Prisoner's Dilemma Game Research Articles

Past-performance-driven strategy updating promote cooperation in the spatial prisoner's dilemma game

Strategy update rules play an important role in repeated Prisoner's Dilemma games. This work proposes a modified strategy update rule based on the traditional Fermi function, in which individual past performance is taken into account in strategy update. Then, the consistency aspiration α serves as a benchmark to measure an individual's past performance, and the past performance score is dynamically adjusted during the evolution process according to the BM reinforcement learning rules. The computational results indicate that the proposed modified strategy update rules can significantly improve system cooperation t than the traditional version, and the network reciprocity effect is enhanced as the result of past performance is coupled into the strategy update rule. Moreover, different temptation to defection b exist a corresponding aspiration α result in maximizing system cooperation. Furthermore, an optimal sensitivity level β can also result in a maximizing system cooperation. As a whole, for α − β phase diagram, different a will correspond to an optimal value β that allows the system to achieve the maximum cooperation. Finally, the proposed mechanism is robust. Hopefully this can help to inspire further research on how to deal with social dilemmas.

Effects of value-driven social learning on cooperation in the prisoner's dilemma games.

Despite the growing attention and research on the impact of Q-learning-based strategy updating on the evolution of cooperation, the joint role of individual learners and social learners in evolutionary games has seldom been considered. Here, we propose a value-driven social learning model that incorporates a shape parameter, β, to characterize the degree of radicalism or conservatism in social learning. Using the prisoner's dilemma game on a square lattice as a paradigm, our simulation results show that the cooperation level has a non-trivial dependence of β, density ρ, and dilemma strength b. We find that both β and ρ have nonmonotonic effects on cooperation; specifically, moderate levels of radicalism in social learning can facilitate cooperation remarkably, and when slightly conservative, can form a favorable cooperation region with the appropriate ρ. Moreover, we have demonstrated that social learners play a key role in the formation of network reciprocity, whereas individual learners play a dual role of support and exploitation. Our results reveal a critical balance between individual learning and social learning that can maximize cooperation and provide insights into understanding the collective behavior in multi-agent systems.

The effect of survival pressure-based defector reward behavior on cooperation in spatial prisoner's dilemma games.

Research has shown that rewarding behavior can greatly facilitate the occurrence of cooperation in social dilemmas. Yet rewards entail costs, making the reward itself an altruistic behavior. The reasons for adopting rewarding behaviors then become an interesting matter, so we propose a game model in which defectors are pressured by survival to adopt rewarding behaviors. Research suggests that defector reward strategies can be used as a transition strategy for defectors to alleviate survival stress and promote cooperation in the spatial prisoner's dilemma. A small survival pressure threshold will make it altogether easier for defectors to adopt a strategy that rewards defection and thus tempts the emergence of cooperators in their neighborhood. In addition, the effect of the payment cost α and the reward amount β of the rewarding behavior on the evolution of the system will be limited by the temptation b of the defector, and the effect of each parameter on the promotion of cooperation in the system is not linear. That is, when α is fixed, b and β can still optimize the level of cooperation in a given combination. The same holds for α and β when b is fixed.

Evolutionary dynamics in voluntary multigames with exploiting factor and interrelated networks

It is well known that multigames with individuals participation voluntarily and interrelated networks can improve cooperation level. Based on these mechanisms, we propose exploitation factor in spatial structured population, which means players in higher network layer will exploit part of payoff from lower layer players. In this work, double and triple interrelated networks population can be divided into two types of Prisoner's dilemma game (PDG) and Snowdrift game (SDG) respectively by sucker's payoff. The results show that exploitation can promote cooperation level, especially for lower layer.

Adjustment of link weights based on tolerance promotes cooperation in spatial prisoner's dilemma game

Understanding and explaining cooperative behavior in human society has become an open question. In this paper, we propose a dynamic adjustment of pair relationships in a spatial prisoner's dilemma game. Unlike previous studies that individuals dynamically adjust the intensity of interaction with their opponents at each step, this work consider tolerance, in which the intensity of interaction is adjusted when the time of successive defections by an individual exceeds a tolerance threshold T. We find that although the proposed mechanism can significantly improve cooperation compared to traditional versions, a higher tolerance for continuous defection behavior is not conducive to the evolution of cooperation. Furthermore, an environmental adaptor that dynamically adjusts the paired relationship with the opponent at all times is beneficial for the evolution of cooperation. And the higher the degree of adjustment in the paired relationship, the lower the probability of continuous exploitation by defector. We hope that our work can provide some insights into explaining the existence and maintenance of cooperation.

Reciprocal-reward–based environmental feedback promotes cooperation in spatial prisoner's dilemma game

Reciprocal rewards for cooperative behavior are seen as a way of solving social dilemmas. In fact, reward values should not be fixed, but rather linked to the level of social development. Thus, this work proposes a mechanism by which reciprocal reward values in the spatial prisoner's dilemma game depend on region environmental feedback. Therefore, the reciprocal reward values are non-linear and inversely correlated with the environmental state. Numerical simulations show that the non-linear and dynamic reciprocal reward can significantly improve cooperation compared to the traditional version, which can also reduce social costs to some extent. In addition, a higher level of sensitivity to changes in the environment is beneficial for cooperative evolution. Specifically, while setting reciprocal rewards based on greater local environmental feedback can increase the threshold of temptation b at which cooperators become extinct, its contribution to encouraging cooperation is limited. Furthermore, different interval ranges of b-values will have a corresponding range of local environment to encourage cooperation to achieve the optimum. Finally, the results show that the system has good robustness.

Evolutionary branching in multi-level selection models.

We study a model of group-structured populations featuring individual-level birth and death events, and group-level fission and extinction events. Individuals play games within their groups, while groups play games against other groups. Payoffs from individual-level games affect birth rates of individuals, and payoffs from group-level games affect group extinction rates. We focus on the evolutionary dynamics of continuous traits with particular emphasis on the phenomenon of evolutionary diversification. Specifically, we consider two-level processes in which individuals and groups play continuous snowdrift or prisoner's dilemma games. Individual game strategies evolve due to selection pressure from both the individual and group level interactions. The resulting evolutionary dynamics turns out to be very complex, including branching and type-diversification at one level or the other. We observe that a weaker selection pressure at the individual level results in more adaptable groups and sometimes group-level branching. Stronger individual-level selection leads to more effective adaptation within each group while preventing the groups from adapting according to the group-level games.

Third party interventions promote cooperation on the interdependent networks: A perspective based on prospect theory.

Third-party intervention is a beneficial means to alleviate conflicts and promote cooperation among disputants. The decision-making of disputants is closely related to the intensity of the impact of third-party intervention on their profits. Actually, disputants often decide whether to adopt cooperative strategies based on their own perceived rather than actual gains or losses brought about by third-party intervention. We, therefore, introduce prospect theory to explore the formation and maintenance of cooperation in a system composed of third parties and disputants, which, respectively, constitute two sub-networks of the interdependent networks. Both interveners and disputants participate in a prisoner's dilemma game, and the third-party intervener will pay a certain price to impose certain punishments on the defectors of the disputed layer. The simulation results show that the introduction of third-party intervention based on the prospect theory alleviates the conflicts in the dispute layer and promotes cooperation among disputants, which indicates that third parties such as governments or organizations should appropriately consider the risk attitudes of disputants when mediating their conflicts. The level of cooperation at the dispute layer is inversely proportional to the intervention cost and directly proportional to the intervention intensity. Our research may shed some new light on the study of the evolution of cooperation under third-party intervention.

Evolution of cooperation with asymmetric rewards

Rewards, as a form of positive reinforcement, effectively encourage cooperation. In this paper, we study a multi-population prisoner's dilemma game with asymmetric rewards, where agents in the same population play prisoner's dilemma game, and agents from the giver population can reward agents from the recipient population only if they make the same choice. In well-mixed populations, asymmetric rewards can facilitate cooperation. Similarly, asymmetric rewards on the regular square lattice can effectively prevent complete defection. In both well-mixed and structured populations, seemingly disadvantageous cooperative givers play an important role in the maintenance and spread of cooperation. Especially on lattice, cooperative givers with asymmetric rewards can be active in the system through diverse cases of cyclic dominance. Our findings provide deeper insights into the impact of asymmetry on cooperative behavior and the development of altruistic behavior in real-world scenarios.

Explaining interference effects in prisoner dilemma games

Explaining interference effects in prisoner dilemma games

The guidance of neutral human populations maintains cooperation in the prisoner's dilemma game

In game theory, the emergence and maintenance of cooperative behavior within a group is a significant topic in evolutionary game theory and complex network theory. However, the limitations of a single mechanism in traditional networks restrict a thorough analysis of the sustenance and development of cooperative behavior, given the challenges posed by the diversity of social groups. To address this issue, this paper combines reinforcement learning game strategies with traditional prisoner's dilemma strategies based on two-layer coupled network to investigate the transmission of cooperative behavior among individuals in games. In our research, we study the evolutionary pattern and phase transitions using the Monte Carlo method. We use the prisoner's dilemma game as a mathematical model, establishing two subpopulations in each layer, with mutually payoff-neutral players between different subpopulations. This configuration results in intriguing spatiotemporal dynamics and patterns, leading to the spontaneous emergence of a cyclic dominance, where defectors from one group become prey for cooperators in another group, and vice versa. By simulating game evolution, we explore individual strategy changes and the impact of individual abilities on cooperative behavior in reinforcement learning. Extensive validations indicate that, in social dilemmas, adjusting the abilities of groups through effective guidance can sustain cooperative behavior. This guidance enables us to comprehend the stability of cooperation under adverse conditions. Simultaneously, the coexistence of two subpopulations greatly amplifies the complexity of evolutionary dynamics, causing a increase in cooperation rate.

Pink-noise dynamics in an evolutionary game on a regular graph.

We consider a multiplayer prisoner's dilemma game on a square lattice and regular graphs based on the pairwise-Fermi update rule, and we obtain heatmaps of the fraction of cooperators and the correlation of neighboring pairs. In the heatmap, we find a mixed region where cooperators and defectors coexist, and the correlation between neighbors is significantly enhanced. Moreover, we observe pink-noise behavior in the mixed region, where the power spectrum can be fitted by a power-law function of frequency. We also find that the pink-noise behavior can be reproduced in a simple random-walk model. In particular, we propose a modified random-walk model which can reproduce not only the pink-noise behavior but also the deviation from it observed in a low-frequency region.

Evolutionary Dynamics of Population Games With an Aspiration-Based Learning Rule.

Agents usually adjust their strategic behaviors based on their own payoff and aspiration in gaming environments. Hence, aspiration-based learning rules play an important role in the evolutionary dynamics in a population of competing agents. However, there exist different options for how to use the aspiration information for specifying the microscopic learning rules. It is also interesting to investigate under what conditions the aspiration-based learning rules can favor the emergence of cooperative behavior in population games. A new learning rule, called as "Satisfied-Cooperate, Unsatisfied-Defect", is proposed here, which is based on aspiration. Under this learning rule, agents prefer to cooperate when their income is satisfied; otherwise, they prefer the strategy of defection. We introduce this learning rule to a population of agents playing a generalized two-person game. We, respectively, obtain the mathematical conditions in which cooperation is more abundant in finite well-mixed, infinite well-mixed, and structured populations under weak selection. Interestingly, we find that these conditions are identical, no matter whether the aspiration levels for cooperators and defectors are the same or not. Furthermore, we consider the prisoner's dilemma game (PDG) as an example and perform numerical calculations and computer simulations. Our numerical and simulation results agree well and both support our theoretical predictions in the three different types of populations. We further find that our aspiration-based learning rule can promote cooperation more effectively than alternative aspiration-based learning rules in the PDG.

Impact of environmental stochastic fluctuations on the evolutionary stability of imitation dynamics.

To show the impact of environmental noise on imitation dynamics, the stochastic stability and stochastic evolutionary stability of a discrete-time imitation dynamics with random payoffs are studied in this paper. Based on the stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model, we extend the concept of stochastic evolutionary stability to the stochastic imitation dynamics, which is defined as a strategy such that, if all the members of the population adopt it, then the probability for any mutant strategy to invade the population successfully under the influence of natural selection is arbitrarily low. Our main results show clearly that the stochastic evolutionary stability of the system depends only on the properties of the mean matrix of the random payoff matrix and is independent of the randomness of the random payoff matrix. Moreover, as two examples, we show also that under the framework of stochastic imitation dynamics, the noise intensity affects the evolution of cooperative behavior in a stochastic prisoner's dilemma game and the system's nonlinear dynamic behavior.

A social monitoring mechanism for third-party judges promotes cooperation in evolutionary games

Corruption of third-party judges seriously undermines the level of cooperation. Without intervention, more corruptors and defectors would emerge, disrupting social harmony. Therefore, introducing an anti-corruption mechanism is crucial for the evolution of cooperation. In this paper, we propose a social monitoring mechanism to monitor third-party judges so that their payoffs are affected by the proportions of cooperators. Monte Carlo simulations on periodic boundary lattices. The results show that the social monitoring mechanism is effective in promoting cooperation and inhibiting corruption, and enhances the effectiveness of zealots in promoting cooperation. This facilitation effect is not only manifested in the Prisoner's Dilemma Game but also in the Snowdrift Game, which confirms the robustness of the results. Our research provides new insights for solving social dilemmas and curbing corruption.

Cooperation dynamics of prisoner's dilemma games on an evolutionary weighted network with heterogeneous preferences.

Relationships between people in real life are dynamically changed with the interaction process, and due to the heterogeneous preferences, this change is different from person to person. Based on this observation, we propose a new spatial and weighted prisoner's dilemma game model with heterogeneous individuals. Two types of tags, namely, tag-F (concerned about social fairness) and tag-W (concerned about personal well-being), are introduced to describe individuals' different preferences. The link weights indicating the interaction strength between individuals are updated based on different rules that depend on their tags. Through simulations, we verify that a large link weight control factor and a high proportion of tag-F individuals favor the emergence and persistence of cooperation. In addition, an increase in the link weight sensitivity factor favors the evolution of cooperation when the link weight control factor is small. Moreover, while the level of cooperation increases with the proportion of tag-F type in the population, contrary to our intuition, when the population consists entirely of tag-F individuals, in some cases, cooperation cannot reach a higher level compared with the situation when they are mixed with tag-W type. However, at high dilemma intensities, cooperators emerge only when the entire population consists of tag-F type. These results may provide some new insights into the impact of the evolutionary weighted network with heterogeneous preferences on collective cooperative behavior.

Memory and target payoff enhance cooperation in evolutionary social dilemmas.

We proposed a neighbor selection mechanism based on memory and target payoff, where the target payoff is the maximum value of the group's average expected payoff. According to this mechanism, individuals prioritize selecting neighbors whose average payoffs in the last M rounds are close to the target payoff for strategy learning, aiming to maximize the group's expected payoff. Simulation results on the grid-based Prisoner's Dilemma and Snowdrift games demonstrate that this mechanism can significantly improve the group's payoff and cooperation level. Furthermore, the longer the memory length, the higher the group's payoff and cooperation level. Overall, the combination of memory and target payoff can lead to the emergence and persistence of cooperation in social dilemmas as individuals are motivated to cooperate based on both their past experiences and future goals. This interplay highlights the significance of taking into account numerous variables in comprehending and promoting cooperation within evolutionary frameworks.

Reputation-based disconnection-reconnection mechanism in Prisoner's Dilemma Game within dynamic networks

Individual reputation significantly influences players’ decision-making. Players tend to disconnect from neighbors with lower reputations so as to mitigate potential losses in the game. Therefore, it is crucial and meaningful to explore how to apply reputation in the game to enhance cooperation. In this study, we propose a reputation-based evolution mechanism in the Prisoner's Dilemma Game (PDG) within dynamic complex networks, including square lattices and small-world networks. We introduce the reputation factor α and the selection intensity σ in order to measure the changes in reputation and their impact objectively. Moreover, considering the impact of information connectivity, we define the environmental degree r and investigate the effects of environmental degree r and rewiring probability p on cooperation rates. Simulation results demonstrate that the reputation factor and information connectivity have a positive promoting effect on cooperative behavior; conversely, the increase in selection intensity affects cooperation negatively.

Third-party arbitration and forgiving strategies increase cooperation when perception errors are common.

Humans cooperate in groups in which mutual monitoring is common, and this provides the possibility of third-party arbitration. Third-party arbitration stabilizes reciprocity in at least two ways: first, when it is accurate, it reduces the frequency of misunderstandings resulting from perception errors, and second, even when it is inaccurate, it provides a public signal that allows pairs to align their expectations about how to behave after errors occur. Here, we describe experiments that test for these two effects. We find that in an iterated, sequential Prisoner's Dilemma game with errors, players with the highest average payoffs are those who make use of third-party arbitration and who also employ forgiving strategies. The combination of these two behaviours reduces the detrimental effects of errors on reciprocity, resulting in more cooperation.

Gender and cooperation in the presence of negative externalities

Social dilemmas often impose negative externalities on third parties. We experimentally analyze gender differences in cooperation in such a setting, i.e., a prisoner's dilemma game, with a passive third party that may be harmed when active players mutually cooperate. Applying a within-subjects setting, we compare cooperation under anonymity and social information, as personal characteristics are commonly known in real-life relations. Results show that the presence of a negative externality particularly affects guilt-averse women, who cooperate less often independently of the degree of information they receive. No gender difference is found absent negative externalities.