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

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.

Impact of social reward on the evolution of cooperation in voluntary prisoner's dilemma

The existence and sustainability of cooperation is a critical issue in nature and social systems. Reward is an essential mechanism to enhance cooperation. Meanwhile, some individuals loathe competition and then choose to escape and become a loner in competition. In this scenario, we propose a four-strategy networked evolutionary game model involving rewarders, loners, cooperators, and defectors. The classical square lattice and the Erdös-Rényi random network are adopted to describe the interaction between individuals. The four-strategy model is an extension of the classic prisoner's dilemma game model. The simulation results show that the introduction of new strategic choices can significantly improve cooperation in the population. The promotion level of cooperation is directly correlated with reward intensity and negatively correlated with reward cost. With regard to the evolution of altruistic behaviors, the fixed income from interactions with loners has an impact that is connected to the temptation to defect. Furthermore, by analyzing characteristic snapshots of four strategies, we further dissect the essence of the evolution of cooperation. As the temptation value increases, cooperators and rewarders first form compact clusters, then more and more loners join to resist the intrusion of defectors. Eventually, the three strategies coexist stably in a spatially structured population. Our research may shed some light on exploring the nature of cooperation and solving social dilemmas in the future.

Costly incentives design from an institutional perspective: cooperation, sustainability and affluence

Incentives are usually introduced by the regulator entity (third-party), to promote cooperation in a market. The implementation of incentives is always costly and thus might fail to be enforced sustainably. This work aims at exploring the effects of incentives from an institutional perspective, while coping with the scenario where the third-party is part of the system but not composed by players. The evolutionary game theory (EGT) framework is applied to identify the incentives that lead to pure cooperation. In contrast to traditional EGT, this paper introduces an elimination mechanism that can reduce the market size. The incentives identified in the EGT analysis are further examined in simulation experiments which measure the market size, affluence and sustainability. The findings show: (1) light punishment leads to a reduction of the market size, yet heavier punishment is beneficial to the market size and wealth; (2) mixed incentives will generally lead to different wealth of the third party and of the participants. While under moderate strength, the wealth of both parties is the same and their overall wealth is maximal; (3) for sustainability, pure punishment (resp. reward) is sustainable (resp. unsustainable), the sustainability of mixed incentives depends on both their strength and agents’ rationality level.

Super-rational aspiration induced strategy updating helps resolve the tragedy of the commons in a cooperation system with exit rights

Avoiding the tragedy of the commons requires altruists to incur some losses to benefit the group. Although specific rules and self-enforcing agreements could help maintain the cooperation system stable, the costly recognition and free-rider problem are still questioned these two cooperation maintenance mechanisms. We here considered the situation of both exit costs and exit benefits in the asymmetric prisoner's dilemma game and introduced a super-rational aspiration induced strategy updating, where players adjust strategies in line with their payoffs and aspirations. If their payoffs reach or exceed the aspiration levels, which may be rational or super-rational, they keep their strategies. Otherwise, they imitate a local neighbor's strategy. We explored this rule in the structured and well-mixed population. The results show that super-rationality and asymmetry could together promote cooperation when exit costs exist. With exit benefit, super-rationality promotes cooperation in both structures and asymmetry only works in the well-mixed population. This suggests that the introduced strategy updating rule could sustain cooperation among egoists with exit rights.

Effect of memory, intolerance, and second-order reputation on cooperation

The understanding of cooperative behavior in social systems has been the subject of intense research over the past few decades. In this regard, the theoretical models used to explain cooperation in human societies have been complemented with a growing interest in experimental studies to validate the proposed mechanisms. In this work, we rely on previous experimental findings to build a theoretical model based on two cooperation driving mechanisms: second-order reputation and memory. Specifically, taking the donation game as a starting point, the agents are distributed among three strategies, namely, unconditional cooperators, unconditional defectors, and discriminators, where the latter follow a second-order assessment rule: shunning, stern judging, image scoring, or simple standing. A discriminator will cooperate if the evaluation of the recipient's last actions contained in his memory is above a threshold of (in)tolerance. In addition to the dynamics inherent to the game, another imitation dynamics, involving much longer times (generations), is introduced. The model is approached through a mean-field approximation that predicts the macroscopic behavior observed in Monte Carlo simulations. We found that, while in most second-order assessment rules, intolerance hinders cooperation, it has the opposite (positive) effect under the simple standing rule. Furthermore, we show that, when considering memory, the stern judging rule shows the lowest values of cooperation, while stricter rules show higher cooperation levels.

Reciprocity-based cooperative phalanx maintained by overconfident players

According to the evolutionary game theory principle, a strategy representing a higher payoff can spread among competitors. But there are cases when a player consistently overestimates or underestimates her own payoff, which undermines proper comparison. Interestingly, both underconfident and overconfident individuals are capable of elevating the cooperation level significantly. While former players stimulate a local coordination of strategies, the presence of overconfident individuals enhances the spatial reciprocity mechanism. In both cases the propagations of competing strategies are influenced in a biased way resulting in a cooperation supporting environment. These effects are strongly related to the nonlinear character of invasion probabilities which is a common and frequently observed feature of microscopic dynamics.

Coevolution of teaching ability and cooperation in spatial evolutionary games

Individuals with higher reputation are able to spread their social strategies easily. At the same time, one’s reputation is changing according to his previous behaviors, which leads to completely different teaching abilities for players. To explore the effect of the teaching ability influenced by reputation, we consider a coevolutionary model in which the reputation score affects the updating rule in spatial evolutionary games. More precisely, the updating probability becomes bigger if his/her partner has a positive reputation. Otherwise, the updating probability becomes smaller. This simple design describes the influence of teaching ability on strategy adoption effectively. Numerical results focus on the proportion of cooperation under different levels of the amplitude of change of reputation and the range of reputation. For this dynamics, the fraction of cooperators presents a growth trend within a wide range of parameters. In addition, to validate the generality of this mechanism, we also employ the snowdrift game. Moreover, the evolution of cooperation on Erdős-Rényi random graph is studied for the prisoner’s dilemma game. Our results may be conducive to understanding the emergence and sustainability of cooperation during the strategy adoptions in reality.

Punishment diminishes the benefits of network reciprocity in social dilemma experiments

Network reciprocity has been widely advertised in theoretical studies as one of the basic cooperation-promoting mechanisms, but experimental evidence favoring this type of reciprocity was published only recently. When organized in an unchanging network of social contacts, human subjects cooperate provided the following strict condition is satisfied: The benefit of cooperation must outweigh the total cost of cooperating with all neighbors. In an attempt to relax this condition, we perform social dilemma experiments wherein network reciprocity is aided with another theoretically hypothesized cooperation-promoting mechanism-costly punishment. The results reveal how networks promote and stabilize cooperation. This stabilizing effect is stronger in a smaller-size neighborhood, as expected from theory and experiments. Contrary to expectations, punishment diminishes the benefits of network reciprocity by lowering assortment, payoff per round, and award for cooperative behavior. This diminishing effect is stronger in a larger-size neighborhood. An immediate implication is that the psychological effects of enduring punishment override the rational response anticipated in quantitative models of cooperation in networks.

Impact of Social Reward on the Evolution of the Cooperation Behavior in Complex Networks

Social reward, as a significant mechanism explaining the evolution of cooperation, has attracted great attention both theoretically and experimentally. In this paper, we study the evolution of cooperation by proposing a reward model in network population, where a third strategy, reward, as an independent yet particular type of cooperation is introduced in 2-person evolutionary games. Specifically, a new kind of role corresponding to reward strategy, reward agents, is defined, which is aimed at increasing the income of cooperators by applying to them a social reward. Results from numerical simulations show that consideration of social reward greatly promotes the evolution of cooperation, which is confirmed for different network topologies and two evolutionary games. Moreover, we explore the microscopic mechanisms for the promotion of cooperation in the three-strategy model. As expected, the reward agents play a vital role in the formation of cooperative clusters, thus resisting the aggression of defectors. Our research might provide valuable insights into further exploring the nature of cooperation in the real world.

A Generalized Spatial Measure for Resilience of Microbial Systems

The emergent property of resilience is the ability of a system to return to an original state after a disturbance. Resilience may be used as an early warning system for significant or irreversible community transition; that is, a community with diminishing or low resilience may be close to catastrophic shift in function or an irreversible collapse. Typically, resilience is quantified using recovery time, which may be difficult or impossible to directly measure in microbial systems. A recent study in the literature showed that under certain conditions, a set of spatial-based metrics termed recovery length, can be correlated to recovery time, and thus may be a reasonable alternative measure of resilience. However, this spatial metric of resilience is limited to use for step-change perturbations. Building upon the concept of recovery length, we propose a more general form of the spatial metric of resilience that can be applied to any shape of perturbation profiles (for example, either sharp or smooth gradients). We termed this new spatial measure “perturbation-adjusted spatial metric of resilience” (PASMORE). We demonstrate the applicability of the proposed metric using a mathematical model of a microbial mat.

Understanding Cooperative Behavior Based on the Coevolution of Game Strategy and Link Weight

In reality, the dependency relationship among individuals is heterogeneous and time-varying. Based on this fact, we present a new mechanism of coevolution of game strategy and link weight when analyzing the evolution of cooperation. In detail, we model the population on a regular network, on which the relationship between players is depicted by a weighted link, and prisoner's dilemma has been applied to describe the interaction of players. Further, the impact of this mechanism on the cooperative behavior has been outlined. By conducting large-scale Monte Carlo simulations, we can easily draw a conclusion that this mechanism can promote cooperation efficiently. Compared with the traditional case, when the temptation of defection b is large, the fraction of cooperation is still able to keep in a high level. With a comprehensive examination of the distribution of stable link weight, it is evident that the coevolution mechanism would deviate the initial distribution. This mechanism induces the heterogeneity of players, which enhances the fraction of cooperation. Numerical simulations also indicate that an intermediate value of Δ/δ warrants an optimal resolution of prisoner's dilemma. The mechanism of coevolution of game strategy and link weight has a practical significance and will provide new insight for the further research.

Heterogeneous Coupling between Interdependent Lattices Promotes the Cooperation in the Prisoner's Dilemma Game

In the research realm of game theory, interdependent networks have extended the content of spatial reciprocity, which needs the suitable coupling between networks. However, thus far, the vast majority of existing works just assume that the coupling strength between networks is symmetric. This hypothesis, to some extent, seems inconsistent with the ubiquitous observation of heterogeneity. Here, we study how the heterogeneous coupling strength, which characterizes the interdependency of utility between corresponding players of both networks, affects the evolution of cooperation in the prisoner's dilemma game with two types of coupling schemes (symmetric and asymmetric ones). Compared with the traditional case, we show that heterogeneous coupling greatly promotes the collective cooperation. The symmetric scheme seems much better than the asymmetric case. Moreover, the role of varying amplitude of coupling strength is also studied on these two interdependent ways. Current findings are helpful for us to understand the evolution of cooperation within many real-world systems, in particular for the interconnected and interrelated systems.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

Effect of resource spatial correlation and hunter-fisher-gatherer mobility on social cooperation in Tierra del Fuego

This article presents an agent-based model designed to explore the development of cooperation in hunter-fisher-gatherer societies that face a dilemma of sharing an unpredictable resource that is randomly distributed in space. The model is a stylised abstraction of the Yamana society, which inhabited the channels and islands of the southernmost part of Tierra del Fuego (Argentina-Chile). According to ethnographic sources, the Yamana developed cooperative behaviour supported by an indirect reciprocity mechanism: whenever someone found an extraordinary confluence of resources, such as a beached whale, they would use smoke signals to announce their find, bringing people together to share food and exchange different types of social capital. The model provides insight on how the spatial concentration of beachings and agents' movements in the space can influence cooperation. We conclude that the emergence of informal and dynamic communities that operate as a vigilance network preserves cooperation and makes defection very costly.

Optimal distribution of incentives for public cooperation in heterogeneous interaction environments

In the framework of evolutionary games with institutional reciprocity, limited incentives are at disposal for rewarding cooperators and punishing defectors. In the simplest case, it can be assumed that, depending on their strategies, all players receive equal incentives from the common pool. The question arises, however, what is the optimal distribution of institutional incentives? How should we best reward and punish individuals for cooperation to thrive? We study this problem for the public goods game on a scale-free network. We show that if the synergetic effects of group interactions are weak, the level of cooperation in the population can be maximized simply by adopting the simplest “equal distribution” scheme. If synergetic effects are strong, however, it is best to reward high-degree nodes more than low-degree nodes. These distribution schemes for institutional rewards are independent of payoff normalization. For institutional punishment, however, the same optimization problem is more complex, and its solution depends on whether absolute or degree-normalized payoffs are used. We find that degree-normalized payoffs require high-degree nodes be punished more lenient than low-degree nodes. Conversely, if absolute payoffs count, then high-degree nodes should be punished stronger than low-degree nodes.

Impact of social punishment on cooperative behavior in complex networks

Social punishment is a mechanism by which cooperative individuals spend part of their resources to penalize defectors. In this paper, we study the evolution of cooperation in 2-person evolutionary games on networks when a mechanism for social punishment is introduced. Specifically, we introduce a new kind of role, punisher, which is aimed at reducing the earnings of defectors by applying to them a social fee. Results from numerical simulations show that different equilibria allowing the three strategies to coexist are possible as well as that social punishment further enhance the robustness of cooperation. Our results are confirmed for different network topologies and two evolutionary games. In addition, we analyze the microscopic mechanisms that give rise to the observed macroscopic behaviors in both homogeneous and heterogeneous networks. Our conclusions might provide additional insights for understanding the roots of cooperation in social systems.

Shared rewarding overcomes defection traps in generalized volunteer's dilemmas

For societies to produce or safeguard public goods, costly voluntary contributions are often required. From the perspective of each individual, however, it is advantageous not to volunteer such contributions, in the hope that other individuals will carry the associated costs. This conflict can be modeled as a volunteer's dilemma. To encourage rational individuals to make voluntary contributions, a government or other social organizations can offer rewards, to be shared among the volunteers. Here we apply such shared rewarding to the generalized N-person volunteer's dilemma, in which a threshold number of volunteers is required for producing the public good. By means of theoretical and numerical analyses, we show that without shared rewarding only two evolutionary outcomes are possible: full defection or coexistence of volunteers and non-volunteers. We show that already small rewards destabilize full defection, stabilizing small fractions of volunteers instead. Furthermore, at these intermediate reward levels, we find a hysteresis effect such that increasing or decreasing group sizes can trigger different social outcomes. In particular, when group size is increased, the fraction of volunteers first increases gradually before jumping up abruptly; when group size is then decreased again, the fraction of volunteers not only remains high, but even continues to increase. As the shared reward is increased beyond a critical level, the bistablitity underlying this hysteresis effect vanishes altogether, and only a single social outcome remains, corresponding to considerable fractions of volunteers. We find that this critical level of shared rewarding is relatively small compared to the total cost of contributing to the public good. These results show that the introduction of shared rewarding is remarkably effective in overcoming defection traps in the generalized volunteer's dilemma.

Inferring reputation promotes the evolution of cooperation in spatial social dilemma games

In realistic world individuals with high reputation are more likely to influence the collective behaviors. Due to the cost and error of information dissemination, however, it is unreasonable to assign each individual with a complete cognitive power, which means that not everyone can accurately realize others’ reputation situation. Here we introduce the mechanism of inferring reputation into the selection of potential strategy sources to explore the evolution of cooperation. Before the game each player is assigned with a randomly distributed parameter p denoting his ability to infer the reputation of others. The parameter p of each individual is kept constant during the game. The value of p indicates that the neighbor possessing highest reputation is chosen with the probability p and randomly choosing an opponent is left with the probability 1−p. We find that this novel mechanism can be seen as an universally applicable promoter of cooperation, which works on various interaction networks and in different types of evolutionary game. Of particular interest is the fact that, in the early stages of evolutionary process, cooperators with high reputation who are easily regarded as the potential strategy donors can quickly lead to the formation of extremely robust clusters of cooperators that are impervious to defector attacks. These clusters eventually help cooperators reach their undisputed dominance, which transcends what can be warranted by the spatial reciprocity alone. Moreover, we provide complete phase diagrams to depict the impact of uncertainty in strategy adoptions and conclude that the effective interaction topology structure may be altered under such a mechanism. When the estimation of reputation is extended, we also show that the moderate value of evaluation factor enables cooperation to thrive best. We thus present a viable method of understanding the ubiquitous cooperative behaviors in nature and hope that it will inspire further studies to resolve social dilemmas.

Cooperation and age structure in spatial games

We study the evolution of cooperation in evolutionary spatial games when the payoff correlates with the increasing age of players (the level of correlation is set through a single parameter, α). The demographic heterogeneous age distribution, directly affecting the outcome of the game, is thus shown to be responsible for enhancing the cooperative behavior in the population. In particular, moderate values of α allow cooperators not only to survive but to outcompete defectors, even when the temptation to defect is large and the ageless, standard α=0 model does not sustain cooperation. The interplay between age structure and noise is also considered, and we obtain the conditions for optimal levels of cooperation.

Coveting thy neighbors fitness as a means to resolve social dilemmas

In spatial evolutionary games the fitness of each individual is traditionally determined by the payoffs it obtains upon playing the game with its neighbors. Since defection yields the highest individual benefits, the outlook for cooperators is gloomy. While network reciprocity promotes collaborative efforts, chances of averting the impending social decline are slim if the temptation to defect is strong. It is, therefore, of interest to identify viable mechanisms that provide additional support for the evolution of cooperation. Inspired by the fact that the environment may be just as important as inheritance for individual development, we introduce a simple switch that allows a player to either keep its original payoff or use the average payoff of all its neighbors. Depending on which payoff is higher, the influence of either option can be tuned by means of a single parameter. We show that, in general, taking into account the environment promotes cooperation. Yet coveting the fitness of one's neighbors too strongly is not optimal. In fact, cooperation thrives best only if the influence of payoffs obtained in the traditional way is equal to that of the average payoff of the neighborhood. We present results for the prisoner's dilemma and the snowdrift game, for different levels of uncertainty governing the strategy adoption process, and for different neighborhood sizes. Our approach outlines a viable route to increased levels of cooperative behavior in structured populations, but one that requires a thoughtful implementation.

Aspiring to the fittest and promotion of cooperation in the prisoner's dilemma game

Strategy changes are an essential part of evolutionary games. Here, we introduce a simple rule that, depending on the value of a single parameter w , influences the selection of players that are considered as potential sources of the new strategy. For positive w players with high payoffs will be considered more likely, while for negative w the opposite holds. Setting w equal to zero returns the frequently adopted random selection of the opponent. We find that increasing the probability of adopting the strategy from the fittest player within reach, i.e., setting w positive, promotes the evolution of cooperation. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process and for different interaction networks. Since the evolution to widespread defection is tightly associated with cooperators having a lower fitness than defectors, the fact that positive values of w facilitate cooperation is quite surprising. We show that the results can be explained by means of a negative feedback effect that increases the vulnerability of defectors although initially increasing their survivability. Moreover, we demonstrate that the introduction of w effectively alters the interaction network and thus also the impact of uncertainty by strategy adoptions on the evolution of cooperation.

Effects of mobility in a population of prisoner's dilemma players

We address the problem of how the survival of cooperation in a social system depends on the motion of the individuals. Specifically, we study a model in which prisoner's dilemma players are allowed to move in a two-dimensional plane. Our results show that cooperation can survive in such a system provided that both the temptation to defect and the velocity at which agents move are not too high. Moreover, we show that when these conditions are fulfilled, the only asymptotic state of the system is that in which all players are cooperators. Our results might have implications for the design of cooperative strategies in motion coordination and other applications including wireless networks.

Evolving learning rules and emergence of cooperation in spatial prisoner's dilemma

In the evolutionary Prisoner's dilemma (PD) game, agents play with each other and update their strategies in every generation according to some microscopic dynamical rule. In its spatial version, agents do not play with every other but, instead, interact only with their neighbours, thus mimicking the existing of a social or contact network that defines who interacts with whom. In this work, we explore evolutionary, spatial PD systems consisting of two types of agents, each with a certain update (reproduction, learning) rule. We investigate two different scenarios: in the first case, update rules remain fixed for the entire evolution of the system; in the second case, agents update both strategy and update rule in every generation. We show that in a well-mixed population the evolutionary outcome is always full defection. We subsequently focus on two-strategy competition with nearest-neighbour interactions on the contact network and synchronised update of strategies. Our results show that, for an important range of the parameters of the game, the final state of the system is largely different from that arising from the usual setup of a single, fixed dynamical rule. Furthermore, the results are also very different if update rules are fixed or evolve with the strategies. In these respect, we have studied representative update rules, finding that some of them may become extinct while others prevail. We describe the new and rich variety of final outcomes that arise from this co-evolutionary dynamics. We include examples of other neighbourhoods and asynchronous updating that confirm the robustness of our conclusions. Our results pave the way to an evolutionary rationale for modelling social interactions through game theory with a preferred set of update rules.

Complex cooperative networks from evolutionary preferential attachment

In spite of its relevance to the origin of complex networks, the interplay between form and function and its role during network formation remains largely unexplored. While recent studies introduce dynamics by considering rewiring processes of a pre-existent network, we study network growth and formation by proposing an evolutionary preferential attachment model, its main feature being that the capacity of a node to attract new links depends on a dynamical variable governed in turn by the node interactions. As a specific example, we focus on the problem of the emergence of cooperation by analyzing the formation of a social network with interactions given by the Prisoner's Dilemma. The resulting networks show many features of real systems, such as scale-free degree distributions, cooperative behavior and hierarchical clustering. Interestingly, results such as the cooperators being located mostly on nodes of intermediate degree are very different from the observations of cooperative behavior on static networks. The evolutionary preferential attachment mechanism points to an evolutionary origin of scale-free networks and may help understand similar feedback problems in the dynamics of complex networks by appropriately choosing the game describing the interaction of nodes.