Evolutionary Prisoner's Dilemma Game Research Articles

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.

Adaptive co-evolution of strategies and network leading to optimal cooperation level in spatial prisoner's dilemma game

We study evolutionary prisoner's dilemma game on adaptive networks where a population of players co-evolves with their interaction networks. During the co-evolution process, interacted players with opposite strategies either rewire the link between them with probability p or update their strategies with probability 1 – p depending on their payoffs. Numerical simulation shows that the final network is either split into some disconnected communities whose players share the same strategy within each community or forms a single connected network in which all nodes are in the same strategy. Interestingly, the density of cooperators in the final state can be maximised in an intermediate range of p via the competition between time scale of the network dynamics and that of the node dynamics. Finally, the mean-field analysis helps to understand the results of numerical simulation. Our results may provide some insight into understanding the emergence of cooperation in the real situation where the individuals' behaviour and their relationship adaptively co-evolve.

Cooperation influenced by the correlation degree of two-layered complex networks in evolutionary prisoner's dilemma games

An evolutionary prisoner's dilemma game is investigated on two-layered complex networks respectively representing interaction and learning networks in one and two dimensions. A parameter q is introduced to denote the correlation degree between the two-layered networks. Using Monte Carlo simulations we studied the effects of the correlation degree on cooperative behaviour and found that the cooperator density nontrivially changes with q for different payoff parameter values depending on the detailed strategy updating and network dimension. An explanation for the obtained results is provided.

A Tag-Based Evolutionary Prisoner's Dilemma Game on Networks with Different Topologies

Researchers from many disciplines have been interested in the maintenance of cooperation in animal and human societies using the Prisoner's Dilemma game. Recent studies highlight the roles of cognitively simple agents in the evolution of cooperation who read tags to interact either discriminately or selectively with tolerably similar partners. In our study on a one-shot Prisoner's Dilemma game, artificial agents with tags and tolerance perceive dissimilarities to local neighbors to cooperate with in-group and otherwise defect. They imitate tags and learn tolerance from more successful neighbors. In terms of efficiency, society-wide cooperation can evolve even when the benefits of cooperation are relatively low. Meanwhile, tolerance however decreases as agents become homogenized. In terms of stability, parochial cooperators are gullible to the deviants - defectors displaying tolerably similar tags. We find that as the benefits of cooperation increase and the dimensions of tag space become larger, emergent societies can be more tolerant towards heterogeneous others. We also identify the effects of clustering and small-world-ness on the dynamics of tag-based parochial cooperation in spite of its fundamental vulnerability to those deviants regardless of network topology. We discuss the issue of tag mutability in search for alternative societies in which tag-based parochial cooperation is not only efficient but also robust.

Evolutionary prisoner's dilemma on Newman–Watts social networks with an asymmetric payoff distribution mechanism

In this paper, we introduce an asymmetric payoff distribution mechanism into the evolutionary prisoner's dilemma game (PDG) on Newman–Watts social networks, and study its effects on the evolution of cooperation. The asymmetric payoff distribution mechanism can be adjusted by the parameter α: if α > 0, the rich will exploit the poor to get richer; if α < 0, the rich are forced to offer part of their income to the poor. Numerical results show that the cooperator frequency monotonously increases with α and is remarkably promoted when α > 0. The effects of updating order and self-interaction are also investigated. The co-action of random updating and self-interaction can induce the highest cooperation level. Moreover, we employ the Gini coefficient to investigate the effect of asymmetric payoff distribution on the the system's wealth distribution. This work may be helpful for understanding cooperative behaviour and wealth inequality in society.

Phase diagrams for three-strategy evolutionary prisoner’s dilemma games on regular graphs

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.

Cooperation enhanced by moderate tolerance ranges in myopically selective interactions

We present a mode of myopically selective interaction to study the evolutionary prisoner's dilemma game in scale-free networks. Each individual has a reputation-based tolerance range and only tends to interact with the neighbors whose reputation is within its tolerance range. Moreover, its reputation is assessed in response to the interactions in the neighborhood. Interestingly, we show that moderate values of tolerance range can result in the best promotion of cooperation due to the emergence of group selection mechanism. Furthermore, we study the effects of weighting factor in the assessment rule of reputation on the evolution of cooperation. We also show how cooperation evolves in some extended situations, where an interaction stimulus payment is considered for individuals, and where the strategy and reputation of individuals can spread simultaneously. Our results may enhance the understanding of evolutionary dynamics in graph-structured populations where individuals conditionally play with their neighbors according to some myopic selection criteria.

Evolutionary Prisoner's Dilemma Game Based on Division of Work

We propose a new two-type-player prisoner's dilemma game based on the division of work on a square lattice, in which a fraction of the population μ are assigned type A and the rest B. In a one-shot two-player game, we let both of their original payoffs be scaled by a same multiplicative factor α > 1, if two neighboring players are of different types; however we leave the payoffs unchanged if they are of the same type. Then we show that combined with the two-type setup, the square lattice can assist to induce different social ranks according to players' abilities to collect payoffs. Simulation results show that the density of cooperation is significantly promoted for a wide range of the temptation to defection parameters and that there are optimal values for both α and μ leading to the maximal cooperation level. We reach these results by analyzing the distribution of the players in the social ranks and we also show some typical snapshots of the system.

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.

Prisoner's Dilemma Game on Clustered Scale-Free Networks under Different Initial Distributions

The evolutionary prisoner's dilemma game is investigated under different initial distributions for cooperators and defectors on scale-free networks with a tunable clustering coefficient. It is found that, on the one hand, cooperation can be enhanced with the increasing clustering coefficient when only the most connected nodes are occupied by cooperators initially. On the other hand, if cooperators just occupy the lowest-degree nodes at the beginning, then the higher the value of the clustering coefficient, the more unfavorable the environment for cooperators to survive for the increment of temptation to defect. Thereafter, we analytically argue these nontrivial phenomena by calculating the cooperation probability of the nodes with different degrees in the steady state, and obtain the critical values of initial frequency of cooperators below which cooperators would vanish finally for the two initial distributions.

Evolutionary prisoner's dilemma game on highly clustered community networks

This paper studies the evolutionary prisoner's dilemma game on a highly clustered community network in which the clustering coefficient and the community size can be tuned. It finds that the clustering coefficient in such a degree- homogeneous network inhibits the emergence of cooperation for the entire range of the payoff parameter. Moreover, it finds that the community size can also have a marked influence on the evolution of cooperation, with a larger community size leading to not only a lower cooperation level but also a smaller threshold of the payoff parameter above which cooperators become extinct.

Diversity of rationality affects the evolution of cooperation

By modifying the Fermi updating rule, we present the diversity of individual rationality to the evolutionary prisoner's dilemma game, and our results shows that this diversity heavily influences the evolution of cooperation. Cluster-forming mechanism of cooperators can either be highly enhanced or severely deteriorated by different distributions of rationality. Slight change in the rationality distribution may transfer the whole system from the global absorbing state of cooperators to that of defectors. Based on mean-field argument, quantitative analysis of the stability of cooperative clusters reveals the critical role played by agents with moderate degree values in the evolution of the whole system. The inspiration from our work may provide us a deeper comprehension toward some social phenomena.

Evolutionary Games on Weighted Newman–Watts Small-World Networks

We investigate the evolutionary prisoner's dilemma game (PDG) on weighted Newman–Watts (NW) networks. In weighted NW networks, the link weight wij is assigned to the link between the nodes i and j as: wij = (κi · κj)β, where κi(κj) is the degree of node i(j) and β represents the strength of the correlations. Obviously, the link weight can be tuned by only one parameter β. We focus on the cooperative behavior and wealth distribution in the system. Simulation results show that the cooperator frequency is promoted by a large range of β and there is a minimal cooperation frequency around β = – 1. Moreover, we also employ the Gini coefficient to study the wealth distribution in the population. Numerical results show that the Gini coefficient reaches its minimum when β ≈ – 1. Our work may be helpful in understanding the emergence of cooperation and unequal wealth distribution in society.

Evolving learning rules and emergence of cooperation in spatial prisoner's dilemma

In the evolutionary Prisoner's dilemma (PD) game, agents play with each other and update their strategies in every generation according to some microscopic dynamical rule. In its spatial version, agents do not play with every other but, instead, interact only with their neighbours, thus mimicking the existing of a social or contact network that defines who interacts with whom. In this work, we explore evolutionary, spatial PD systems consisting of two types of agents, each with a certain update (reproduction, learning) rule. We investigate two different scenarios: in the first case, update rules remain fixed for the entire evolution of the system; in the second case, agents update both strategy and update rule in every generation. We show that in a well-mixed population the evolutionary outcome is always full defection. We subsequently focus on two-strategy competition with nearest-neighbour interactions on the contact network and synchronised update of strategies. Our results show that, for an important range of the parameters of the game, the final state of the system is largely different from that arising from the usual setup of a single, fixed dynamical rule. Furthermore, the results are also very different if update rules are fixed or evolve with the strategies. In these respect, we have studied representative update rules, finding that some of them may become extinct while others prevail. We describe the new and rich variety of final outcomes that arise from this co-evolutionary dynamics. We include examples of other neighbourhoods and asynchronous updating that confirm the robustness of our conclusions. Our results pave the way to an evolutionary rationale for modelling social interactions through game theory with a preferred set of update rules.

Evolutionary prisoner's dilemma games with one-dimensional local interaction and imitation

In this paper we explore the impact of imitation rules on players' long-run behaviors in evolutionary prisoner's dilemma games. All players sit sequentially and equally spaced around a circle. Players are assumed to interact only with their neighbors, and to imitate either their successful neighbors and/or themselves or the successful actions taken by their neighbors and/or themselves. In the imitating-successful-player dynamics, full defection is the unique long-run equilibrium as the probability of players' experimentations (or mutations) tend to 0. By contrast, full cooperation could emerge in the long run under the imitating-successful-action dynamics. Moreover, it is discovered that the convergence rate to equilibrium under local interaction could be slower than that under global interaction.

Evolutionary prisoner's dilemma games with one-dimensional local interaction and imitation

In this paper we explore the impact of imitation rules on players' long-run behaviors in evolutionary prisoner's dilemma games. All players sit sequentially and equally spaced around a circle. Players are assumed to interact only with their neighbors, and to imitate either their successful neighbors and/or themselves or the successful actions taken by their neighbors and/or themselves. In the imitating-successful-player dynamics, full defection is the unique long-run equilibrium as the probability of players' experimentations (or mutations) tend to 0. By contrast, full cooperation could emerge in the long run under the imitating-successful-action dynamics. Moreover, it is discovered that the convergence rate to equilibrium under local interaction could be slower than that under global interaction.

PRISONER'S DILEMMA GAMES WITH SOME TROUBLEMAKERS IN A RANDOM NETWORK

We further study the competition among loner, cooperative and defective strategies by considering an evolutionary prisoner's dilemma (PD) game with some troublemakers in the random network by means of Monte Carlo simulations. It can be observed that the frequency of cooperators (fc) decreases with increasing of loners' (e) and defectors' payoffs (b). In particular, fc has a sharp decline when the loners' payoffs approximate 1.0. In addition, we find that introduction of troublemaker strategies prevents the uniform defection in the spatial evolutionary PD games for large b values and small e values. In practice, it also prevents cooperative strategies and defective strategies from vanishing for large e values.

Evolutionary Prisoner's Dilemma Game Based on Pursuing Higher Average Payoff

We investigate the prisoner's dilemma game based on a new rule: players will change their current strategies to opposite strategies with some probability if their neighbours' average payoffs are higher than theirs. Compared with the cases on regular lattices (RL) and Newman–Watts small world network (NW), cooperation can be best enhanced on the scale-free Barabási–Albert network (BA). It is found that cooperators are dispersive on RL network, which is different from previously reported results that cooperators will form large clusters to resist the invasion of defectors. Cooperative behaviours on the BA network are discussed in detail. It is found that large-degree individuals have lower cooperation level and gain higher average payoffs than that of small-degree individuals. In addition, we find that small-degree individuals more frequently change strategies than do large-degree individuals.

EFFECTS OF LEARNING ACTIVITY ON COOPERATION IN EVOLUTIONARY PRISONER'S DILEMMA GAME

We study the evolutionary Prisoner's Dilemma game under individual learning activity mechanism on small-world and scale-free networks, respectively. Each player updates its strategy with quenched learning rate which characterizes the strength of individual learning activity during the evolutionary process. Simulation results show that the mechanism of learning activity presents nontrivial phenomena on both the two networks: optimal intermediate levels of learning activity can promote or sustain cooperation. Specifically, there exists an interesting resonance-like manner in the evolutionary game when the intermediate level of learning activity promotes cooperation, and there exists a plateau for cooperation in the favorable moderate region of learning activity when the intermediate level of learning activity sustains cooperation. Moreover, these interesting phenomena are not sensitive to the two networks with other topological parameters. Our work can be helpful in reflecting the effects of individual learning mechanism on cooperative behavior in social systems.

Evolutionary prisoner’s dilemma game on Newman-Watts networks

Maintenance of cooperation was studied for a two-strategy evolutionary prisoner's dilemma game where the players are located on a one-dimensional chain and their payoff comes from games with the nearest- and next-nearest-neighbor interactions. The applied host geometry makes it possible to study the impacts of two conflicting topological features. The evolutionary rule involves some noise affecting the strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the phase diagram as a function of noise level and a payoff parameter. The peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by Newman and Watts.