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.     

Impacts of memory on a regular lattice for different population sizes with asynchronous update in spatial snowdrift game

Memory is an important factor on the evolution of cooperation in spatial structure. For evolutionary biologists, the problem is often how cooperation acts can emerge in an evolving system. In the case of snowdrift game, it is found that memory can boost cooperation level for large cost-to-benefit ratio r, while inhibit cooperation for small r. Thus, how to enlarge the range of r for the purpose of enhancing cooperation becomes a hot issue recently. This paper addresses a new memory-based approach and its core lies in: Each agent applies the given rule to compare its own historical payoffs in a certain memory size, and take the obtained maximal one as virtual payoff. In order to get the optimal strategy, each agent randomly selects one of its neighbours to compare their virtual payoffs, which can lead to the optimal strategy. Both constant-size memory and size-varying memory are investigated by means of a scenario of asynchronous updating algorithm on regular lattices with different sizes. Simulation results show that this approach effectively enhances cooperation level in spatial structure and makes the high cooperation level simultaneously emerge for both small and large r. Moreover, it is discovered that population sizes have a significant influence on the effects of cooperation.     

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.     

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.     

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.     

Evolutionary games on scale-free networks with a preferential selection mechanism

Considering the heterogeneity of individuals’ influence in the real world, we introduce a preferential selection mechanism to evolutionary games (the Prisoner’s Dilemma Game and the Snowdrift Game) on scale-free networks and focus on the cooperative behavior of the system. In every step, each agent chooses an individual from all its neighbors with a probability proportional to k^α indicating the influence of the neighbor, where k is the degree. Simulation results show that the cooperation level has a non-trivial dependence on α. To understand the effect of preferential selection mechanism on the evolution of the system, we investigate the time series of the cooperator frequency in detail. It is found that the cooperator frequency is greatly influenced by the initial strategy of hub nodes when α>0. This observation is confirmed by investigating the system behavior when some hub nodes’ strategies are fixed.     

Cooperative behavior in evolutionary snowdrift game with bounded rationality

An evolutionary snowdrift game (SG) that incorporates bounded rationality and limited information in the evolutionary process is proposed and studied. Based on SG in a well-mixed population and defining the winning action at a turn to be the one that gets a higher payoff, the most recent m winning actions can be used as a public information based on which the competing agents decide their next actions. This defines a strategy pool from which each agent picks a number of strategies as their tool in adapting to the competing environment. The payoff parameter r in SG serves to set the maximum number of winners per turn. Due to the bounded rationality and limited information, the cooperative frequency shows steps and plateaux as a function of r and these features tend to be smoothed out as m increases. These features are results of an interplay between a restricted subset of m-bit histories that the system can visit at a value of r and the limited capacity that agents can adapt. The standard deviation in the number of agents taking the cooperative action is also studied. For general values of r, our model generates a realization of the binary-agent-resource model. The idea of introducing bounded rationality into a two-person game to realize the minority game or binary-agent-resource model could be a useful tool for future research.     

Integrating neighborhoods in the evaluation of fitness promotes cooperation in the spatial prisoner’s dilemma game

A fundamental question of human society is the evolution of cooperation. Many previous studies explored this question via setting spatial background, where players obtain their payoffs by playing game with their nearest neighbors. Another undoubted fact is that the environment plays an important role in the individual development. Inspired by these phenomena, we reconsider the definition of individual fitness which integrates the environment, denoted by the average payoff of all individual neighbors, with the traditional individual payoffs by introducing a selection parameter u. Tuning u equal to zero returns the traditional version, while increasing u bears the influence of environment. We find that considering the environment, i.e., integrating neighborhoods in the evaluation of fitness, promotes cooperation. If we enhance the value of u, the invasion of defection could be resisted better. We also provide quantitative explanations and complete phase diagrams presenting the influence of the environment on the evolution of cooperation. Finally, the universality of this mechanism is testified for different neighborhood sizes, different topology structures and different game models. Our work may shed light on the emergence and persistence of cooperation in our life.     

Understanding cooperative behavior in structurally disordered populations

The effects of an inhomogeneous competing environment on the extent of cooperation are studied within the context of a site-diluted evolutionary snowdrift game on a square lattice, with the occupied sites representing the players, both numerically and analytically. The frequency of cooperation ?_C generally shows a non-monotonic dependence on the fraction of occupied sites ρ, for different values of the payoff parameter r. Slightly diluting a lattice leads to a lower cooperation for small and high values of r. For a range of r, however, dilution leads to an enhanced cooperation. An analytic treatment is developed for?_C^I + ?_C^II, with?_C^I emphasizing the importance of the small clusters of players especially for?_C^II from the other players is shown to be inadequate. A local configuration approximation (LCA) that treats the local competing configurations as the variables and amounts to include spatial correlation up to the neighborhood of a player’s neighbors is developed. Results of ?_C (ρ) and the number of different local configurations from LCA are in good agreement with simulation results. A transparent physical picture of the dynamics stemming from LCA is also presented. The theoretical approach provides a framework that can be readily applied to competing agent-based models in structurally ordered and disordered populations.     

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.     

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.     

Evaluating fitness by integrating the highest payoff within the neighborhood promotes cooperation in social dilemmas

In this paper, we propose a modified fitness evaluation mechanism, which integrates the environmental factors into the focal player’s fitness calculation, to investigate the evolution of cooperative behaviors in the prisoner’s dilemma game. Here, the fitness of a player is computed by combining the individual raw payoff and the highest payoff within the neighborhood, which is regulated by a single parameter termed as trust level η$\eta$. We show, compared to the traditional version (η=0$\eta =0$), that the cooperation level can be highly enhanced for η>0$\eta >0$. Meanwhile, we illustrate the dynamical evolution of cooperators on the square lattice, and for different defection parameters b$b$ the _FC−K$F_C-K$ curves are utilized to investigate the impact of noise during the strategy updates. Likely, the role of pursuing the highest payoff within the neighborhood also favors the survival of cooperators in the spatial snowdrift game. In addition, the sensibility of knowing the external factors is often not identical for all individuals and we consider the distributed trust level in which η$\eta$ is a distributed parameter, and the results indicate that pursuing the highest payoff in the neighborhood is also inspiring as a consequence of its positive effect on cooperation. The current results are highly instructive for us to further understand the maintenance and emergence of cooperation under the framework of evolutionary game theory.     

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.     

Promotion of cooperation by aspiration-induced migration

In this paper, we study an aspiration-induced migration model, in which each individual plays the prisoner’s dilemma game with those being within a circle of radius r centered on himself/herself. An individual will migrate to a randomly chosen place with the velocity v if his/her payoff is below the aspiration level. We report that cooperative behavior is favored when the aspiration level and interaction radius are moderate, and the migration velocity is slow.     

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.     

On evolutionary spatial heterogeneous games

How cooperation between self-interested individuals evolve is a crucial problem, both in biology and in social sciences, that is far from being well understood. Evolutionary game theory is a useful approach to this issue. The simplest model to take into account the spatial dimension in evolutionary games is in terms of cellular automata with just a one-parameter payoff matrix. Here, the effects of spatial heterogeneities of the environment and/or asymmetries in the interactions among the individuals are analysed through different extensions of this model. Instead of using the same universal payoff matrix, bimatrix games in which each cell at site (i, j) has its own different ‘temptation to defect’ parameter T(i,j) are considered. First, the case in which these individual payoffs are constant in time is studied. Second, an evolving evolutionary spatial game such that T=T(i,j;t), i.e. besides depending on the position evolves (by natural selection), is used to explore the combination of spatial heterogeneity and natural selection of payoff matrices.     

Uncertainty of cooperation in random scale-free networks

We study the spatial prisoner’s dilemma game where the players are located on the nodes of a random scale-free network. The prisoner’s dilemma game is a powerful tool and has been used for the study of mutual trust and cooperation among individuals in structured populations. We vary the structure of the network and the payoff values for the game, and show that the specific conditions can greatly influence the outcome of the game. A variety of behaviors are reproduced and the percentage of cooperating agents fluctuates significantly, even in the absence of irrational behavior. For example, the steady state of the game may be a configuration where either cooperators or defectors dominate, while in many cases the solution fluctuates between these two limiting behaviors.     

The effect of asymmetric payoff mechanism on evolutionary networked prisoner’s dilemma game

In social and biological systems, there are obvious individual divergence and asymmetric payoff phenomenon due to the strength, power and influence differences. In this paper, we introduce an asymmetric payoff mechanism to evolutionary Prisoner’s Dilemma Game (PDG) on scale-free networks. The co-effects of individual diversity and asymmetric payoff mechanism on the evolution of cooperation and the wealth distribution under different updating rules are investigated. Numerical results show that the cooperation is highly promoted when the hub nodes are favored in the payoff matrix, which seems to harm the interest of the majority. But the inequality of social wealth distribution grows with the unbalanced payoff rule. However, when the node difference is eliminated in the learning strategy, the asymmetric payoff rule will not affect the cooperation level. Our work may sharpen the understanding of the cooperative behavior and wealth inequality in the society.     

The effective activation energy Ueff(T,B,J) in Hg-1223 single phase superconductors

We review the methods of calculating the effective activation energy U_eff(T,B,J) for both transport measurements and magnetic decay, together with some theoretical models. Then, we apply these methods to our Hg-1223 single-phase superconductor to obtain the activation energy. Transport results give that the magnetic field and temperature dependence of the U_eff can be well described as U₀B^−α(1−T/T_c)^m. Magnetic relaxation shows that the current density dependence of U(J) can be scaled onto a single curve, which can be considered as the activation energy at some temperature T₀. The pinning mechanism in the measured temperature range does not change, and the activation energy depends separately on the three variables: T, B, and J, are responsible for the magnetic decay data scaling onto a single curve at various temperatures. As temperatures close to zero and near T_c, thermally assisted flux motion model is no longer valid since other processes dominate.