Evolutionary prisoner’s dilemma on Newman-Watts social networks with an asymmetric payoff distribution mechanism

In this paper, we introduce an asymmetric payoff distribution mechanism into the evolutionary prisoner's dilemma game (PDG) on Newman--Watts social networks, and study its effects on the evolution of cooperation. The asymmetric payoff distribution mechanism can be adjusted by the parameter α: if α>0, the rich will exploit the poor to get richer; if α<0, the rich are forced to offer part of their income to the poor. Numerical results show that the cooperator frequency monotonously increases with α and is remarkably promoted when α>0. The effects of updating order and self-interaction are also investigated. The co-action of random updating and self-interaction can induce the highest cooperation level. Moreover, we employ the Gini coefficient to investigate the effect of asymmetric payoff distribution on the the system's wealth distribution. This work may be helpful for understanding cooperative behaviour and wealth inequality in society.     

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.     

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.     

Coevolution of game and network structure with adjustable linking

Most papers about the evolutionary game on graph assume the statistic network structure. However, in the real world, social interaction could change the relationship among people. And the change of social structure will also affect people’s strategies. We build a coevolution model of prisoner’s dilemma game and network structure to study the dynamic interaction in the real world. Differing from other coevolution models, players rewire their network connections according to the density of cooperation and other players’ payoffs. We use a parameter α to control the effect of payoff in the process of rewiring. Based on the asynchronous update rule and Monte Carlo simulation, we find that, when players prefer to rewire their links to those who are richer, the temptation can increase the cooperation density.     

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.     

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.     

Cooperation and community structure in social networks

Situations of conflict giving rise to social dilemmas are widespread in society. One way of studying these important phenomena is by using simplified models of individual behavior under conflicting situations such as evolutionary game theory. Starting from the observation that individuals interact through networks of acquaintances, we study the evolution of cooperation on model and real social networks through well known paradigmatic games. Using a new payoff scheme which leaves replicator dynamics invariant, we find that cooperation is sustainable in such networks, even in the difficult case of the prisoner’s dilemma. The evolution and stability of cooperation implies the condensation of game strategies into the existing community structures of the social network in which clusters of cooperators survive thanks to their higher connectivity towards other fellow cooperators.     

Diversity of contribution promotes cooperation in public goods games

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

Effects of expectation and noise on evolutionary games

Considering the difference between the actual and expected payoffs, we bring a stochastic learning updating rule into an evolutionary Prisoners Dilemma game and the Snowdrift game on scale-free networks, and then investigate how the expectation level A and environmental noise κ influence cooperative behavior. Interestingly, numerical results show that the mechanism of promoting cooperation exhibits a resonance-like fashion including the coaction of A, κ and the payoff parameters. High cooperator frequency is induced by some optimal parameter regions. The variation of time series has also been investigated. This work could be of particular interest in the evolutionary game dynamics of biological and social systems.     

Socioeconomic implications of donation distributions

Individual donation depends on personal wealth and individual willingness to donate. On the basis of a donation model proposed in our previous study, a simplified version of an individual donation model is derived by relaxing the restrictions of the maximum wealth in the economy. Thus, the whole distribution is determined by only two parameters. One of them relates to the exponent of the distribution of society wealth and the other refers to the donation amount of the kindest poorest person. The parameters reflect the degree of wealth inequality and the charitable enthusiasm of society, respectively. Using actual donation data, we develop a specific parameter estimation method combining linear regression and the Kolmogorov–Smirnov (KS) statistic to get the value of two socioeconomic indicators. Applications to Chinese individual donations in response to the 2004 Indian Ocean tsunami and the 2008 Wenchuan earthquake indicate a rising inequality in social wealth distribution in China. Also, more charitable enthusiasm is observed in the response to the 2008 Wenchuan earthquake.     

An Improved Fitness Evaluation Mechanism with Memory in Spatial Prisoner's Dilemma Game on Regular Lattices

To deeply understand the emergence of cooperation in natural,social and economical systems,we present an improved fitness evaluation mechanism with memory in spatial prisoner's dilemma game on regular lattices.In our model,the individual fitness is not only determined by the payoff in the current game round,but also by the payoffs in previous round bins.A tunable parameter,termed as the memory strength(μ),which lies between 0 and 1,is introduced into the model to regulate the ratio of payoffs of current and previous game rounds in the individual fitness calculation.When μ = 0,our model is reduced to the standard prisoner's dilemma game;while μ = 1 represents the case in which the payoff is totally determined by the initial strategies and thus it is far from the realistic ones.Extensive numerical simulations indicate that the memory effect can substantially promote the evolution of cooperation.For μ 1,the stronger the memory effect,the higher the cooperation level,but μ = 1 leads to a pathological state of cooperation,but can partially enhance the cooperation in the very large temptation parameter.The current results are of great significance for us to account for the role of memory effect during the evolution of cooperation among selfish players.     

Evolution of cooperation among mobile agents

We study the effects of mobility on the evolution of cooperation among mobile players, which imitate collective motion of biological flocks and interact with neighbors within a prescribed radius R. Adopting the the prisoner’s dilemma game and the snowdrift game as metaphors, we find that cooperation can be maintained and even enhanced for low velocities and small payoff parameters, when compared with the case that all agents do not move. But such enhancement of cooperation is largely determined by the value of R, and for modest values of R, there is an optimal value of velocity to induce the maximum cooperation level. Besides, we find that intermediate values of R or initial population densities are most favorable for cooperation, when the velocity is fixed. Depending on the payoff parameters, the system can reach an absorbing state of cooperation when the snowdrift game is played. Our findings may help understanding the relations between individual mobility and cooperative behavior in social systems.     

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

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

Evolutionary prisoner’s dilemma game on weighted scale-free networks

This paper investigates the evolutionary prisoner’s dilemma on weighted scale-free networks. The weighted networks are generated by adopting Barabási-Albert scale-free network and assigning link weight with w_ij=(k_i×k_j)^β. Simulation results show that the cooperation frequency has a strong dependence on β. The value of β which is associated with the maximal cooperation frequency has been sought out. Moreover, Gini coefficient and Pareto exponent of the system’s wealth distribution are investigated. The inequality of wealth distribution is minimized at β≈−1.     

A multi-agent dynamic model based on different kinds of bequests

We investigate how wealth transfer that happens at the end of an agent’s life affects its final distribution based on a multi-agent dynamic model. We discuss two kinds of wealth transfers: to a single agent and to charities. The first kind of bequest is common in our realistic world and is always regarded by the public as unequal inheritance. The bequests to charities will be gathered and then equally redistributed among the survivors in our model. We find that when all the decedents choose the second kind of bequest, the final distribution is the Gibbs exponential function. When all the decedents choose the first kind of bequest, the result is condensation that a single individual accumulates all the available wealth. When an increasing portion of decedents choose the one-heir bequests, the exponential distribution evolves towards a power law shape (accompanied by deteriorating inequality). This shape firstly appears from the intermediate range of wealth and extends towards the top end of the simulated distribution, while the distribution remains exponential for high values of the wealth. At the same time, the Gini coefficient increases and the wealth accumulation becomes serious. At last, we analyze the source of the inequality. We find that not only unequal inheritances, but also equal division of the charity’s wealth can relatively contribute to an inequality of wealth distribution.     

Promoting cooperation by local contribution under stochastic win-stay-lose-shift mechanism

We introduce a stochastic win-stay-lose-shift (WSLS) mechanism into evolutionary Prisoner’s Dilemma on small-world networks. At each time step, after playing with all its immediate neighbors, each individual gets a score to evaluate its performance in the game. The score is a linear combination of an individual’s total payoff (i.e., individual gain from the group) and local contribution to its neighbors (i.e., individual donation to the group). If one’s actual score is not larger than its desired score aspiration, it switches current strategy to the opposite one with the probability depending on the difference between the two scores. Under this stochastic WSLS regime, we assume that each focal individual gains its fixed score aspiration under the condition of full cooperation in its neighborhood, and find that cooperation is significantly enhanced under some certain parameters of the model by studying the evolution of cooperation. We also explore the influences of different values of learning rate and intensity of deterministic switch on the evolution of cooperation. Simulation results show that cooperation level monotonically increases with the relative weight of the local contribution to the score. For much low intensity of deterministic switch, cooperation is to a large extent independent of learning rate, and full cooperation can be reached when relative weight is not less than 0.5. Otherwise, cooperation level is affected by the value of learning rate. Besides, we find that the cooperation level is not sensitive to the topological parameters. To explain these simulation results, we provide corresponding analytical results based on mean-field approximation, and find out that simulation results are in close agreement with the analytical ones. Our work may be helpful in understanding the cooperative behavior in social systems based on this stochastic WSLS mechanism.     

The mechanism of alliance promotes cooperation in the spatial multi-games

In reality, individuals often tend to enhance their competitiveness by forming alliances with the partners sharing the same view. Motivated by this phenomenon, we propose a novel model considering the mechanism of alliance to investigate the evolution of cooperation in the spatial multi-games, in which the intensity of alliance could be adjusted by a parameter μ. In detail, the focal individual will contribute the ratio μ of his own payoff as a cost of the alliance, and then expect to gain the equal portion of average benefits of the neighbors holding the same strategy, which means that the individual fitness can be tuned by the intensity of alliance. Through extensive numerical simulations, one could find that the mechanism of alliance can significantly promote the prosperity of cooperation. Moreover, the larger the value of μ, the stronger the intensity of alliance, which could lead to the higher the level of cooperation. In particular, the evolutionary dynamics presents a complex situation when the co-changes of the intensity of alliance μ and sucker's payoff δ are commonly considered. Furthermore, we also verify the robustness of the proposed mechanism by using the diversity of the intensity of alliance and different network topologies. The current results are conducive to comprehending the emergence and persistence of cooperation within the real-world population.     

Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games

Many previous studies have shown that the environment plays an important role for social individuals. In this paper, we integrate the environmental factor, which is defined as the average payoff of all a player's neighbours, with the standard Fermi updating rule by introducing a tunable parameter, ω. It is found that the level of cooperation increases remarkably, and that the cooperators can better resist the invasion of defection with an increase in ω. This interesting phenomenon is then explained from a microscopic view. In addition, the universality of this mechanism is also proved with the help of the small-world network and the random regular graph. This work may be helpful in understanding cooperation behaviour in species from unicellular organisms up to human beings.     

Equivalence between Bell inequalities and quantum minority games

We show that, for a continuous set of entangled four-partite states, the task of maximizing the payoff in the symmetric-strategy four-player quantum Minority game is equivalent to maximizing the violation of a four-particle Bell inequality. We conclude the existence of direct correspondences between (i) the payoff rule and Bell inequalities, and (ii) the strategy and the choice of measured observables in evaluating these Bell inequalities. We also show that such a correspondence is unique to minority-like games.     

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.