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.     

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.     

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.     

Analyzing phase diagrams and phase transitions in networked competing populations

Phase diagrams exhibiting the extent of cooperation in an evolutionary snowdrift game implemented in different networks are studied in detail. We invoke two independent payoff parameters, unlike a single payoff often used in most previous works that restricts the two payoffs to vary in a correlated way. In addition to the phase transition points when a single payoff parameter is used, phase boundaries separating homogeneous phases consisting of agents using the same strategy and a mixed phase consisting of agents using different strategies are found. Analytic expressions of the phase boundaries are obtained by invoking the ideas of the last surviving patterns and the relative alignments of the spectra of payoff values to agents using different strategies. In a Watts-Strogatz regular network, there exists a re-entrant phenomenon in which the system goes from a homogeneous phase into a mixed phase and re-enters the homogeneous phase as one of the two payoff parameters is varied. The non-trivial phase diagram accompanying this re-entrant phenomenon is quantitatively analyzed. The effects of noise and cooperation in randomly rewired Watts-Strogatz networks are also studied. The transition between a mixed phase and a homogeneous phase is identify to belong to the directed percolation universality class. The methods used in the present work are applicable to a wide range of problems in competing populations of networked agents.     

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.     

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 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 $p>0$.     

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.     

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.     

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.     

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.     

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.     

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.     

Adjusting learning motivation to promote cooperation

An evolutionary prisoner’s dilemma game with players adjusting their learning motivation is studied. At each time step, each player can adjust his/her learning motivation according to the difference between the current payoff and payoff aspiration. Greater payoff aspiration means stronger learning motivation, and vice versa. We find that the density of cooperation in a spatial prisoner’s dilemma game is enhanced when the learning motivation mechanism is considered. Meanwhile, we show that proper noise can not only induce the highest cooperation level but also can maintain the cooperation phenomenon even though there is more temptation to defect.     

Cooperation in N-person evolutionary snowdrift game in scale-free Barabási-Albert networks

Cooperation in the N-person evolutionary snowdrift game (NESG) is studied in scale-free Barabási-Albert (BA) networks. Due to the inhomogeneity of the network, two versions of NESG are proposed and studied. In a model where the size of the competing group varies from agent to agent, the fraction of cooperators drops as a function of the payoff parameter. The networking effect is studied via the fraction of cooperative agents for nodes with a particular degree. For small payoff parameters, it is found that the small-k agents are dominantly cooperators, while large-k agents are of non-cooperators. Studying the spatial correlation reveals that cooperative agents will avoid to be nearest neighbors and the correlation disappears beyond the next-nearest neighbors. The behavior can be explained in terms of the networking effect and payoffs. In another model with a fixed size of competing groups, the fraction of cooperators could show a non-monotonic behavior in the regime of small payoff parameters. This non-trivial behavior is found to be a combined effect of the many agents with the smallest degree in the BA network and the increasing fraction of cooperators among these agents with the payoff for small payoffs.     

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.     

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.     

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.     

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.     

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.