Spatial Social Dilemmas Research Articles

The impact of mobility costs on cooperation and welfare in spatial social dilemmas

From over-exploitation of resources to urban pollution, sustaining well-being requires solving social dilemmas of cooperation. Often such dilemmas are studied assuming that individuals occupy fixed positions in a network or lattice. In spatial settings, however, agents can move, and such movements involve costs. Here we investigate how mobility costs impact cooperation dynamics. To this end, we study cooperation dilemmas where individuals are located in a two-dimensional space and can be of two types: cooperators–or cleaners, who pay an individual cost to have a positive impact on their neighbours–and defectors–or polluters, free-riding on others’ effort to sustain a clean environment. Importantly, agents can pay a cost to move to a cleaner site. Both analytically and through agent-based simulations we find that, in general, introducing mobility costs increases pollution felt in the limit of fast movement (equivalently slow strategy revision). The effect on cooperation of increasing mobility costs is non-monotonic when mobility co-occurs with strategy revision. In such scenarios, low (yet non-zero) mobility costs minimise cooperation in low density environments; whereas high costs can promote cooperation even when a minority of agents initially defect. Finally, we find that heterogeneity in mobility cost affects the final distribution of strategies, leading to differences in who supports the burden of having a clean environment.

Double cyclic dominance promotes cooperation in spatial social dilemmas

The emergence of cooperation is a challenging and important problem. In the realistic world, individuals usually make their opponents cooperate by strengthening the relationship between cooperative neighbors and themselves. Inspired by this phenomenon, a coevolution mechanism is introduced where some individuals can adjust the strength of the relationship between their neighbors and themselves, while others maintain the strength of the relationship regardless of their neighbors’ strategies. Through agent-based simulations, we find that cooperative behavior is greatly promoted under the proposed mechanism, and there is an optimal ratio of co-evolutionary individuals for the evolution of cooperation. The potential reason is that under the optimal ratio, the system evolves into three classes which leads to double cycles of individuals with different strategies and weights.

Impact of resource-based conditional interaction on cooperation in spatial social dilemmas

Due to the contradiction between the finiteness of resources and people’s infinite demand for them, we cannot deny the impact of the limited resources on human behavior. To this end, we construct a novel resource-based conditional interaction model from a tiny perspective, in which not only can limited resources be redistributed among the population, but resources owned by players also affect whether they can interact with each other or not. To be specific, a player who successfully imitates his neighbor’s strategy will have to pay ɛ proportion of his resources to the opponent as the learning cost. In addition, if and only if the resource difference between the focal player x and one of his neighbors y is within an acceptable tolerance interval τ, they will indisputably interact with each other. We mainly resort to the prisoner’s dilemma game and asynchronous strategy update to verify the effectiveness of the model. By resorting to extensive Monte Carlo simulations, we find that there exists an optimal acceptable tolerance interval τ, varying with the value of cost-to-benefit ratio u, to make the promotion of cooperation the most obvious. We also confirm that players’ irrational behavior can be influenced by this resource-based partner selection. However, if we introduce one kind of minimal resource protection mechanism into our proposed model, the level of cooperation cannot be further elevated, or even be hindered when compared with the case without the minimal resource protection mechanism. In the end, we further verify the robustness and effectiveness of the proposed model through other social dilemmas, network topologies, and synchronous strategy update pattern. To a certain extent, we wish that our efforts can wipe out some barriers for researching the evolution of cooperation within the selfish population.

Spatial social dilemmas promote diversity

Cooperative investments in social dilemmas can spontaneously diversify into stably coexisting high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation, we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual-based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavorable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

Reputational preference-based payoff punishment promotes cooperation in spatial social dilemmas

Reputational preference-based payoff punishment promotes cooperation in spatial social dilemmas

Asymmetrical influence promotes cooperation in the spatial social dilemmas

Asymmetrical influence of population is ubiquitous in human and animal societies. Based on this fact, we construct a setup of asymmetric influence of structured population, which is often put with the combination mechanism of tolerance and punishment when the dispute of unfair outcome happens, to investigate the evolution of cooperation. In detail, the structured populations are classified into two types: A-type players (strong players) and B-type players (weak players). A-type players are capable to decide whether or not to lower the fitness of B-type neighbors with punitive action due to the tolerance limits. This action affects the behavior of B-type neighbors in the evolutionary game. Based on our simulations, it is noted that the mechanism of punishment combining with tolerance in an asymmetrical environment can promote the evolution of cooperation. Furthermore, the lower threshold of tolerance with the looser punishment, the higher the level of cooperation; and the higher degree of tolerance with the stricter punishment will maintain the survival of cooperators. It is also worth mentioning that if A-type population is the key minority impacting the majority of B-type, they can favor affecting and further promoting the evolution of cooperation in the evolutionary game.

Heterogeneity of reputation increment driven by individual influence promotes cooperation in spatial social dilemma

Cooperation is ubiquitous, but our innate selfishness greatly challenge our motivation to cooperate since natural selection favors the fittest individuals in all the ecosystems. In addition, cooperation is costly, implementing it weighs down the individual wealth and the prosperity of human society. Therefore, how to deal with social dilemma has attracted numerous scholars’ attentions. Among previous researches, indirect reciprocity acts a crucial role in the promotion of cooperation. However, scholars focus more on the consistency and constancy of all the players’ reputation fluctuation, and ignore its potential features, the heterogeneity and dynamic of the reputation increment. In real world, such a scenario can reflect it, for example, a famous person and a notorious person have a totally distinct reputation variation even if they have coincident actions. Inspired by aforesaid particularity of reputation variation, a new mechanism, heterogeneity of reputation increment driven by individual influence is introduced, in which players who have more payoffs than the average of their neighbors’ would be more influential, and their actions would lead to a bigger scale of reputation fluctuation due to more attentions from others. Simulation results show that cooperation is facilitated effectively by our new mechanism, compared with traditional model with constant reputation variation.

Emergence of cooperation in spatial social dilemmas with expulsion

In social dilemmas cooperators pay costs for other individuals to receive benefits, whereas defectors are the benefactors but do not provide anything in return. What is more, except in the snowdrift game, defective strategies are evolutionarily stable against invasion by cooperative strategies under strong selection. It is thus of interest to determine under which conditions the spontaneous emergence of cooperation is possible in a population consisting entirely of defectors. To that effect, we here want to concentrate on exploring how expulsion influences the emergent dynamics of cooperation in the context of the spatial social dilemmas. Interestingly, expulsion can indeed support the emergence of cooperators for all classes of social dilemma games. By analyzing the spatial expansion and invasion processes of cooperative individuals, we further reveal that merely a 1×2 (or 2×1) cluster of expulsive cooperators, introduced for example by means of mutations, is required for invading and completely dominating a spatial population of expulsive defectors provided that the cost of cooperation is no larger than a constant value.

Reputational preference and other-regarding preference based rewarding mechanism promotes cooperation in spatial social dilemmas* *Project supported by the National Natural Science Foundation of China (Grant No. 62062049), the Social Science Project of the Ministry of Education of China (Grant No. 20YJCZH212), and the Natural Science Foundation of Gansu Province, China (Grant No. 20JR5RA390).

To study the incentive mechanisms of cooperation, we propose a preference rewarding mechanism in the spatial prisoner’s dilemma game, which simultaneously considers reputational preference, other-regarding preference and the dynamic adjustment of vertex weight. The vertex weight of a player is adaptively adjusted according to the comparison result of his own reputation and the average reputation value of his immediate neighbors. Players are inclined to pay a personal cost to reward the cooperative neighbor with the greatest vertex weight. The vertex weight of a player is proportional to the preference rewards he can obtain from direct neighbors. We find that the preference rewarding mechanism significantly facilitates the evolution of cooperation, and the dynamic adjustment of vertex weight has powerful effect on the emergence of cooperative behavior. To validate multiple effects, strategy distribution and the average payoff and fitness of players are discussed in a microcosmic view.

Robust cooperation against mutations via costly expulsion

Research has shown that spatial selection can lead to the evolution of cooperation in social dilemmas. However, the effectiveness of spatial reciprocity among cooperative players is often diminished by mutations. Here, we report the evolution of robust cooperation against mutations, which is brought about by costly expulsion in the spatial public goods game. We find that costly expulsion is able to enhance the robustness of positive assortment among cooperators against mutations. Moreover, we also show that moderate mutation rates are beneficial for the expansion of spatial clusters among cooperative individuals once the spontaneous formation of segregation patterns is completed by costly expulsion. Our results thus communicate the counterintuitive conclusions that mutations are not always detrimental for the evolution of cooperation in spatial social dilemma games, and what is more, can even be beneficial for faster spreading of cooperator clusters.

Behavioral observability and reputational-preference–based rewarding mechanism promotes cooperation in spatial social dilemmas

To explore the incentive mechanisms of cooperation, inspired by preference for reputation in indirect reciprocity and the influence of behavioral observability on fitness, we present a new rewarding mechanism by incorporating these two impact factors into the evaluation of fitness in the spatial prisoner's dilemma game (PDG), under which a dynamically changing reward is established for cooperative neighbors whose reputation is higher than the average score of all neighbors. Simulation results reveal that the proposed rewarding mechanism favors the evolution of cooperation, under the joint effects of behavioral observability and reputational preference, cooperators can gradually agglomerate and form close clusters to defend the invasion of defectors. Moreover, we have investigated the characteristic snapshots and strategy transitions during the evolutionary process, which further validate the above outcome.

Reputation-based strategy persistence promotes cooperation in spatial social dilemma

Understanding the emergence and maintenance of cooperation in social dilemma has received a lot of attention. In previous research, many scholars found that the reputation mechanism can promote cooperation, and the variation of reputation is consistent. However, in reality, according to both one's current action and past experiences, every individual's impression from others is modified to varying extent everyday. In other words, the length of duration of the same performance influences the diverse scale of their own reputation fluctuation. Therefore, a reputation-based strategy persistence mechanism, in which the increment of current reputation is determined by the persistence of last strategy, is proposed. Moreover, we introduce a parameter α to illustrate the impact of strategy persistence on reputation variation. The results of simulation show that the new mechanism paves the way for cooperation in evolutionary game, and the smaller α is, the better the mechanism performs.

Effects of Reciprocal Rewarding on the Evolution of Cooperation in Voluntary Social Dilemmas

Voluntary participation, as an effective mechanism to promote cooperation in game theory, has been widely concerned. In the meanwhile, reciprocal rewarding plays an important role in motivating individual initiative. Inspired by this phenomenon, we investigate the effect of reciprocal rewarding on the evolutionary cooperation in spatial social dilemmas, including prisoner's dilemma game and the snowdrift game with voluntary participation. In our model, a cooperative individual fitness will be redefined if one could obtain additional incentive bonus which is proportional to the number of cooperative neighbors. Moreover, each individual is a pure strategist in the spatial structured population and could only choose one of three strategies - cooperation, defection and being a loner. Through numerical simulations, we have confirmed that, compared with the traditional situation, reciprocal rewarding and the payoff of loner can significantly promote the cooperative behavior among the population, and the greater the contribution of reciprocal rewarding/payoff of loner, the more obvious the promoting effect on cooperation. In addition, we also find that there is a condition for loner to make the system fall into the three-strategy cyclic dominance, that is, the payoff of loner can not be too small or too large, which will destroy the situation of cyclic dominance. With regard to these results, it is strongly unveiled that reciprocal rewarding has a positive role to resolve the social dilemmas in the evolution of cooperation.

Reputation-based adaptive adjustment of link weight among individuals promotes the cooperation in spatial social dilemmas

The relationships among individuals on the network are often dynamic and adaptive adjustment with the interactions, which have a strong effect on the cooperation behavior. In this paper, we propose a new spatial evolutionary game model to further investigate the emergence and persistence of cooperation, in which the link weights among individuals are dynamically and adaptively adjusted by comparing between their own reputation and the average one of their nearest neighbors. If the reputation of the focal player is greater than the average one of his/her nearest neighbors, the nearest neighbors around the focal player will follow him/her, thus the link weight between the nearest neighbors and the focal player will increase by one unit as a reward. In contrast, the corresponding link weight will be decreased by one unit as a punishment. In addition, we use another parameter ε to control the possible range of link weight. By conducting extensive Monte Carlo numerical simulations in the prisoner’s dilemma and the snowdrift games, it is clearly verified that the adaptive adjustment of link weight can dramatically promote the evolution of cooperation, and the cooperation can be further elevated into a higher level when compared with the standard spatial dilemma game. The current results are helpful for us to better understand the influence of dynamic link weight on the collective cooperation behavior.

Imitate or innovate: Competition of strategy updating attitudes in spatial social dilemma games

Evolution is based on the assumption that competing players update their strategies to increase their individual payoffs. However, while the applied updating method can be different, most of previous works proposed uniform models where players use identical way to revise their strategies. In this work we explore how imitation-based or learning attitude and innovation-based or myopic best-response attitude compete for space in a complex model where both attitudes are available. In the absence of additional cost the best response trait practically dominates the whole snow-drift game parameter space which is in agreement with the average payoff difference of basic models. When additional cost is involved then the imitation attitude can gradually invade the whole parameter space but this transition happens in a highly nontrivial way. However, the role of competing attitudes is reversed in the stag-hunt parameter space where imitation is more successful in general. Interestingly, a four-state solution can be observed for the latter game which is a consequence of an emerging cyclic dominance between possible states. These phenomena can be understood by analyzing the microscopic invasion processes, which reveals the unequal propagation velocities of strategies and attitudes.

Environmental feedback drives cooperation in spatial social dilemmas

Exploiting others is beneficial individually but it could also be detrimental globally. The reverse is also true: a higher cooperation level may change the environment in a way that is beneficial for all competitors. To explore the possible consequence of this feedback we consider a coevolutionary model where the local cooperation level determines the payoff values of the applied prisoner's dilemma game. We observe that the coevolutionary rule provides a significantly higher cooperation level comparing to the traditional setup independently of the topology of the applied interaction graph. Interestingly, this cooperation supporting mechanism offers lonely defectors a high surviving chance for a long period hence the relaxation to the final cooperating state happens logarithmically slow. As a consequence, the extension of the traditional evolutionary game by considering interactions with the environment provides a good opportunity for cooperators, but their reward may arrive with some delay.

Effects of compassion on the evolution of cooperation in spatial social dilemmas

Cooperation plays an essential role in the evolution of social species, chief among all in humans. In this paper, we study the effects of compassion on the evolution of cooperation in spatial social dilemmas by introducing a payoff redistribution mechanism. In particular, a player whose payoff is larger than the average in its neighborhood will share some it with its comparatively poor neighbors. We find that such a simple redistribution mechanism, which we interpret as a form of compassion, significantly promotes the evolution of cooperation. While traditional network reciprocity already supports the formation of compact cooperative clusters, an in-depth analysis of payoff transfer events between players reveals an enhanced form of this phenomenon through the reinforcement of payoffs of cooperators that reside along the borders of such clusters. This significantly enhances the resilience of cooperative clusters, who are in turn able to survive even at adverse conditions where traditional network reciprocity alone would long fail. We show that the observed positive effects of compassion on the evolution of cooperation are robust to changes of the interaction network and to changes in the type of the governing social dilemma.

Incorporating the information from direct and indirect neighbors into fitness evaluation enhances the cooperation in the social dilemmas

We propose an improved fitness evaluation method to investigate the evolution of cooperation in the spatial social dilemmas. In our model, a focal player’s fitness is calculated as the linear combination of his own payoff, the average payoffs of direct and indirect neighbors in which two independent selection parameters (α and β) are used to control the proportion of various payoff contribution to the current fitness. Then, the fitness-based strategy update rule is still Fermi-like, and asynchronous update is adopted here. A large plethora of numerical simulations are performed to validate the behaviors of the current model, and the results unambiguously demonstrate that the cooperation level is greatly enhanced by introducing the payoffs from the surrounding players. In particular, the influence of direct neighbors become more evident when compared with indirect neighbors since the correlation between focal players and their direct neighbors is much closer. Current outcomes are significant for us to further illustrate the origin and emergence of cooperation within a wide variety of natural and man-made systems.

Emotional Multiagent Reinforcement Learning in Spatial Social Dilemmas.

Social dilemmas have attracted extensive interest in the research of multiagent systems in order to study the emergence of cooperative behaviors among selfish agents. Understanding how agents can achieve cooperation in social dilemmas through learning from local experience is a critical problem that has motivated researchers for decades. This paper investigates the possibility of exploiting emotions in agent learning in order to facilitate the emergence of cooperation in social dilemmas. In particular, the spatial version of social dilemmas is considered to study the impact of local interactions on the emergence of cooperation in the whole system. A double-layered emotional multiagent reinforcement learning framework is proposed to endow agents with internal cognitive and emotional capabilities that can drive these agents to learn cooperative behaviors. Experimental results reveal that various network topologies and agent heterogeneities have significant impacts on agent learning behaviors in the proposed framework, and under certain circumstances, high levels of cooperation can be achieved among the agents.

Spatial social dilemmas: Dilution, mobility and grouping effects with imitation dynamics

We present an extensive, systematic study of the Prisoner’s Dilemma and Snowdrift games on a square lattice under a synchronous, noiseless imitation dynamics. We show that for both the occupancy of the network and the (random) mobility of the agents there are intermediate values that may increase the amount of cooperators in the system and new phases appear. We analytically determine the transition lines between these phases and compare with the mean field prediction and the observed behavior on a square lattice. We point out which are the more relevant microscopic processes that entitle cooperators to invade a population of defectors in the presence of mobility and discuss the universality of these results.