Prisoner's Dilemma Game Research Articles

The effect of historical payoffs on individual strategy updates in the Prisoner's Dilemma

Understanding the evolution of cooperation among selfish individuals remains a big challenge. In this paper, we introduce a new update rule in which individuals make decision according to their historical performance in the Prisoner's Dilemma game on a square lattice. Comparing our results with the Prisoner's Dilemma experiment, such an update rule can explain the experiment well, suggesting that this update rule may be useful for understanding the behavior of individuals in the experiment. Importantly, we can use this rule to explain the moody conditional cooperation.

Super-rational aspiration promotes cooperation in the asymmetric game with peer exit punishment and reward

Super-rational aspiration induced strategy updating with exit rights has been considered in some previous studies, in which the players adjust strategies in line with their payoffs and aspirations, and they have access to exit the game. However, exit payoffs for exiting players are automatically allocated, which is clearly contrary to reality. In this study, evolutionary cooperation dynamics with super-rational aspiration and asymmetry in the Prisoner's Dilemma game is investigated, where exit payoffs are implemented by local peers. The results show that for different population structures, the asymmetry of the system is always contributive to the participation of the players. Furthermore, we show that under different exit payoffs, super-rationality and asymmetry are conductive to the evolution of cooperation.

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.

Investigating the effectiveness of interaction-efficiency-driven strategy updating under progressive-interaction for the evolution of the prisoner's dilemma game

In reality, interactive friend is added may one after another, instead of interacting with all at first. This work has proposed a progressive-interaction behavior driven by consistent aspiration, in which individuals interact with replacement neighbor one after another. Furthermore, payoff and interaction-efficiency are two factors considered in a modified Fermi rule, which particularly emphasizes that individuals achieving identical aspirations with less interaction frequency per unit of time will have higher interaction-efficiency and are more likely to be imitated. Thus, this work introduces the interaction-efficiency-driven strategy updating rules under progressive-interaction behavior in a spatial prisoner's dilemma game. As shown by numerical simulations, the progressive-interaction significantly enhances cooperation more than traditional versions, which may be caused by a heterogeneous number of interacting neighbors. When the interaction-efficiency-driven strategy update is taken into account, the cooperation is elevated to a higher level. Additionally, a specific point is employed to simulate the relationship between the aspiration and the b. As indicated by results, different b values have appropriate aspiration levels to promote the system to achieve the optimal cooperative level. More specifically, an obvious phase transition phenomenon can be found for the promotion results of individual aspiration level on cooperation. Besides, the phase transition phenomenon is existent when we simulate the relationship between the aspiration and the parameter delta for a given b value. In short, this work puts forward a new perspective on understanding cooperative evolution.

Data-driven evolutionary game models for the spread of fairness and cooperation in heterogeneous networks.

Unique large-scale cooperation and fairness norms are essential to human society, but the emergence of prosocial behaviors is elusive. The fact that heterogeneous social networks prevail raised a hypothesis that heterogeneous networks facilitate fairness and cooperation. However, the hypothesis has not been validated experimentally, and little is known about the evolutionary psychological basis of cooperation and fairness in human networks. Fortunately, research about oxytocin, a neuropeptide, may provide novel ideas for confirming the hypothesis. Recent oxytocin-modulated network game experiments observed that intranasal administration of oxytocin to a few central individuals significantly increases global fairness and cooperation. Here, based on the experimental phenomena and data, we show a joint effect of social preference and network heterogeneity on promoting prosocial behaviors by building evolutionary game models. In the network ultimatum game and the prisoner's dilemma game with punishment, inequality aversion can lead to the spread of costly punishment for selfish and unfair behaviors. This effect is initiated by oxytocin, then amplified via influential nodes, and finally promotes global cooperation and fairness. In contrast, in the network trust game, oxytocin increases trust and altruism, but these effects are confined locally. These results uncover general oxytocin-initiated mechanisms underpinning fairness and cooperation in human networks.

Unequal but not separate: Emergence of rich–poor cooperation in resource exchange

AbstractAddressing inequality is a critical challenge for society as a whole and requires sustained efforts to promote fairness and opportunity for all. Filling in the resource gap across social classes is essential in reducing inequality. Previous studies have revealed that ingroup favouritism hinders the rich from cooperating with the poor and expands the wealth gap. Nevertheless, how to find ways of narrowing the gap between the rich and the poor remains an open question. Inducing rich–poor cooperation could be effective in eliciting resource flow across classes. In this study, two experiments using a modified Prisoner's Dilemma game were conducted in Japan (N = 213) and China (N = 150) to examine whether highlighting partners' cooperativeness under permeable group boundaries induces rich–poor cooperation. All participants were allocated to the rich group and participated in the game with programmed rich‐ or poor‐group partner bots, whose cooperation rates were manipulated. Despite the initial resource disparities between participants and the partner bots, participants were more likely to select and cooperate with cooperative poor‐group partners than with non‐cooperative rich‐group partners. Generalized trust played an important role in the enhancement of rich–poor cooperation. The findings shed light on the possibility of reducing inequality and promoting social mobility in society.

Complex evolutionary interactions in multiple populations.

In competitive settings that entail several populations, individuals often engage in intra- and interpopulation interactions that determine their fitness and evolutionary success. With this simple motivation, we here study a multipopulation model where individuals engage in group interactions within their own population and in pairwise interactions with individuals from different populations. We use the evolutionary public goods game and the prisoner's dilemma game to describe these group and pairwise interactions, respectively. We also take into account asymmetry in the extent to which group and pairwise interactions determine the fitness of individuals. We find that interactions across multiple populations reveal new mechanisms through which the evolution of cooperation can be promoted, but this depends on the level of interaction asymmetry. If inter- and intrapopulation interactions are symmetric, the sole presence of multiple populations promotes the evolution of cooperation. Asymmetry in the interactions can further promote cooperation at the expense of the coexistence of the competing strategies. An in-depth analysis of the spatiotemporal dynamics reveals loop-dominated structures and pattern formation that can explain the various evolutionary outcomes. Thus, complex evolutionary interactions in multiple populations reveal an intricate interplay between cooperation and coexistence, and they also open up the path toward further explorations of multipopulation games and biodiversity.

Neural basis of cooperation in the Prisoner's Dilemma in a loss context

Using functional magnetic resonance imaging (fMRI) technology, we conducted an experiment to explore the effects of gain and loss contexts on cooperation and its brain activation patterns in the Prisoner's Dilemma game. The results showed that even in a loss context, participants still chose cooperation in nearly half of the experiment trials, but the cooperation rate was higher in a gain context. The results of fMRI showed that the right ventromedial prefrontal cortex, right dorsomedial prefrontal cortex, right temporoparietal junction, and right precuneus were activated more in the decision process of making a cooperative rather than a defection choice, and also that the activation of the bilateral precuneus and bilateral inferior parietal lobule induced by a cooperative choice was greater in a loss context than in a gain context. Thus, compared with when they were making a defection decision, in the process of making a cooperative decision, participants might reason and process more on the psychological activities of others, especially in a loss context.

Promoting effect of adaptive interaction based on random neighbors to cooperation in the spatial prisoner's dilemma game

Interaction is an important step in cooperation evolution, and non-deterministic interactions are common in complex and changeable environments. Therefore, the number of interactive neighbors reflects the individual's response to the environment. Inspired by this, this study introduces an adaptively adjusted number of interactions based on random neighbors into a spatial prisoner's dilemma game, in which individuals adaptively adjust the number of interactive neighbors based on the satisfaction of returns. It is noteworthy that the interactive neighbors are randomly selected corresponding to the practical phenomenon that our friends will change over time, thereby resulting in un-fixed friends. As shown by numerous simulations, adaptively adjusting the number of interactive neighbors based on random neighbors can significantly enhance cooperation, and it performs better in strengthening cooperation than traditional deterministic interactions. Apart from that, cooperation can be promoted by an appropriate interactive neighborhood. Overall, this study is expected to help inspire more research on addressing social dilemmas.

Reproductive variance can drive behavioral dynamics

The concept of fitness is central to evolution, but it quantifies only the expected number of offspring an individual will produce. The actual number of offspring is also subject to demographic stochasticity-that is, randomness associated with birth and death processes. In nature, individuals who are more fecund tend to have greater variance in their offspring number. Here, we develop a model for the evolution of two types competing in a population of nonconstant size. The fitness of each type is determined by pairwise interactions in a prisoner's dilemma game, and the variance in offspring number depends upon its mean. Although defectors are preferred by natural selection in classical population models, since they always have greater fitness than cooperators, we show that sufficiently large offspring variance can reverse the direction of evolution and favor cooperation. Large offspring variance produces qualitatively new dynamics for other types of social interactions, as well, which cannot arise in populations with a fixed size or with a Poisson offspring distribution.

Which is better for competing firms with quality increasing: behavior-based price discrimination or uniform pricing?

With the prevalence of subscription service model (i.e. Software-as-a-service, SaaS), firms are able to continuously improve their service quality online. Besides, subscribers need to renew their services after each subscription period, which provides more convenience for firms to implement behavior-based price discrimination (BBPD), namely, charging different prices for the repeat customers and for the rival's customers. Firms can respond to the rival's BBPD by adopting BBPD or by adopting uniform pricing (UP) and with a higher service quality. In this paper, we study the choices of competing firms’ pricing strategies in a two-period duopoly market when both firms can improve their quality in the second period. We consider firms’ asymmetric capability to improve their service quality and find that this asymmetry plays a critical role in determining firms’ pricing strategies. With exogenous quality improvement levels, we find that when two firms’ quality improvement difference is small, multiple equilibria prevail, namely, both adopt BBPD or both adopt UP in equilibrium. When the quality improvement difference is moderate, the asymmetric case where the disadvantaged firm adopts BBPD while the advantaged one adopts UP prevails in equilibrium, which is a unique finding absent in the previous literature. When the quality improvement difference is large, both firms will adopt UP in equilibrium, but both firms can be better off by adopting BBPD when the quality improvement difference exceeds a threshold, which is a win-win situation but a prisoner's dilemma game different from the previous studies. By examining industry profit, consumer surplus and social welfare, we find that firms’ interests are usually not aligned with the social planner's interests, and the social planner's interests are not always aligned with consumers’ interests. With endogenous quality improvement choices, the firm with R&D cost advantage tends to make a greater quality improvement, and the asymmetric case where firms adopt different pricing strategies disappears due to the impact of endogeneity of quality improvement. More specifically, when the R&D efficiency difference is small or moderate, both firms will adopt either BBPD or UP in equilibrium; but when two firms’ R&D efficiency difference is large, both firms will tend to adopt only UP in equilibrium. Comparing with the case under exogenous quality improvement levels, the social planner's interests are better aligned with the overall industry's interests but are worse aligned with consumers’ interests.

Schizophrenia and Increased Distrust-Based Competitiveness in Interpersonal Interactions: A Serial Process Model.

Game theory paradigms, such as the Prisoner's Dilemma Game (PDG), have been used to study nonclinical paranoia, though research using clinical populations has been scarce. We test our novel theoretical model that schizophrenia leads to competitiveness in interpersonal interactions, and that this link is serially mediated by trait paranoia, state paranoia, and distrust. In this quasi-experimental study, individuals with schizophrenia spectrum diagnoses with current persecutory delusions (n = 46) and a nonclinical control group (n = 43) played the PDG, and completed measures of trait paranoia, state paranoia, and distrust. Individuals with schizophrenia competed more in the PDG than the control group. Supporting our theoretical model, all direct effects were significant: schizophrenia was associated with higher trait paranoia (H1); trait paranoia predicted state paranoia in the PDG (H2); state paranoia in the PDG predicted distrust of the opponent in the PDG (H3); and distrust predicted competition in the PDG (H4). The hypothesized indirect effect of schizophrenia on competition in the PDG via trait paranoia, state paranoia, and distrust was supported in a serial mediation model (H5). The findings make clear theoretical and methodological contributions. We provide the first evidence for a theoretical process model by which schizophrenia leads to competitiveness in interpersonal interactions via trait paranoia, state paranoia, and distrust. Game theory paradigms, and the PDG in particular, are important for advancing theory and research on paranoia as it occurs in both clinical and nonclinical populations.

Moral reasoning and moral competence as predictors of cooperative behavior in a social dilemma

The level of moral development may be crucial to understand behavior when people have to choose between prioritizing individual gains or pursuing general social benefits. This study evaluated whether two different psychological constructs, moral reasoning and moral competence, are associated with cooperative behavior in the context of the prisoner's dilemma game, a two-person social dilemma where individuals choose between cooperation or defection. One hundred and eighty-nine Mexican university students completed the Defining Issues Test (DIT-2; measuring moral reasoning) and the Moral Competence Test (MCT) and played an online version of the prisoner’s dilemma game, once against each participant in a group of 6–10 players. Our results indicate that cooperative behavior is strongly affected by the outcomes in previous rounds: Except when both participants cooperated, the probability of cooperation with other participants in subsequent rounds decreased. Both the DIT-2 and MCT independently moderated this effect of previous experiences, particularly in the case of sucker-outcomes. Individuals with high scores on both tests were not affected when in previous rounds the other player defected while they cooperated. Our findings suggest that more sophisticated moral reasoning and moral competence promote the maintenance of cooperative behaviors despite facing adverse situations.

Intuitionistic Fuzzy Approach to Three Player Prisoner's Dilemma Game

Intuitionistic Fuzzy Approach to Three Player Prisoner's Dilemma Game

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

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

The influence of own historical information and environmental historical information on the evolution of cooperation

This article has studied the influence of own historical information and environmental historical information on the evolution of cooperation in the three common social dilemma games. The results show that only the past two-step information can effectively promote cooperation. In the prisoner's dilemma game, cooperation can be improved by using only the individual own information. In the stag hunt game, using only environmental information can promote cooperation. In the snowdrift game, when the parameter values are different, the results are different. The cooperation ratio can be maximized by using either its own information or environmental information only. This article provides a new way to promote and understand cooperation.

Optimization of institutional incentives for cooperation in structured populations.

The application of incentives, such as reward and punishment, is a frequently applied way for promoting cooperation among interacting individuals in structured populations. However, how to properly use the incentives is still a challenging problem for incentive-providing institutions. In particular, since the implementation of incentive is costly, to explore the optimal incentive protocol, which ensures the desired collective goal at a minimal cost, is worthy of study. In this work, we consider the positive and negative incentives for a structured population of individuals whose conflicting interactions are characterized by a Prisoner's Dilemma game. We establish an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks. We find that both types of optimal incentive protocols are identical and time-invariant. Moreover, we compare the optimal rewarding and punishing schemes concerning implementation cost and provide a rigorous basis for the usage of incentives in the game-theoretical framework. We further perform computer simulations to support our theoretical results and explore their robustness for different types of population structures, including regular, random, small-world and scale-free networks.

Adaptive exit facilitates the evolution of cooperation in the spatial prisoner's dilemma game with punishment

The enforcement of costly punishment can facilitate cooperation. However, it is unclear how punishment would influence cooperation if individuals are free to exit an interaction. In this work, individuals are allowed to exit with a certain probability, which is adaptively adjusted based on the difference between expectations and payoffs. According to simulation results, adaptive exit can significantly improve cooperation even when punishment is weak; moreover, there is an optimal level of aspiration that causes adaptive exit to best promote the evolution of cooperation. Because of reciprocity, cooperative individuals maintain the lowest level of exit probability, allowing the relationship between them to be maintained, and the interaction between cooperative individuals and defectors is affected by adaptive exit processing, which is related to aspiration levels. Results show that cooperation prevails under moderate aspiration levels, because the exit mechanism protects cooperative individuals from exploitation by defectors; however, the exit mechanism does not prevent the invasion of cooperators by defectors when the aspiration level is low. The negative impact of sanctions causes defectors to increase their exit probability, and the exit of the defectors prevents nearby punishers from exploitation as well as reduces the implementation of punishment, thereby easing the punishment burden on the punisher. Therefore, the adaptive exit mechanism helps to eliminate second-order free-riding at a lower aspiration level. The study identifies the significant effects of exit on punishment, and highlights the important role of their association in the evolution of cooperation.

Managing strategic inventory under retailer competition

Besides carrying the operational inventory, retailers may invest in the strategic inventory for obtaining the discount of future wholesale price, although this behavior can induce the holding cost. In this paper, we adopt a game theoretical method to check the role of strategic inventory under the retailer-to-retailer competition. The impact of strategic inventory on a two-tier supply chain model is investigated under the retailer competition. A supplier sells products to customers through two competing retailers who may choose to carry the strategic inventory in a two-period setting. The results indicate that the retailer-to-retailer competition may affect the decisions of supply chain parties about the strategic inventory and pricing as follows. First, the competition weakens the incentives of retailers in holding the strategic inventory. Second, the lower inventory holding expense would still stimulate retailers to hold the strategic inventory for more profits. Third, facing two competing and asymmetric retailers, the prisoner's dilemma and chicken game equilibrium may occur between the two retailers when the inventory holding cost is intermediate under the uniform wholesale price strategy. However, under the differentiated wholesale price strategy, the prisoner's dilemma and chicken game equilibrium can be eliminated. For similar operational styles including store styles and locations, retailers can break through the dilemma by simultaneously holding the strategic inventory, and can obtain more profits from carrying strategic inventory under differentiated wholesale price strategy. Therefore, the differentiated wholesale price strategy can promote the two retailers to cooperate with each other and can benefit all parties.

Nash equilibrium realization of population games based on social learning processes.

In the two-population game model, we assume the players have certain imitative learning abilities. To simulate the learning process of the game players, we propose a new swarm intelligence algorithm by combining the particle swarm optimization algorithm, where each player can be considered a particle. We conduct simulations for three typical games: the prisoner's dilemma game (with only one pure-strategy Nash equilibrium), the coin-flip game (with only one fully-mixed Nash equilibrium), and the coordination game (with two pure-strategy Nash equilibria and one fully-mixed Nash equilibrium). The results show that when the game has a pure strategy Nash equilibrium, the algorithm converges to that equilibrium. However, if the game does not have a pure strategy Nash equilibrium, it exhibits periodic convergence to the only mixed-strategy Nash equilibrium. Furthermore, the magnitude of the periodical convergence is inversely proportional to the introspection rate. After conducting experiments, our algorithm outperforms the Meta Equilibrium Q-learning algorithm in realizing mixed-strategy Nash equilibrium.