Evolutionary Spatial Prisoner Research Articles

The promoting effect of adaptive persistence aspiration on the cooperation based on the consideration of payoff and environment in prisoner's dilemma game

This work explores whether holding the last aspiration for a period of time can promote cooperation. Specifically, an evolutionary spatial prisoner's dilemma game mode is proposed, in which the players adjust strategies and aspirations by considering the payoff and environment. Therefore, the core is to allow players to hold the current aspiration for a period of time. Through numerical calculation, this study finds that the existence of an appropriate duration of aspiration can promote cooperation when b is less than a certain value. Moreover, the cooperation is gradually enhanced with the increase of T-max (maximum aspiration duration) when b is greater than it, but the enhancing effect is limited. It is also found that an appropriate value α (sensitivity to environmental change) can promote cooperation at different b intervals. Besides, this system indicates good robustness. Overall, this study provides a new perspective on exploring cooperative evolution based on aspiration.

Does the conformity save us when information advantage fails?

Evolutionary spatial Prisoner’s Dilemma Game (PDG) has been widely applied to study the possible mechanisms of cooperation promotion. In this paper, we discuss the conformity, information advantage and the coupling between them. We assume there is global conformity for all the players when no decision in the best-takes-all strategy provides a better payoff. By assuming conformity, the neighborhood diversity is then introduced by allowing normal (advantage) players to see 4 (8) neighbors. With these modifications in place, the model is still a classical game theory model with a deterministic property, and is capable of describing how different types of players with different conformities can interact among themselves without the violation of payoff priority. We show that the global conformity, though subtle in the pursuing the payoff, significantly changes the cooperation. On comparing 4 extreme cases with different combinations of conformity and advantage, the results suggest that conformity is more important in the promotion of cooperation. In addition, the simulation of conformal conflict presents a scenario wherein the cooperators may extend to the other side of the conformal boundary.

Influence of bolstering network reciprocity in the evolutionary spatial Prisoner’s Dilemma game: a perspective

Many recent studies on evolutionary spatial Prisoner’s Dilemma (SPD) games have provided insights into the mechanisms and frameworks that bolster the effect of network reciprocity. In this article, we provide a concise and comprehensive review of previous studies on evolutionary games and network reciprocity. Subsequently, we evaluate and compare the results from such studies in a unified manner to answer an open question in evolutionary SPD games: What are the factors underlying network reciprocity and what effect do these factors have on the emergence and promotion of cooperation? As a first step, we introduce a novel indicator to quantitatively evaluate the effectiveness (contribution) of a final fraction of cooperators via the introduction of the associated mechanisms into a simple evolutionary SPD game. In this game, the players are located on a two-dimensional square lattice with the Moore neighborhood and update their strategies by imitating the strategy of the best performing player among their neighbors, and the dynamics are separated into two periods: the enduring (END) period and the expanding (EXP) period. The initial fraction of cooperators is decreased transiently via the invasion and exploitation of defectors in the END period, and over the period, the fraction of cooperators is increased to expand cooperative clusters in the EXP period. Moreover, we also evaluate the statistical validity of our indicator by performing regression analyses. Our results indicate that two factors bolster the effect of network reciprocity: (1) the shape of the cooperative cluster (C-cluster) formed in the END period and (2) the ability to expand a single “perfect C-cluster,” which is the smallest patch, to increase the opportunity for interactions between cooperators and defectors and reduce exploitation by defectors in the EXP period.

The evolution of cooperation in spatial prisoner’s dilemma game with dynamic relationship-based preferential learning

The relationships in human society are heterogeneous and dynamically change with interactions, which have a strong effect on individual’s learning behaviors. In this light we present a new mechanism of preferential learning based on dynamic relationship into evolutionary spatial prisoner’s dilemma game to further investigate the incentive mechanisms of cooperative behaviors. In detail, we consider that the strength of relationship between pairwise individuals adaptively changes according to their mutual strategies and the adjusting rate is related to individuals’ sensitivity to interactions. Based on the heterogeneous and dynamic relationship, individuals prefer neighbors with stronger relationship to learn from instead of learning randomly The learning preference is measured by the preference intensity. By means of Monte Carlo simulations, we find that both the sensitivity factor and the preference intensity have multiple effects on the evolution of cooperation. Furthermore, to validate the multiple effects in a microcosmic view, strategy transitions during the evolution are also discussed. Interestingly, we find that there exists a trade-off between the influence of the sensitivity factor and the preference intensity on the evolution of cooperation. Presented results are robust to variations of the network structures and may provide a new understanding to the emergence of cooperative behaviors.

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.

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).

Diverse strategy-learning styles promote cooperation in evolutionary spatial prisoner's dilemma game

Observational learning and practice learning are two important learning styles and play important roles in our information acquisition. In this paper, we study a spacial evolutionary prisoner's dilemma game, where players can choose the observational learning rule or the practice learning rule when updating their strategies. In the proposed model, we use a parameter p controlling the preference of players choosing the observational learning rule, and found that there exists an optimal value of p leading to the highest cooperation level, which indicates that the cooperation can be promoted by these two learning rules collaboratively and one single learning rule is not favor the promotion of cooperation. By analysing the dynamical behavior of the system, we find that the observational learning rule can make the players residing on cooperative clusters more easily realize the bad sequence of mutual defection. However, a too high observational learning probability suppresses the players to form compact cooperative clusters. Our results highlight the importance of a strategy-updating rule, more importantly, the observational learning rule in the evolutionary cooperation.

Diverse roles of the reduced learning ability of players in the evolution of cooperation

Individual heterogeneity in the reproductive rate is found to play an important role in the emergence and persistence of cooperation. Most of the existing literature focused mainly on the enhancement of cooperation by the introduction of inhomogeneous teaching capability of the individuals. It is far from clear how the heterogeneous learning ability of the individuals affects the evolution of cooperation. To fill this research gap, we make comparative studies of the evolutionary spatial prisoner's dilemma game with reduced learning or teaching ability of the players, under both synchronous and asynchronous strategy updating schemes. By carrying out extensive computer simulations, we show that cooperation can always be facilitated if the inhomogeneous teaching ability of the players is considered, irrespectively of the strategy updating manner. By contrast, cooperation is promoted (inhibited) in the case of synchronous (asynchronous) strategy updating, if heterogeneous learning ability is considered, which is attributed to the reduced ability of cooperators to expand their domains.

Social dilemma alleviated by sharing the gains with immediate neighbors

We study the evolution of cooperation in the evolutionary spatial prisoner's dilemma game (PDG) and snowdrift game (SG), within which a fraction α of the payoffs of each player gained from direct game interactions is shared equally by the immediate neighbors. The magnitude of the parameter α therefore characterizes the degree of the relatedness among the neighboring players. By means of extensive Monte Carlo simulations as well as an extended mean-field approximation method, we trace the frequency of cooperation in the stationary state. We find that plugging into relatedness can significantly promote the evolution of cooperation in the context of both studied games. Unexpectedly, cooperation can be more readily established in the spatial PDG than that in the spatial SG, given that the degree of relatedness and the cost-to-benefit ratio of mutual cooperation are properly formulated. The relevance of our model with the stakeholder theory is also briefly discussed.

The evolutionary spatial prisoner's dilemma on a cycle

In this paper we consider the Evolutionary Spatial Prisoner's Dilemma (ESPD) in which players are modelled by the vertices of a cycle representing a spatial or organisational structure amongst the players. During each round of the ESPD every pair of adjacent players in the cycle play a classical prisoner's dilemma against each other, and they update their strategies from one round to the next based on the perceived success achieved by the strategies of neighbouring players during the previous round. In this way players are able to adapt and learn from each other's strategies as the game progresses without being able to rationalise good strategies. We characterise all steady states of the game as well as the structures of those initial states that lead to the emergence of persistent substates of cooperation over time. We finally determine analytically (i.e. without using simulation) the probability that the game's states will evolve from a randomly generated initial state towards a steady state which accommodates some form of persistent cooperation. More specifically, we show that there exists a range of game parameter values for which the likelihood of the emergence of persistent cooperation increases to almost certainty as the length of the cycle increases.

An asymptotic analysis of the evolutionary spatial prisoner’s dilemma on a path

In this paper, we consider the Evolutionary Spatial Prisoner’s Dilemma (ESPD) in which players are modelled by the vertices of an underlying graphG representing some spatial organisational structure amongst the players. During each round of the ESPD every pair of adjacent players in G play a classical prisoner’s dilemma against each other, and they update their strategies from one round to the next based on the perceived success achieved by the strategies of neighbouring players during the previous round. In this way, players are able to adapt and learn from each other’s strategies as the game progresses without being able to rationalise good strategies. We characterise all steady states of the ESPD for the case where G is a path, and we also characterise the structures of those initial states that lead to the emergence of persistent substates of cooperation over time. We finally determine analytically (i.e. without using simulation) the probability that the game’s states will evolve from a randomly generated initial state towards a steady state which accommodates some form of persistent cooperation. More specifically, we show that there exists a range of game parameters for which the likelihood of the emergence of persistent cooperation increases to almost certainty as the length of the path increases.

Repeated thinking promotes cooperation in spatial prisoner's dilemma game

Inspired by the realistic process of taking decisions in social life, we have proposed a repeated thinking mechanism in the evolutionary spatial prisoner's dilemma game where players are denoted by the vertices and play games with their direct neighbors. Under our mechanism, a player i will randomly select a neighbor j and then deliberate for M times before strategy updating. It will remain unchanged if not all M considerations suggest it to learn the strategy of j. We mainly focus on the evolution of cooperation in the systems. Interestingly, we find that the cooperation level fC is remarkably promoted and fC has a monotonic dependence on the caution parameter M, indicating that being cautious facilitates the emergence and persistence of cooperation. We give a simple but clear explanation for this cooperation promotion via detecting the cooperator–defector transition process. Moreover, the robustness of this mechanism is also examined on different noise levels and game models.

Effect of information transmission on cooperative behavior

Considering the fact, in the real world, that information is transmitted with a time delay, we study an evolutionary spatial prisoner's dilemma game where agents update strategies according to certain information that they have learned. In our study, the game dynamics are classified by the modes of information learning as well as game interaction, and four different combinations, i.e. the mean-field case, case I, case II and local case, are studied comparatively. It is found that the time delay in case II smoothes the phase transition from the absorbing states of C (or D) to their mixing state, and promotes cooperation for most parameter values. Our work provides insights into the temporal behavior of information and the memory of the system, and may be helpful in understanding the cooperative behavior induced by the time delay in social and biological systems.

Diversity of reproduction time scale promotes cooperation in spatial prisoner’s dilemma games

We study an evolutionary spatial prisoner's dilemma game where the fitness of the players is determined by both the payoffs from the current interaction and their history. We consider the situation where the selection time scale is slower than the interaction time scale. This is done by implementing probabilistic reproduction on an individual level. We observe that both too fast and too slow reproduction rates hamper the emergence of cooperation. In other words, there exists an intermediate selection time scale that maximizes cooperation. Another factor we find to promote cooperation is a diversity of reproduction time scales.

Effects of strategy-migration direction and noise in the evolutionary spatial prisoner’s dilemma

Spatial games are crucial for understanding patterns of cooperation in nature (and to some extent society). They are known to be more sensitive to local symmetries than, e.g., spin models. This paper concerns the evolution of the prisoner's dilemma game on regular lattices with three different types of neighborhoods--the von Neumann, Moore, and kagomé types. We investigate two kinds of dynamics for the players to update their strategies (that can be unconditional cooperator or defector). Depending on the payoff difference, an individual can adopt the strategy of a random neighbor [a voter-model-like dynamics (VMLD)] or impose its strategy on a random neighbor, i.e., invasion-process-like dynamics (IPLD). In particular, we focus on the effects of noise, in combination with the strategy dynamics, on the evolution of cooperation. We find that VMLD, compared to IPLD, better supports the spreading and sustaining of cooperation. We see that noise has nontrivial effects on the evolution of cooperation: maximum cooperation density can be realized either at a medium noise level, in the limit of zero noise or in both these regions. The temptation to defect and the local interaction structure determine the outcome. Especially, in the low noise limit, the local interaction plays a crucial role in determining the fate of cooperators. We elucidate these both by numerical simulations and mean-field cluster approximation methods.

Cooperation enhanced by the difference between interaction and learning neighborhoods for evolutionary spatial prisoner’s dilemma games

We study an evolutionary prisoner's dilemma game with two layered graphs, where the lower layer is the physical infrastructure on which the interactions are taking place and the upper layer represents the connections for the strategy adoption (learning) mechanism. This system is investigated by means of Monte Carlo simulations and an extended pair-approximation method. We consider the average density of cooperators in the stationary state for a fixed interaction graph, while varying the number of edges in the learning graph. According to the Monte Carlo simulations, the cooperation is modified substantially in a way resembling a coherence-resonance-like behavior when the number of learning edges is increased. This behavior is reproduced by the analytical results.