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.     

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.     

Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions

When a population structure is modelled as a square lattice,the cooperation may be improved for an evolutionary prisoner dilemma game or be inhibited for an evolutionary snowdrift game.In this work,we investigate cooperation in a population on a square lattice where the interaction among players contains both prisoner dilemma game and snowdrift game.The heterogeneity in interaction is introduced to the population in two different ways:the heterogenous character of interaction assigned to every player(HCP) or the heterogenous character of interaction assigned to every link between any two players(HCL).The resonant enhancement of cooperation in the case of HCP is observed while the resonant inhibition of cooperation in the case of HCL is prominent.The explanations on the enhancement or inhibition of cooperation are presented for these two cases.     

Spatial evolutionary game with bond dilution

In this paper, we study numerically the prisoner’s dilemma game (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched and annealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at the critical occupation probability q^∗, which is higher than the bond percolation transition point for a square lattice. In the critical region, the defined order parameter has a scaling form as P_e∼(q−q^∗)^β for q<q^∗ with the critical exponents β=1.42 for PDG and β=1.52 for SG, which differ from those with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperative behavior. We find that the cooperation is much enhanced in the range of small payoff parameters on a lattice with slightly annealed bond dilution.     

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.     

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.     

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.     

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 $p>0$.     

Clustering Effect on the Evolution of Cooperation in a Herding Snowdrift Game

We study the evolution of cooperation for two cluster breaking mechanisms in a herding snowdrift game. The cooperative behavior is observed to be related to the duster size. A negative dependence of the payoff parameter r on cooperative behavior is discovered. For a low r, herding helps promote the cooperation, whereas for a high r, herding tends to prevent cooperative behavior.     

Spatial snowdrift game in heterogeneous agent systems with co-evolutionary strategies and updating rules

We propose an evolutionary snowdrift game model for heterogeneous systems with two types of agents, in which the inner-directed agents adopt the memory-based updating rule while the copycat-like ones take the unconditional imitation rule; moreover, each agent can change his type to adopt another updating rule once the number he sequentially loses the game at is beyond his upper limit of tolerance. The cooperative behaviors of such heterogeneous systems are then investigated by Monte Carlo simulations. The numerical results show the equilibrium cooperation frequency and composition as functions of the cost-to-benefit ratio r are both of plateau structures with discontinuous steplike jumps, and the number of plateaux varies non-monotonically with the upper limit of tolerance vTas well as the initial composition of agents fa0.Besides, the quantities of the cooperation frequency and composition are dependent crucially on the system parameters including vT, fa0, and r. One intriguing observation is that when the upper limit of tolerance is small, the cooperation frequency will be abnormally enhanced with the increase of the cost-to-benefit ratio in the range of 0 < r < 1/4. We then probe into the relative cooperation frequencies of either type of agents, which are also of plateau structures dependent on the system parameters. Our results may be helpful to understand the cooperative behaviors of heterogenous agent systems.     

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.     

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.     

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.     

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.     

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.     

Evacuation flow of pedestrians considering compassion effect

By means of game theory, the effect of compassion mechanism on the evacuation dynamics of pedestrians from a room is studied based on a cellular automaton model. Pedestrians can choose to cooperate or defect in a snowdrift game during the movement. With the compassion mechanism, pedestrians share their payoff to the poorest peer when several pedestrians compete for the same empty cell. Simulation results show that the escape time grows with fear degree r of the snowdrift game, and the compassion mechanism will have a different effect on the system compared with the situation of a spatial game with fixed population. By payoff redistribution, the compassion can help the minor strategy to survive. When the fear degree r is large, the compassion can sustain the cooperative behavior, and spontaneously decreases the escape time. When the fear degree r is small, the compassion will decrease the cooperation frequency, and slightly increase the escape time. The phenomenon is explained by the evolution and competition of defectors and cooperators in the system.Finally, the effect of initial cooperator proportion, the effect of two exits, and the effect of "Richest-Following" strategy,and the effect of initial density are also discussed.     

Limitation of network inhomogeneity in improving cooperation in coevolutionary dynamics

Cooperative behavior is common in nature even if selfishness is sometimes better for an individual. Empirical and theoretical studies have shown that the invasion and expansion of cooperators are related to an inhomogeneous connectivity distribution. Here we study the evolution of cooperation on an adaptive network, in which an individual is able to avoid being exploited by rewiring its link(s). Our results indicate that the broadening of connectivity distribution is not always beneficial for cooperation. Compared with the Poisson-like degree distribution, the exponential-like degree distribution is detrimental to the occurrence of a higher level of cooperation in the continuous snowdrift game (CSG).     

Randomness Effect on Cooperation in Memory-Based Snowdrift Game

A memory-based snowdrift game (MBSG) on spatial small-world networks is investigated. It is found that cooperation rate versus temptation shows some step structures on small-world networks, similar to the case on regular lattices. With the increment of rewiring probability based on four-neighbourregular lattices, more steps are observable. Interestingly, it is observed that cooperation rate peaks at a specific value of temptation, which indicates that properly encouraging selfish actions may lead to better cooperative behaviours in the MBSG on small-world networks. Memory effects are also discussed for different rewiring probabilities. Furthermore, optimal regions arefound in the parameter planes. The strategy-related average degrees of individuals are helpful to understand the obtained results.     

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.