Cooperation In Spatial Games Research Articles

Effect of mixed random and directional migration on cooperation in the spatial prisoner’s dilemma

The migration behavior of the population plays a significant role in the evolution of cooperation in spatial games. During the migration process, individuals may be affected by uncertain and irrational factors, and these will affect their direction of migration. In previous studies, the migration direction has been determined either entirely randomly or using entirely deterministic factors, referred to as random and directional migration, respectively. However, in real-world situations, a mixture of both random and deterministic factors will be included in individuals’ migration decisions. In this work, we examined the effects of a mixture of directional and random migration on cooperation. In our model, for the directional migration, the average migration direction of an individual’s neighbors is used. We then introduce the parameter α to adjust the proportion of directional and random migration. The results show that a small value of α and a lower speed are beneficial to cooperation. Furthermore, we find that there is always an optimal range of α that will best promote the evolution of cooperation. We explain these phenomena based on time series of the cooperation level and the order parameter, and spatial population snapshots. We also investigate the effects of the interaction radius between individuals and population density on the evolution of cooperation. The robustness of this migration mode is further verified with respect to another game model, and we find that the conclusions remain the same.

How Lévy Flights Triggered by the Presence of Defectors Affect Evolution of Cooperation in Spatial Games.

Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini, who showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighboring defectors. However, the optimal condition for neighboring defectors and how the condition changes with the intensity of Lévy flights are still unclear. Here, we developed an agent-based model in a square lattice where agents perform Lévy flights depending on the fraction of neighboring defectors. We systematically studied the relationships among three factors for cooperation: sensitivity to defectors, the intensity of Lévy flights, and population density. Results of evolutionary simulations showed that moderate sensitivity most promoted cooperation. Then, we found that the shortest movements were best for cooperation when the sensitivity to defectors was high. In contrast, when the sensitivity was low, longer movements were best for cooperation. Thus, Lévy flights, the balance between short and long jumps, promoted cooperation in any sensitivity, which was confirmed by evolutionary simulations. Finally, as the population density became larger, higher sensitivity was more beneficial for cooperation to evolve. Our study highlights that Lévy flights are an optimal searching strategy not only for foraging but also for constructing cooperative relationships with others.

The impact of truncation selection and diffusion on cooperation in spatial games

Natural Selection is frequently modelled via proportional selection where survival is proportional to the average payoff differential. There has been little attention devoted to modelling truncation selection where replicators below a threshold are culled and survivors reproduce. Here, we systematically explore truncation selection for two strategy games in a spatial setting. We employ two variations of truncation selection: independent, where the threshold is fixed; and dependent, where the proportion culled is fixed. Further, we explore the effects of diffusion with the algorithms: contest-diffusion-offspring (CDO), and diffusion-contest-offspring (DCO). CDO and DCO frequently facilitate and diminish cooperation, respectively. For independent truncation, there are three qualitative regimes determined by the payoff threshold: cooperation decreases as the threshold rises; polymorphisms are stable; and extinction is frequent. Further, an intermediate payoff to cooperators playing defectors can maximize cooperation for the DCO algorithm with a high payoff threshold. Dependent truncation affects games differently; lower levels reduce cooperation for the Hawk Dove game and increase it for the Stag Hunt, and higher levels produce the opposite effects. Comparing these truncation methods to proportional selection, we show how they impact the prevalence of cooperation.

Neighbor-considered migration facilitates cooperation in prisoner’s dilemma games

Migration (e.g. between cities and nations) has been shown to be an effective mechanism in facilitating the evolution of cooperation in spatial games. In contingent migration (e.g. success-driven migration), individuals choose the relocation place based on their expected payoffs. In other words, success-driven migration assumes that individuals make decisions about where to migrate strategically rather than randomly. Existing behavioral experiments have shown that human have other-regarding preference. In this paper, we study individuals’ cooperation behaviors in the prisoner’s dilemma game on a two-dimensional square lattice, where individuals have other-regarding migration preference. We introduce a neighbor-considered migration strategy, which considers both benefits of individuals and their neighbors. During the migration process, an individual always moves to a reachable empty site with the highest fairness payoff, which takes the benefit of all relevant stakeholders (including the particular individual and the neighbors) into consideration. We explore the effect of the different fairness, while considering the individuals when they weigh their own interests and their neighbors’ interests. Our simulation results indicate that neighbor-considered migration can effectively promote the level of cooperation by helping cooperative clusters evade the invasion of defectors.

The influence of migration speed on cooperation in spatial games

Migration is a common phenomenon in human society which provides a person an opportunity to search for a new life from one area to another. In the framework of game theory, people may migrate to escape from a current adverse environment (evading defection). Since people may migrate at different speeds, it is interesting to figure out the influence of migration speed on the evolution of cooperative behavior. In an attempt to discover the influence, we propose here a model based on an adaptive migration mechanism. In this model, an individual migrates or updates his/her strategy asynchronously, which is tuned by migration frequency. Firstly, it is found that an appropriate migration speed may evoke an effective mechanism, which enables cooperators dominate even in highly adverse conditions. Secondly, we check how migration speed alters the paradigm of cooperation quantitatively in the conditions of different migration frequency. When migration frequency is high, cooperation is promoted only at a small migration speed. However, when migration frequency is low, cooperation is always promoted at any migration speed. In addition, we also investigated the influence of temptation to defect on cooperation for the case of different migration speeds and migration frequencies. Our results may provide a fresh perspective on the understanding of how human behavior affects cooperation.

Quantifying the impact of noise on macroscopic organization of cooperation in spatial games

Spatial games have been extensively studied in recent years. It is known that spatial structure could have a differing role in promoting cooperation, if comparing prisoner’s dilemma to snowdrift game. Less is known about whether there also exists qualitative difference of the emerging macroscopic spatial pattern of cooperation in the two respective games, in particular when noise is present in individual strategy learning. To address this issue, we quantify and compare the impact of noise on spatial organization of cooperation in the two games. In our simulations, individuals are located in a spatial lattice with the von Neumann neighbourhood. They play games with their immediate neighbors and update their strategies based on a Fermi function-like rule, 1/[1+exp(ΔP)/K], where ΔP is the payoff difference between the focal individual and one randomly chosen neighbour, and K represents the noise level (temperature) in strategy learning. Small K values represent the tendency of imitating better performing individuals while high K values increase the likelihood that individuals adopt worse performing strategies. Our results reveal striking differences between the two types of games in regard to the resulting macroscopic pattern of cooperation. Assortment between cooperators is much weaker in the snowdrift game than in the prisoner’s dilemma. Intermediate levels of noise maximize the local assortment in the prisoner’s dilemma while most weakening assortment in the snowdrift game. The cluster size of cooperators has a peak in the prisoner’s dilemma but shows a valley in the snowdrift game for varying level of noise. With increasing the noise level cooperators break into small clusters in the prisoner’s dilemma while cooperators tend to be clustered into larger lumps in the snowdrift game. These quantitative results highlight the underlying microscopic interaction, whether being the prisoner’s dilemma or the snowdrift type, nontrivially determines how noise affects the spatial pattern of cooperation.

Scale-Free Relationships Facilitate Cooperation in Spatial Games with Sequential Strategy

Recently, the area of study of spatial game continuously has extended, and researchers have especially presented a lot of works of coevolutionary mechanism. We have recognized coevolutionary mechanism as one of the factors for the promotion of cooperation like five rules by Nowak. However, those studies still deal with the optimal response (best decision). The best decision is persuasive in most cases, but does not apply to all situations in the real world. Contemplating that question, researchers have presented some works discussing not only the best decision but also the second-best decision. Those studies compare the results between the best and the second-best, and also state the applicability of the second-best decision. This study, considering that trend, has extended the match between two groups to spatial game with the second-best decision. This extended model expresses relationships of groups as a spatial network, and every group matches other groups of relationships. Then, we examine how mutual cooperation changes in each case where either we add probabilistic perturbation to relationships or ties form various types of the structure. As a result, unlike most results utilizing the best decision, probabilistic perturbation does not induce any change. On the other hand, when ties are the scale-free structure, mutual cooperation is enhanced like the case of the best decision. When we probe the evolution of strategies in that case, groups with many ties play a role for leading the direction of decision as a whole. This role appears without explicit assignment. In the discussion, we also state that the presented model has an analogy to the real situation, collusive tendering.

Role of adaptive migration in promoting cooperation in spatial games

Recent work has revealed that success-driven migration can promote cooperation among selfish individuals in evolutionary games. This migration mechanism relies, however, on nonlocal information about the states of the individuals and their computational capabilities for prediction. We investigate the role of adaptive migration in cooperative behavior in the framework of spatial game by proposing an alternative migration strategy that requires only local information obtainable through game interactions. Our results demonstrate that adaptive migration can be effective in promoting cooperation in two ways. First, there exists an optimal degree of migration associated with the density of empty sites and migration speed, which leads to the highest level of cooperation. Second, adaptive migration can induce an outbreak of cooperation from an environment dominated by defectors. These findings hold for common types of evolutionary games that involve pairwise interactions.

Spatial evolutionary game theory: Hawks and Doves revisited

We consider a spatial generalization of evolutionary game theory in which strategies are distributed over a spatial array of sites. We assume that the strategy corresponding to a given site has local interactions with the strategies sitting on neighbouring sites, and that the strategies change if neighbouring strategies are doing better. After briefly setting the stage with a formal definition of spatial evolutionary game theory, we consider the spatial extension of the Hawk-Dove game, and we show that the results are qualitatively different from those obtained from classical evolutionary game theory. For example, the proportion of Hawks in the population is in general lower in the spatial game than in the classical one. We also consider spatial generalizations of the extensions of the Hawk-Dove game obtained by including strategies such as Retaliator and Bully. Here, too, the results from the spatial game are very different from the classical results. In particular, with space Retaliator is a much more successful strategy than one would expect from classical considerations. This suggests that, in general, spatial structure may facilitate the evolution of strategies such as Retaliator, which do not necessarily prosper classically, and which are reminiscent of the \`nice', \`provokable' and `forgiving' strategies which seem to play a central role in the evolution of cooperation. The results indicate that including spatial structure in evolutionary game theory is a fruitful extension.