The emergence of coordination in public good games

In physical models it is well understood that the aggregate behaviour of a system is not in one to one correspondence with the behaviour of the average individual element of that system. Yet, in many economic models the behaviour of aggregates is thought of as corresponding to that of an individual. A typical example is that of public goods experiments. A systematic feature of such experiments is that, with repetition, people contribute less to public goods. A typical explanation is that people “learn to play Nash” or something approaching it. To justify such an explanation, an individual learning model is tested on average or aggregate data. In this paper we will examine this idea by analysing average and individual behaviour in a series of public goods experiments. We analyse data from a series of games of contributions to public goods and as is usual, we test a learning model on the average data. We then look at individual data, examine the changes that this produces and see if some general model such as the EWA (Expected Weighted Attraction) with varying parameters can account for individual behaviour. We find that once we disaggregate data such models have poor explanatory power. Groups do not learn as supposed, their behaviour differs markedly from one group to another, and the behaviour of the individuals who make up the groups also varies within groups. The decline in aggregate contributions cannot be explained by resorting to a uniform model of individual behaviour. However, the Nash equilibrium of such a game is a total payment for all the individuals and there is some convergence of the group in this respect. Yet the individual contributions do not converge. How the individuals “self-organsise” to coordinate, even in this limited way remains to be explained.     

The coevolution of partner switching and strategy updating in non-excludable public goods game

Spatial public goods game is a popular metaphor to model the dilemma of collective cooperation on graphs, yet the non-excludable property of public goods has seldom been considered in previous models. Based upon a coevolutionary model where agents play public goods games and adjust their partnerships, the present model incorporates the non-excludable property of public goods: agents are able to adjust their participation in the games hosted by others, whereas they cannot exclude others from their own games. In the coevolution, a directed and dynamical network which represents partnerships among autonomous agents is evolved. We find that non-excludable property counteracts the positive effect of partner switching, i.e., the equilibrium level of cooperation is lower than that in the situation of excludable public goods game. Therefore, we study the effect of individual punishment that cooperative agents pay a personal cost to decrease benefits of those defective neighbors who participate in their hosted games. It is found that the cooperation level in the whole population is heightened in the presence of such a costly behavior.     

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.     

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.     

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.     

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.     

Other-regarding preference and the evolutionary prisoner’s dilemma on complex networks

Prevailing models of the evolutionary prisoner’s game on networks always assume that agents are pursuing their own profit maximization. But the results from experimental games show that many agents have other-regarding preference. In this paper, we study the emergence of cooperation from the prisoner’s dilemma game on complex networks while some agents exhibit other-regarding preference such as inequality aversion, envious and guilty emotions. Contrary to common ideas, the simulation results show that the existence of inequality aversion agents does not promote cooperation emergence on a BA (Barabási and Albert) scale-free network in most situations. If the defection attraction is big and agents exhibit strong preference for inequality aversion, the frequency of cooperators will be lower than in situations where no inequality aversion agents exist. In some cases, the existence of the inequality agents will even induce the frequency of cooperators to zero, a feature which is not observed in previous research on the prisoner’s dilemma game when the underlying interaction topology is a BA scale-free network. This means that if an agent cares about equality too much, it will be difficult for cooperation to emerge and the frequency of cooperators will be low on BA networks. The research on the effect of envy or guilty emotions on the emergence of cooperation in the prisoner’s dilemma game on BA networks obtains similar results, though some differences exist. However, simulation results on a WS (Watts and Strogatz) small-world network display another scenario. If agents care about the inequality of agents very much, the WS network favors cooperation emergence in the prisoners’ dilemma game when other-regarding agents exist. If the agent weight on other-regarding is lowered, the cooperation frequencies emerging on a WS network are not much different from those in situations without other-regarding agents, although the frequency of cooperators is lower than those of the situation without other-regarding preference agents sometimes. All the simulation results imply that inequality aversion and its variations can have important effects on cooperation emergence in the prisoner’s dilemma game, and different network topologies have different effects on cooperation emergence in the prisoner’s dilemma game played on complex networks.     

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.     

The effect of recommended role models in prisoner’s dilemma game

Previous studies concerning the prisoner’s dilemma game on graphs conventionally assume that individuals select role models from their replacement graphs at random. We propose a extended prisoner’s dilemma game model to study the impact of recommended role models on the evolution of cooperation in a homogeneous population. Individuals are endowed with the capacity to recommend the ones they imitated in the past to their neighbors for strategy updating. Numerical simulations show that cooperation can be improved significantly when recommendation is allowed. Our results might be helpful in explaining the widespread cooperation in the real world.     

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.     

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.     

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.     

Evolution of cooperation in a heterogeneous population with influential individuals

Influential individuals are introduced and integrated with the public goods game (PGG) to investigate their influence on the emergence and evolution of cooperation. In the model, some influential individuals whose behaviors can be controlled by us are introduced into a homogeneous population on a square lattice. The influential individuals can play three kinds of roles: I. exemplar, II. supervisor with the power to punish defectors, and III. supervisor with the power to reward cooperative co-players. It is found that the existence of influential individuals who play Role I turns out to be detrimental to cooperation and that the larger the number of influential individuals is, the more difficult it is for cooperation to be maintained. For those playing supervisory roles, both punishment and reward are found to be effective ways for the influential individuals to promote and stabilize cooperative behavior. By comparing the critical costs and the mean payoffs for a low multiplication factor under the role of punishment and the role of reward, it is found that reward is a more effective intervention measure than punishment for influential individuals seeking to improve cooperation and that reward leads to a higher mean payoff.     

Effects of aspiration on public cooperation in structured populations

We introduce a deterministic win-stay-lose-shift rule into the spatial public goods game, according to which a player will change its current strategy only if its payoff is below a predefined aspiration level. Simulation results on the square lattice and scale-free network indicate that the aspiration level greatly affects the evolution of cooperation. For small multiplication factors, the frequency of cooperation increases to 0.5 as the aspiration level increases. For large multiplication factors, intermediate levels of aspiration prove optimal for the successful evolution of public cooperation. Some qualitative analyses are provided to explain the above results. Besides, we have found that there exists a ping-pong vibration of cooperation at some specific values of multiplication factors and aspiration levels.     

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.     

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.     

Evolutionary games defined at the network mesoscale: the Public Goods game

The evolutionary dynamics of the Public Goods game addresses the emergence of cooperation within groups of individuals. However, the Public Goods game on large populations of interconnected individuals has been usually modeled without any knowledge about their group structure. In this paper, by focusing on collaboration networks, we show that it is possible to include the mesoscopic information about the structure of the real groups by means of a bipartite graph. We compare the results with the projected (coauthor) and the original bipartite graphs and show that cooperation is enhanced by the mesoscopic structure contained. We conclude by analyzing the influence of the size of the groups in the evolutionary success of cooperation.     

The evolution of cooperation in public good game with deposit

The emergence of cooperation still remains a fundamental conundrum in the social and behavior sciences. We introduce a new mechanism, deposit mechanism, into theoretical model to explore how this mechanism promotes cooperation in a well-mixed population. Firstly, we extend the common binary-strategy combination of cooperation and defection in public good game by adding a third strategy, namely, deposit cooperation. The players with deposit cooperation strategy pay a deposit in advance to obtain the benefits of public good at a lower contributions compared with the players with cooperation strategy, when the provision of public good is successful. Then, we explore the evolution of cooperation in the public good game with deposit by means of the replicator dynamics. Theoretical computations and stimulations show that the deposit mechanism can promote cooperation in a well-mixed population, and the numbers of equilibrium point are determined by variables of public good game. On the one hand, when the coexistence of cooperators and defectors is the stable equilibrium point in the evolutionary system, increasing the threshold of public good and adopting the weak altruism way for share benefits can enhance the level of cooperation in the population. On the other hand, if the coexistence of deposit cooperators and defectors is the stable equilibrium point, it is effective to promote the deposit cooperation by lowering the values of discount and deposit, and raising the threshold of public good.     

Spatial Games Based on Pursuing the Highest Average Payoff

We propose a strategy updating mechanism based on pursuing the highest average payoff to investigate the prisoner＇s dilemma game and the snowdrift game. We apply the new rule to investigate cooperative behaviours on regular, small-world, scale-free networks, and find spatial structure can maintain cooperation for the prisoner＇s dilemma game. fn the snowdrift game, spatial structure can inhibit or promote cooperative behaviour which depends on payoff parameter. We further study cooperative behaviour on scale-free network in detail. Interestingly, non-monotonous behaviours observed on scale-free network with middle-degree individuals have the lowest cooperation level. We also find that large-degree individuals change their strategies more frequently for both games.     

Reputation-based network selection mechanism using game theory

Current and future wireless environments are based on the coexistence of multiple networks supported by various access technologies deployed by different operators. As wireless network deployments increase, their usage is also experiencing a significant growth. In this heterogeneous multi-technology multi-application multi-terminal multi-user environment users will be able to freely connect to any of the available access technologies. Network selection mechanisms will be required in order to keep mobile users “always best connected” anywhere and anytime. In such a heterogeneous environment, game theory techniques can be adopted in order to understand and model competitive or cooperative scenarios between rational decision makers. In this work we propose a theoretical framework for combining reputation-based systems, game theory and network selection mechanism. We define a network reputation factor which reflects the network’s previous behaviour in assuring service guarantees to the user. Using the repeated Prisoner’s Dilemma game, we model the user–network interaction as a cooperative game and we show that by defining incentives for cooperation and disincentives against defecting on service guarantees, repeated interaction sustains cooperation.