Prisoner's Dilemma Game Research Articles

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.

The impact of dynamic linking on cooperation on complex networks.

In complex social systems, individual relationships and the surrounding environment are constantly changing, allowing individuals to interact on dynamic networks. This study aims to investigate how individuals in a dynamic network engaged in a prisoner's dilemma game adapt their competitive environment through random edge breaks and reconnections when faced with incomplete information and adverse local conditions, thereby influencing the evolution of cooperative behavior. We find that random edge breaks and reconnections in dynamic networks can disrupt cooperative clusters, significantly hindering the development of cooperation. This negative impact becomes more pronounced over larger time scales. However, we also observe that nodes with higher degrees of connectivity exhibit greater resilience to this cooperation disruption. Our research reveals the profound impact of dynamic network structures on the evolution of cooperation and provides new insights into the mechanisms of cooperation in complex systems.

Promotion of cooperation by payoff-based view radii in different interaction modes

In this letter, we introduce the payoff-based view radii into evolutionary prisoner's dilemma games performed in a two-dimensional plane and study how the adaptive view radii affect cooperation. Two types of feedback are considered, positive feedback and negative one. In the case of positive feedback, high-payoff (low-payoff) agents have large (small) view radii. In the case of negative feedback, the things are opposite. Meanwhile, three different interaction modes are considered, one-way visual interaction, proactive visual interaction, and two-way visual interaction. Our results show that the payoff-based view radii could promote cooperation effectively in all cases. Especially, there exist optimal behaviors of cooperation for both positive feedback and negative feedback. When agents are allowed to move, we find that the cooperation level could be further improved by slow migration. Our results shed light on the promotion of cooperation by the adaptive view radii and suggest different ways to adjust view radii to achieve high cooperation levels in different interaction modes.

Endogenous monitoring through voluntary reporting in an infinitely repeated prisoner's dilemma game: experimental evidence

AbstractExogenous reputational information is known to improve cooperation. This study experimentally investigates how people create such information by reporting their partner's action choices, and whether endogenous monitoring helps to sustain cooperation, in an indefinitely repeated prisoner's dilemma game with random matching. The experimental results show that most subjects report their opponents' action choices, thereby successfully cooperating when reporting does not involve costs. However, when reporting is costly, participants are strongly discouraged from doing so. Consequently, they fail to achieve strong cooperative norms when the reported information is conveyed privately only to their next‐round interaction partners. Costly reporting occurs only occasionally even when there is a public record whereby all future partners can check the reported information, but significantly more frequently relative to the condition in which it is sent to the next partner only. With public records, groups can foster cooperative norms aided by reported information that gradually accumulates and becomes more informative over time.

Game-theoretic agent-based modelling of micro-level conflict: Evidence from the ISIS-Kurdish war.

This article delves into the dynamics of a dyadic political violence case study in Rojava, Northern Syria, focusing on the conflict between Kurdish rebels and ISIS from January 1, 2017, to December 31, 2019. We employ agent-based modelling and a formalisation of the conflict as an Iterated Prisoner's Dilemma game. The study provides a nuanced understanding of conflict dynamics in a highly volatile region, focusing on microdynamics of an intense dyadic strategic interaction between two near-equally- powered actors. The choice of using a model based on the Iterated Prisoner's Dilemma, though a classical approach, offers substantial insights due to its ability to model dyadic, equally-matched strategic interactions in conflict scenarios effectively. The investigation primarily reveals that shifts in territorial control are more critical than geographical or temporal factors in determining the conflict's course. Further, the study observes that the conflict is characterised by periods of predominantly one-sided violence. This pattern underscores that the distribution of attacks, and target choices are a more telling indicator of the conflict nature than specific behavioural patterns of the actors involved. Such a conclusion aligns with the strategic implications of the underlying model, which emphasises the outcome of interactions based on differing aggression levels. This research not only sheds light on the conflict in Rojava but also reaffirms the relevance of this type of game-theoretical approach in contemporary conflict analysis.

Empirically testing a relationship between cooperation and the prime numbers.

Theoretical models suggest a relationship between cooperation and the prime numbers. In environments where agents play multiple one-shot prisoner's dilemma games per generation, cooperators evolve to fixation more frequently when cooperating on a cyclical schedule with a prime-number period length. This finding parrots classic predator-prey models showing selection for prime-number prey life cycles. Here, I report an empirical test of the former models using previously published data concerning humans playing one-shot public goods games across multiple time points-i.e. an analogue to multiple one-shot prisoner's dilemma games. I find very modest evidence of cyclicality at prime-numbered time intervals, though results indicate rough agreement between theoretical predictions and observed rates of full cooperation across time points. Analyses of individual decisions find increased contributions to the public good at prime-number time points and separate placebo tests indicate a 4-in-1000 chance of spuriously estimating this effect. However, when exploratory analyses exclude low-value prime-numbered time points, the magnitude of the estimated effect decreases and the hypothesis of no effect cannot be rejected, implying that low-value, prime-number time points drive estimates, contrary to theoretical model predictions. These findings cast doubt on the hypothesis of increased cooperation at prime-number time points-at least among humans playing public goods games.

Exploring cyclic dominance in rock-paper-scissor strategies within the Optional Prisoner's Dilemma: A quantitative analysis with environmental feedback

This research investigation delves into the concept of cyclic dominance and its interplay with environmental feedback within the framework of the Optional Prisoner's Dilemma (OPD) game. A quantitative model has been meticulously constructed to encompass both the OPD and Rock-Paper-Scissors (RPS) games, wherein the outcomes are contingent upon environmental feedback. The intricate relationship between player populations and their surrounding environment is elucidated through the utilization of replicator dynamics. The empirical findings of this study shed light on the existence of distinct equilibria contingent upon the prevailing environmental circumstances. Under scenarios characterized by resource depletion, an equilibrium state emerges wherein only solitary individuals (referred to as "loners") persist. Conversely, under conditions of resource replenishment, an equilibrium is established where various strategic approaches coexist in equal proportions, akin to the dynamics of the Rock-Paper-Scissors game. Notably, cyclic dominance manifests itself during periods characterized by intermediate environmental conditions. Several pivotal factors have been identified as determinants of these dynamic phenomena, including the rate of environmental replenishment, the strength of the dilemma presented, the payoff for loners, and the reward for a draw. It has been empirically demonstrated that the presence of extreme values in the replenishment rate, irrespective of whether exceptionally high or exceptionally low, serves as an impediment to the emergence of cyclic dominance. This model underscores the substantial influence of environmental feedback on social dilemmas, particularly in the context of the OPD game, ultimately leading to the emergence of competitive cyclic dominance games reminiscent of the Rock-Paper-Scissors game. In doing so, it elucidates the intricate interplay between evolutionary game incentives and the dynamics of the surrounding environment.