivity, because of the heterogeneity of real social ties. This difference is characterized by the weight of the link between paired individuals, with an adjustable parameter precisely controlling the heterogeneous level of ties. By numerical simulations, it is found that allocating both too much and too little funds to diverse neighbors can remarkably improve the cooperation level. However, there exists a worst mode of funds allocation leading to the most unfavorable cooperation induced by the moderate values of the parameter. In order to better reveal the potential causes behind these nontrivial phenomena we probe the microscopic characteristics including the average payoff and the cooperator density for individuals of different degrees. It demonstrates rather different dynamical behaviors between the modes of these two types of cooperation promoter. Besides, we also investigate the total link weights of individuals numerically and theoretically for negative values of the parameter, and conclude that the payoff magnitude of middle-degree nodes plays a crucial role in determining the cooperators’ fate.     

Maintenance of cooperation induced by punishment in public goods games

In this paper, we study the public goods games with punishment by adopting the well-known approximate best response dynamics. It shows that the evolution of cooperation is affected by two aspects when other parameters are fixed. One is the punishment mechanism which can avoid the dilemma of lacking investment, and the other is the degree of rationality. Theoretical analysis and numerical results indicate that the existence of punishment mechanism and distribution of rationality are the keys to the enhancement of cooperation level. We also testify that they can heavily influence the payoffs of system as well. The findings in this paper may provide a deeper understanding of some social dilemmas.     

Effect of habitat destruction on cooperation in public goods games

We explore the evolution of cooperation in a public goods game where the habitat destruction is taken into account. A model of ordinary differential equations is proposed, in which incorporate the habitat destroyed. And we focus on the impact of different levels of habitat destruction on cooperation. Our results show that the fraction of cooperation receives the biggest boost for moderate habitat destruction, and decreases to zero very quickly for a higher degree of habitat destruction. Similarly, our results suggest that low or moderate habitat degradation can promote the evolution of cooperators, and once habitat degradation is too severe, the fraction of cooperators will decline dramatically. Furthermore, we investigate the cooperation level with different multiplication factor, and results demonstrate that the cooperation increases monotonously with the increasing of the multiplication factor. Our findings may provide many more insights for understanding the emergence and maintenance of cooperation in the complex biological and social systems.     

Coevolving agent strategies and network topology for the public goods games

Much of human cooperation remains an evolutionary riddle. Coevolutionary public goods games in structured populations are studied where players can change from an unproductive public goods game to a productive one, by evaluating the productivity of the public goods games. In our model, each individual participates in games organized by its neighborhood plus by itself. Coevolution here refers to an evolutionary process entailing both deletion of existing links and addition of new links between agents that accompanies the evolution of their strategies. Furthermore, we investigate the effects of time scale separation of strategy and structure on cooperation level. This study presents the following: Foremost, we observe that high cooperation levels in public goods interactions are attained by the entangled coevolution of strategy and structure. Presented results also confirm that the resulting networks show many features of real systems, such as cooperative behavior and hierarchical clustering. The heterogeneity of the interaction network is held responsible for the observed promotion of cooperation. We hope our work may offer an explanation for the origin of large-scale cooperative behavior among unrelated individuals.     

Pool expulsion and cooperation in the spatial public goods game

We study the evolution of cooperation in the spatial public goods game with cooperation, defection, and pool expulsion as the three competing strategies. Using the Monte Carlo method, we show that the evolution of pool expulsion and cooperation can be maintained even if the synergistic effects are not high enough to sustain cooperation based on spatial reciprocity alone, and even if the cost of pool expulsion is not negligible. Interestingly, pool expellers are protected against, or even prevail over, defectors as a result of spatial pattern formation, by means of which vacant sites form an active layer around them. Moreover, we observe continuous and discontinuous phase transitions between frozen coexistence, stationary coexistence, absorbing states, and oscillatory states in the phase diagrams. Our results indicate that pool expulsion might play an important role in the resolution of social dilemmas that unfold in groups.     

Preferential selection promotes cooperation in a spatial public goods game

We introduce a preferential selection mechanism into a spatial public goods game where players are located on a square lattice. Each individual chooses one of its neighbors as a reference with a probability proportional to , where P_y is the neighbor’s payoff and A (≥0) is a tunable parameter. It is shown that the introduction of such a preferential selection can remarkably promote the emergence of cooperation over a wide range of the multiplication factor. We find that the mean payoffs of cooperators along the boundary are higher than that of defectors and cooperators form larger clusters as A increases. The extinction thresholds of cooperators and defectors for different values of noise are also investigated.     

Evolutionary public goods games on scale-free networks with unequal payoff allocation mechanism

In this paper, we bring an unequal payoff allocation mechanism into evolutionary public goods game on scale-free networks and focus on the cooperative behavior of the system. The unequal mechanism can be tuned by one parameter α: if α>0, the hub nodes can use its degree advantage to collect more payoff; if α<0, numerous non-hub nodes will obtain more payoff in a single round game. Simulation results show that the cooperation level has a non-trivial dependence on α. For the small enhancement factor r, the cooperator frequency can be promoted by both negative and positive α. For large r, there exists an optimal α that can obtain the highest cooperation level. Our results may sharpen the understanding of the emergence of cooperation induced by the unequal payoff allocation mechanism.     

Dynamic allocation of investments promotes cooperation in spatial public goods game

In this paper, we introduce a dynamic allocation investment mechanism into the spatial public goods game. In the model, if a cooperator i$i$ collects more payoffs from one group centered on the player j$j$ in the last round, he will put more investments to the group in the next round. The intensity of investment is controlled by a tunable parameter α$\alpha$. Simulation results show that the emergence of cooperation is remarkably promoted, and a larger α$\alpha$ induces a higher cooperation level. These interesting phenomena are then extensively explained by pairwise investments, average payoffs and evolution snapshots.     

Emergence of cooperation in spatial public goods game with conditional participation

Conditional interactions are common in both human and animal societies. To understand the impacts of this feature on the evolution of cooperation, we propose a modified public goods game combined with conditional interactions in terms of the aspiration payoffs. Through simulations, we find that the function of the fraction of cooperators and the synergy factor is non-monotonic. This indicates that a large synergy factor is not always in favor of the promotion of cooperation. In addition, for a high aspiration, the typical coexistence state of cooperators and defectors could disappear, and the system demonstrates a sharp transition from the complete defection state to the complete cooperation state as the synergy factor increases. Furthermore, an interesting critical phenomenon is found in a finite system, i.e., the system can randomly evolve into a complete defection state or a complete cooperation state. An explanation of these evolutionary outcomes is provided in this paper, which is in agreement with the simulation results.     

Interaction stochasticity may hinder cooperation in the spatial public goods game

We propose an analytic model to explore the effect of interaction stochasticity on the evolution of cooperation in spatial public goods game. The results show that whether cooperation can dominate in populations crucially depends on the player's probability of opting-out. Stochastic opting-out enhances cooperation as long as the probability of opting-out is less than a threshold depending on the graph's degree. Otherwise the promoting effect of spatial structures on cooperation is hindered even neutralized by stochastic opting-out. Moreover, there exists an intermediate optimal probability with which the advantage of cooperation over defection is maximized in the evolutionary race. Interestingly, the optimal probability is related to the percolation threshold of the underlying graph. Our findings illustrate that spatial structures may not facilitate cooperation when stochastic opting-out is allowed, and provide a link between physics and social sciences.     

Cooperation and charity in spatial public goods game under different strategy update rules

Human cooperation can be influenced by other human behaviors and recent years have witnessed the flourishing of studying the coevolution of cooperation and punishment, yet the common behavior of charity is seldom considered in game-theoretical models. In this article, we investigate the coevolution of altruistic cooperation and egalitarian charity in spatial public goods game, by considering charity as the behavior of reducing inter-individual payoff differences. Our model is that, in each generation of the evolution, individuals play games first and accumulate payoff benefits, and then each egalitarian makes a charity donation by payoff transfer in its neighborhood. To study the individual-level evolutionary dynamics, we adopt different strategy update rules and investigate their effects on charity and cooperation. These rules can be classified into two global rules: random selection rule in which individuals randomly update strategies, and threshold selection rule where only those with payoffs below a threshold update strategies. Simulation results show that random selection enhances the cooperation level, while threshold selection lowers the threshold of the multiplication factor to maintain cooperation. When charity is considered, it is incapable in promoting cooperation under random selection, whereas it promotes cooperation under threshold selection. Interestingly, the evolution of charity strongly depends on the dispersion of payoff acquisitions of the population, which agrees with previous results. Our work may shed light on understanding human egalitarianism.