Public Goods Game Model Research Articles

When multi-group selection meets mystery of cooperation in structured public goods games.

Cooperation is the cornerstone of social stability and human development. In order to promote mutual cooperation among individuals, some researchers analyzed the important factors influencing individual behavior from the perspective of group selection, while others revealed the evolutionary mechanism of cooperative behavior in groups from the perspective of network reciprocity. However, group selection and network reciprocity actually work together and simultaneously drive individuals to cooperate with each other. Analyzing each mechanism in isolation provides an incomplete understanding of the interaction process. Inspired by this, we integrate the coupled effects of both group selection and network reciprocity on the behavior of individuals. We develop a structured public goods game model to study the evolution of individual cooperative behavior in multiple groups, where each individual can interact not only with intra-group individuals but also with inter-group individuals. Based on the fixed probabilities of multi-group selection, including intra-group and inter-group selection, we derive a general condition that promotes cooperation among individuals. Besides, we discuss the effects of the number of neighbors in a group, group size, and group size on the selection of cooperative behavior. Finally, we systematically compare our model with the well-mixed case, and the results show that a structured population enhances cooperation. Increasing the number of populations boosts the fixation probability of cooperation. To the best of our knowledge, this paper is the first to study the cooperative evolutionary dynamics of multi-group selection in structured populations through public goods games.

A double-edged sword: diverse interactions in hypergraphs

Abstract Interaction diversity and interaction identity, as two classic interaction patterns in real-world systems, are frequently employed to probe into the cooperative dilemmas among participants with diverse interaction identities within groups. Considering the influence of factors such as hyperlink configurations on the evolution of cooperation within complex networks, this study particularly focuses on the organic integration of hypergraphs and diverse interaction modes. Employing the public goods game model in uniformly stochastic hypergraphs, we explore the effects of synergy factors and diverse interacting individuals on the cooperation rate and strategy consistency within hyperlinks. We emphasize that the diverse interacting individuals are a double-edged sword, and the controlled number of such individuals is conducive to the evolution of cooperation. The findings of this study reveal the significance of these two typical interaction patterns and provide valuable insights for coordinating them to achieve social optimality in cooperation.

Free-rider or contributor: A dilemma in spatial threshold public goods games

Late movers refer to individuals who adjust their strategies after observing the choices of others in a public goods game, can significantly impact the likelihood of public goods provision. This creates a social dilemma: late movers act as free-riders to rely on others to contribute, yet they also risk the public good not being provided if contributions fall short. To address this dilemma, we utilize a spatial threshold public goods game model to examine how the behaviors of late movers influence overall cooperative levels within a population. We propose a probabilistic rule for late movers that captures the dilemma between cooperating to enhance the collective good and free-riding to maximize individual benefits. Our findings indicate that strategic late movers can significantly enhance cooperation by adapting to local cooperation levels. However, an excessively high proportion of late movers may reduce the overall level of cooperation. Additionally, the formation of clusters by late movers results in higher cooperative levels compared to random distribution. Our study provides novel insights into the dynamics of cooperation within the framework of spatial threshold public goods games. By incorporating the role of late movers, we highlight how adaptive strategies can mitigate the free-rider problem and promote cooperative behaviors.

Replicator-mutator dynamics with evolutionary public goods game-environmental feedbacks.

Feedback loops between strategies and the environment are commonly observed in socio-ecological, evolution-ecological, and psychology-economic systems. However, the impact of mutations in these feedback processes is often overlooked. This study proposes a novel model that integrates the public goods game with environmental feedback, considering the presence of mutations. In our model, the enhancement factor of the public goods game combines positive and negative incentives from the environment. By employing replicator-mutator (RM) equations, we provide an objective understanding of the system's evolutionary state, focusing on identifying conditions that foster cooperation and prevent the tragedy of the commons. Specifically, mutations play a crucial role in the RM dynamics, leading to the emergence of an oscillatory tragedy of the commons. By verifying the Hopf bifurcation condition, we establish the existence of a stable limit cycle, providing valuable insights into sustained oscillation strategies. Moreover, the feedback mechanism inherent in the public goods game model offers a fresh perspective on effectively addressing the classic dilemma of the tragedy of the commons.

Bonus-based mercenary punishment promotes cooperation in public goods games

Various regions often adopt punish strategies to solve traffic congestion problems. Punishing defectors is an effective strategy to solve the first-order free-rider problem in a public goods game. But this behavior is costly because the punisher is often also involved in the original joint venture and therefore vulnerable, which jeopardizes the effectiveness of this incentive. As an option, we could hire special players whose sole duty would be to monitor the population and punish defectors. The fines collected by various regions will also be used to subsidize the construction of public transportation. Thereby, we derive inspiration, and propose an improved public goods game model based on bonus and mercenary punishment. Research has shown that after cooperator gives the punisher an appropriate bonus, cooperators can strengthen the punisher, thereby weakening the defector's advantage and indirectly promoting cooperation by stabilizing the punisher's position in the system. In addition, the mechanism of reusing the fines collected from defectors and then subsidize to other players in the system can directly promote the emergence of cooperation.

Discreet investors exert greater influence on cooperation in the public goods game

This paper studies the evolutionary process of cooperative behavior in a public goods game model with heterogeneous investment strategies in square lattices. In the proposed model, players are divided into defectors, cooperators and discreet investors. Among these, defectors do not participate in investing, discreet investors make heterogeneous investments based on the investment behavior and cooperation value of their neighbors, and cooperators invest equally in each neighbor. In real life, heterogeneous investment is often accompanied by time or economic costs. The discreet investors in this paper pay a certain price to obtain their neighbors’ investment behavior and cooperation value, which quantifies the time and economic costs of the heterogeneous investment process. The results of Monte Carlo simulation experiments in this study show that discreet investors can effectively resist the invasion of the defectors, form a stable cooperative group and expand the cooperative advantage in evolution. However, when discreet investors pay too high a price, they lose their strategic advantage. The results in this paper help us understand the role of heterogeneous investment in promoting and maintaining human social cooperation.

Heterogeneous reputation promotes cooperation in spatial public goods game

Many properties of things are heterogeneous, which promotes the reproduction and development of human society. Naturally, the reputation has heterogeneity, and the heterogeneity of reputation can effectively promote cooperation in evolutionary systems. However, how to measure the heterogeneity of reputation and reveal the evolutionary mechanisms by which it fosters cooperation is not fully understood. In this paper, a modified public goods game (PGG) model is proposed based on co-evolution mode, in which a player's reputation is measured by the number of times the player adheres to the same strategy. To better understand the influence of heterogeneous reputation on individual's decision, we propose two evolutionary mechanisms, namely the absolute mechanism and the relative mechanism based on reputation. Simulation verifies that two proposed mechanisms are easier to promote cooperation compared with the traditional PGG model, and the relative mechanism is easier to promote the evolution of cooperation than the absolute mechanism.

Optimization of supervision mechanism for public cooperation considering limited supervision ability and heterogeneous preferences

Punishment promotes cooperation among selfish agents. Unlike previous studies, we propose a new supervision (heterogeneous preference supervision, HPS) mechanism based on the original random supervision (ORS) mechanism considering regulators’ limited supervision ability and agents’ heterogeneous preferences. The concepts of exemption list capacity, observation period, and removal time are introduced as the variables under the HPS mechanism. A public goods game model is built to verify the supervision effects under the ORS and HPS mechanisms. The simulation results show that the HPS mechanism can promote cooperation more than the ORS mechanism. Increasing the exemption list capacity can make regulators pay more attention to defectors and improve the cooperation level. Setting a relatively moderate observation period benefits a better supervision effect, while a too-small or too-large observation period leads to the collapse of cooperation. Shortening the removal time can increase the updating speed of the exemption list and enhance the role of the exemption list, resulting in improving the fraction of cooperators.

The impact of current and historical reputation with non-uniform change on cooperation in spatial public goods game

In the complex human social environment, reputation plays an important role in interpersonal communication. Normally, individuals pay attention to their neighbors’ current and historical reputations during strategy learning, and the variation of individual reputation over time is often non-uniform. This paper proposes a new spatial public goods game model that involves current and historical reputation. An exponential function of the rounds in which the individual continuously maintains the cooperative or defective strategy unchanged is defined to reflect the change of the individual reputation. And two discount coefficients involving five parameters are introduced to the imitation probability of strategy learning. Simulation results indicate that high sensitivities of individuals to her/his own reputation, to the reputation of her/his neighbors, and to changes in neighbors’ historical reputations are all beneficial for the cooperative behavior of the system. And individual behavior of paying attention to changes in neighbors’ historical reputation is also beneficial for the cooperation, but attention to more historical reputation changes in neighbors does not further promote cooperative behavior. The sensitivity of the population to individual historical information about strategy has little impact on the cooperative behavior of the system.

Inertia in spatial public goods games under weak selection

Due to limited cognitive skills for perceptual error or other emotional reasons, players may keep their current strategies even if there is a more promising choice. Such behavior inertia has already been studied, but its consequences remained unexplored in the weak selection limit. To fill this gap, we consider a spatial public goods game model where inertia is considered during the imitation process. By using the identity-by-descent method, we present analytical forms of the critical synergy factor $r^\star$, which determines when cooperation is favored. We find that inertia hinders cooperation, which can be explained by the decelerated coarsening process under weak selection. Interestingly, the critical synergy conditions for different updating protocols, including death-birth and birth-death rules, can be formally linked by the extreme limits of the inertia factor. To explore the robustness of our observations, calculations are made for different lattices and group sizes. Monte Carlo simulations also confirm the results.

Co-evolution of regional integration and green innovation under two-layer network

With global low-carbon development, green innovation has becoming a key factor for countries to obtain and maintain core competitive advantages. However, the division of scientific research resources often fails to meet the requirements of economic development. Therefore, regional integration cooperation has become an inevitable choice for innovation and development. However, the effect of the impact, direction, and interaction of regional integration policies on green innovation decision-making remains uncertain. To study the synergistic development mechanism between the two, this study constructs a two-layer network model that includes the decision and influence layers and uses a graph neural network model to map the relationship between the two-layer networks. With the help of opinion dynamics model and public goods game model, we draw the following conclusions. (1) Limited communication between regions can encourage all parties to adopt active green innovation strategies. However, excessive communication can lead to disunity of opinions among all parties, which may not be conducive to green innovation and result in an increase in free-rider behavior. (2) Regional integration development can effectively prompt all parties to choose active green innovation strategies. (3) Increasing the breadth of regional integration cooperation is more effective in promoting the active participation of all parties in green innovation, compared to increasing the depth of regional integration cooperation. This study provides theoretical support for decision-making in regional integration and green innovation development.

A New Strategy to Solve “the Tragedy of the Commons” in Sustainable Grassland Ecological Compensation: Experience from Inner Mongolia, China

Based on the principle of “who protects and benefits will compensate”, in recent years, many Chinese herders have begun to voluntarily participate in sustainable grassland ecological compensation by donating money. However, this traditional strategy may have brought about “the tragedy of the commons”. A new strategy described as “wealthy herders pay money, ordinary herders participate in supervision, and relevant departments post a list of participants at the end of each month”, which was trialed in the Keshiketeng Banner, Inner Mongolia, China, attempts to solve “the tragedy of the commons”. This new strategy is neither “Leviathan” nor “Privatization”; it creates a third way for grassland herders to achieve spontaneous cooperation in protecting grassland ecology. This article presents a theoretical analysis framework and experimental simulation method using this new strategy. Considering the importance of face culture and gossip in herders’ social lives, this study used a public goods game (PGG) model to analyze and experimentally simulate the effect of this new strategy. The simulated data show the following: (1) Compared with the traditional strategy, this new strategy promotes cooperation more effectively. It requires less money, time and human capital from herders and can mobilize more herders to participate in sustainable grassland ecological compensation, leading to the emergence of a cooperation equilibrium among herders. (2) In this new strategy, the total contributions of herders are inversely proportional to herders’ “reputation tolerance”, and they are directly proportional to herders’ “income level”, “total budget” and “satisfaction and emotion”. The cooperation level is inversely proportional to “reputation tolerance” and “income level” and directly proportional to “total budget” and “satisfaction and emotion”. (3) The advantage of this new strategy is that it is robust to changes in the exogenous coefficient. Our research contributes to the application of the PGG model in the sustainable development of various resources and developing new approaches to mitigating “the tragedy of the commons”.

Evolution of cooperation with nonlinear environment feedback in repeated public goods game

The mutual feedback between the environment and the strategy is a significant mechanism in the evolution of cooperation. A feedback-evolving repeated public goods game model with nonlinear effects is explored. The players' behaviors in the current round may influence the environment, leading the environment feedback on the behavior by the modulation of payoffs in the form of positive or negative. Under the former case, the groups realize the synergy effect of payoffs in the next round. Vice versa, the discount effect will be instead. The introduction of nonlinear effects enables heterogeneous feedback of the environment on different groups without introducing too many environmental states. The results show the enhancing role of the nonlinear effects on the evolution of cooperation compared to linear switching of enforcement factors. Compared to static game environments, the dynamic feedback scenario can realize better performance in the evolution of cooperation. The effect is more pronounced as the synergy factor increases or the discount factor decreases. The results are verified when the game environment changes between two or three states and under different environment transition structures. In addition, under the scenarios of probabilistic transition, the results are explored by taking into account the cases of state-independent and state-dependent, where both the homogeneity and heterogeneity transition probability are considered in the state-independent case, respectively. The probabilistic transition results further emphasized the transition probability's significant role in the evolution of cooperation when cooperation behaviors are more likely incentivized to interact in the valuable environment.

The coordination of collective and individual solutions in risk-resistant scenarios.

Human societies are constantly coping with global risks. In the face of these risks, people typically have two options, that is, to respond together as a whole (collective solution) or to respond independently (individual solution). Based on these two solutions, individuals have a variety of behavioral strategies. On the other hand, various regulatory bodies supported by the population limit people's choices and punish individuals who do not contribute to collective solutions. So with different risks, how do the two solutions, the various individual strategies, and the constraints from regulators affect the group's response to risk? This paper proposes an extended public goods game model involving opportunists and the regulator to explore the effectiveness of collective and individual solutions against risks. The results show that requiring individuals to invest more in the collective solution reduces the group' s success in resisting risk. To improve the group's ability to resist risk, investment in individual solution should be at least no less than that in collective solution. The establishment fund and punishment intensity of the regulatory agency have no significant effect on the success of collective and individual solutions. This inspires us to contemplate the role and measures of various types of authorities in coping with global risks.

Redistribution based on willingness to cooperate promotes cooperation while intensifying equality in heterogeneous populations

Public resource redistribution is an important means of social income regulation, which maintains social income equality by transferring payoffs to low-income groups, usually in the form of taxes and social services. Originally studied as a means of maintaining social equality, redistribution has also been used to address social dilemmas through evolutionary game theory in recent years. However, the definition of low-income groups in previous studies is not comprehensive. They are used to studying the inequality generated by participants during the game but ignore the inequality caused by initial endowment differences. At the same time, is it beneficial to the whole that redistribution requires the rich to subsidize the poor, but leads the rich to choose defection (tax evasion, etc.)? How should the redistribution mechanism work to reduce inequality while facilitating the spread of altruistic behavior? To explore whether the redistribution mechanism can reach a balance between promoting cooperation and maintaining equality of payoffs, we develop a generic public resource redistribution framework based on heterogeneous populations by a spatial public goods game model. We control redistribution preferences for willingness to cooperate and initial endowments by parameters α and β. With a heterogeneous population, we find that helping the poor through redistribution mechanisms does not have significant positive effects and may even result in exploitation of the rich; focusing on participants’ willingness to cooperate rather than total investment can promote cooperation, enhance overall payoffs and reduce inequality.

A generalized public goods game model based on Nash bargaining

According to the actual investment and return situation, we here propose a generalized public goods game (PGG) model combining with the Nash bargaining. The effects of the segmental synergy factor and investment threshold on the evolution of individuals’ behavior are analyzed. The Nash equilibrium and Pareto optimality related to the PGG are theoretically calculated firstly, and then the influences of different factors on individuals’ strategy choose, the difference of individuals’ strategy and individuals’ gain are explored according to the numerical simulation. Results show that with the increase of investment threshold T, the difference of individuals’ strategy and individuals’ investment increase first and then rapidly decrease to 0. Therefore, once when the investment threshold T is larger than a certain value of Tc, all individuals will adopt the free-ride strategy. And there exists an optimal T that can maximize the average individuals’ investment. With the increase of the game round, individuals’ investment strategies tend to be stable, and the average investment of individuals in a heterogeneous network is greater than that in a homogeneous network. Our work tries to provide a deeper insight to network reciprocity.

Reputation-based probabilistic punishment on the evolution of cooperation in the spatial public goods game

Low-reputation defectors are more likely to be punished than defectors with high reputations. Motivated by this reality, this paper proposes a new mechanism, the reputation-based probabilistic punishment, into the spatial public goods game model. In this mechanism, players with time-dependent reputations are divided into two types, good players with reputations higher than the reputation threshold and bad players with reputations lower than the threshold. A defector considered a good player is less likely to be punished than a defector considered a bad player. Based on these assumptions, we systematically explore how this mechanism influences the evolution of cooperation. Through extensive simulations, we verify that a higher value of the reputation threshold is more conducive to promoting and maintaining cooperation. Moreover, increasing the cost of being punished could effectively encourage players to take cooperative behaviors. Simulation results show that both increasing the punishment intensity and increasing the punishment fine could increase the cost of being punished and are beneficial to the promotion of cooperation. Additionally, in the structured population, the distributions of strategies, reputation, and payoff in the evolutionary stable state are mainly present in the form of clusters.

When costly migration helps to improve cooperation.

Motion is a typical reaction among animals and humans trying to reach better conditions in a changing world. This aspect has been studied intensively in social dilemmas where competing players' individual and collective interests are in conflict. Starting from the traditional public goods game model, where players are locally fixed and unconditional cooperators or defectors are present, we introduce two additional strategies through which agents can change their positions of dependence on the local cooperation level. More importantly, these so-called sophisticated players should bear an extra cost to maintain their permanent capacity to evaluate their neighborhood and react accordingly. Hence, four strategies compete, and the most successful one can be imitated by its neighbors. Crucially, the introduction of costly movement has a highly biased consequence on the competing main strategies. In the majority of parameter space, it is harmful to defectors and provides a significantly higher cooperation level when the population is rare. At an intermediate population density, which would be otherwise optimal for a system of immobile players, the presence of mobile actors could be detrimental if the interaction pattern changes slightly, thereby blocking the optimal percolation of information flow. In this parameter space, sophisticated cooperators can also show the co-called Moor effect by first avoiding the harmful vicinity of defectors; they subsequently transform into an immobile cooperator state. Hence, paradoxically, the additional cost of movement could be advantageous to reach a higher general income, especially for a rare population when subgroups would be isolated otherwise.

Exploring the driving mechanism of environmental mass incidents

Environmental mass incidents caused by pollution of enterprises seriously threaten the government's credibility and social stability. Thus, this paper tries to explore the driving mechanism of environmental mass incidents and to master critical factors for coping with it. Firstly, the spatial public goods game model on complex networks are applied to study the cooperation drive mechanism among individuals within the surrounding people. Secondly, a tripartite evolutionary game model including the government departments, polluting enterprises, and surrounding people is established to learn the complex conflict-of-interest relationships and behavior mechanism among them. Finally, the influence of key factors on system evolution is simulated. The results are as follows: In a complex network, individuals in the surrounding people work together for the common good. The driving factors of collective protest are common goal, strategic interaction, emotional contagion and the payoff of collective protest. The economic benefits related to environmental subsidies generated positive effect on surrounding people's collective action. In addition, the economic punishment mechanism of government departments has not shown influence on the strategic choices of surrounding people and polluting enterprises. Furthermore, the rectification burden of polluting enterprises and profit of collective protest are the key factors affecting the equilibrium of system evolution.

A Strategy Adaptive Evolution Approach Based on the Public Goods Game

Cluster behavior is prevalent in nature. Many individuals change their behavior to adapt to a dynamically changing environment by following simple rules of behavior and interacting with information from neighboring individuals. In this study, the traditional public goods game model is improved by combining the advantages of game theory and interactive learning. A strategy adaptive evolution method based on a public goods game is proposed. The emergence of cooperative behavior in weighted networks under the co-evolution of game strategies and node weights is explored in conjunction with multi-agent interactive learning. The results show that in a public goods game with strategic adaptation, a person’s influence becomes greater if their level of adaptation exceeds the desired level, and less otherwise. This weight adjustment is defined by the intensity parameter δ. A moderate δ value can effectively facilitate the occurrence of cooperative evolution. The level of cooperation depends mainly on the weight distribution of participants, which leads to the formation of cooperative clusters controlled by high-weighted cooperators. Even with the great temptation to defect, these cooperators can prevail over defectors. The adjustment of node weights increases the heterogeneity of individuals. This research provides a viable pathway to solve social dilemmas and will further promote the application of multi-agent intelligent decision making.