Network Reciprocity Research Articles

The existence of fence-sitters relaxes the spatial prisoner’s dilemma and enhances network reciprocity

We introduce a new spatial prisoner’s dilemma (SPD) model in which the so-called network reciprocity is enhanced more than in the conventional model. In addition to the usual binary strategies—perfect cooperator and perfect defector—we introduce “fence-sitters”, who either cooperate or defect with equal probability, as a third strategy. In regions with larger Stag-Hunt-type dilemmas but smaller Chicken-type dilemmas, the existence of the fence-sitters works as a buffer, hampering the exploitation of cooperators by neighboring defectors and significantly enhancing the cooperative phase. Conversely, in regions with larger Chicken-type dilemmas but smaller Stag-Hunt-type dilemmas, the existence of the fence-sitters expands the critical dilemma strength for the survival of cooperation, but it reduces the cooperating fraction more than in the conventional model. What we have found may justify the conclusion that the existence of people with neutral opinions—often regarded betwixt and between—absorbs the severe competition between two extreme groups, bringing a more accommodationist situation to our society, backed with greater cooperation.

The dynamics of cooperation in asymmetric sub-populations

Sacrificing personal benefits for a common good is at odds with the fundamental principle of Darwinian evolution: if only the fittest survives, then there should be no place for cooperation. But cooperative behavior actually abounds, and constitutes one of the most persistent and fascinating puzzles of nature. One solution to this puzzle is network reciprocity, where the collective dynamics of cooperators affords them protection against invading defectors. Commonly, however, such a competition does not unfold in isolation. Populations are often divided into sub-populations, with different evolutionary rules describing the interactions between them. Here we propose and study a paradigmatic model that captures the essence of this setup. Specifically, if two players belong to the same sub-population, they play the prisoner’s dilemma game. If not, they play either the harmony game, the snowdrift game, the stag-hunt game, or the prisoner’s dilemma game. Due to such an asymmetry in the interactions across sub-populations, a fascinating evolutionary dynamics sets up that greatly expands the survivability of cooperators. For instance, when the harmony game applies, cyclic dominance spontaneously emerges, wherein cooperators in one sub-population become predators of defectors in the other sub-population. One also may observe self-organized segregation, wherein both sub-populations maintain a mixed state of cooperators and defectors. As a general rule, we show that the lower the dilemma strength between sub-populations, the more abundant the cooperative strategy in the entire population. Results are confirmed by means of Monte Carlo simulations with pair approximation method, which reveals a rich plethora of novel and generally valid paths to cooperation.

Evolution of cooperation driven by collective interdependence on multilayer networks

Group interactions, formulated in terms of a public goods game, can not be deduced by the corresponding sum of pairwise interactions. This study proposes a collective interdependence characterizing the functioning of interdependent groups between subnetworks. Via the establishment of global group interactions across subnetwork layers, we show a double-edged sword of collective interdependence in promoting cooperation. Enhancement of collective interdependence hinders the evolution of cooperation whenever global synergy factor is small, while an optimal collective interdependence emerges most favoring the evolution of cooperation for high global synergy factors. However, for such high global synergy factors a low level of interdependence puts cooperators to a most disadvantaged place. A combination of low levels of collective interdependence and high local synergy factors shows that a moderate global synergetic effect most favors the evolution of cooperation. Our work reveals that collective interdependence impacts interdependent network reciprocity significantly and highlights the importance of network reciprocity in enhancing the evolution of cooperation.

The evolution of cooperation affected by aspiration-driven updating rule in multi-games with voluntary participation

Abstract We have demonstrated multi-games with voluntary participation can promote cooperation and found aspiration-driven updating rule can also facilitate cooperation to some degree. Stimulated by these, we combine the two mechanisms and investigate the evolution of cooperation. In our work, the population is randomly divided into two types, one type players play the PDG, while the other type players play SG. At the same time, a part of players randomly select to use the aspiration-driven updating rule, and the other part of players choose to use the Fermi-function based strategy adoption rule. Besides, players have three strategies - cooperation, defection and going it alone- to choose. We reveal that both the diversity of sucker's payoff and updating strategy can facilitate cooperation to some degree. Moreover, larger magnitude of sucker's payoff and less players using the aspiration-driven rule can improve cooperation resulting from spatial distribution of strategies. And players playing SG play a more important role to enhance cooperation than players playing PDG, which is due to the special network reciprocity. Furthermore, the easily satisfied players intend to cooperate, while the greedy players intend to be loners to avoid obtaining nothing.

Evolution of Cooperation in the Presence of Higher-Order Interactions: From Networks to Hypergraphs.

Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions going beyond pairwise models, such as graphs, in which all the interactions are considered as involving only two individuals at a time. Hypergraphs respond to this need, providing a mathematical representation of a system allowing from pairs to larger groups. In this work, through the use of different hypergraphs, we study how group interactions influence the evolution of cooperation in a structured population, by analyzing the evolutionary dynamics of the public goods game. Here we show that, likewise to network reciprocity, group interactions also promote cooperation. More importantly, by means of an invasion analysis in which the conditions for a strategy to survive are studied, we show how, in heterogeneously-structured populations, reciprocity among players is expected to grow with the increasing of the order of the interactions. This is due to the heterogeneity of connections and, particularly, to the presence of individuals standing out as hubs in the population. Our analysis represents a first step towards the study of evolutionary dynamics through higher-order interactions, and gives insights into why cooperation in heterogeneous higher-order structures is enhanced. Lastly, it also gives clues about the co-existence of cooperative and non-cooperative behaviors related to the structural properties of the interaction patterns.

How does conformity promote the enhancement of cooperation in the network reciprocity in spatial prisoner's dilemma games?

A new spatial prisoner's dilemma game model considering the conformity is proposed. A series of simulations confirms the present model bringing more enhanced network reciprocity than the conventional model. The observed enhancement is mechanically justified by relying the concept of END and EXP period. In line with more enhanced network reciprocity, we built a new spatial prisoner's dilemma (SPD) game considering conformity with presumption of lattice for underlying network and pairwise Fermi (PW-Fermi) for update rule. The key protocol is that a neighbor with more weight resulting from the conformity would be more likely selected as a pairwise opponent in the PW-Fermi updating process instead of random selection the conventional model presuming. A series of systematic simulations confirms that our model realizes more enhanced network reciprocity than the conventional SPD model. We elucidated its mechanism by means of considering on the concept of END period (meaning an initial period in which the global-cooperation fraction decreases from its initial value) and EXP periods (meaning the period following to END period in which global-cooperation increases) that substantially explaining how cooperative clusters survive in initial period and extend to defective zones afterwards.

Exit Option Induced by Win-Stay-Lose-Leave Rule Provides Another Route to Solve the Social Dilemma in Structured Populations

Theoretical have shown that cooperation can be promoted by the so-called reciprocity, where cooperation can coexist with defection via the compact cooperative clusters. However, this mechanism has one assumption interpreted as players have no chance to exit from the game even the situation is extremely bad, which is in sharp contrast with real-life situations. Here, we relax this assumption in the spatial prisoner's dilemma game by giving players the right to choose one of their state between active (participate in the game) and inactive (exit from the game) in each round of the game determined by win-stay-lose-leave rule. This new rule attracts a winner whose payoff is larger than the average payoff of its neighbors to stay in its current state, thus retains its current advantage to its neighbors. Conversely, a loser is pushed to leave from its current state and thus increase its chance to obtain a higher payoff. Specifically, we incorporate exit costs into consideration by assuming anyone who decides to exit from the game must pay a cost $\gamma$. Extensively numerical simulation shows that for the requirement of exit cost is neither too high or too small, a full cooperation plateau is achieved, where cooperation evolves with the support of enhanced network reciprocity. In fact, inactive players can only exist at the boundary of cooperative clusters during its evolution dynamics, which create a crucial buffer area for the endangered cooperators. Finally, the joint effect of this protective film composed of inactive players and cooperative clusters forming the foundation of enhanced network reciprocity. Our findings may provide new insight into network reciprocity.

Cooperation on Interdependent Networks by Means of Migration and Stochastic Imitation.

Evolutionary game theory in the realm of network science appeals to a lot of research communities, as it constitutes a popular theoretical framework for studying the evolution of cooperation in social dilemmas. Recent research has shown that cooperation is markedly more resistant in interdependent networks, where traditional network reciprocity can be further enhanced due to various forms of interdependence between different network layers. However, the role of mobility in interdependent networks is yet to gain its well-deserved attention. Here we consider an interdependent network model, where individuals in each layer follow different evolutionary games, and where each player is considered as a mobile agent that can move locally inside its own layer to improve its fitness. Probabilistically, we also consider an imitation possibility from a neighbor on the other layer. We show that, by considering migration and stochastic imitation, further fascinating gateways to cooperation on interdependent networks can be observed. Notably, cooperation can be promoted on both layers, even if cooperation without interdependence would be improbable on one of the layers due to adverse conditions. Our results provide a rationale for engineering better social systems at the interface of networks and human decision making under testing dilemmas.

Direct Reciprocity and Model-Predictive Strategy Update Explain the Network Reciprocity Observed in Socioeconomic Networks

Network reciprocity has been successfully put forward (since M. A. Nowak and R. May’s, 1992, influential paper) as the simplest mechanism—requiring no strategical complexity—supporting the evolution of cooperation in biological and socioeconomic systems. The mechanism is actually the network, which makes agents’ interactions localized, while network reciprocity is the property of the underlying evolutionary process to favor cooperation in sparse rather than dense networks. In theoretical models, the property holds under imitative evolutionary processes, whereas cooperation disappears in any network if imitation is replaced by the more rational best-response rule of strategy update. In social experiments, network reciprocity has been observed, although the imitative behavior did not emerge. What did emerge is a form of conditional cooperation based on direct reciprocity—the propensity to cooperate with neighbors who previously cooperated. To resolve this inconsistency, network reciprocity has been recently shown in a model that rationally confronts the two main behaviors emerging in experiments—reciprocal cooperation and unconditional defection—with rationality introduced by extending the best-response rule to a multi-step predictive horizon. However, direct reciprocity was implemented in a non-standard way, by allowing cooperative agents to temporarily cut the interaction with defecting neighbors. Here, we make this result robust to the way cooperators reciprocate, by implementing direct reciprocity with the standard tit-for-tat strategy and deriving similar results.

Heterogeneous aspiration resolves social dilemma in structured populations

In this paper, we study the impact of heterogeneous aspiration-driven strategy update rules on the evolution of cooperation in spatial prisoner's dilemma game. Individuals on network are divided into two categories: one part is the traditional players who update their strategies by imitating their neighbors’ strategies, the remaining of the individuals update their strategies based on heterogeneous aspiration. In the latter case, the probability of aspiration-driven updating is closely related to the payoffs of neighbors. Through Monte Carlo simulation, it found that heterogeneous aspiration can efficiently promote the network reciprocity. From the micro perspective, we find that such a mechanism can reserve the classical spatial distribution of cooperators, where cooperator clusters will be exploited by defectors. Based on the classical distribution, the strategies transitions show the essential reason for promotion of cooperation. In order to verify the generality of the conclusion, we further introduce the proposed mechanism into the snowdrift game and the public goods game, and obtain the observations consistent with the prisoner's dilemma game.

Analysis Method of Enterprise Network Relationship Based on Graph Theory

Network relationship is positively correlated with enterprise development and performance, which is an important basis for studying enterprise development and judging enterprise development situation. In this paper, complex network theory and methods are used to study enterprise network relationships. Based on graph theory, a graph model of enterprise network relationships is constructed, and the Laplacian matrix of the graph model is used to analyze important indicators of network relationships, relationship strength, stability, reciprocity, centrality and other indicators of numerical analysis methods.

Learning continuous and consistent strategy promotes cooperation in prisoner’s dilemma game with mixed strategy

Research on cooperative evolution behavior based on memory mechanism has been a hot topic of many scholars in recent years. However, most previous studies have considered neighbors’ historical payoffs when individuals choose role models whom they will learn from, but paid less attention to the stability of neighbors’ strategy which indicates how frequently they change their strategies in memory length. In this paper, we study the memory length M and the strategy persistence level u of individuals and the change rate of cooperative tendency δ on the cooperative evolution under the mixed strategy of the prisoner’s dilemma. Strategy persistence level is measured by the number of times which the strategy in neighbor’s memory length is continuous and consistent with neighbor’s current strategy, and can determine the probability that the individual learns the neighbor’s strategy when updating the strategy; the change rate of cooperative tendency δ is described by the standard deviation of the normal distribution. We investigate the effects of M and δ on the evolutionary of cooperation. The results show that the persistence strategy mechanism which we proposed in the memory length can improve network reciprocity and facilitate the generation and maintenance of cooperation, the larger the memory length is, the smaller the standard deviation value is, the more conducive to cooperation.

Cooperator-driven and defector-driven punishments: How do they influence cooperation?

Economic studies have shown that there are two types of regulation schemes which can be considered as a vital part of today's global economy: self-regulation enforced by self-regulation organizations to govern industry practices and government regulation which is considered as another scheme to sustain corporate adherence. An outstanding problem of particular interest is to understand quantitatively the role of these regulation schemes in evolutionary dynamics. Typically, punishment usually occurs for enforcement of regulations. Taking into account both types of punishments to influence the regulations, we develop a game model where six evolutionary situations with corresponding combinations of strategies are considered. Furthermore, a semianalytical method is developed to allow us to give accurate estimations of the boundaries between the phases of full defection and nondefection. We find that, associated with the evolutionary dynamics, for an infinite well-mixed population, the mix of both punishments performs better than one punishment alone in promoting public cooperation, but for a networked population the cooperator-driven punishment turns out to be a better choice. We also reveal the monotonic facilitating effects of the synergy effect, punishment fine, and feedback sensitivity on the public cooperation for an infinite well-mixed population. Conversely, for a networked population an optimal intermediate range of feedback sensitivity is needed to best promote punishers' populations. Overall, a networked structure is overall more favorable for punishers and further for public cooperation, because of both network reciprocity and mutualism between punishers and cooperators who do not punish defectors. We provide physical understandings of the observed phenomena, through a detailed statistical analysis of frequencies of different strategies and spatial pattern formations in different evolution situations. These results provide valuable insights into how to select and enforce suitable regulation measures to let public cooperation remain prevalent, which has potential implications not only for self-regulation, but also for other topics in economics and social science.

Close spatial arrangement of mutants favors and disfavors fixation.

Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in the expansion of mutants. However, the clustering of mutants and the game dynamics are typically coupled. It is still unclear how the clustering of mutants alone alters the evolutionary dynamics. To this end, we employ a minimal model with frequency independent fitness on a circle. It disentangles the clustering of mutants from game dynamics. The distance between two mutants on the circle is adopted as a natural indicator for the clustering of mutants or assortment. We find that the assortment is an amplifier of the selection for the connected mutants compared with the separated ones. Nevertheless, as mutants are separated, the more dispersed mutants are, the greater the chance of invasion is. It gives rise to the non-monotonic effect of clustering, which is counterintuitive. On the other hand, we find that less assortative mutants speed up fixation. Our model shows that the clustering of mutants plays a non-trivial role in fixation, which has emerged even if the game interaction is absent.

Experiment research on the evolution of cooperation and network game theory

<p indent=0mm>Focusing on the mechanisms of the evolution and emergence of cooperative behavior, this paper reviews the recent development of direct reciprocity and network reciprocity from perspectives of both theoretical and experimental research based on two-player Prisoner’s Dilemma and multi-player Public Goods games. Repeated interactions provide the necessary condition for the emergence of direct reciprocity. The zero-determinant strategy, which is a new theory in direct reciprocity, has been deeply investigated from both theory and experiment. The zero-determinant strategy is a kind of memory-one strategies, which can unilaterally determine the payoff relationship in repeated games and may provide a novel tool to explore human decision behavior. Investigations on networked game experiments revealed that dynamic networks can facilitate and maintain human cooperation, while the influence of static networks on cooperation remains controversial. Furthermore, it is summarized that networked game experiments provide an efficient approach to studying the formation mechanism of network structure and the emergence of cooperation in complex systems.

The evolution of cooperation in multi-games with aspiration-driven updating rule

Abstract We introduce multi-games with aspiration-driven updating rule and investigate the evolution of cooperation. In our work, a percentage of players randomly choose to play the PDG, while the remaining players play SG. Meanwhile, a part of players randomly select to use the aspiration-driven updating rule, and the other part of players select to use the Fermi-function updating rule. We find that both the diversity of sucker's payoff and updating rule have a positive effect on the cooperation. Besides, when the contribution of the sucker's payoff is larger or more players use the aspiration-driven rule to update the strategy, more cooperators appear derived from the spatial distribution of strategies and special network reciprocity. What is more, when more players choose to play SG and simultaneously use the aspiration-driven rule, cooperators gradually flourish. In addition, we also reveal that the easily contented players prefer to cooperate.

The effect of asymmetric reproductive ability on the evolution of cooperation on interdependent networks

In this paper, we consider an asymmetric reproductive ability on interdependent networks and investigate how this setting affects the evolution of cooperation. In detail, players decide to update their strategies at each step on main network (network B), while for sub network (network A), players update their strategies with a fixed probability p. Obviously, the system restores the traditional case when p = 1, where cooperation can survive by interdependent network reciprocity. And our asymmetric set-up comes into play when p < 1. Numerical simulation results show that our asymmetric coupling will hinder the overall cooperation level for small p. In detail, the introduction of asymmetric reproductive ability urges the formation of symmetry breaking and further weakens the positive impact by location synchronous effect. And the root cause is entirely distinct situation of utility differences on two networks. These observations further demonstrate a class of phenomena on interdependent networks that it would have catastrophic consequences on one network even if a unrelated change only occurs seemingly on another network.

A cooperative game model for the multimodality of coauthorship networks

This study provided a game model to simulate the evolution of coauthorship networks, which is a geometric hypergraph built on a circle. A fraction of nodes are randomly selected to attach an arc to express their reputation. The cooperation condition of a new node and existing nodes depends on their distance and the existing nodes’ reputation. The condition gives an expression of kin selection and network reciprocity, two typical mechanisms of cooperation. The size of a node’s reputation is expressed by the length of its arc, which is defined by a function of time and hyperdegree. The function describes the heterogeneity in the size of reputation on nodes and that in the fading speed of reputation on hyperdegrees. The model reveals that the heterogeneities can reproduce the dichotomy of node clustering and degree assortativity, as well as the trichotomy of degree and hyperdegree distributions: generalized Poisson, power-law, and exponential cutoff.

A stochastic Pairwise Fermi rule modified by utilizing the average in payoff differences of neighbors leads to increased network reciprocity in spatial prisoner's dilemma games

In a 2 × 2 prisoner's dilemma (PD) game, network reciprocity is one of the mechanisms for increasing social viscosity, which leads to a cooperative equilibrium. The Pairwise Fermi (PW-Fermi) rule has been accepted as an updating protocol, as its stochasticity is similar to the real-world human decision-making process. In this paper, we elucidated a modification to the PW-Fermi rule by utilizing the averaged payoff difference instead of the simple payoff difference between a focal agent and their neighbors. This led to a significantly enhanced level of network reciprocity. The mechanism of this enhancement is clarified by discussing the concepts of the enduring period (END) and the expanding period (EXP).

Direct reciprocity and model-predictive rationality explain network reciprocity over social ties

Since M. A. Nowak & R. May’s (1992) influential paper, limiting each agent’s interactions to a few neighbors in a network of contacts has been proposed as the simplest mechanism to support the evolution of cooperation in biological and socio-economic systems. The network allows cooperative agents to self-assort into clusters, within which they reciprocate cooperation. This (induced) network reciprocity has been observed in several theoreticalmodels and shown to predict the fixation of cooperation under a simple rule: the benefit produced by an act of cooperation must outweigh the cost of cooperating with all neighbors. However, the experimental evidence among humans is controversial: though the rule seems to be confirmed, the underlying modeling assumptions are not. Specifically, models assume that agents update their strategies by imitating better performing neighbors, even though imitation lacks rationality when interactions are far from all-to-all. Indeed, imitation did not emerge in experiments. What did emerge is that humans are conditioned by their own mood and that, when in a cooperative mood, they reciprocate cooperation. To help resolve the controversy, we design a model in which we rationally confront the two main behaviors emerging from experiments—reciprocal cooperation and unconditional defection—in a networked prisoner’s dilemma. Rationality is introduced by means of a predictive rule for strategy update and is bounded by the assumed model society. We show that both reciprocity and a multi-step predictive horizon are necessary to stabilize cooperation, and sufficient for its fixation, provided the game benefit-to-cost ratio is larger than a measure of network connectivity. We hence rediscover the rule of network reciprocity, underpinned however by a different evolutionary mechanism.