Role of population density and increasing neighborhood in the evolution of cooperation on diluted lattices

We investigate the evolution of cooperative behaviors with increasing neighborhood size on diluted lattices. For three typical pairwise game models which include prisoner’s dilemma, snowdrift and stag hunt games, all numerical results indicate that cooperation can persist or emerge around the optimal population density which is dictated by the percolation threshold on the square lattice. Meanwhile, the neighborhood size determines the interaction ranges of focal players and then dominates the percolation threshold, and extensive numerical simulations demonstrate that the intermediate neighborhood size is the most beneficial to the evolution of cooperation in the current lattice setup. The current findings can help to deeply understand the sustenance and emergence of collective cooperation in many natural, social and economic systems.     

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.     

A novel snowdrift game model with edge weighting mechanism on the square lattice

We propose a novel snowdrift game model with edge weighting mechanism to explore the cooperative behaviors among the players on the square lattice. Based on the assumption of three types of weight distribution including uniform, exponential and power-law schemes, the cooperation level is largely boosted in contrast with the traditional snowdrift game on the unweighted square lattice. Extensive numerical simulations indicate that the fraction of cooperators greatly augments, especially for the intermediate range of cost-to-benefit ratio r. Furthermore, we investigate how the cooperative behaviors are affected by the undulation amplitude of weight distribution and noise strength of strategy selection, respectively. The simulation results will be conducive to further understanding and analyzing the emergence of cooperation, which is a ubiquitous phenomenon in social and biological science.     

Evolutionary Games in Two-Layer Networks with the Introduction of Dominant Strategy

We study evolutionary games in two-layer networks by introducing the correlation between two layers through the C-dominance or the D-dominance. We assume that individuals play prisoner's dilemma game(PDG) in one layer and snowdrift game(SDG) in the other. We explore the dependences of the fraction of the strategy cooperation in different layers on the game parameter and initial conditions. The results on two-layer square lattices show that, when cooperation is the dominant strategy; initial conditions strongly influence cooperation in the PDG layer while have no impact in the SDG layer. Moreover,in contrast to the result for PDG in single-layer square lattices, the parameter regime where cooperation could be maintained expands significantly in the PDG layer. We also investigate the effects of mutation and network topology. We find that different mutation rates do not change the cooperation behaviors. Moreover,similar behaviors on cooperation could be found in two-layer random networks.     

Evolutionary snowdrift game with disordered environments in mobile societies

We investigate an evolutionary snowdrift game on a square $N=L\times L$ lattice with periodic boundary conditions, where a population of $n_{0}$ ($n_{0}\leq N$) players located on the sites of this lattice can either cooperate with or defect from their nearest neighbours. After each generation, every player moves with a certain probability $p$ to one of the player's nearest empty sites. It is shown that, when $p=0$, the cooperative behaviour can be enhanced in disordered structures. When $p>0$, the effect of mobility on cooperation remarkably depends on the payoff parameter $r$ and the density of individuals $\rho$ ($\rho=n_{0}/N$). Compared with the results of $p=0$, for small $r$, the persistence of cooperation is enhanced at not too small values of $\rho$; whereas for large $r$, the introduction of mobility inhibits the emergence of cooperation at any $\rho<1$; for the intermediate value of $r$, the cooperative behaviour is sometimes enhanced and sometimes inhibited, depending on the values of $p$ and $\rho$. In particular, the cooperator density can reach its maximum when the values of $p$ and $\rho$ reach their respective optimal values. In addition, two absorbing states of all cooperators and all defectors can emerge respectively for small and large $r$ in the case of $p>0$.     

Evolution of cooperation among interacting individuals through molecular dynamics simulations

Using molecular dynamics (MD) simulation and evolutionary game theory, we incorporate the spacial structure of individuals into the study of the behaviors of cooperation, by adopting the prisoner's dilemma and snowdrift game as metaphors of cooperation between unrelated individuals. The results show that the introduction of spacial structure enhances cooperation using the strategy of prisoner's dilemma while does not make much changes to the cooperation if the strategy of snowdrift game is used. It is also found that our model is a meta-phase between regular ring graph model and complex network model. And the “activity of players” T^* we introduced makes our simulation much more closer to real world problems.     

Evolution of cooperation in the snowdrift game among mobile players with random-pairing and reinforcement learning

The evolutionary spatial game in a mobile population has attracted many researchers of biological, social and economic sciences. Considering some facts observed in the real world, this paper proposes a novel spatial evolutionary snowdrift game model with movable players. In this model, one player interacts only with the nearest neighbor in each turn, and makes decision in a reinforcement learning way. In a very large range of the parameters moving ability enhances cooperation, but under some special condition, velocity heavily depresses cooperation. Some explanations have also been given out by investigating the strategy-change behavior of players. The findings may be helpful in understanding cooperative behavior in natural and social systems consisting of mobile agents.     

Dynamic Evolution with Limited Learning Information on a Small-World Network

This paper investigates the dynamic evolution with limited learninginformation on a small-world network. In the system, the information amongthe interaction players is not very lucid, and the players are not allowedto inspect the profit collected by its neighbors, thus the focal playercannot choose randomly a neighbor or the wealthiest one and compare itspayoff to copy its strategy. It is assumed that the information acquaintedby the player declines in the form of the exponential with the geographicaldistance between the players, and a parameter V is introduced to denote theinspect-ability about the players. It is found that under the hospitableconditions, cooperation increases with the randomness and is inhibited bythe large connectivity for the prisoner's dilemma; however, cooperation ismaximal at the moderate rewiring probability and is chaos with theconnectivity for the snowdrift game. For the two games, the acuminous sightis in favor of the cooperation under the hospitable conditions; whereas, themyopic eyes are advantageous to cooperation and cooperation increases withthe randomness under the hostile condition.     

Behavior of Collective Cooperation Yielded by Two Update Rules in Social Dilemmas: Combining Fermi and Moran Rules

We combine the Fermi and Moran update rules in the spatial prisoner's dilemma and snowdrift games to investigate the behavior of collective cooperation among agents on the regular lattice. Large-scale simulations indicate that, compared to the model with only one update rule, the cooperation behavior exhibits the richer phenomena, and the role of update dynamics should be paid more attention in the evolutionary game theory. Meanwhile, we also observe that the introduction of Moran rule, which needs to consider all neighbor's information, can markedly promote the aggregate cooperation level, that is, randomly selecting the neighbor proportional to its payoff to imitate will facilitate the cooperation among agents. Current results will contribute to further understand the cooperation dynamics and evolutionary behaviors within many biological, economic and social systems.     

Cooperative dynamics of snowdrift game on spatial distance-dependent small-world networks

We investigate the evolution of cooperative behaviors of small-world networking agents in a snowdrift game mode, where two agents (nodes) are connected with probability depending on their spatial Euclidean lattice distance in the power-law form controlled by an exponent α. Extensive numerical simulations indicate that the game dynamics crucially depends on the spatial topological structure of underlying networks with different values of the exponent α. Especially, in the distance-independent case of α=0, the small-world connectivity pattern contributes to an enhancement of cooperation compared with that in regular lattices, even for the case of having a high cost-to-benefit ratio r. However, with the increment of α>0, when r≥0.4, the spatial distance-dependent small-world (SDSW) structure tends to inhibit the evolution of cooperation in the snowdrift game.     

Behavior of Collective Cooperation Yielded by Two Update Rules in Social Dilemmas:Combining Fermi and Moran Rules

We combine the Fermi and Moran update rules in the spatial prisoner's dilemma and snowdrift games to investigate the behavior of collective cooperation among agents on the regular lattice.Large-scale simulations indicate that,compared to the model with only one update rule,the cooperation behavior exhibits the richer phenomena,and the role of update dynamics should be paid more attention in the evolutionary game theory.Meanwhile,we also observe that the introduction of Moran rule,which needs to consider all neighbor's information,can markedly promote the aggregate cooperation level,that is,randomly selecting the neighbor proportional to its payoff to imitate will facilitate the cooperation among agents.Current results will contribute to further understand the cooperation dynamics and evolutionary behaviors within many biological,economic and social systems.     

Scale-free networks provide a unifying framework for the emergence of cooperation

We study the evolution of cooperation in the framework of evolutionary game theory, adopting the prisoner's dilemma and snowdrift game as metaphors of cooperation between unrelated individuals. In sharp contrast with previous results we find that, whenever individuals interact following networks of contacts generated via growth and preferential attachment, leading to strong correlations between individuals, cooperation becomes the dominating trait throughout the entire range of parameters of both games, as such providing a unifying framework for the emergence of cooperation. Such emergence is shown to be inhibited whenever the correlations between individuals are decreased or removed. These results are shown to apply from very large population sizes down to small communities with nearly 100 individuals.     

Coexistence of cooperators and defectors in well mixed populations mediated by limiting resources

Traditionally, resource limitation in evolutionary game theory is assumed just to impose a constant population size. Here we show that resource limitations may generate dynamical payoffs able to alter an original prisoner's dilemma, and to allow for the stable coexistence between unconditional cooperators and defectors in well-mixed populations. This is a consequence of a self-organizing process that turns the interaction payoff matrix into evolutionary neutral, and represents a resource-based control mechanism preventing the spread of defectors. To our knowledge, this is the first example of coexistence in well-mixed populations with a game structure different from a snowdrift game.     

Effects of Dimers on Cooperation in the Spatial Prisoner's Dilemma Game

We investigate the evolutionary prisoner's dilemma game in structured populations by introducing dimers, which are defined as that two players in each dimer always hold a same strategy. We find that influences of dimers on cooperation depend on the type of dimers and the population structure. For those dimers in which players interact with each other, the cooperation level increases with the number of dimers though the cooperation improvement level
depends on the type of network structures. On the other hand, the dimers, in which there are not mutual interactions, will not do any good to the cooperation level in a single community, but interestingly, will improve the cooperation level in a population with two communities. We explore the relationship between dimers and self-interactions and find that the effects of dimers are similar to that of self-interactions. Also, we find that the dimers, which are established over two communities in a multi-community network, act as one type of interaction through which information between communities is communicated by the requirement that two players in a dimer hold a same strategy.     

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.     

Effects of Inertia on Evolutionary Prisoner's Dilemma Game

Considering the inertia of individuals in real life, we propose a modified Fermi updating rule, where the inertia of players is introduced into evolutionary prisoner's dilemma game (PDG) on square lattices. We mainly focus on how the inertia affects the cooperative behavior of the system. Interestingly, we find that the cooperation level has a nonmonotonic dependence on the inertia: with small inertia, cooperators will soon be invaded by defectors; with large inertia, players are unwilling to change their strategies and the cooperation level remains the same as the initial state; while a moderate inertia can induce the highest cooperation level. Moreover, effects of environmental noise and individual inertia are studied. Our work may be helpful in understanding the emergence and persistence of cooperation in nature and society.     

Benefit community promotes evolution of cooperation in prisoners' dilemma game

Exploring the emergence and maintenance of cooperation in social dilemma is valuable and it arises considerable concerns of many researchers. In this paper, we propose a mechanism to promote cooperation, called benefit community,in which cooperators linking together form a common benefit community and all their payoffs obtained from game are divided coequally. The robustness of conclusions is tested for the PDG(prisoners' dilemma game) on square lattice and WS small world network. We find that cooperation can be promoted by this typical mechanism, especially, it can diffuse and prevail more easily and rapidly on the WS small world network than it on the square lattice, even if a big temptation to defect b. Our research provides a feasible direction to resolve the social dilemma.     

Effects of Inertia on Evolutionary Prisoner's Dilemma Game

Considering the inertia of individuals in real life,we propose a modified Fermi updating rule,where the inertia of players is introduced into evolutionary prisoner's dilemma game(PDG) on square lattices.We mainly focus on how the inertia affects the cooperative behavior of the system.Interestingly,we find that the cooperation level has a nonmonotonic dependence on the inertia:with small inertia,cooperators will soon be invaded by defectors;with large inertia,players are unwilling to change their strategies and the cooperation level remains the same as the initial state;while a moderate inertia can induce the highest cooperation level.Moreover,effects of environmental noise and individual inertia are studied.Our work may be helpful in understanding the emergence and persistence of cooperation in nature and society.     

Spatial snowdrift game with myopic agents

We have studied a spatially extended snowdrift game, in which the players are located on the sites of two-dimensional square lattices and repeatedly have to choose one of the two strategies, either cooperation (C) or defection (D). A player interacts with its nearest neighbors only, and aims at playing a strategy which maximizes its instant pay-off, assuming that the neighboring agents retain their strategies. If a player is not content with its current strategy, it will change it to the opposite one with probability p next round. Here we show through simulations and analytical approach that these rules result in cooperation levels, which differ to large extent from those obtained using the replicator dynamics.     

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.