Spatial Public Goods Game Research Articles

History loyalty-based reward promotes cooperation in the spatial public goods game

Reward has been proved to be an effective mechanism to sustain cooperation among selfish individuals. In this paper, we propose a history loyalty-based reward in which a cooperator can gain additional reward if the time he sticks to the cooperation strategy is over a loyalty threshold. Accordingly, defectors have to bear the cost of reward subsequently. The results on the spatial public goods game show that the cooperation could be immensely enhanced when the loyalty threshold and the reward factor are suitable. Besides, the time evolution of cooperator density and the spatial distribution of cooperators and defectors are investigated. Our work extends the form of reward in the evolution of spatial public goods game.

Effects of enhancement level on evolutionary public goods game with payoff aspirations

In this work, we study the spatial evolutionary public goods game by introducing payoff aspirations for players, and players update their strategies with a stochastic probability depending on the difference between their collecting payoffs from neighbors and own payoff aspirations. A striking finding is that the cooperation level is a nonmonotonic function of the enhancement factor for a fixed aspiration level, which means a proper enhancement factor leads to the optimal cooperation level, whereas too high or too low enhancement factors will impede the evolution of cooperation for a particular aspiration level. This phenomenon is contradictory to the previous finding that a larger enhancement factor always leads a higher cooperation level. We explain this anomaly with a comprehensive analysis of the probabilities transitions of C players to D players as well as the reverse. The presented results may be not only helpful in understanding the cooperative behavior induced by the aspiration level in economic or animal society, but also have important implications for the cooperation regulation in system by enhancement factors and payoff aspirations.

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.

Evolutionary Cooperation in Networked Public Goods Game with Dependency Groups

Either in microlevel organizations or macrolevel societies, the individuals acquire benefits or payoffs by forming interdependency groups linked by common interests. Conducting research on the effects of interdependency groups on the evolution of cooperation could have a better understanding of the social dilemma problem. In this paper, we studied a spatial public goods game with nonlocal interdependency groups where each of participants is located in a two‐dimensional square lattice or Watts–Strogatz small‐world network with payoffs obtaining from the interactions with nearest neighbors. In terms of the enhancement factor, the effects of group density on the evolutionary cooperation can be quite different. For a low enhancement factor, the cooperation level is a nonmonotonic function with the varying density of interdependency groups in the system, which means a proper density of interdependency groups can best promote the cooperative level. For a moderate enhancement factor, a higher density of interdependency groups can always correspond to a higher cooperative level. However, if the enhancement factor is too high, a high density of interdependency groups can impede the evolutionary cooperation. We give the explanations for the different roles of group density of interdependency by using the transition probabilities of C players into D players as well as the reverse. Our findings are very helpful for the understanding of emergence cooperation as well as the cooperation regulation in the selfish individuals.

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.

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.

Understanding spatial public goods games on three-layer networks

Cooperation abounds in all biological systems. Spatial public goods game (PGG) serves as a baseline model when investigating the evolution of cooperation in collective interactions. Since analyzing collective interactions and dynamics in spatial structures is very complicated, a full theoretical understanding of spatial PGGs is still deficient. Here we generalize spatial PGGs in a three-layer weighted network—investment, benefit allocation, and strategy dispersal layers—which are not necessarily identical and thus cover a wide class of population structures and interaction scenarios. We provide an analytic formula that accurately predicts when cooperation is favored over defection, and that is applicable in populations of any size under weak selection. We prove that in regular networks investment and benefit allocation are essentially symmetric—that exchanging structures of investment and benefit allocation layers does not affect the evolutionary dynamics at all. The success of cooperators relies heavily on the correlation between an individual’s investment in a game and its benefit allocated from the same game. In most cases, the positive correlation, i.e., a greater investment in games with a larger share of benefits, facilitates a cooperative society. Importantly, we also show that diversifying the amounts of investment in different games or benefit allocation to different participants, if implementing improperly, might impede the global cooperation.

Locality based wealth rule favors cooperation in costly public goods games

Most previous investigations on spatial Public Goods Games (PGGs) assume that individuals are self-organized and all are engaged in every available groups, which is in sharp contrast with realistic situations, where players may follow some local rules of whether they can participant or not. In this paper we embed the locality based wealth rule into self-adaptive region based mechanism to study the evolution of cooperation. When deciding about participation, both individual wealth of a player and qualification threshold of its local region are considered. Player whose wealth exceeds its local limit is allowed to participant in the joint venture by paying some entrance fee, which is independent of its strategy of whether cooperate or not. Otherwise, they are forbidden in the current round. Our result shows that the proposed mechanism can boost the emergence and maintenance of cooperation, with moderate values of controlling strength, and when the entrance fee is not harsh. Furthermore, small region segmentation scheme favors cooperation better with small controlling factor, whereas larger regional size can slow down the extinction process of cooperators when controlling factor is relatively high. We hope that our findings could shed light on better understanding of the emergence of cooperation among structured populations.

Public cooperation in two-layer networks with asymmetric interaction and learning environments

Strategy updating is generally based on payoff comparison and strategy learning within the interaction pairs on networks in evolutionary games. In many previous works, the interaction and learning environments are assumed to be the same networks. However, in the real world, they might be different. In this work, we consider the spatial public goods game on two-layer networks, where the interaction and learning environments are represented by two asymmetric layers, respectively. We focus on the effects of edge overlap ω between the interaction and learning networks on the evolution of cooperation. The simulation results show that, the effects of ω on the evolution of cooperation depend on the synergy factor r. For relatively small r, higher overlap between the interaction and learning environments will be more favorable for cooperation. However, the situation is reverse for relatively large r, where the lower overlap between the interaction and learning environments results in higher level of cooperation. We also find that the asymmetry between the interaction and learning environments inhibits the coexistence of the cooperators and defectors. Furthermore, we show that the results of the model are robust to the underlying networks with different node degrees.

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.

Spatial public goods game with continuous contributions based on Particle Swarm Optimization learning and the evolution of cooperation

How to explain the evolution of cooperation in social dilemmas has plagued the theorists of various disciplines. In this paper, we study the public goods game on two structured populations, assuming that each individual can make a continuous contribution in its strategy space. In our model, individuals have memory and imitation ability. They can remember their best history strategy and learn their best neighbors by obtaining the local information. We use the particle swarm optimization (PSO) algorithm to simulate the learning process of individuals. Two network structures of the square lattice and the nearest-neighbor coupled network are considered respectively. We investigate the combined effects of memory, the invest enhancement and the network structure on the average cooperation level of the population. By simulation experiments, we find that the PSO learning mechanism can promote the evolution of cooperation in a large range of parameters under both of the two kinds of networks. Compared with the square lattice network, the nearest-neighbor coupled network can promote the evolution of cooperation in a larger parameter scope. Compared with the Fermi rule and the Genetic Algorithm learning, the PSO learning can induce the population achieve a wider range of cooperation level, which makes it possible to achieve a higher level of cooperation. These results are conducive to a better understanding of the emergence of cooperation in the real world.

Tolerance-based punishment and cooperation in spatial public goods game

Punishment, as a remarkable way, has been proposed to explain the emergence and persistence of cooperation in the human species. Inspired by the fact that people have a certain tolerance for free-riders before punishment, therefore, we study the effect of tolerance-based punishment on the evolution of cooperation in spatial public goods game. Cooperators punish defectors on the basis of the tolerance threshold during the evolutionary process and have to bear the relevant costs of sanction subsequently. Different from previous works, the new mechanism can reduce the frequency of punishing by controlling the tolerance for punishment. We find that this mechanism can lead to synergistic effects, and it can stabilize the circumstance of full cooperation under adverse conditions. By means of analysis of the emergence of cooperative clusters, we demonstrate that the tolerance-based punishment can promote cooperation through enhancing spatial reciprocity. In addition, the readiness of cooperation increases obviously by adjusting this kind of punishment. Our work extends the form of punishment in the evolution of spatial public goods game and the results are conducive to a better understanding of punishment.

Stochastic win-stay-lose-learn promotes cooperation in the spatial public goods game

In this paper, we propose a stochastic win-stay-lose-learn in which an individual is more likely to imitate one of its nearest neighbor’s strategy if its aspiration is not achieved. The results on the spatial public good game show that the cooperation can be greatly enhanced when the aspiration is moderate. Besides, we have studied the time evolution of the spatial distribution of strategies and the probability that cooperators and defectors choose to learn respectively.

Strategy intervention in spatial voluntary public goods games

It is well known that punishment has been considered to enhance cooperation based on empirical and theoretical studies. An important question arises over the existence of the “second-order” problems, which result from the interactions between individuals. In this paper, we propose strategy intervention as a new mechanism in spatial voluntary public goods game, in which individuals only know their own payoffs. By virtue of centralized institution, the defectors may reserve amounts of probabilities to contribute to the common pool with a certain amount of investments. The centralized institution is established costly by all the participants to enforce the intervention rather than peer punishment. We find that the number of cooperators (defectors) decreases (increases) with weak intervention, which contrasts our intuition. Loners vanish and cooperation emerges significantly, as the level of intervention reaches a threshold. We give an accurate range of intervention leading to full cooperation, and demonstrate that at partial cooperation state, proper intervention can remarkably increase income accumulations of individuals, between which there exists smaller income gap in contrast to typical models. We highlight the significance of intervention enforced by a higher authority for maintaining social stability.

Group formation in the spatial public goods game with continuous strategies

We study group formation in the spatial public goods game with continuous strategies. An active group will become inactive with high (low) probability if the collective contribution is below (above) a threshold value. Meanwhile, an inactive group recovers to be active with some probability. We have found that the cooperation level and the average payoff of players are maximized at moderate values of the threshold and the recovery rate. Spatial distributions of strategies and active groups are plotted to understand the evolution of cooperation.

Contribution inequality in the spatial public goods game: Should the rich contribute more?

Scale-free property exists in resource distribution, and the rich pay more in public goods is commonplace in reality. What will happen if the rich are expected to contribute more in the spatial public goods game? This paper therefore proposes a new contribution paradigm, in which individual contribution is determined by his payoff in the last evolution step. Tunable parameter w is used to characterize the contribution rate of nodes whose payoff is larger than the average. The results of simulations reveal that the impact of w on cooperation is associated with the enhancement factor r. When r is low, the higher w is, the lower the cooperation rate is. With the increment of r, the value of w to optimize cooperation rate increases with r. The relationship of cooperation rate and wealth on the network has also been investigated. Interestingly, higher cooperation rate does not always lead to higher wealth. Finally, the impact of w on the wealth distribution on the network is explored. The higher w reduces the inequality in wealth distribution by the shrinking of lower-class, which is enhanced by a higher r.

Peer pressure and incentive mechanisms in social networks

Cooperation can be viewed as a social norm that is expected in our society. In this work, a framework based on spatial public goods game theory is established to study how peer pressure and incentive mechanisms can influence the evolution of cooperation. A unified model with adjustable parameters is developed to represent the effects of pure Personal Mechanism, Personal Mechanism with peer pressure and Social Mechanism, which demonstrates that when the sum of rewards plus the peer pressure felt by defectors is larger than the effective cost of cooperation, cooperation can prevail. As the peer pressure is caused by other cooperators in a game, group size and network structure play an important role. In particular, larger group size and more heterogeneous structured population can make defectors feel more peer pressure, which will promote the evolution and sustainment of cooperation.

Stability of subsystem solutions in agent-based models

The fact that relatively simple entities, such as particles or neurons, or even ants or bees or humans, give rise to fascinatingly complex behaviour when interacting in large numbers is the hallmark of complex systems science. Agent-based models are frequently employed for modelling and obtaining a predictive understanding of complex systems. Since the sheer number of equations that describe the behaviour of an entire agent-based model often makes it impossible to solve such models exactly, Monte Carlo simulation methods must be used for the analysis. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among agents that describe systems in biology, sociology or the humanities often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. This begets the question: when can we be certain that an observed simulation outcome of an agent-based model is actually stable and valid in the large system-size limit? The latter is key for the correct determination of phase transitions between different stable solutions, and for the understanding of the underlying microscopic processes that led to these phase transitions. We show that a satisfactory answer can only be obtained by means of a complete stability analysis of subsystem solutions. A subsystem solution can be formed by any subset of all possible agent states. The winner between two subsystem solutions can be determined by the average moving direction of the invasion front that separates them, yet it is crucial that the competing subsystem solutions are characterised by a proper composition and spatiotemporal structure before the competition starts. We use the spatial public goods game with diverse tolerance as an example, but the approach has relevance for a wide variety of agent-based models.

Impact of individual difference and investment heterogeneity on the collective cooperation in the spatial public goods game

A novel PGG model with individual difference and investment heterogeneity is proposed.The individual difference is implemented by classifying the individuals into two classes.The investment heterog...

Promotion of cooperation based on swarm intelligence in spatial public goods games

In this paper, we introduce the swarm intelligence methods into the evolutionary dynamics, and have studied the impact of swarm intelligent algorithm on the evolution of cooperation among selfish individuals in the continuous version of spatial public goods games (PGG). We update an individual’s strategy according to the memory which records the most successful individual strategy in the past (referred to as its personal best strategy) as well as the knowledge of the best current strategy found by its nearest neighbors (referred to as the neighborhood best strategy). Through extensive simulations, we find that the introduction of swarm intelligence into PGG can promote cooperation strongly. Other pertinent quantities such as the time evolution of cooperator density, the spatial distribution of strategies and the updated velocities are also investigated.