Spatial Prisoner Research Articles

Emergence of an unpredictable evolution in a spatial prisoner’s dilemma via a player’s multiple perspectives

Spatial prisoner’s dilemma (SPD) has attracted researchers’ attention as a model of conflict for players. In SPD, players have two different strategies, namely, defectors and cooperators. A defector earns a high payoff from an opponent co-operator while getting nothing from an opponent defector. On the contrary, cooperators promote a win–win relationship between the two cooperators. These mechanisms influence population dynamics in SPD, and many SPD models have been developed. However, little is known about the emergence of an unstable or unpredictable evolution in population dynamics using an SPD model, which may be observed in living systems. In addressing this issue, two SPD models were proposed. In both models, players change the neighborhood definition in accordance with their strategies and sometimes select the rule for this change using probability or local information. Result showed that our models generated characteristic population patterns that may be linked to a self-organized criticality (SOC), a term referring to many systems of interconnected, nonlinear elements that evolve over time into a critical state. In fact, the second model could be spontaneously close to the critical point using local information.

Asymmetric population promotes and jeopardizes cooperation in spatial prisoner’s dilemma game

The evolution of cooperation in asymmetric populations has recently garnered widespread attention. In this study, we propose a new asymmetric model to explore the impact of asymmetric behavior on cooperation evolution. Our model assumes that weak players are always forced to cooperate when interacting with strong players, while strong players always receive the temptation payoff. When encountering peers from the same category, players engage in a prisoner’s dilemma game, differing only in the intensity of the dilemma between population categories. Results suggest that in scenarios where traditional network reciprocity can support cooperation, asymmetric interactions between strong and weak players undermine this reciprocity and jeopardize the maintenance of cooperation. Conversely, in scenarios where traditional network reciprocity fails to support cooperation, asymmetric interactions enable strong players to leverage weak cooperative clusters to establish cooperation. However, as the prevalence of asymmetric interactions increases, it becomes detrimental to cooperation within the weak player subgroups, thus undermining overall cooperation. These findings contribute to a deeper understanding of how asymmetry impacts cooperation.

Spontaneous symmetry breaking of cooperation between species.

In mutualistic associations, two species cooperate by exchanging goods or services with members of another species for their mutual benefit. At the same time, competition for reproduction primarily continues with members of their own species. In intra-species interactions, the prisoner's dilemma is the leading mathematical metaphor to study the evolution of cooperation. Here we consider inter-species interactions in the spatial prisoner's dilemma, where members of each species reside on one lattice layer. Cooperators provide benefits to neighbouring members of the other species at a cost to themselves. Hence, interactions occur across layers but competition remains within layers. We show that rich and complex dynamics unfold when varying the cost-to-benefit ratio of cooperation, r. Four distinct dynamical domains emerge that are separated by critical phase transitions, each characterized by diverging fluctuations in the frequency of cooperation: (i) for large r cooperation is too costly and defection dominates; (ii) for lower r cooperators survive at equal frequencies in both species; (iii) lowering r further results in an intriguing, spontaneous symmetry breaking of cooperation between species with increasing asymmetry for decreasing r; (iv) finally, for small r, bursts of mutual defection appear that increase in size with decreasing r and eventually drive the populations into absorbing states. Typically, one species is cooperating and the other defecting and hence establish perfect asymmetry. Intriguingly and despite the symmetrical model set-up, natural selection can nevertheless favour the spontaneous emergence of asymmetric evolutionary outcomes where, on average, one species exploits the other in a dynamical equilibrium.

Preferential selection based on aspiration and memory in spatial evolutionary prisoner’s dilemma game

When studying the spontaneous emergence of cooperation within the framework of evolutionary game theory, the process of strategy selection plays a crucial role in evolutionary dynamics. In most previous studies, it has been assumed that individuals randomly select learning targets from their neighbors. However, by considering the heterogeneity of individuals’ influence in real-life social networks, a preference for more realistic choices should be taken into account. Here, we propose a preferential selection mechanism based on individuals’ payoff aspiration in memory-based evolutionary game on square lattice networks. For each iteration, players in the regular lattices will prioritize selecting neighbors with the highest satisfaction rate over the most recent runs. The simulation results demonstrate that both the level of aspiration and the length of memory can elevate the level of cooperation to a certain extent. For the moderate aspiration level, when the value of betrayal temptation is less than threshold, cooperation level monotonously decreases with the increment of memory length. However, when betrayal temptation is above threshold, memory mechanism can facilitates the evolution of cooperation, while the absence of such a mechanism leads collaborators toward extinction.

Memory–based adaptive interaction willingness enhances cooperation in spatial prisoner's dilemma

As an important part of the evolution of cooperation, interaction diversity has attracted extensive attention. Traditional studies mainly focus exclusively on either time or space, and the comprehensive impact of these two dimensions on the evolution of cooperation deserves further exploration. In this paper, the interaction willingness is abstracted as the network weight, and an adaptive mechanism for interaction willingness based on the delay effect is proposed by combining the network weight and accumulated payoff. Specifically, individuals adjust their willingness to participate in the game with a delay, considering the change in accumulated payoffs within a certain period, rather than the change in immediate payoffs. The degree of delay is reflected in the length of individuals' memory of historical payoffs. Simulation results deviate from the intuition that more information facilitates cooperation, and in certain modes of interaction willingness adjustment, the correlation between cooperation level and memory–length is non–monotonic. Long–memory is conducive to promoting cooperation under the condition of a relatively large range adjustment of interaction willingness. Conversely, with a narrow adjustment range of interaction willingness, cooperation can be effectively established with short–memory. These results hold substantial implications for both theory and practice, as slight adjustment often corresponds to low loss and high operability. In addition, the underlying mechanism is furthermore explored, revealing that the ‘isolation zones’ composed of individuals with low interaction willingness play an important role in promoting cooperation.

The universal probabilistic reward based on the difference of payoff realizes the evolution of cooperation

Many previous studies regarding the theoretical models of social dilemmas have described that rewarding opponents can help players to cooperate with each other. Those studies deal with reward not for defectors, but for cooperators. However, many prior researches concerning punishment that is also considered to be necessary for the evolution of cooperation discuss punishment on cooperators as well as defectors. Considering the group level, defectors who have too much payoff due to many cooperators around them will lose their superiority over cooperators by rewarding not only cooperators but also defectors surrounding them. As a result, cooperation among players can be realized.Therefore, this study proposes the new universal probabilistic reward based on the difference of payoff, where the greater the payoff of each player is, the higher the probability of giving reward to their opponents becomes regardless of the strategy of their opponents. This paper shows that by introducing the reward of this study into the framework of the spatial prisoner's dilemma game utilized in previous studies, the high-level evolution of cooperation can be achieved especially in both the relatively sparse regular and scale-free topology of connections. In addition, the programming method utilized to construct the simulator for this study will serve as teaching materials for specialized education at Kanagawa Institute of Technology.

Granting leaders priority exit options promotes and jeopardizes cooperation in social dilemmas

Cooperation in human interactions often involves voluntary participation rather than obligatory requirements. This paper investigates the impact of voluntary exit strategies on cooperation within scale-free networks, employing spatial prisoner’s dilemma games as the framework. We analyze three distinct exit strategies: leader priority exit, fringe priority exit, and random exit, across varying dilemma strengths. Our findings reveal that in scenarios of weak dilemma strength, leader priority exit significantly diminishes cooperation, particularly when only a small subset of players is granted exit rights. In contrast, under strong dilemma strength conditions, both leader priority and random exits serve to unexpectedly bolster cooperation. This enhancement is primarily facilitated by high-degree nodes opting out, thereby enabling low-degree nodes to establish stable cooperation. Furthermore, our study underscores the pivotal role of network structure. In assortative networks, random exit emerges as the most effective strategy in promoting cooperation. On the other hand, disassortative networks exhibit a substantial increase in cooperation levels when employing leader priority exit. These results highlight the intricate interplay between exit strategies, dilemma strength, and network topology, offering critical insights for promoting cooperation in varied social contexts.

The emergence of cooperative behavior based on random payoff and heterogeneity of concerning social image

The heterogeneity among individuals in a group usually leads to the volatility of the individual’s income in the game and the difference in the attention paid to the individual’s social image. This paper proposes a spatial prisoner’s dilemma model with random fluctuations of temptation returns and the social images of individuals considered. The payoff of a defector in the game with a cooperator is set as a random variable. And the social image value of an individual who holds the cooperative strategy is set to a positive value, while the social image value of an individual who holds defective strategy is set to zero. Some individuals in the group care about the social images, while others do not. Social image values and benefits are combined into the fitness of individual. The learning strategy adopted is the combination of optimal fitness and Fermi rule based on return difference.Numerical experiments indicate that the volatility of the temptation value is beneficial for cooperation under low temptation, but the volatility of the defector’s income is unfavorable for cooperation under high temptation. And the volatility increases the complexity of system evolution, thereby prolonging the evolution time required to reach steady state. The presence of individuals in the system who care about social image can promote cooperation under low temptation, but the proportion of individuals concerned with social image has little effect on the cooperation behavior. When the social image value of individuals with cooperative strategy increases, the cooperative behavior characteristics of the system do not change qualitatively.

Restructuring of neighborhood definition based on strategies will enhance the cooperation in a spatial prisoner's dilemma

Defectors often survive in spatial game theory, prompting researchers to devise models for cooperators' survival. However, most models overlook players' perspectives based on their strategies. This study introduces an agent-based model considering players' varied neighborhood conditions. In the first model, losing a competition alters the neighborhood definition, randomly adjusting the number of neighborhoods. In the second model, the number of neighborhoods changes based on the player's strategy after losing a competition. Compared to the other proposed model and a traditional spatial prisoner's dilemma model, the second model facilitates easier maintenance of cooperator density.

Spatial game with multiple interaction patterns in constrained interaction environment: A computational method based on opponent’s ability

In previous studies of spatial game, it is usually assumed that all individuals in a population play the game in the same interaction pattern. However, in real life, a variety of interaction patterns often emerge spontaneously, especially in a constrained interaction environment. Furthermore, some scenarios reflect the fact that individuals are generally more inclined to choose the capable partners for interactions. For these reasons, we propose a computational method on evolutionary game with multiple interaction patterns based on opponent’s ability and investigate the cooperative dynamics of spatial prisoner’s dilemma. For simplicity, in this work, we just consider the case of two interaction patterns, which are conditional interaction and stochastic interaction. For the individual who adopt the conditional interaction pattern, if the opponent is strong in ability, it will be perfectly willing to interact with opponent, while if the opponent is weak in ability, it will not do that. For the individual who adopt the stochastic interaction pattern, the willingness of interaction with other party varies with the opponent’s abilities, the more powerful the opponent, the stronger the willingness of interaction. Notably, for paired two individuals, regardless of the combined configuration of interaction patterns, the final interaction intensity between them depends on the result of information fusion of their unilateral interaction willingness. Experimental results show that the mechanism with multiple interaction patterns can effectively propagate cooperative behavior in networked population, and the moderate estimation level can lead to the flourish of cooperation. Besides, we find that the more the straightforward individuals (who adopt the conditional interaction pattern) in the population, the more prosperous the propagation of cooperative behavior, while the more the euphemistic individuals (who adopt the stochastic interaction pattern), the easier the emergence of selfish behavior. Considering such a real environment with multiple interaction patterns can provide a brand new perspective for further study on the dynamics of evolutionary game in structured population.

The cooperative evolution in the spatial prisoner's dilemma game with the local loyalty of two-strategy

The loyalty of individual strategies is an important factor affecting cooperative behavior in human society. Previous researches on the loyalty have mostly focused on the loyalty of individual to a single strategy, that is, the loyalty of individual to cooperative strategy. But from the perspective of defective strategy, individual holding defective strategy can also be seen as so-called loyalty to defective strategy.This paper presents a spatial prisoner's dilemma game model with the local loyalty of two-strategy, in which the loyalty to cooperative strategy and loyalty to defective strategy are considered simultaneously. To distinguish the loyalties of these two types of strategies, the loyalty value corresponding to defective strategy is represented as a negative number. A fitness of an individual is defined as weighted average of individual's payoff and power of average loyalty of local nearest neighbors. In strategy learning, if the loyalty of an individual and the loyalty of the target neighbor are different signs, a discount coefficient is added to the strategy transition probability function to reduce the probability.Numerical simulations show that considering individual loyalty to the two strategies can promote cooperative behavior in the system when the strength index of local loyalty in fitness function is positive, and there are optimal values for the weighting coefficient and the strength index of local loyalty, which can enable the system to reach full cooperators state. The discount coefficient to the fitness of neighbor can impact on the frequency of cooperators surviving and the proportion of surviving cooperators during phase transition.

The influence of environment-based reputation on cooperation in spatial prisoner's dilemma game

Research has shown that reputation mechanism can be effective in facilitating the generation and maintenance of cooperation in social dilemmas. In the traditional reputation mechanism, the amount of change in reputation is fixed. However, in real life, people's impression scores of a player are often influenced by the environment in which he lives. In other words, we prefer to celebrate the cooperative heroes who are surrounded by defectors and spurn the defectors who are hidden among the many cooperators. Thus, this paper proposes an environment-based reputation model, in which the increment of a player's reputation will be influenced by the local and global environment. The results of the research indicate that the greater the intensity of the environmental influence, the more it promotes the evolution of cooperation. It is worth noting that the local and global environmental focus shows peculiarities. Focusing on the global environment can improve the fraction of cooperators at low defection temptations while focusing on the local environment will facilitate cooperators’ survival at higher defection temptations.

Twisted strategy may enhance the evolution of cooperation in spatial prisoner’s dilemma

Twisted strategy may enhance the evolution of cooperation in spatial prisoner’s dilemma

Information accuracy of migration and imitation influences the evolution of cooperation in spatial prisoner's dilemma

Migration is ubiquitous in both biological worlds and human societies for seeking more suitable environments or greater benefits. Intensive studies have revealed that it may provide an effective mechanism to promote cooperation in the context of spatial prisoner' dilemma. As an attempt to give a more generic analysis on how migration influences the evolution of cooperation, we in this work propose a dynamic model that takes into account the agents' different tendencies on migration and imitation. More importantly, we introduce two information accuracy parameters that respectively affect the agents' decisions on the migration destination and on their imitation in the current neighborhood. The simulation results show that the higher accuracy of the information on imitation and migration contributes to a higher cooperation level. In particular, the imitation-oriented mode is more conducive to cooperation when the imitation information accuracy is high. Likewise, the migration-oriented mode is more conducive to cooperation with higher accuracy of the migration information. When imitation and migration information accuracies are both high, cooperation is effectively maintained in two different ways: the imitation-dominant pattern and the migration-dominant pattern, while the scenario in which the imitation tendency is close to the migration tendency has an inhibitory effect on cooperation. Finally, we analyse the robustness of the results under different population densities.

Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game

Many recent studies have used reinforcement learning methods to investigate the behavior of agents in evolutionary games. Q-learning, in particular, has become a mainstream method during this development. Here we introduce Q-learning agents into the evolutionary prisoner's dilemma game on a square lattice. Specifically, we associate the state space of Q-learning agents with the strategies of their neighbors, and we introduce a neighboring reward information sharing mechanism. We thus provide Q-learning agents with the payoff information of their neighbors, in addition to their strategies, which has not been done in previous studies. Through simulations, we show that considering neighborhood payoff information can significantly promote cooperation in the population. Moreover, we show that for an appropriate strength of neighborhood payoff information sharing, a chessboard pattern emerges on the lattice. We analyze in detail the reasons for the emergence of the chessboard pattern and the increase in cooperation frequency, and we also provide a theoretical analysis based on the pair approximation method. We hope that our research will inspire effective approaches for resolving social dilemmas by means of sharing more information among reinforcement learning agents during evolutionary games.

Adaptive interaction driven by learning effect in spatial prisoner’s dilemma

Abstract We propose a computing model in which the individual can automatically adjust the interaction intensity with his mentor according to the learning effect to investigate the cooperative dynamics of spatial prisoner’s dilemma. More specifically, when the cumulative payoff of a learner is more than his reference earning, he will strengthen the interaction with his mentor; otherwise, he will reduce that. The experimental results indicate that this mechanism can improve the emergence of cooperation in networked population, and the driving coefficient of interaction intensity plays an important role in promoting cooperation. Interestingly, under a certain social dilemma condition, there exists the smallest driving coefficient, resulting in optimal cooperation due to the positive feedback effect between the individual’s satisfaction frequency and the number of the effective neighbor. Moreover, we find the experimental results in accord with the ones of theoretical prediction obtained from an extended of the classical pair-approximation. These conclusions obtained by considering the relationship with mentor can provide a new perspective for further study on the dynamics of evolutionary game in structured population.

Asymmetry of individual activity promotes cooperation in the spatial prisoner's dilemma game.

We consider an aspiration-based asymmetric individual activity co-evolutionary prisoner's dilemma game model on the square lattice. In detail, each player has an individual weight that evolves with its strategy. We introduce an asymmetric setting that only μ fractions of players in the network chosen to be active can update their individual weights according to whether their payoffs satisfy their aspirations. Therefore, our individual weights can be regarded as a type of intrinsic motivation satisfaction. Through Monte Carlo simulations on the square lattice, our co-evolution mechanism has been proven to significantly promote cooperation. Furthermore, we find that the moderate μ leads to the highest cooperation level. By dividing the players into four types according to their strategies and weights, we prove that the inversion of the dominant relationship between cooperators and defectors with low weights leads to a conversion cycle of the four types of players, which enhances spatial reciprocity to promote cooperation. Moderate μ results in an appropriate number of low-weight players in this cycle. Interestingly, our results show that higher weight heterogeneity does not lead to higher levels of cooperation, which is contrary to the intuition formed in the previous work.

Antisocial behavior-based environmental feedback in spatial prisoner's dilemma game

The environment, people, and behavior are interdependent and always interactive influence. In reality, a higher cooperation level may change the environment in a way that is beneficial for all competitors. Herein, this work proposed a coevolutionary model in the prisoner's dilemma game where the community environment determines the anti-social revenue structure. Moreover, to capture this phenomenon that behavioral changes induced by the individual's perception of the external environment, the memory effect is also taken into account in this work. It suggests that individuals' stored information about community cooperative level (referred to as indirect experience) can trigger antisocial behavior by influencing their perceptions of antisocial revenue, and that their own past strategic information (referred to as direct experience) also plays a similar role. Numerous simulations show that the proposed mechanism significantly improves cooperation, especially when sensitivity to the environment is taken into account. Furthermore, when memory is considered, a lack of experience and insufficient environmental information may result in different optimal cooperative levels corresponding to different b. In addition, the impact of the sufficient indirect experience and the immediate environmental feedback on cooperation are nearly identical. When individuals have sufficient direct and indirect experiences, their roles can be interchangeably in decision-making. Finally, we discovered that the proposed mechanism exhibits good robustness. Overall, this study demonstrates an innovative use of quantitative calculations to describe the conclusions of qualitative analysis, which have theoretical and practical implications.

Heterogeneity of strategy persistence promotes cooperation in spatial prisoner’s dilemma game

Heterogeneity plays a critical role in promoting cooperation for various evolutionary systems, which has received widespread attention and research from relevant scholars. Additionally, reputation has a significant impact on an individual’s decision-making, and scholars have proposed reputation-based evolutionary mechanisms to promote cooperation. Most of these mechanisms measure reputation as part of an individual’s payoff, and influencing the individual’s decision-making. However, the influence of reputation on the individual’s decision-making should be more through heterogeneous way. Here we propose the heterogeneity of strategy persistence evolutionary mechanism, in which the heterogeneity factor and the number of strategy persistence together influence the evolution of cooperation. We measure the reputation of individuals by the number of strategy persistence. The proposed mechanism effectively highlights the heterogeneity of an individual’s reputation and has shown to be successful in promoting cooperation.

Emotion-based renewal strategies to promote cooperation in spatial prisoner’s dilemma games

Individual interactions are accompanied by changes in emotions, and emotion can influence people’s decisions and thus the evolution of cooperation in a group. Therefore, it is very important and meaningful to consider how to apply emotions in the game so as to promote cooperation. However, emotions are not affected unilaterally in the interaction. It includes both the individual’s own emotions and their acceptance of the external, as well as the shock brought to the individual by the external emotions. In this paper, we propose a new model based on the prisoner’s dilemma to fully consider the influence of both internal and external factors in the change of emotion and the proportion of both sides on cooperation. The simulation shows that emotional stability is more effective in fostering cooperation. In other words, when the proportion of their own emotional continuity is higher and individuals are less accepting of their neighbors’ emotions, the more cooperation is facilitated and the former has a greater degree of influence on cooperation. Taken together, this results maybe provide a perspective, to some extent, on how to apply the condition of emotional change to facilitate the evolution of cooperation in groups in real-life situations.