Spatial Prisoner Research Articles

Impact of punishment on the evolution of cooperation in spatial prisoner’s dilemma game

The role of punishment in the evolution of cooperation has been disputed for a long time because its effectiveness is challenged by antisocial punishment. Here we introduce neutral punishment, which is executed only if personal payoff is less than the average payoff of neighbors, into spatial prisoner’s dilemma game. In addition, we consider memory length to control the time span of recovery. It is shown that cooperative behavior is remarkably promoted and even dominant with increment of punishment rate and memory length in spite of the high value of temptation. Moreover, we draw a conclusion that this facilitation phenomenon of cooperation is due to heterogeneity of individual fitness coefficient, thus verifying the positive effect of heterogeneity on the evolution of cooperation through network reciprocity.

Donation of richer individual can support cooperation in spatial voluntary prisoner's dilemma game

The compassionate behavior is present throughout the human society, and rich people always could not help having sympathy for poor individual. Inspired by this fact, we consider a donation model to describe the emergency and maintenance of cooperation with voluntary participate in spatial prisoner's dilemma game and we study this model on a square lattice. In detail, when the focal player has the least income in the group which includes his nearest four neighbors and himself, one of his neighbors who has the highest income will donate some proportion of his extra money to him. On the other hand, if focal individual is not the poorest, he will donate some incomes to his poorest neighbor. Through numeric simulation, we conclude that our donation model can promote the evolution of cooperation monotonously. Especially, the larger proportion payoff rich people can contribute, the higher level of cooperation we can get.

Influence of strategy continuity on cooperation in spatial prisoner’s dilemma games with migrating players

The phenomenon of cooperation is prevalent in both nature and human society. In this paper a simulative model is developed to examine how the strategy continuity influences cooperation in the spatial prisoner’s games in which the players migrate through the success-driven migration mechanism. Numerical simulations illustrate that the strategy continuity promotes cooperation at a low rate of migration, while impeding cooperation when the migration rate is higher. The influence of strategy continuity is also dependent on the game types. Through a more dynamic analysis, the different effects of the strategy continuity at low and high rates of migration are explained by the formation, expansion, and extinction of the self-assembled clusters of “partial- cooperators” within the gaming population.

Imitating Contributed Players Promotes Cooperation in the Prisoner’s Dilemma Game

Recently, researchers become increasingly interested in studying cooperation in evolutionary game theory by learning people’s behavior patterns in real world. One of the patterns is that when some people contribute more to a group, their behaviors are more likely to be imitated by their neighbors. Inspired by this, we introduce a preferential selection mechanism that players have higher probability in learning from their contributed counterparts. We follow the lattice arrangement of players and conduct spatial prisoner’s dilemma game. We define the contribution of individual by the payoffs of his four neighbors, and a larger value corresponds to greater contribution. In strategy updating stage, we first calculate the contribution of all players, and then, for each player, we propose one of his neighbors to imitate according to their contribution. At last, we decide to imitate or not by Femi’s dynamics. The simulations have justified our acclaim that imitating contributed players can prompt cooperation. This paper helps us understand why and how cooperation emerges in real world.

Asymmetric learning ability promotes cooperation in structured populations

In this paper, we consider an asymmetric learning ability in structured population. The rate of strategy adoption from one randomly neighbor is controlled by the payoff difference and a factor α. Players of type B will think over α (α > 1) times when he decides whether to adopt his opponent's strategy. While players of type A just consider only once. The concentration of players A and B are denoted by v and 1−v and remains unchanged during the simulations. Through numerous computing simulations, we find that our new setup of asymmetric learning ability can promote the evolution cooperation in the spatial prisoner's dilemma game. Furthermore, in order to explore the generality of this finding, we have tested the results on spatial public good games and random regular graphs.

Effect of noise-perturbing intermediate defense measures in voluntary vaccination games

Recently, a new vaccination game model was proposed, where an intermediate defense measure besides two fundamental strategies; committing vaccination that leads to a perfect immunity and not committing vaccination, was introduced as third strategy. We explore what happens if both effectiveness and cost of an intermediate defense measure stochastically perturbing on the viewpoint of whether or not the third strategy helping to improve total social payoff. We found that unlike resonance effect by adding noise to payoff matrix in case of spatial prisoner's dilemma (SPD) games, adding time-varying noise on both effectiveness and cost does not make difference from the default setting without perturbation to the third strategy. However, if the noise initially given to each agent is frozen, we found the third strategy becoming robust to survive. In particular, if the strategy updating rule allows a more advantageous third strategy can be more commonly shared among agents through copying, the total social payoff is significantly improved.

Realistic decision-making processes in a vaccination game

Previous studies of vaccination games have nearly always assumed a pairwise comparison between a focal and neighboring player for the strategy updating rule, which comes from numerous compiled studies on spatial versions of 2-player and 2-strategy (2 × 2) games such as the spatial prisoner’s dilemma (SPD). We propose, in this study, new update rules because the human decision-making process of whether to commit to a vaccination is obviously influenced by a “sense of crisis” or “fear” urging him/her toward vaccination, otherwise they will likely be infected. The rule assumes that an agent evaluates whether getting a vaccination or trying to free ride should be attempted based on observations of whether neighboring non-vaccinators were able to successfully free ride during the previous time-step. Compared to the conventional updating rule (standard pairwise comparison assuming a Fermi function), the new rules generally realize higher vaccination coverage and smaller final epidemic sizes. One rule in particular shows very good performance with significantly smaller epidemic sizes despite comparable levels of vaccination coverage. This is because the specific update rule helps vaccinators spread widely in the domain, which effectively hampers the spread of epidemics.

Heterogeneous fitness promotes cooperation in the spatial prisoner's dilemma game

Many literatures suggest that incorporating the environment of a focal player (denoted by the average payoff of its all immediate neighbors) into its fitness can promote cooperation in spatial evolutionary games. However, the immediate neighbors influence the focal one to varying degree. Inspired by these, we quantify the focal player's environment with a weighted average payoff of its all immediate neighbors via two interdependent parameters. Numerous simulations show that two moderate parameter pairs favor cooperation, in addition, when the contribution of all immediate neighbors’ payoffs to the environment is negative, the cooperation is promoted remarkably. The generality of this mechanism is verified on different networks and more games. Our work might shed light on the understanding of the evolution of cooperative behaviors in real life.

Effect of the migration mechanism based on risk preference on the evolution of cooperation

Individual migration is an effective means of promoting cooperation in the spatial structure. We can think of migration as a kind of risky investment, so risk attitudes can produce an effect on the migration decision. In order to understand the relationship between risk preference and the evolution of cooperation, the spatial prisoner’s dilemma game model with individual migration based on risk preference is established. By introducing the homogeneity of risk preference, we find that lower risk aversion values keep a high level of cooperation under a larger defection parameter, while the cooperation level can be raised when the whole population is risk-seeking and at lower risk aversion values under a smaller defection parameter. Under the heterogeneous risk preference assumption, simulation results indicate that the cooperation strategy is a winning strategy in a steady state for a wide parameter space and the cooperation level decreases with increasing in the variance of risk preference. From typical snapshots, we can see that co-evolution of the network structure and cooperation strategy has been realized. Cooperative clusters can also be found in the typical snapshots, which have proved the migration mechanism based on risk to be effective in favoring the evolution of cooperation.

Evolution of cooperation through adaptive interaction in a spatial prisoner’s dilemma game

In this paper, we study the effect of adaptive interaction on the evolution of cooperation in a spatial prisoner’s dilemma game. The connections of players are co-evolutionary with cooperation; whether adjacent players can play the prisoner’s dilemma game is associated with the strategies they took in the preceding round. If a player defected in the preceding round, his neighbors will refuse to play the prisoner’s dilemma game with him in accordance with a certain probability distribution. We use the disconnecting strength to represent this probability. We discuss the evolution of cooperation with different values of temptation to defect, sucker’s payoff and disconnecting strength. The simulation results show that cooperation can be significantly enhanced through increasing the value of the disconnecting strength. In addition, the increase in disconnecting strength can improve the cooperators’ ability to resist the increase in temptation and the decrease in reward. We study the parameter ranges for three different evolutionary results: cooperators extinction, defectors extinction, cooperator and defector co-existence. Meanwhile, we recruited volunteers and designed a human behavioral experiment to verify the theoretical simulation results. The punishment of disconnection has a positive effect on cooperation. A higher disconnecting strength will enhance cooperation more significantly. Our research findings reveal some significant insights into efficient mechanisms of the evolution of cooperation.

Preferential selection based on degree difference in the spatial prisoner's dilemma games

Strategy evolution in spatial evolutionary games is generally implemented through imitation processes between individuals. In most previous studies, it is assumed that individuals pick up one of their neighbors randomly to learn from. However, by considering the heterogeneity of individuals' influence in the real society, preferential selection is more realistic. Here, we introduce a preferential selection mechanism based on degree difference into spatial prisoner's dilemma games on Erdös-Rényi networks and Barabási-Albert scale-free networks and investigate the effects of the preferential selection on cooperation. The results show that, when the individuals prefer to choose the neighbors who have small degree difference with themselves to imitate, cooperation is hurt by the preferential selection. In contrast, when the individuals prefer to choose those large degree difference neighbors to learn from, there exists optimal preference strength resulting in the maximal cooperation level no matter what the network structure is. In addition, we investigate the robustness of the results against variations of the noise, the average degree and the size of network in the model, and find that the qualitative features of the results are unchanged.

Co-evolution of cooperation and limited resources on interdependent networks

In this article, based on interdependent networks, cooperation in spatial prisoner's dilemma game (PDG) with coevolving resources is studied. By means of a strategy-independent rule, limited resources can be continually re-allocated among different players in the same network or across different networks. The coevolution of dynamics is discussed respectively for two cases: game circumstances on two coupled networks are identical or asymmetry. In the first case, we obtain that the involvement of resources can be significantly beneficial for cooperative behaviors, and the heterogeneous distribution of resources can positively enhance interdependent network reciprocity. Furthermore, an optimal value (ρ ≈ 0.8) of the interdependent strength exists for cooperation, which is obviously larger than the previous results (ρ ≈ 0.5) without coevolving resources. Besides, we also find resources follow the power law distribution, where cooperators and interconnected players tend to obtain more resources. In the second case, we mainly focus on the flow of resources between networks as well as the intertwined effect of the interdependent strength and resources distribution on cooperation. Microscopic dynamical properties of the flow of resources jointly caused by various factors have been discussed. In certain conditions, instead of promoting the cooperation, the influx of resources could even be positive for defection.

The role of emotions in spatial prisoner’s dilemma game with voluntary participation

This paper develops an extended emotional model in the voluntary prisoner’s dilemma game. It fills a gap in the traditional imitation mechanism by assuming that players do not simply imitate pure strategies, but imitate the emotional profiles of one another instead. The relationship between emotional profiles and strategy are constructed and Monte Carlo simulations are performed on a square lattice. Simulation results reveal that with an increase in the temptation parameter T (1⩽T⩽2), the order-chaos-order transition occurs. When T is around 1.2, we find that a bifurcation occurs. From a social system perspective, as T increases, the system changes from a benign one (respect for the ‘successful’ people and sympathy towards the weak one) to one that is of vicious nature (that is, bully the weak and fear the strong).

Heterogeneous game resource distributions promote cooperation in spatial prisoner’s dilemma game

In social networks, individual abilities to establish interactions are always heterogeneous and independent of the number of topological neighbors. We here study the influence of heterogeneous distributions of abilities on the evolution of individual cooperation in the spatial prisoner’s dilemma game. First, we introduced a prisoner’s dilemma game, taking into account individual heterogeneous abilities to establish games, which are determined by the owned game resources. Second, we studied three types of game resource distributions that follow the power-law property. Simulation results show that the heterogeneous distribution of individual game resources can promote cooperation effectively, and the heterogeneous level of resource distributions has a positive influence on the maintenance of cooperation. Extensive analysis shows that cooperators with large resource capacities can foster cooperator clusters around themselves. Furthermore, when the temptation to defect is high, cooperator clusters in which the central pure cooperators have larger game resource capacities are more stable than other cooperator clusters.

Environment promotes the evolution of cooperation in spatial voluntary prisoner's dilemma game

In reality fitness can be affected by the environment. We explore the evolution of cooperation with the influence of environment on prisoners’ dilemma game with voluntary participation. An individual's fitness is redefined to involve one's own payoff and the average performance of neighbors via preference level u. When u equals zero, the game falls back to its traditional form in which the fitness of an individual simply reflects one's own benefit. When u is larger than 0, the environment plays a role. Numerical simulations show that, for small b, increasing u enables the frequency of cooperation to increase monotonously and even dominate the whole population. For large b, although cooperators are exploited by defectors, the existence of loners protects them from getting wiped out. Finally three strategies start to exhibit cyclic dominance.

Evolutionary fate of memory-one strategies in repeated prisoner’s dilemma game in structured populations

We study evolutionary spatial prisoner’s dilemma game involving a one-step memory mechanism of the individuals whenever making strategy updating. In particular, during the process of strategy updating, each individual keeps in mind all the outcome of the action pairs adopted by himself and each of his neighbors in the last interaction, and according to which the individuals decide what actions they will take in the next round. Computer simulation results imply that win-stay-lose-shift like strategy win out of the memory-one strategy set in the stationary state. This result is robust in a large range of the payoff parameter, and does not depend on the initial state of the system. Furthermore, theoretical analysis with mean field and quasi-static approximation predict the same result. Thus, our studies suggest that win-stay-lose-shift like strategy is a stable dominant strategy in repeated prisoner’s dilemma game in homogeneous structured populations.

Does information of how good or bad your neighbors are enhance cooperation in spatial Prisoner's games?

Network reciprocity is one of the key mechanisms to solve social dilemmas, and has attracted many researchers for the last decade. Here, we explore what happens if network reciprocity is dovetailed with indirect reciprocity. This is motivated by the idea that a player may utilize observed information to evaluate his neighbors. Simulations based on our minimal model reveal that adding indirect reciprocity does not always increase the level of cooperation beyond the level of model without indirect reciprocity. This implies that the combination of two different reciprocity mechanisms, each enhancing cooperation if applied independently, can lead negative interference effect on cooperation. The details of this depend on type of action assessment system determining what is good and bad. Interestingly, we found that a high level of information is not always superior to low levels of information.

Enhancement of cooperation through conformity-driven reproductive ability

We propose a conformity-driven reproductive ability in which an individual i is more (less) likely to imitate a neighbor j’s strategy if j’s strategy is majority (minority) in i’s neighborhood. The results on the evolutionary spatial prisoner’s dilemma game show that, compared to homogeneous reproductive ability, conformity-driven reproductive ability can greatly enhance cooperation. This finding is robust with respect to different types of network structures (including square lattice and scale-free network) and to different ways of strategy updating (including synchronous and asynchronous strategy updating).

Properties of a new small-world network with spatially biased random shortcuts

This paper introduces a small-world (SW) network with a power-law distance distribution that differs from conventional models in that it uses completely random shortcuts. By incorporating spatial constraints, we analyze the divergence of the proposed model from conventional models in terms of fundamental network properties such as clustering coefficient, average path length, and degree distribution. We find that when the spatial constraint more strongly prohibits a long shortcut, the clustering coefficient is improved and the average path length increases. We also analyze the spatial prisoner’s dilemma (SPD) games played on our new SW network in order to understand its dynamical characteristics. Depending on the basis graph, i.e., whether it is a one-dimensional ring or a two-dimensional lattice, and the parameter controlling the prohibition of long-distance shortcuts, the emergent results can vastly differ.

Heterogeneous preference selection promotes cooperation in spatial prisoners’ dilemma game

Adopting the strategy of neighbor who performs better is crucial for the evolution of cooperation in evolutionary games, in that such an action may help you get higher benefit and even evolutionary advantages. Inspired by this idea, here we introduce a parameter α to control the selection of preferred opponents between the most successful neighbor and one random neighbor. For α equaling to zero, it turns to the traditional case of random selection, while positive α favors the player that has high popularity. Besides, considering heterogeneity as one important factor of cooperation promotion, in this work, the population is divided into two types. Players of type A, whose proportion is v, select opponent depending on the parameter α, while players of type B, whose proportion is 1−v, select opponent randomly. Through numerous computing simulations, we find that popularity-driven heterogeneous preference selection can truly promote cooperation, which can be attributed to the leading role of cooperators with type A. These players can attract cooperators of type B forming compact clusters, and thus lead to a more beneficial situation for resisting the invasion of defectors.