Different perceptions of social dilemmas: evolutionary multigames in structured populations

Cooperation dynamics in multi-issue repeated social dilemma games with correlated strategy

In the real world, individuals are often involved in collaboration on multiple issues, and these issues may interact with each other. Given the complexity of the interaction, we establish a multi-issue repeated game model, in which individuals participate in multiple social dilemma games simultaneously and repeatedly, and strategies in different issue games are correlated and reactive. We explore the cooperation dynamics of strategies in the population from a multiobjective perspective, in which an individual's preference for each issue is described by a weight vector, and heterogeneous preferences of individuals in the population are also considered. Through simulations on two-issue games, we find that compared to the uncorrelated case, the correlated strategy can significantly promote cooperation in both games regardless of which issue players prefer. Under the condition of homogeneous preference, an increase in the payoff weight of a given issue decreases the level of cooperation in that issue, and the optimal condition to sustain cooperation to the maximum extent is when the payoff weights of all issues are equal. Moreover, under the condition of heterogeneous preference, there exists an optimal proportion of players with different preferences under which the cooperation rate can reach its highest level in the population. This work highlights individual trade-offs on different issues when engaging in multiple games simultaneously and further enriches the research of evolutionary games from a multiobjective and correlated strategy perspective.

Evolutionary game selection creates cooperative environments

The emergence of collective cooperation in competitive environments is a well-known phenomenon in biology, economics, and social systems. While most evolutionary game models focus on the evolution of strategies for a fixed game, how strategic decisions coevolve with the environment has so far mostly been overlooked. Here, we consider a game selection model where not only the strategies but also the game can change over time following evolutionary principles. Our results show that coevolutionary dynamics of games and strategies can induce novel collective phenomena, fostering the emergence of cooperative environments. When the model is taken on structured populations the architecture of the interaction network can significantly amplify pro-social behavior, with a critical role played by network heterogeneity and the presence of clustered groups of similar players, distinctive features observed in real-world populations. By unveiling the link between the evolution of strategies and games for different structured populations, our model sheds new light on the origin of social dilemmas ubiquitously observed in real-world social systems.

Complex pathways to cooperation emergent from asymmetry in heterogeneous populations

Cooperation within asymmetric populations has garnered significant attention in evolutionary games. This paper explores cooperation evolution in populations with weak and strong players, using a game model where players choose between cooperation and defection. Asymmetry stems from different benefits for strong and weak cooperators, with their benefit ratio indicating the degree of asymmetry. Varied rankings of parameters including the asymmetry degree, cooperation costs, and benefits brought by weak players give rise to scenarios including the prisoner's dilemma (PDG) for both player types, the snowdrift game (SDG), and mixed PDG-SDG interactions. Our results indicate that in an infinite well-mixed population, defection remains the dominant strategy when strong players engage in the prisoner's dilemma game. However, if strong players play snowdrift games, global cooperation increases with the proportion of strong players. In this scenario, strong cooperators can prevail over strong defectors when the proportion of strong players is low, but the prevalence of cooperation among strong players decreases as their proportion increases. In contrast, within a square lattice, the optimum global cooperation emerges at intermediate proportions of strong players with moderate degrees of asymmetry. Additionally, weak players protect cooperative clusters from exploitation by strong defectors. This study highlights the complex dynamics of cooperation in asymmetric interactions, contributing to the theory of cooperation in asymmetric games.

Evolution of cooperation in multigame with environmental space and delay

Replicator dynamics is widely used in evolutionary game theory, however, most previous studies on replicator dynamics focus on single games and ignore multiple social dilemmas encountered by individuals in a population. This paper uses replicator dynamics to construct a multigame system with environmental space and delay based on three social dilemmas. For the non-delayed and delayed multigame systems, rich dynamics for stability, bistability, transcritical bifurcation, Hopf bifurcation, and the direction, stability and periodic variation of periodic solutions are comprehensively investigated. Accordingly, we use numerical simulations to assist in exploring the effects of multigame, environmental space, and time delay on strategic dynamics. The results show that large proportions of snowdrift game and stag hunt game are conducive to the prosperity of cooperators, and defectors are easy to survive when the proportion of prisoner's dilemma is large. The cooperator gains the advantage of benefit distribution from environmental space, or the defector gets less benefit distribution as punishment, which will make pure cooperation the dominant strategy. Furthermore, environmental space can allow cooperation and defection to coexist oscillatingly. Interestingly, large delays reverse the coexistence of cooperation and defection to a situation dominated by the purely cooperative strategy.

Interaction between games give rise to the evolution of moral norms of cooperation

In many biological populations, such as human groups, individuals face a complex strategic setting, where they need to make strategic decisions over a diverse set of issues and their behavior in one strategic context can affect their decisions in another. This raises the question of how the interaction between different strategic contexts affects individuals’ strategic choices and social norms? To address this question, I introduce a framework where individuals play two games with different structures and decide upon their strategy in a second game based on their knowledge of their opponent’s strategy in the first game. I consider both multistage games, where the same opponents play the two games consecutively, and reputation-based model, where individuals play their two games with different opponents but receive information about their opponent’s strategy. By considering a case where the first game is a social dilemma, I show that when the second game is a coordination or anti-coordination game, the Nash equilibrium of the coupled game can be decomposed into two classes, a defective equilibrium which is composed of two simple equilibrium of the two games, and a cooperative equilibrium, in which coupling between the two games emerge and sustain cooperation in the social dilemma. For the existence of the cooperative equilibrium, the cost of cooperation should be smaller than a value determined by the structure of the second game. Investigation of the evolutionary dynamics shows that a cooperative fixed point exists when the second game belongs to coordination or anti-coordination class in a mixed population. However, the basin of attraction of the cooperative fixed point is much smaller for the coordination class, and this fixed point disappears in a structured population. When the second game belongs to the anti-coordination class, the system possesses a spontaneous symmetry-breaking phase transition above which the symmetry between cooperation and defection breaks. A set of cooperation supporting moral norms emerges according to which cooperation stands out as a valuable trait. Notably, the moral system also brings a more efficient allocation of resources in the second game. This observation suggests a moral system has two different roles: Promotion of cooperation, which is against individuals’ self-interest but beneficial for the population, and promotion of organization and order, which is at both the population’s and the individual’s self-interest. Interestingly, the latter acts like a Trojan horse: Once established out of individuals’ self-interest, it brings the former with itself. Importantly, the fact that the evolution of moral norms depends only on the cost of cooperation and is independent of the benefit of cooperation implies that moral norms can be harmful and incur a pure collective cost, yet they are just as effective in promoting order and organization. Finally, the model predicts that recognition noise can have a surprisingly positive effect on the evolution of moral norms and facilitates cooperation in the Snow Drift game in structured populations.

How do moral norms spontaneously evolve in the presence of selfish incentives? An answer to this question is provided by the observation that moral systems have two distinct functions: Besides encouraging self-sacrificing cooperation, they also bring organization and order into the societies. In contrast to the former, which is costly for the individuals but beneficial for the group, the latter is beneficial for both the group and the individuals. A simple evolutionary model suggests this latter aspect is what makes a moral system evolve based on the individuals’ self-interest. However, a moral system behaves like a Trojan horse: Once established out of the individuals’ self-interest to promote order and organization, it also brings self-sacrificing cooperation.

Eco-evolutionary dynamics of multigames with mutations

Most environments favor defection over cooperation due to natural selection. Nonetheless, the emergence of cooperation is omnipresent in many biological, social, and economic systems, quite contrary to the well-celebrated Darwinian theory of evolution. Much research has been devoted to better understanding how and why cooperation persists among self-interested individuals despite their competition for limited resources. Here we go beyond a single social dilemma since individuals usually encounter various social challenges. In particular, we propose and study a mathematical model incorporating both the prisoner’s dilemma and the snowdrift game. We further extend this model by considering ecological signatures like mutation and selfless one-sided contribution of altruist free space. The nonlinear evolutionary dynamics that results from these upgrades offer a broader range of equilibrium outcomes, and it also often favors cooperation over defection. With the help of analytical and numerical calculations, our theoretical model sheds light on the mechanisms that maintain biodiversity, and it helps to explain the evolution of social order in human societies.

Dynamical reciprocity in interacting games: Numerical results and mechanism analysis

We study the evolution of two mutually interacting pairwise games on different topologies. On two-dimensional square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction becomes very strong. A mean-field theory is developed that points out dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either the individual or bulk scenario: invasion, neutral, and catalyzed types; their combination puts cooperators at a persistent advantage position, which boosts the cooperation. The robustness of the revealed reciprocity is strengthened by the studies of model variants, including the public goods game, asymmetrical or time-varying interactions, games of different types, games with timescale separation, different updating rules, etc. The structural complexities of the underlying population, such as Newman-Watts small world networks, Erdős-Rényi random networks, and Barabási-Albert networks, also do not alter the working of the dynamical reciprocity. In particular, as the number of games engaged increases, the cooperation level continuously improves in general. However, our analysis shows that the dynamical reciprocity works only in structured populations, otherwise the game-game interaction has no any impact on the cooperation at all. In brief, our work uncovers a cooperation mechanism in the structured populations, which indicates the great potential for human cooperation since concurrent issues are so often seen in the real world.

Evolutionary multigame with conformists and profiteers based on dynamic complex networks

Evolutionary game on complex networks provides a new research framework for analyzing and predicting group decision-making behavior in an interactive environment, in which most researchers assumed players as profiteers. However, current studies have shown that players are sometimes conformists rather than profit-seeking in society, but most research has been discussed on a simple game without considering the impact of multiple games. In this paper, we study the influence of conformists and profiteers on the evolution of cooperation in multiple games and illustrate two different strategy-updating rules based on these conformists and profiteers. Different from previous studies, we introduce a similarity between players into strategy-updating rules and explore the evolutionary game process, including the strategy updating, the transformation of players' type, and the dynamic evolution of the network structure. In the simulation, we implement our model on scale-free and regular networks and provide some explanations from the perspective of strategy transition, type transition, and network topology properties to prove the validity of our model.

Criticality and Griffiths phases in random games with quenched disorder

The perceived risk and reward for a given situation can vary depending on resource availability, accumulated wealth, and other extrinsic factors such as individual backgrounds. Based on this general aspect of everyday life, here we use evolutionary game theory to model a scenario with randomly perturbed payoffs in a prisoner's dilemma game. The perception diversity is modeled by adding a zero-average random noise in the payoff entries and a Monte Carlo simulation is used to obtain the population dynamics. This payoff heterogeneity can promote and maintain cooperation in a competitive scenario where only defectors would survive otherwise. In this work, we give a step further, understanding the role of heterogeneity by investigating the effects of quenched disorder in the critical properties of random games. We observe that payoff fluctuations induce a very slow dynamic, making the cooperation decay behave as power laws with varying exponents, instead of the usual exponential decay after the critical point, showing the emergence of a Griffiths phase. We also find a symmetric Griffiths phase near the defector's extinction point when fluctuations are present, indicating that Griffiths phases may be frequent in evolutionary game dynamics and play a role in the coexistence of different strategies.

Complex evolutionary dynamics due to punishment and free space in ecological multigames

The concurrence of ecological and evolutionary processes often arises as an integral part of various biological and social systems. We here study eco-evolutionary dynamics by adopting two paradigmatic metaphors of social dilemmas with contrasting outcomes. We use the Prisoner’s Dilemma and Snowdrift games as the backbone of the proposed mathematical model. Since cooperation is a costly proposition in the face of the Darwinian theory of evolution, we go beyond the traditional framework by introducing punishment as an additional strategy. Punishers bare an additional cost from their own resources to try and discourage or prohibit free-riding from selfish defectors. Our model also incorporates the ecological signature of free space, which has an altruistic-like impact because it allows others to replicate and potentially thrive. We show that the consideration of these factors has broad implications for better understanding the emergent complex evolutionary dynamics. In particular, we report the simultaneous presence of different subpopulations through the spontaneous emergence of cyclic dominance, and we determine various stationary points using traditional game-theoretic concepts and stability analysis.

Cooperation and competition between pair and multi-player social games in spatial populations

The conflict between individual and collective interests is in the heart of every social dilemmas established by evolutionary game theory. We cannot avoid these conflicts but sometimes we may choose which interaction framework to use as a battlefield. For instance some people like to be part of a larger group while other persons prefer to interact in a more personalized, individual way. Both attitudes can be formulated via appropriately chosen traditional games. In particular, the prisoner's dilemma game is based on pair interaction while the public goods game represents multi-point interactions of group members. To reveal the possible advantage of a certain attitude we extend these models by allowing players not simply to change their strategies but also let them to vary their attitudes for a higher individual income. We show that both attitudes could be the winner at a specific parameter value. Interestingly, however, the subtle interplay between different states may result in a counterintuitive evolutionary outcome where the increase of the multiplication factor of public goods game drives the population to a fully defector state. We point out that the accompanying pattern formation can only be understood via the multipoint or multi-player interactions of different microscopic states where the vicinity of a particular state may influence the relation of two other competitors.

Strategy equilibrium in dilemma games with off-diagonal payoff perturbations

We analyze the strategy equilibrium of dilemma games considering a payoff matrix affected by small and random perturbations on the off-diagonal. Notably, a recent work [Proc. R. Soc. A 476, 20200116 (2020)1364-502110.1098/rspa.2020.0116] reported that while cooperation is sustained by perturbations acting on the main diagonal, a less clear scenario emerges when perturbations act on the off-diagonal. Thus, the second case represents the core of this investigation, aimed at completing the description of the effects that payoff perturbations have on the dynamics of evolutionary games. Our results, achieved by analyzing the proposed model under a variety of configurations as different update rules, suggest that off-diagonal perturbations actually constitute a nontrivial form of noise. In particular, the most interesting effects are detected near the phase transition, as perturbations tend to move the strategy distribution towards nonordered states of equilibrium, supporting cooperation when defection is pervading the population, and supporting defection in the opposite case. To conclude, we identified a form of noise that, under controlled conditions, could be used to enhance cooperation and greatly delay its extinction.

Heterogeneity in evolutionary games: an analysis of the risk perception

In this study, we analyse the relationship between heterogeneity and cooperation. Previous investigations suggest that this relation is non-trivial, as some authors found that heterogeneity sustains cooperation, while others obtained different results. Among the possible forms of heterogeneity, we focus on the individual perception of risks and rewards related to a generic event, which can appear in a number of social and biological systems. The modelling approach is based on the framework of evolutionary game theory. To represent this kind of heterogeneity, we implement small and local perturbations on the pay-off matrix of simple two-strategy games, such as the Prisoner's Dilemma. So, while usually the pay-off is considered to be a global and time-invariant structure, i.e. it is the same for all individuals of a population at any time, in our model its value is continuously affected by small variations, in both time and space (i.e. position on a lattice). We found that such perturbations can be beneficial or detrimental to cooperation, depending on their setting. Notably, cooperation is strongly supported when perturbations act on the main diagonal of the pay-off matrix, whereas when they act on the off-diagonal the resulting effect is more difficult to quantify. To conclude, the proposed model shows a rich spectrum of possible equilibria, whose interpretation might offer insights and enrich the description of several systems.

Seasonal payoff variations and the evolution of cooperation in social dilemmas

Varying environmental conditions affect relations between interacting individuals in social dilemmas, thus affecting also the evolution of cooperation. Oftentimes these environmental variations are seasonal and can therefore be mathematically described as periodic changes. Accordingly, we here study how periodic shifts between different manifestations of social dilemmas affect cooperation. We observe a non-trivial interplay between the inherent spatiotemporal dynamics that characterizes the spreading of cooperation in a particular social dilemma type and the frequency of payoff changes. In particular, we show that periodic changes between two available games with global ordering best be fast, while periodic changes between global and local ordering games best be slow for cooperation to thrive. We also show that the frequency of periodic changes between two local ordering social dilemmas is irrelevant, because then the process is fast and simply the average cooperation level of the two is returned. The structure of the interaction network plays an important role too in that lattices promote local ordering, whilst random graphs hinder the formation of compact cooperative clusters. Conversely, for local ordering the regular structure of the interaction network is only marginally relevant as role-separating checkerboard patterns do not rely on long-range order.

An experimental study of network effects on coordination in asymmetric games

Network structure has often proven to be important in understanding the decision behavior of individuals or agents in different interdependent situations. Computational studies predict that network structure has a crucial influence on behavior in iterated 2 by 2 asymmetric ‘battle of the sexes’ games. We test such behavioral predictions in an experiment with 240 human subjects. We found that as expected the less ‘random’ the network structure, the better the experimental results are predictable by the computational models. In particular, there is an effect of network clustering on the heterogeneity of convergence behavior in the network. We also found that degree centrality and having an even degree are important predictors of the decision behavior of the subjects in the experiment. We thus find empirical validation of predictions made by computational models in a computerized experiment with human subjects.

Coevolution of Environmental Perception and Cooperative Behavior in Evacuation Crowd

For the evacuation crowd of social agents, environment plays a big effect on the behavior and decision of the agents. When facing the uncertain environment, the behavior and decision of agents depend heavily on the perception of environment. Therefore, the cooperation between agents and their perception of environment may coexist during evacuation. Here we establish a mechanism to analyze the coevolution between the cooperation of agents and the perception of environment. In detail, we use a regular square lattice with periodic boundaries, where two payoff matrices are used to describe two kinds of games between neighbors in the safe and dangerous environments. For individual agent, its perception can be adjusted by interacting with neighboring agents. When the environment is generally considered dangerous, the fraction of cooperative agents keeps at a high level, even if the value of b is very large. When all the agents think that the environment is safe, the fraction of cooperation will decrease as the value of b increases.

Cooperation dynamics of generalized reciprocity in state-based social dilemmas

We introduce a framework for studying social dilemmas in networked societies where individuals follow a simple state-based behavioral mechanism based on generalized reciprocity, which is rooted in the principle "help anyone if helped by someone." Within this general framework, which applies to a wide range of social dilemmas including, among others, public goods, donation, and snowdrift games, we study the cooperation dynamics on a variety of complex network examples. By interpreting the studied model through the lenses of nonlinear dynamical systems, we show that cooperation through generalized reciprocity always emerges as the unique attractor in which the overall level of cooperation is maximized, while simultaneously exploitation of the participating individuals is prevented. The analysis elucidates the role of the network structure, here captured by a local centrality measure which uniquely quantifies the propensity of the network structure to cooperation by dictating the degree of cooperation displayed both at the microscopic and macroscopic level. We demonstrate the applicability of the analysis on a practical example by considering an interaction structure that couples a donation process with a public goods game.

Effects of external forcing on evolutionary games in complex networks

How did cooperation evolve in a complex network is an intensely investigated problem. Many mechanisms that promote cooperation have been proposed within the framework of the evolutionary game theory. Motivated by the fact that people in society or even a certain group are often controlled by a variety of simple rules, we present an external forcing mechanism to analyze the underlying reasons of widespread cooperation in this paper. In detail, we model the agents on a simple regular network, on which the learning method is controlled by external forcing mechanism, and prisoner's dilemma has been applied to describe the interaction of agents. By conducting large-scale Monte Carlo simulations, we can easily draw a conclusion that this mechanism can promote cooperation efficiently. In addition, we also show that the proposed mechanism is effective for the cooperation promotion for other game models, such as snowdrift game and multigames. Taken together, the mechanism of external forcing on the evolutionary game is a strong promoter of cooperation even under a severe temptation condition, which has a practical significance and will provide new insight into the analysis and control of cooperative strategy in the complex network for the further research.

Cyclic dominance emerges from the evolution of two inter-linked cooperative behaviours in the social amoeba

Evolution of cooperation has been one of the most important problems in sociobiology, and many researchers have revealed mechanisms that can facilitate the evolution of cooperation. However, most studies deal only with one cooperative behaviour, even though some organisms perform two or more cooperative behaviours. The social amoeba Dictyostelium discoideum performs two cooperative behaviours in starvation: fruiting body formation and macrocyst formation. Here, we constructed a model that couples these two behaviours, and we found that the two behaviours are maintained because of the emergence of cyclic dominance, although cooperation cannot evolve if only either of the two behaviours is performed. The common chemoattractant cyclic adenosine 3',5'-monophosphate (cAMP) is used in both fruiting body formation and macrocyst formation, providing a biological context for this coupling. Cyclic dominance emerges regardless of the existence of mating types or spatial structure in the model. In addition, cooperation can re-emerge in the population even after it goes extinct. These results indicate that the two cooperative behaviours of the social amoeba are maintained because of the common chemical signal that underlies both fruiting body formation and macrocyst formation. We demonstrate the importance of coupling multiple games when the underlying behaviours are associated with one another.

Network effects on coordination in asymmetric games

Network structure can have an important effect on the behavior of players in an iterated 2 × 2 game. We study the effect of network structure on global and local behavior in asymmetric coordination games using best response dynamics. We find that global behavior is highly dependent on network topology. Random (Erdös-Rényi) networks mostly converge to homogeneous behavior, but the higher the clustering in the network the more heterogeneous the behavior becomes. Behavior within the communities of the network is almost exclusively homogeneous. The findings suggest that clustering of networks facilitates self-organization of uniform behavior within clusters, but heterogeneous behavior between clusters. At the local level we find that some nodes are more important in determining the equilibrium behavior than other nodes. Degree centrality is for most networks the main predictor for the behavior and nodes with an even degree have an advantage over nodes with an uneven degree in dictating the behavior. We conclude that the behavior is difficult to predict for (Erdös-Rényi) networks and that the network imposes the behavior as a function of clustering and degree heterogeneity in other networks.

A Robust Method for Inferring Network Structures

Inferring the network structure from limited observable data is significant in molecular biology, communication and many other areas. It is challenging, primarily because the observable data are sparse, finite and noisy. The development of machine learning and network structure study provides a great chance to solve the problem. In this paper, we propose an iterative smoothing algorithm with structure sparsity (ISSS) method. The elastic penalty in the model is introduced for the sparse solution, identifying group features and avoiding over-fitting, and the total variation (TV) penalty in the model can effectively utilize the structure information to identify the neighborhood of the vertices. Due to the non-smoothness of the elastic and structural TV penalties, an efficient algorithm with the Nesterov's smoothing optimization technique is proposed to solve the non-smooth problem. The experimental results on both synthetic and real-world networks show that the proposed model is robust against insufficient data and high noise. In addition, we investigate many factors that play important roles in identifying the performance of ISSS.

Evolutionary mixed games in structured populations: Cooperation and the benefits of heterogeneity

Evolutionary games on networks traditionally involve the same game at each interaction. Here we depart from this assumption by considering mixed games, where the game played at each interaction is drawn uniformly at random from a set of two different games. While in well-mixed populations the random mixture of the two games is always equivalent to the average single game, in structured populations this is not always the case. We show that the outcome is, in fact, strongly dependent on the distance of separation of the two games in the parameter space. Effectively, this distance introduces payoff heterogeneity, and the average game is returned only if the heterogeneity is small. For higher levels of heterogeneity the distance to the average game grows, which often involves the promotion of cooperation. The presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin, and they also emphasize the importance of the population structure in amplifying facilitators of cooperation.