Social experiments in the mesoscale: humans playing a spatial prisoner's dilemma

Background

The evolutionary origin of cooperation among unrelated individuals remains a key unsolved issue across several disciplines. Prominent among the several mechanisms proposed to explain how cooperation can emerge is the existence of a population structure that determines the interactions among individuals. Many models have explored analytically and by simulation the effects of such a structure, particularly in the framework of the Prisoner''s Dilemma, but the results of these models largely depend on details such as the type of spatial structure or the evolutionary dynamics. Therefore, experimental work suitably designed to address this question is needed to probe these issues.

Methods and Findings

We have designed an experiment to test the emergence of cooperation when humans play Prisoner''s Dilemma on a network whose size is comparable to that of simulations. We find that the cooperation level declines to an asymptotic state with low but nonzero cooperation. Regarding players'' behavior, we observe that the population is heterogeneous, consisting of a high percentage of defectors, a smaller one of cooperators, and a large group that shares features of the conditional cooperators of public goods games. We propose an agent-based model based on the coexistence of these different strategies that is in good agreement with all the experimental observations.

Conclusions

In our large experimental setup, cooperation was not promoted by the existence of a lattice beyond a residual level (around 20%) typical of public goods experiments. Our findings also indicate that both heterogeneity and a “moody” conditional cooperation strategy, in which the probability of cooperating also depends on the player''s previous action, are required to understand the outcome of the experiment. These results could impact the way game theory on graphs is used to model human interactions in structured groups.     

How Feeling Betrayed Affects Cooperation

For a population of interacting self-interested agents, we study how the average cooperation level is affected by some individuals'' feelings of being betrayed and guilt. We quantify these feelings as adjusted payoffs in asymmetric games, where for different emotions, the payoff matrix takes the structure of that of either a prisoner''s dilemma or a snowdrift game. Then we analyze the evolution of cooperation in a well-mixed population of agents, each of whom is associated with such a payoff matrix. At each time-step, an agent is randomly chosen from the population to update her strategy based on the myopic best-response update rule. According to the simulations, decreasing the feeling of being betrayed in a portion of agents does not necessarily increase the level of cooperation in the population. However, this resistance of the population against low-betrayal-level agents is effective only up to some extend that is explicitly determined by the payoff matrices and the number of agents associated with these matrices. Two other models are also considered where the betrayal factor of an agent fluctuates as a function of the number of cooperators and defectors that she encounters. Unstable behaviors are observed for the level of cooperation in these cases; however, we show that one can tune the parameters in the function to make the whole population become cooperative or defective.     

Natural selection of cooperation and degree hierarchy in heterogeneous populations

One of the current theoretical challenges to the explanatory powers of Evolutionary Theory is the understanding of the observed evolutionary survival of cooperative behavior when selfish actions provide higher fitness (reproductive success). In unstructured populations natural selection drives cooperation to extinction. However, when individuals are allowed to interact only with their neighbors, specified by a graph of social contacts, cooperation-promoting mechanisms (known as lattice reciprocity) offer to cooperation the opportunity of evolutionary survival. Recent numerical works on the evolution of Prisoner's Dilemma in complex network settings have revealed that graph heterogeneity dramatically enhances the lattice reciprocity. Here we show that in highly heterogeneous populations, under the graph analog of replicator dynamics, the fixation of a strategy in the whole population is in general an impossible event, for there is an asymptotic partition of the population in three subsets, two in which fixation of cooperation or defection has been reached and a third one which experiences cycles of invasion by the competing strategies. We show how the dynamical partition correlates with connectivity classes and characterize the temporal fluctuations of the fluctuating set, unveiling the mechanisms stabilizing cooperation in macroscopic scale-free structures.     

Moving away from nasty encounters enhances cooperation in ecological prisoner's dilemma game

We study the role of migration in the evolution of cooperation. Individuals spatially located on a square lattice play the prisoner's dilemma game. Dissatisfied players, who have been exploited by defectors, tend to terminate interaction with selfish partners by leaving the current habitats, and explore unknown physical niches available surrounding them. The time scale ratio of game interaction to natural selection governs how many game rounds occur before individuals experience strategy updating. Under local migration and strong selection, simulation results demonstrate that cooperation can be stabilized for a wide range of model parameters, and the slower the natural selection, the more favorable for the emergence of cooperation. Besides, how the selection intensity affects cooperators' evolutionary fate is also investigated. We find that increasing it weakens cooperators' viability at different speeds for different time scale ratios. However, cooperation is greatly improved provided that individuals are offered with enough chance to agglomerate, while cooperation can always establish under weak selection but vanishes under very strong selection whenever individuals have less odds to migrate. Whenever the migration range restriction is removed, the parameter area responsible for the emergence of cooperation is, albeit somewhat compressed, still remarkable, validating the effectiveness of collectively migrating in promoting cooperation.     

Gender Differences in Cooperation: Experimental Evidence on High School Students

The emergence of cooperation among unrelated human subjects is a long-standing conundrum that has been amply studied both theoretically and experimentally. Within the question, a less explored issue relates to the gender dependence of cooperation, which can be traced back to Darwin, who stated that "women are less selfish but men are more competitive". Indeed, gender has been shown to be relevant in several game theoretical paradigms of social cooperativeness, including prisoner''s dilemma, snowdrift and ultimatum/dictator games, but there is no consensus as to which gender is more cooperative. We here contribute to this literature by analyzing the role of gender in a repeated Prisoners'' Dilemma played by Spanish high-school students in both a square lattice and a heterogeneous network. While the experiment was conducted to shed light on the influence of networks on the emergence of cooperation, we benefit from the availability of a large dataset of more 1200 participants. We applied different standard econometric techniques to this dataset, including Ordinary Least Squares and Linear Probability models including random effects. All our analyses indicate that being male is negatively associated with the level of cooperation, this association being statistically significant at standard levels. We also obtain a gender difference in the level of cooperation when we control for the unobserved heterogeneity of individuals, which indicates that the gender gap in cooperation favoring female students is present after netting out this effect from other socio-demographics factors not controlled for in the experiment, and from gender differences in risk, social and competitive preferences.     

An institutional mechanism for assortment in an ecology of games

Recent research has revived Long's "ecology of games" model to analyze how social actors cooperate in the context of multiple political and social games. However, there is still a paucity of theoretical work that considers the mechanisms by which large-scale cooperation can be promoted in a dynamic institutional landscape, in which actors can join new games and leave old ones. This paper develops an agent-based model of an ecology of games where agents participate in multiple public goods games. In addition to contribution decisions, the agents can leave and join different games, and these processes are de-coupled. We show that the payoff for cooperation is greater than for defection when limits to the number of actors per game ("capacity constraints") structure the population in ways that allow cooperators to cluster, independent of any complex individual-level mechanisms such as reputation or punishment. Our model suggests that capacity constraints are one effective mechanism for producing positive assortment and increasing cooperation in an ecology of games. The results suggest an important trade-off between the inclusiveness of policy processes and cooperation: Fully inclusive policy processes reduce the chances of cooperation.     

Heterogeneous aspirations promote cooperation in the prisoner's dilemma game

To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter that interpolates between these two extreme options. Setting equal to zero returns the random selection of the opponent, while positive favor the fitter players. In addition, we divide the population into two groups. Players from group select their opponents as dictated by the parameter , while players from group do so randomly irrespective of . We denote the fraction of players contained in groups and by and , respectively. The two parameters and allow us to analyze in detail how aspirations in the context of the prisoner''s dilemma game influence the evolution of cooperation. We find that for sufficiently positive values of there exist a robust intermediate for which cooperation thrives best. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process and for different interaction networks. We also provide complete phase diagrams depicting the dependence of the impact of and for different values of , and contrast the validity of our conclusions by means of an alternative model where individual aspiration levels are subject to evolution as well. Our study indicates that heterogeneity in aspirations may be key for the sustainability of cooperation in structured populations.     

Evolution of Cooperation in Spatial Traveler's Dilemma Game

Traveler''s dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler''s dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors'' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.     

Evolutionary dynamics of continuous strategy games on graphs and social networks under weak selection

Understanding the emergence of cooperation among selfish individuals has been a long-standing puzzle, which has been studied by a variety of game models. Most previous studies presumed that interactions between individuals are discrete, but it seems unrealistic in real systems. Recently, there are increasing interests in studying game models with a continuous strategy space. Existing research work on continuous strategy games mainly focuses on well-mixed populations. Especially, little theoretical work has been conducted on their evolutionary dynamics in a structured population. In the previous work (Zhong et al., BioSystems, 2012), we showed that under strong selection, continuous and discrete strategies have significantly different equilibrium and game dynamics in spatially structured populations. In this paper, we further study evolutionary dynamics of continuous strategy games under weak selection in structured populations. By using the fixation probability based stochastic dynamics, we derive exact conditions of natural selection favoring cooperation for the death–birth updating scheme. We also present a network gain decomposition of the game equilibrium, which might provide a new view of the network reciprocity in a quantitative way. Finally, we make a detailed comparison between games using discrete and continuous strategies. As compared to the former, we find that for the latter (i) the same selection conditions are derived for the general 2 × 2 game; especially, the rule b/c > k in a simplified Prisoner's Dilemma is valid as well; however, (ii) for a coordination game, interestingly, the risk-dominant strategy is disfavored. Numerical simulations have also been conducted to validate our results.     

Competition and evolutionary stability of plants in a spatially structured habitat

We modelled the population dynamics of two types of plants with limited dispersal living in a lattice structured habitat. Each site of the square lattice model was either occupied by an individual or vacant. Each individual reproduced to its neighbors. We derived a criterion for the invasion of a rare type into a population composed of a resident type based on a pair-approximation method, in which the dynamics of both average densities and the nearest neighbor correlations were considered. Based on this invasibility criterion, we showed that, when there is a tradeoff between birth and death rates, the evolutionarily stable type is the one that has the highest ratio of birth rate to mortality. If these types are different species, they form segregated spatial patterns in the lattice model in which intraspecific competitive interactions occur more frequently than interspecific interactions. However, stable coexistence is not possible in the lattice model contrary to results from completely mixed population models. This clearly shows that the casual conclusion, based on traditional well mixed population models, that different species can coexist if intraspecific competition is stronger than interspecific competition, does not hold for spatially structured population models.     

Cooperation and conflict: field experiments in Northern Ireland

The idea that cohesive groups, in which individuals help each other, have a competitive advantage over groups composed of selfish individuals has been widely suggested as an explanation for the evolution of cooperation in humans. Recent theoretical models propose the coevolution of parochial altruism and intergroup conflict, when in-group altruism and out-group hostility contribute to the group''s success in these conflicts. However, the few empirical attempts to test this hypothesis do not use natural groups and conflate measures of in-group and unbiased cooperative behaviour. We conducted field experiments based on naturalistic measures of cooperation (school/charity donations and lost letters'' returns) with two religious groups with an on-going history of conflict—Catholics and Protestants in Northern Ireland. Conflict was associated with reduced donations to out-group schools and the return of out-group letters, but we found no evidence that it influences in-group cooperation. Rather, socio-economic status was the major determinant of cooperative behaviour. Our study presents a challenge to dominant perspectives on the origins of human cooperation, and has implications for initiatives aiming to promote conflict resolution and social cohesion.     

Dependency Links Can Hinder the Evolution of Cooperation in the Prisoner’s Dilemma Game on Lattices and Networks

Networks with dependency links are more vulnerable when facing the attacks. Recent research also has demonstrated that the interdependent groups support the spreading of cooperation. We study the prisoner’s dilemma games on spatial networks with dependency links, in which a fraction of individual pairs is selected to depend on each other. The dependency individuals can gain an extra payoff whose value is between the payoff of mutual cooperation and the value of temptation to defect. Thus, this mechanism reflects that the dependency relation is stronger than the relation of ordinary mutual cooperation, but it is not large enough to cause the defection of the dependency pair. We show that the dependence of individuals hinders, promotes and never affects the cooperation on regular ring networks, square lattice, random and scale-free networks, respectively. The results for the square lattice and regular ring networks are demonstrated by the pair approximation.     

Multilevel selection analysis of a microbial social trait

The study of microbial communities often leads to arguments for the evolution of cooperation due to group benefits. However, multilevel selection models caution against the uncritical assumption that group benefits will lead to the evolution of cooperation. We analyze a microbial social trait to precisely define the conditions favoring cooperation. We combine the multilevel partition of the Price equation with a laboratory model system: swarming in Pseudomonas aeruginosa. We parameterize a population dynamics model using competition experiments where we manipulate expression, and therefore the cost‐to‐benefit ratio of swarming cooperation. Our analysis shows that multilevel selection can favor costly swarming cooperation because it causes population expansion. However, due to high costs and diminishing returns constitutive cooperation can only be favored by natural selection when relatedness is high. Regulated expression of cooperative genes is a more robust strategy because it provides the benefits of swarming expansion without the high cost or the diminishing returns. Our analysis supports the key prediction that strong group selection does not necessarily mean that microbial cooperation will always emerge.     

Stochastic Heterogeneous Interaction Promotes Cooperation in Spatial Prisoner's Dilemma Game

Previous studies mostly investigate player''s cooperative behavior as affected by game time-scale or individual diversity. In this paper, by involving both time-scale and diversity simultaneously, we explore the effect of stochastic heterogeneous interaction. In our model, the occurrence of game interaction between each pair of linked player obeys a random probability, which is further described by certain distributions. Simulations on a 4-neighbor square lattice show that the cooperation level is remarkably promoted when stochastic heterogeneous interaction is considered. The results are then explained by investigating the mean payoffs, the mean boundary payoffs and the transition probabilities between cooperators and defectors. We also show some typical snapshots and evolution time series of the system. Finally, the 8-neighbor square lattice and BA scale-free network results indicate that the stochastic heterogeneous interaction can be robust against different network topologies. Our work may sharpen the understanding of the joint effect of game time-scale and individual diversity on spatial games.     

Exploring the Link between Genetic Relatedness r and Social Contact Structure k in Animal Social Networks

Our understanding of how cooperation can arise in a population of selfish individuals has been greatly advanced by theory. More than one approach has been used to explore the effect of population structure. Inclusive fitness theory uses genetic relatedness r to express the role of population structure. Evolutionary graph theory models the evolution of cooperation on network structures and focuses on the number of interacting partners k as a quantity of interest. Here we use empirical data from a hierarchically structured animal contact network to examine the interplay between independent, measurable proxies for these key parameters. We find strong inverse correlations between estimates of r and k over three levels of social organization, suggesting that genetic relatedness and social contact structure capture similar structural information in a real population.     

Adaptive Dynamics of Altruistic Cooperation in a Metapopulation: Evolutionary Emergence of Cooperators and Defectors or Evolutionary Suicide?

We investigate the evolution of public goods cooperation in a metapopulation model with small local populations, where altruistic cooperation can evolve due to assortment and kin selection, and the evolutionary emergence of cooperators and defectors via evolutionary branching is possible. Although evolutionary branching of cooperation has recently been demonstrated in the continuous snowdrift game and in another model of public goods cooperation, the required conditions on the cost and benefit functions are rather restrictive, e.g., altruistic cooperation cannot evolve in a defector population. We also observe selection for too low cooperation, such that the whole metapopulation goes extinct and evolutionary suicide occurs. We observed intuitive effects of various parameters on the numerical value of the monomorphic singular strategy. Their effect on the final coexisting cooperator–defector pair is more complex: changes expected to increase cooperation decrease the strategy value of the cooperator. However, at the same time the population size of the cooperator increases enough such that the average strategy does increase. We also extend the theory of structured metapopulation models by presenting a method to calculate the fitness gradient in a general class of metapopulation models, and try to make a connection with the kin selection approach.     

Maintenance affects the stability of a two-tiered microbial 'food chain'?

Recent studies have shown that constraints on available resources may play an important role in the evolution of cooperation, especially when individuals do not posses the capacity to recognize other individuals, memory or other developed abilities, as it is the case of most unicellular organisms, algae or even plants. We analyze the evolution of cooperation in the case of a limiting resource, which is necessary for reproduction and survival. We show that, if the strategies determine a prisoner's dilemma, the outcome of the interactions may be modified by the limitation of resources allowing cooperators to invade the entire population. Analytic expressions for the region of cooperation are provided. Furthermore we derive expressions for the connection between fitness, as understood in evolutionary game theory, and resource exchanges, which may be of help to link evolutionary game theoretical results with resource based models.     

ASSORTMENT AND THE EVOLUTION OF GENERALIZED RECIPROCITY

Reciprocity is often invoked to explain cooperation. Reciprocity is cognitively demanding, and both direct and indirect reciprocity require that individuals store information about the propensity of their partners to cooperate. By contrast, generalized reciprocity, wherein individuals help on the condition that they received help previously, only relies on whether an individual received help in a previous encounter. Such anonymous information makes generalized reciprocity hard to evolve in a well‐mixed population, as the strategy will lose out to pure defectors. Here we analyze a model for the evolution of generalized reciprocity, incorporating assortment of encounters, to investigate the conditions under which it will evolve. We show that, in a well‐mixed population, generalized reciprocity cannot evolve. However, incorporating assortment of encounters can favor the evolution of generalized reciprocity in which indiscriminate cooperation and defection are both unstable. We show that generalized reciprocity can evolve under both the prisoner's dilemma and the snowdrift game.     

Emergence and resilience of cooperation in the spatial prisoner's dilemma via a reward mechanism

We study the problem of the emergence of cooperation in the spatial Prisoner's Dilemma. The pioneering work by Nowak and May [1992. Evolutionary games and spatial chaos. Nature 415, 424-426] showed that large initial populations of cooperators can survive and sustain cooperation in a square lattice with imitate-the-best evolutionary dynamics. We revisit this problem in a cost-benefit formulation suitable for a number of biological applications. We show that if a fixed-amount reward is established for cooperators to share, a single cooperator can invade a population of defectors and form structures that are resilient to re-invasion even if the reward mechanism is turned off. We discuss analytically the case of the invasion by a single cooperator and present agent-based simulations for small initial fractions of cooperators. Large cooperation levels, in the sustainability range, are found. In the conclusions we discuss possible applications of this model as well as its connections with other mechanisms proposed to promote the emergence of cooperation.     

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.