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.     

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.     

Selective investment promotes cooperation in public goods game

Most previous investigations on spatial Public Goods Game assume that individuals treat neighbors equivalently, which is in sharp contrast with realistic situations, where bias is ubiquitous. We construct a model to study how a selective investment mechanism affects the evolution of cooperation. Cooperators selectively contribute to just a fraction among their neighbors. According to the interaction result, the investment network can be adapted. On selecting investees, three patterns are considered. In the random pattern, cooperators choose their investees among the neighbors equiprobably. In the social-preference pattern, cooperators tend to invest to individuals possessing large social ties. In the wealth-preference pattern, cooperators are more likely to invest to neighbors with higher payoffs. Our result shows robustness of selective investment mechanism that boosts emergence and maintenance of cooperation. Cooperation is more or less hampered under the latter two patterns, and we prove the anti-social-preference or anti-wealth-preference pattern of selecting investees can accelerate cooperation to some extent. Furthermore, the theoretical analysis of our mechanism on double-star networks coincides with simulation results. We hope our finding could shed light on better understanding of the emergence of cooperation among adaptive populations.     

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.     

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.     

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.     

Cooperation in the spatial public goods game with the second-order reputation evaluation

The existence of reputation can significantly promote the level of cooperation within the human population. In the recent years, most of the researches were based on binary image score or first-order evaluation standard. In this paper, we propose a second-order reputation evaluation model, in which the individual's image score will change not only in accordance with his own strategy, but also the reputation value of neighbors. Individuals try to enhance their reputation to cooperate with the surrounding high-reputation individuals, and then becomes an influential individual in the population. The existence of this mechanism renders the individuals who at the edge of the clusters formed by the cooperators to rapidly accumulate their reputation values through the adoption of cooperative strategy, and then the cooperative strategy can be spread widely and rapidly in the whole population. Through extensive numerical simulations, it is clearly indicated that the population cooperation behavior will be obviously improved when the individual's influence factor becomes smaller or the reputation step length increases. The current results are further conducive to understanding the emergence of cooperation in many real world systems.     

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.     

Evolution of cooperation in the spatial public goods game with the third-order reputation evaluation

In this letter, in order to deeply explore the role of individual reputation in the evolutionary game dynamics, we present a new third-order reputation evaluation model to discuss the evolution of cooperation in the spatial public goods game. In the current model, we should not only consider the strategy (cooperation, C or defection, D) of a focal player, but also take his own reputation and his opponent's reputation status into account. Among them, the individual reputation will be divided into being good and bad according to the specified threshold, and the good player will be endowed with the more influential strategy transfer ability, which further helps to create the clusters of cooperative and good players within the population and then fosters the cooperation. A large plethora of experimental simulation results indicate that four rules under the third-order reputation mechanism can lead to the promotion of cooperation when compared to the traditional public goods game model. The current work is conductive to a better understanding of the persistence and emergence of collective cooperation in real-world systems.     

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.     

Tolerance-based punishment in continuous public goods game

Altruistic punishment for defectors is considered as a key motive for the explanation of cooperation. However, there is no clear border between the cooperative and defective behaviors in a continuous strategy game. We propose a model to study the effect of punishment on the evolution of cooperation in continuous public goods game, wherein individuals have the traits to punish the co-players based on social tolerance. We show that a reasonable punishment with a uniform tolerance can spur individuals to make more investments. Additionally, for a fixed punishment cost and a fixed fine, a moderate value of tolerance can result in the best promotion of cooperation. Furthermore, we investigate the coevolutionary dynamics of investment and tolerance. We find that the population splits into two branches: high-tolerance individuals who make high investments and low-tolerance individuals who make low investments. A dynamic equilibrium is achieved between these two types of individuals. Our work extends punishment to continuous cooperative behaviors and the results may enhance the understanding of altruistic punishment in the evolution of human cooperation.     

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.     

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.     

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.     

The evolutionary public goods game on scale-free networks with heterogeneous investment

The public goods game (PGG) is generally considered as a suitable paradigm to explain ubiquitous cooperative behavior. In this study, we investigated the evolutionary PGG on scale-free networks and studied the effect of individual heterogeneity by setting the cooperator x an investment value correlated to its degree as I_x=N⋅k_x^β/∑_jk_j^β, where k_x is the degree of x, j runs over all players and β is a tunable parameter. It is shown that the cooperation level is remarkably promoted by negative values of β whereas it is highly depressed by positive values of β. Moreover, the effect of environmental noise has also been investigated. Our result may sharpen the understanding of cooperation induced by the individual diversity.     

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.     

The roles of heterogeneous investment mechanism in the public goods game on scale-free networks

The public goods game is an important theoretical model for investigating the emergence of cooperation in the multi-player social dilemma. It has been proven that scale-free networks can significantly promote cooperation, but fail to sustain cooperation when the player obtains the normalized payoff. In this paper, we introduce heterogeneous investment mechanism into the public goods game on scale-free networks, and study the evolution of cooperation in both cases of accumulated and normalized payoff. Our research reveals that the heterogeneous investment mechanism can obviously facilitate cooperation as the adjusted parameter α increases. The increase of α allows cooperators to emerge under lower values of r. In the case of accumulated payoff, cooperators always firmly occupy the hubs, and the population keeps high cooperation level. In the case of normalized payoff, the increase of α changes the situation that the hubs are easily invaded by defectors, and inhibits the spread of defectors.     

Continuous spatial public goods game with self and peer punishment based on particle swarm optimization

How cooperative behavior emerges and evolves in human society remains a puzzle. It has been observed that the sense of guilt rooted from free-riding and the sense of justice for punishing the free-riders are prevalent in the real world. Inspired by this observation, two punishment mechanisms have been introduced in the spatial public goods game which are called self-punishment and peer punishment respectively in this paper. In each situation, we have introduced a corresponding parameter to describe the level of individual tolerance or social tolerance. For each individual, whether to punish others or whether it will be punished by others depends on the corresponding tolerance parameter. We focus on the effects of the two kinds of tolerance parameters on the cooperation of the population. The particle swarm optimization (PSO)-based learning rule is used to describe the strategy updating process of individuals. We consider both of the memory and the imitation in our model. Via simulation experiments, we find that both of the two punishment mechanisms could facilitate the promotion of cooperation to a large extent. For the self-punishment and for most parameters in the peer punishment, the smaller the tolerance parameter, the more conducive it is to promote cooperation. These results can help us to better understand the prevailing phenomenon of cooperation in the real world.     

Promotion of cooperation induced by nonuniform payoff allocation in spatial public goods game

A nonuniform payoff allocation mechanism is proposed for spatial public goods games where individuals are nodes on a scale-free network. Each individual is assigned a weight k_i ^α, where k_i is the degree of individual i and α is an adjustable parameter that controls the degree of diversity in individuals’ profits. During the evolution progress, the allocation of payoff on individual i is assumed to be proportional to its weight. Individuals synchronously update their strategies according to the stochastic rule with a fixed noise level. It is found that there exists an optimal value of α which yields the highest level of cooperation. Other pertinent quantities, including the payoff and the probability of finding a node playing as cooperator versus the degree, are also investigated computationally and analytically. Our results suggest that a suitable degree of diversity among individuals can promote the emergence of cooperation.