Weak Prisoner Research Articles

Evolutionary dynamics of stochastic games in set-structured populations.

In structured populations, the ecology of games may vary over neighborhoods. The effect of the ecological variations on population dynamics remains largely unknown. We here incorporate the ecological variations into the set-structured populations to explore the coevolutionary dynamics of the ecology and cooperation. Individuals of a population are distributed over sets. Interactions occur in the form of evolutionary games. When two individuals share more common sets, they play the weak prisoner's dilemma. Otherwise, they play the strong prisoner's dilemma. Both the set memberships and the strategy update in the evolutionary process. Changes in set memberships hold sway over the games to be played, which, in turn, influences the performance of strategies. Combining evolutionary set theory and random walks on graphs, we derived the conditions for cooperation to be selected under the weak selection limit. We find that a denser set-structured population increases the probability of individuals participating in a weak prisoner's dilemma, and thereby promoting the spread of cooperation. Properly modulating the population structure and the payoff feedback can further lower the critical benefit-cost ratio required for cooperation to be selected. Our results may help better understand the effects of ecological variations in enhancing cooperative behavior in set-structured populations.

Evolution of cooperation with the diversity of cooperation tendencies

The complete cooperation and the complete defection are two typical strategies considered in evolutionary games in many previous works. However, in real life, strategies of individuals are full of variety rather than only two complete ones. In this work, the diversity of strategies is introduced into the weak prisoners’ dilemma game, which is measured by the diversity of the cooperation tendency. A greater diversity means more cooperation tendencies are provided. The complete cooperation strategy is the full cooperation tendency and the complete defection strategy is without any cooperation tendency. Agents with other cooperation tendencies are partial cooperative and defective. The numerical simulation shows that increasing the diversity of the cooperation tendency promotes the cooperation level, not only the number of cooperators but also the average tendency over the whole population, until the diversity reaches its saturated value. Furthermore, our work points out maintaining cooperation is based on the cooperation efficiency. When cooperator clusters can survive the cooperation efficiency slightly oscillates around its equilibrium. When cooperator clusters cannot resist the invasion of defectors the cooperation efficiency oscillates and quickly decreases to zero. The observations above is not only applied on the weak prisoner’s dilemma but also the snowdrift game. When the effect of the noise for the Femi update mechanism is considered, a greater diversity of strategies not only improves the cooperation level of the whole population but also supports the survival of more rational agents.

The evolution of cooperation and global synchronization in the evolutionary Kuramoto dilemma combined with the prisoner's dilemma

In this work, we have studied the evolution of the cooperation and the global synchronization on the Kuramoto model upon random networks with two games, in which one is the evolutionary Kuramoto dilemma and the other is the weak prisoner's dilemma. Agents can gain their payoffs from these two games synchronously. In the evolutionary Kuramoto game, cooperators prefer the formation of local synchronization but pay less cooperative cost. Cooperators in weak prisoner's dilemma can form cooperator clusters because of the network reciprocity. Combining these two games shows a mutual promotion effect upon the order of the global synchronization but a competition effect on the cooperation level. The global synchronization begins to rise only when most agents choose cooperation. It is manifested by the fact that the optimal coupling strength for the highest cooperation level is smaller than that for the highest synchronization level. The high cooperation cost makes defection the most advantageous strategy when the coupling strength reaches rather high, which diminishes the effect of cooperator clusters, and ultimately results in the absence of both cooperators and global synchronization. When agents have more neighbors in the random network, it is observed that cooperation is consistently hindered unless the coupling strength is relatively small. Additionally, there is always an optimal average degree that allows for global synchronization.

Two-stage strategy update rule based on learning cost in weak prisoner's dilemma.

When players are dissatisfied with their actual payoffs, they will change the actuality by learning strategy of neighbors. The more effort players put in, the more likely they are to succeed in learning. Inspired by this, this paper proposes a two-stage strategy update rule based on learning cost. The players first decide whether to learn strategy according to the updating willingness. If the players imitate the strategy of neighbors, they need to pay the learning cost. Results show that for the well-mixed population, if the updating willingness is homogeneous and remains unchanged, reducing the updating willingness or increasing the learning cost can extend the life cycle of cooperators. If the updating willingness is heterogeneous and dynamically adjusted based on the difference between the actual payoff and the expected payoff, increasing aspiration value and learning cost promotes cooperation. For the structured population, if the updating willingness is homogeneous and remains unchanged, the moderate learning cost is beneficial for cooperators to resist the temptation of defection, and reducing updating willingness makes the system maintain cooperation within a larger parameter range. If the updating willingness is heterogeneous and dynamically adjusted, the larger learning cost and the appropriate aspiration value promote cooperation. This study highlights the complex dynamics of cooperation in paid strategy learning, contributing to the theory of cooperation in the evolutionary game.

Coevolution of relationship and interaction in cooperative dynamical multiplex networks.

While actors in a population can interact with anyone else freely, social relations significantly influence our inclination toward particular individuals. The consequence of such interactions, however, may also form the intensity of our relations established earlier. These dynamical processes are captured via a coevolutionary model staged in multiplex networks with two distinct layers. In a so-called relationship layer, the weights of edges among players may change in time as a consequence of games played in the alternative interaction layer. As an reasonable assumption, bilateral cooperation confirms while mutual defection weakens these weight factors. Importantly, the fitness of a player, which basically determines the success of a strategy imitation, depends not only on the payoff collected from interactions, but also on the individual relationship index calculated from the mentioned weight factors of related edges. Within the framework of weak prisoner's dilemma situation, we explore the potential outcomes of the mentioned coevolutionary process where we assume different topologies for relationship layer. We find that higher average degree of the relationship graph is more beneficial to maintain cooperation in regular graphs, but the randomness of links could be a decisive factor in harsh situations. Surprisingly, a stronger coupling between relationship index and fitness discourage the evolution of cooperation by weakening the direct consequence of a strategy change. To complete our study, we also monitor how the distribution of relationship index vary and detect a strong relation between its polarization and the general cooperation level.

Inhibition and activation of interactions in networked weak prisoner's dilemma.

In the framework of the coevolution dynamics of the weak prisoner's dilemma, inspired by prior empirical research, we present a coevolutionary model with local network dynamics in a static network framework. Viewing the edges of the network as social interactions between individuals, when individuals play the weak prisoner's dilemma game, they accumulate both payoffs and social interaction willingness based on a payoff matrix of the social interaction willingness we constructed. The edges are then inhibiting or activating based on the social interaction willingness of the two individuals, and individuals only interact with others through activated edges, resulting in local network dynamics in a static network framework. Individuals who receive more cooperation will be more likely to activate the edges around them, meaning they will participate in more social interactions. Conversely, individuals who receive more defects will do the opposite. Specifically, we investigate the evolutionary dynamics of cooperation under different levels of sensitivity to social interaction willingness and the temptation to defect. Through the simulation, we find that sparse cooperator clusters can expand greatly when social interaction sensitivity and temptation to defect are low. In contrast, dense cooperator clusters form rapidly in a high social interaction sensitivity, which protects the cooperation from high temptation.

EVOLUTIONARY ANALYSIS OF PRISONER’S DILEMMA GAMES BASED ON MIXED RANDOM-CONFORMITY SELECTING MODEL

Inspired by the conformity phenomenon in human society, we develop a mixed neighbor selecting model adopting random-conformity rule to explore the evolutionary weak prisoner’s dilemma game. The neighbor selection rule of nodes is adjusted based on their fitness and collective influence. Under the degree-normalized payoff framework, the findings derived from Monte Carlo simulations reveal that this mixed selecting model can contribute to an impressive improvement in the Barabási-Albert network’s cooperation. In addition, experimental data obtained by investigating the game-learning skeleton indicate that, in this mixed random-conformity selecting model, normalized collective influence at moderate depth length enables influential nodes to maintain a cooperative strategy for an extended period of time. This can promote the emergence of cooperative strategies at low-degree nodes by facilitating the formation of stable cooperation-clusters centered on high-degree nodes. In addition, the normalized collective influence at excessive depth length increases the likelihood that influential nodes become defectors, thereby inhibiting the growth of cooperation-clusters and limiting cooperation.

The combination of social reward and punishment is conducive to the cooperation and heterogeneity of social relations.

Individual behaviors and social relations influence each other. However, understanding the underlying mechanism remains challenging. From social norms controlling human behavior to individual management of interpersonal relationships, rewards and punishments are some of the most commonly used measures. Through simulating the weak prisoner's dilemma in finite populations, we find that neither a simple reward measure nor a pure punishment mechanism can extensively promote cooperation. Instead, a combination of appropriate punishment and reward mechanisms can promote cooperation's prosperity regardless of how large or small the temptation to defect is. In addition, the combination spontaneously produces inhomogeneities in social relations and individual influence, which support the continued existence of cooperative behavior. Finally, we further explain how cooperators establish a sustainable existence under the combination by investigating the social relations at different moments in a small system. These results demonstrate that dispensing rewards and punishments impartially in society is essential to social harmony.

Spatial evolution of cooperation with variable payoffs.

In the evolution of cooperation, the individuals' payoffs are commonly random in real situations, e.g., the social networks and the economic regions, leading to unpredictable factors. Therefore, there are chances for each individual to obtain the exceeding payoff and risks to get the low payoff. In this paper, we consider that each individual's payoff follows a specific probability distribution with a fixed expectation, where the normal distribution and the exponential distribution are employed in our model. In the simulations, we perform the models on the weak prisoner's dilemmas (WPDs) and the snowdrift games (SDGs), and four types of networks, including the hexagon lattice, the square lattice, the small-world network, and the triangular lattice are considered. For the individuals' normally distributed payoff, we find that the higher standard deviation usually inhibits the cooperation for the WPDs but promotes the cooperation for the SDGs. Besides, with a higher standard deviation, the cooperation clusters are usually split for the WPDs but constructed for the SDGs. For the individuals' exponentially distributed payoff, we find that the small-world network provides the best condition for the emergence of cooperators in WPDs and SDGs. However, when playing SDGs, the small-world network allows the smallest space for the pure cooperative state while the hexagon lattice allows the largest.

Aspiration depends on environmental factors and aspiration duration promotes cooperation in the spatial prisoner’s dilemma game

This paper mainly investigates the impact of the duration of aspiration and environmental factors on cooperation using the weak prisoner’s dilemma game. In real life, people’s aspiration levels are often related to the development levels of areas where they work or live. They will attempt to change the status quo when their income is below the aspiration or expectation level, and learn neighbors’ behaviors based on differences in payoffs, moreover, duration levels of aspiration are heterogeneous for different people. This research used this problem as basis to study the impact of people’s duration levels of aspiration on cooperation using the Monte Carlo simulation method. Numerous simulations show that a dynamic adjustment of aspiration at each Monte Carlo step will lead to the complete disappearance of cooperation, appropriately raising the duration level of aspiration to enhance cooperation. We also find that appropriate reference to the external environment in setting aspirations can promote cooperation. Our analytical conclusions and methods provide a new perspective for exploring the emergence and maintenance of human cooperation.

The spatial inheritance enhances cooperation in weak prisoner’s dilemmas with agents’ exponential lifespan

The spatial evolutionary games have been studied abundantly in the past two decades. Recently, there is research showing that the emergence of cooperators on temporal networks varies from static networks. However, few research takes players’ interaction lifespans into account, which can be regarded as one of the individuals’ characteristics in real networks. In this paper, we introduce a network model based on the M/M/∞ queueing system and perform the weak prisoner’s dilemmas, where each new individual arrives at the network by Poisson process and has an exponential lifespan. In addition, we propose a neighborhood similarity inheritance mechanism to maintain the players’ interaction requirements and reduce the impact of leaving players to the network. We suggest that cooperators emerge in the aforementioned network model on weak prisoner’s dilemmas, and the inheritance mechanism brings more cooperators to the network, which helps to overcome social dilemmas. Besides, by analyzing the evolution process of the degree distribution and the mean clustering coefficient, it is shown that the inheritance mechanism guarantees the degree heterogeneity and higher mean clustering coefficients, which are shown to promote cooperative behaviors in previous studies.

Social dilemma based on reputation and successive behavior

We propose a cooperative evolution model in which successive behaviors influence reputation. There are two mechanisms to form reputation. One is enhancing the reputation of players who cooperate insistently, and reducing it when they defect continuously. The other is increasing reputation when defection is transformed into cooperation, and decreasing when cooperation is turned into defection. These two reputation formation mechanisms are simulated on the weak prisoner's dilemma game. We use a weighted average Fermi equation to describe the probability of players learning the strategy of a randomly selected neighbor. The results show that, under these two mechanisms, both the reward for reputation and emphasis on it have an impact on the cooperation. We find that encouraging “C after C” is better for cooperation than encouraging “C after D”, when people attach importance to reputation to a certain extent. Before reaching that threshold, however, encouraging “C after D” is better when reputation reward is large.

The evolution of cooperation in spatial multigame with voluntary participation

In the real life, we often simultaneously encounter various social dilemmas, which are also inclined to be voluntarily participated in, instead of previous assumption's compulsory participation in. Accounting on this realistic scenario, we have introduced the mechanism that the individuals have access to different payoff matrices corresponding to different social dilemmas to participate in the multigame with three strategies to choose, including cooperation, defection, going it alone. Furthermore, we set a proportion ψ/2 of the population to play the Prisoner's Dilemma, a proportion ψ/2 of the population to play Snowdrift and a proportion 1−ψ of the population to play the weak Prisoner's Dilemma, which results in the fact that the mean payoff matrix returns to the basic weak PD. Though numerical simulations, we find that for the smaller temptation to defect, the cooperation can be enhanced by the diversity of the sucker's payoff in the multigame contrast to the basic case. In addition, when the contribution of sucker's payoff is larger or more players choose to play the Prisoner's Dilemma and Snowdrift, the cooperators become more dominated.

From degree-correlated to payoff-correlated activity for an optimal resolution of social dilemmas.

An active participation of players in evolutionary games depends on several factors, ranging from personal stakes to the properties of the interaction network. Diverse activity patterns thus have to be taken into account when studying the evolution of cooperation in social dilemmas. Here we study the weak prisoner's dilemma game, where the activity of each player is determined in a probabilistic manner either by its degree or by its payoff. While degree-correlated activity introduces cascading failures of cooperation that are particularly severe on scale-free networks with frequently inactive hubs, payoff-correlated activity provides a more nuanced activity profile, which ultimately hinders systemic breakdowns of cooperation. To determine optimal conditions for the evolution of cooperation, we introduce an exponential decay to payoff-correlated activity that determines how fast the activity of a player returns to its default state. We show that there exists an intermediate decay rate at which the resolution of the social dilemma is optimal. This can be explained by the emerging activity patterns of players, where the inactivity of hubs is compensated effectively by the increased activity of average-degree players, who through their collective influence in the network sustain a higher level of cooperation. The sudden drops in the fraction of cooperators observed with degree-correlated activity therefore vanish, and so does the need for the lengthy spatiotemporal reorganization of compact cooperative clusters. The absence of such asymmetric dynamic instabilities thus leads to an optimal resolution of social dilemmas, especially when the conditions for the evolution of cooperation are strongly adverse.

Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices.

We study the spatial evolutionary prisoner's dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoner's dilemma game with a single parameter b. Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z. From the numerical analysis, we find six different regimes on the triangular lattice (z=6). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b. In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b, both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b. On the hexagonal lattice (z=3), there exist two distinctive regimes. For small b, both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b. On the square lattice (z=4), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.

Dynamic instability of cooperation due to diverse activity patterns in evolutionary social dilemmas

Individuals might abstain from participating in an instance of an evolutionary game for various reasons, ranging from lack of interest to risk aversion. In order to understand the consequences of such diverse activity patterns on the evolution of cooperation, we study a weak prisoner's dilemma where each player's participation is probabilistic rather than certain. Players that do not participate get a null payoff and are unable to replicate. We show that inactivity introduces cascading failures of cooperation, which are particularly severe on scale-free networks with frequently inactive hubs. The drops in the fraction of cooperators are sudden, while the spatiotemporal reorganization of compact cooperative clusters, and thus the recovery, takes time. Nevertheless, if the activity of players is directly proportional to their degree, or if the interaction network is not strongly heterogeneous, the overall evolution of cooperation is not impaired. This is because inactivity negatively affects the potency of low-degree defectors, who are hence unable to utilize on their inherent evolutionary advantage. Between cascading failures, the fraction of cooperators is therefore higher than usual, which lastly balances out the asymmetric dynamic instabilities that emerge due to intermittent blackouts of cooperative hubs.

Emergence of super cooperation of prisoner's dilemma games on scale-free networks.

Recently, the authors proposed a quantum prisoner’s dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner’s dilemma (GPD, for short) games based on the weak Prisoner’s dilemma game, the full prisoner’s dilemma game and the normalized Prisoner’s dilemma game, written by GPDW, GPDF and GPDN respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter γ to measure the entangled relationship between the two players. We found that our generalised prisoner’s dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner’s dilemma games.

Coevolutionary success-driven multigames

Wealthy individuals may be less tempted to defect than those with comparatively low payoffs. To take this into consideration, we introduce coevolutionary success-driven multigames in structured populations. While the core game is always the weak prisoner's dilemma, players whose payoffs from the previous round exceed a threshold adopt only a minimally low temptation to defect in the next round. Along with the strategies, the perceived strength of the social dilemma thus coevolves with the success of each individual player. We show that the lower the threshold for using the small temptation to defect, the more the evolution of cooperation is promoted. Importantly, the promotion of cooperation is not simply due to a lower average temptation to defect, but rather due to a dynamically reversed direction of invasion along the interfaces that separate cooperators and defectors on regular networks. Conversely, on irregular networks, in the absence of clear invasion fronts, the promotion of cooperation is due to intermediate-degree players. At sufficiently low threshold values, these players accelerate the erosion of defectors and significantly shorten the fixation time towards more cooperative stationary states. Coevolutionary multigames could thus be the new frontier for the swift resolution of social dilemmas.

Different perceptions of social dilemmas: evolutionary multigames in structured populations.

Motivated by the fact that the same social dilemma can be perceived differently by different players, we here study evolutionary multigames in structured populations. While the core game is the weak prisoner's dilemma, a fraction of the population adopts either a positive or a negative value of the sucker's payoff, thus playing either the traditional prisoner's dilemma or the snowdrift game. We show that the higher the fraction of the population adopting a different payoff matrix the more the evolution of cooperation is promoted. The microscopic mechanism responsible for this outcome is unique to structured populations, and it is due to the payoff heterogeneity, which spontaneously introduces strong cooperative leaders that give rise to an asymmetric strategy imitation flow in favor of cooperation. We demonstrate that the reported evolutionary outcomes are robust against variations of the interaction network, and they also remain valid if players are allowed to vary which game they play over time. These results corroborate existing evidence in favor of heterogeneity-enhanced network reciprocity, and they reveal how different perceptions of social dilemmas may contribute to their resolution.

Co-evolution of strategies and update rules in the prisoner's dilemma game on complex networks

In this paper, we study a weak prisoner's dilemma (PD) game in which both strategies and update rules are subjected to evolutionary pressure. Interactions among agents are specified by complex topologies, and we consider both homogeneous and heterogeneous situations. We consider deterministic and stochastic update rules for the strategies, which in turn may consider single links or the full context when selecting agents to copy from. Our results indicate that the co-evolutionary process preserves heterogeneous networks as a suitable framework for the emergence of cooperation. Furthermore, on those networks the update rule leading to a larger fraction, which we call replicator dynamics, is selected during co-evolution. On homogeneous networks, we observe that even if the replicator dynamics again turns out to be the selected update rule, the cooperation level is greater than on a fixed update rule framework. We conclude that for a variety of topologies, the fact that the dynamics co-evolves with the strategies leads, in general, to more cooperation in the weak PD game.