In a recent Letter [F.C. Santos and J. M. Pacheco Phys. Rev. Lett. \textbf{95}, 098104 (2005)], the scale-free networks are found to be advantageous for the emergence of cooperation. In the present work an evolutionary prisoner's dilemma game with players located on Barabási-Albert scale-free networks is studied in detail. The players are pure strategist and can follow two strategies: either to defect or to cooperate. Serval alternative update rules determining the evolution of each player's strategy are considered. Using systematic Monte Carlo simulations we have calculated the average density of cooperators as a function of the temptation to defect. It is shown that the results obtained by numerical experiments depend strongly on the dynamics of the game, which could lower the important of scale-free topology on the persistence of the cooperation. Particularly, the system exhibits a phase transition, from active state (coexistence of cooperators and defectors) to absorbing state (only defectors surviving) when allowing `` worse\rq\rq strategy to be imitated in the evolution of the game.
Assessing quantitatively the state and dynamics of a social system is a very difficult problem. It is of great importance for both practical and theoretical reasons such as establishing the efficiency of social action programs, detecting possible community needs or allocating resources. In this paper we propose a new general theoretical framework for the study of social complexity, based on the relation of complexity and entropy in combination with evolutionary dynamics to asses the dynamics of the system. Imposing the second law of thermodynamics, we study the conditions under which cooperati…
We explore the emergence of cooperation in the framework of evolutionary game theory. First we introduce the cooperation problem in a novel way that we believe it have important consequences in how problem is addressed. Then we present a minimal model for the emergence of cooperation in growing systems with cultural reproduction where topological structure and the evolution of strategies are decoupled instead a coevolution dynamic. We show that when system grows, there exists cultural reproduction and a nonzero probability that individuals take cooperation as first strategy; there are conditio…
We study in detail the role of short-cuts in promoting the emergence of cooperation in a network of agents playing the Prisoner's Dilemma Game (PDG). We introduce a model whose topology interpolates between the one-dimensional euclidean lattice (a ring) and the complete graph by changing the value of one parameter (the probability p to add a link between two nodes not already connected in the euclidean configuration). We show that there is a region of values of p in which cooperation is largely enhanced, whilst for smaller values of p only a few cooperators are present in the final state, and …
The evolutionary Prisoner's Dilemma Game (PDG) and the Snowdrift Game (SG) with preferential learning mechanism are studied in the Barabási-Albert network. Simulation results demonstrate that the preferential learning of individuals remarkably promotes the cooperative behavior for both two games over a wide range of payoffs. To understand the effect of preferential learning on the evolution of the systems, we investigate the time series of the cooperator density for different preferential strength and payoffs. It is found that in some specific cases two games both show the $1/f$-scaling behavi…
By means of extensive computer simulations, the authors consider the entangled coevolution of actions and social structure in a new version of a spatial Prisoner's Dilemma model that naturally gives way to a process of social differentiation. Diverse social roles emerge from the dynamics of the system: leaders are individuals getting a large payoff who are imitated by a considerable fraction of the population, conformists are unsatisfied cooperative agents that keep cooperating, and exploiters are defectors with a payoff larger than the average one obtained by cooperators. The dynamics generat…
Evolutionary game theory is one of the key paradigms behind many scientific disciplines from science to engineering. Previous studies proposed a strategy updating mechanism, which successfully demonstrated that the scale-free network can provide a framework for the emergence of cooperation. Instead, individuals in random graphs and small-world networks do not favor cooperation under this updating rule. However, a recent empirical result shows the heterogeneous networks do not promote cooperation when humans play a Prisoner's Dilemma. In this paper, we propose a strategy updating rule with payo…
We study a condition of favoring cooperation in a Prisoner's Dilemma game on complex networks. There are two kinds of players: cooperators and defectors. Cooperators pay a benefit b to their neighbors at a cost c, whereas defectors only receive a benefit. The game is a death-birth process with weak selection. Although it has been widely thought that b/c> <k> is a condition of favoring cooperation, we find that b/c> <knn> is the condition. We also show that among three representative networks, namely, regular, random, and scale-free, a regular network favors cooperation the most, whereas a scal…
The emergence and prevalence of cooperative behavior within a group of selfish individuals remains a puzzle for \text{evolutionary game theory} precisely because it conflicts directly with the central idea of natural selection. Accordingly, in recent years, the search for an understanding of how cooperation can be stimulated, even when it conflicts with individual interest, has intensified. We investigate the emergence of cooperation in an age-structured evolutionary spatial game. In it, players age with time and the payoff that they receive after each round \text{depends on} their age. \text{…
According to evolutionary game theory, cooperation in public goods games is eliminated by free-riders, yet in nature, cooperation is ubiquitous. Artificial models resolve this contradiction via the mechanism of network reciprocity. However, existing research only addresses pre-existing networks and does not specifically consider their origins. Further, much work has focused on scale-free networks and so pre-supposes attachment mechanisms which may not exist in nature. We present a coevolutionary model of public goods games in networks, growing by random attachment, from small founding populati…
Models of cooperation grounded on social networks and on the ability of individuals to choose actions and partners aim to describe human social behavior. Extensive computer simulations of these models give important insight in the social mechanisms behind emerging collective behavior. We consider the entangled co-evolution of actions and social structure in a new version of a spatial Prisoner's Dilemma model that naturally gives way to a process of social differentiation. Diverse social roles emerge from the dynamics of the system: leaders -individuals getting a large payoff who are imitated b…
Predicting how cooperative behavior arises in the thermodynamic limit is one of the outstanding problems in evolutionary game theory. For two player games, cooperation is seldom the Nash equilibrium. However, in the thermodynamic limit cooperation is the natural recourse regardless of whether we are dealing with humans or animals. In this work, we use the analogy with the Ising model to predict how cooperation arises in the thermodynamic limit.
Cooperation is observed widely in nature and is thought an essential component of many evolutionary processes, yet the mechanisms by which it arises and persists are still unclear. Among several theories, network reciprocity -- a model of inhomogeneous social interactions -- has been proposed as an enabling mechanism to explain the emergence of cooperation. Existing evolutionary models of this mechanism have tended to focus on highly heterogeneous (scale-free) networks, hence typically assume preferential attachment mechanisms, and consequently the prerequisite that individuals have global net…
In this Letter, we study how cooperation is organized in complex topologies by analyzing the evolutionary (replicator) dynamics of the Prisoner's Dilemma, a two-players game with two available strategies, defection and cooperation, whose payoff matrix favors defection. We show that, asymptotically, the population is partitioned into three subsets: individuals that always cooperate ({\em pure cooperators}), always defect ({\em pure defectors}) and those that intermittently change their strategy. In fact the size of the latter set is the biggest for a wide range of the "stimulus to defect" param…
We study the problem of the emergence of cooperation in the spatial Prisoner's Dilemma. The pioneering work by Nowak and May showed that large initial populations of cooperators can survive and sustain cooperation in a square lattice with imitate-the-best evolutionary dynamics. We revisit this problem in a cost-benefit formulation suitable for a number of biological applications. We show that if a fixed-amount reward is established for cooperators to share, a single cooperator can invade a population of defectors and form structures that are resilient to re-invasion even if the reward mechanis…
Understanding the emergence and sustainability of cooperation is a fundamental problem in evolutionary biology and is frequently studied by the framework of evolutionary game theory. A very powerful mechanism to promote cooperation is network reciprocity, where the interaction patterns and opportunities for strategy spread of agents are constrained to limited sets of permanent interactions partners. Cooperation survives because it is possible for close-knit communities of cooperation to be shielded from invasion by defectors. Here we show that parameter ranges in which cooperation can survive …
Indirect reciprocity in which players cooperate with unacquainted other players having good reputations is a mechanism for cooperation in relatively large populations subjected to social dilemma situations. When the population has group structure, as is often found in social networks, players in experiments are considered to show behavior that deviates from existing theoretical models of indirect reciprocity. First, players often show ingroup favoritism (i.e., cooperation only within the group) rather than full cooperation (i.e., cooperation within and across groups), even though the latter is…
Peer punishment of free-riders (defectors) is a key mechanism for promoting cooperation in society. However, it is highly unstable since some cooperators may contribute to a common project but refuse to punish defectors. Centralized sanctioning institutions (for example, tax-funded police and criminal courts) can solve this problem by punishing both defectors and cooperators who refuse to punish. These institutions have been shown to emerge naturally through social learning and then displace all other forms of punishment, including peer punishment. However, this result provokes a number of que…
It is known that the small-world structure constitutes sufficient conditions to sustain cooperation and thus enhances cooperation. On the contrary, the network with a very long average distance is usually thought of as suppressing the emergence of the cooperation. In this paper we show that the fractal structure, of which the average distance is very long, does not always play a negative role in the organization of cooperation. Compared to regular networks, the fractal structure might even facilitate the emergence of cooperation. This mainly depends on the existence of locally compact clusters…
We study the combined influence of selection and random fluctuations on the evolutionary dynamics of two-strategy ("cooperation" and "defection") games in populations comprising cooperation facilitators. The latter are individuals that support cooperation by enhancing the reproductive potential of cooperators relative to the fitness of defectors. By computing the fixation probability of a single cooperator in finite and well-mixed populations that include a fixed number of facilitators, and by using mean field analysis, we determine when selection promotes cooperation in the important classes …
In social dilemmas, cooperation among randomly interacting individuals is often difficult to achieve. The situation changes if interactions take place in a network where the network structure jointly evolves with the behavioral strategies of the interacting individuals. In particular, cooperation can be stabilized if individuals tend to cut interaction links when facing adverse neighborhoods. Here we consider two different types of reaction to adverse neighborhoods, and all possible mixtures between these reactions. When faced with a gloomy outlook, players can either choose to cut and rewire …
We study an evolutionary spatial prisoner's dilemma game where the fitness of the players is determined by both the payoffs from the current interaction and their history. We consider the situation where the selection timescale is slower than the interaction timescale. This is done by implementing probabilistic reproduction on an individual level. We observe that both too fast and too slow reproduction rates hamper the emergence of cooperation. In other words, there exists an intermediate selection timescale that maximizes cooperation. Another factor we find to promote cooperation is a diversi…
We present insights and empirical results from an extensive numerical study of the evolutionary dynamics of the iterated prisoner's dilemma. Fixation probabilities for Moran processes are obtained for all pairs of 164 different strategies including classics such as TitForTat, zero determinant strategies, and many more sophisticated strategies. Players with long memories and sophisticated behaviours outperform many strategies that perform well in a two player setting. Moreover we introduce several strategies trained with evolutionary algorithms to excel at the Moran process. These strategies ar…
Much research has been carried out to understand the emergence of cooperation in simulated social networks of competing individuals. Such research typically implements a population as a single connected network. Here we adopt a more realistic premise; namely that populations consist of multiple networks, whose members migrate from one to another. Specifically, we isolate the key elements of the scenario where a minority of members from a cooperative network migrate to a network populated by defectors. Using the public goods game to model group-wise cooperation, we find that under certain circu…
We consider the problem of explaining the emergence and evolution of cooperation in dynamic network-structured populations. Building on seminal work by Poncela et al, which shows how cooperation (in one-shot prisoner's dilemma) is supported in growing populations by an evolutionary preferential attachment (EPA) model, we investigate the effect of fluctuations in the population size. We find that the fluctuating model is more robust than Poncela et al's in that cooperation flourishes for a wide variety of initial conditions. In terms of both the temptation to defect, and the types of strategies…
In this Brief Report we study the evolutionary dynamics of the Public Goods Game in a population of mobile agents embedded in a 2-dimensional space. In this framework, the backbone of interactions between agents changes in time, allowing us to study the impact that mobility has on the emergence of cooperation in structured populations. We compare our results with a static case in which agents interact on top of a Random Geometric Graph. Our results point out that a low degree of mobility enhances the onset of cooperation in the system while a moderate velocity favors the fixation of the full-c…
In this manuscript we explore the onset of cooperative traits in the Public Goods game. This well-known game involves N-agent interactions and thus reproduces a large number of social scenarios in which cooperation appears to be essential. Many studies have recently addressed how the structure of the interaction patterns influences the emergence of cooperation. Here we study how information about the payoffs collected by each individual in the different groups it participates in, influences the decisions made by its group partners. Our results point out that cross-information plays a fundament…
Emergence of cooperation: A reputation-modulated reinforcement learning
School of Physics, East China Normal University（华东师范大学物理系）；School of Physics, Ningxia University（宁夏大学物理学院）；School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理与信息工程学院）
In many social dilemmas, individuals tend to generate a situation with low payoffs instead of a system optimum ("tragedy of the commons"). Is the routing of traffic a similar problem? In order to address this question, we present experimental results on humans playing a route choice game in a computer laboratory, which allow one to study decision behavior in repeated games beyond the Prisoner's Dilemma. We will focus on whether individuals manage to find a cooperative and fair solution compatible with the system-optimal road usage. We find that individuals tend towards a user equilibrium with …
The effects of networking on the extent of cooperation emerging in a competitive setting are studied. The evolutionary snowdrift game, which represents a realistic alternative to the well-known Prisoner's Dilemma, is studied in the Watts-Strogatz network that spans the regular, small-world, and random networks through random re-wiring. Over a wide range of payoffs, a re-wired network is found to suppress cooperation when compared with a well-mixed or fully connected system. Two extinction payoffs, that characterize the emergence of a homogeneous steady state, are identified. It is found that, …
Emergence of cooperation in nonlinear higher-order public goods games
Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（交叉物理与复杂系统研究所）；Department of Network and Data Science, Central European University Vienna（中欧大学维也纳网络与数据科学系）；Human Evolutionary Ecology Group, Department of Evolutionary Anthropology, University of Zurich（苏黎世大学生物人类学系人类进化生态组）；Center for Networks and Complexity, Luiss University of Rome（罗马LUISS大学网络与复杂性中心）；Department of AI, Data and Decision Sciences, Luiss University of Rome（罗马LUISS大学人工智能、数据与决策科学系）
The hypothesis that living systems can benefit from operating at the vicinity of critical points has gained momentum in recent years. Criticality may confer an optimal balance between exceedingly ordered and too noisy states. We here present a model, based on information theory and statistical mechanics, illustrating how and why a community of agents aimed at understanding and communicating with each other converges to a globally coherent state in which all individuals are close to an internal critical state, i.e. at the borderline between order and disorder. We study --both analytically and c…
We study the effects of mobility on the evolution of cooperation among mobile players, which imitate collective motion of biological flocks and interact with neighbors within a prescribed radius $R$. Adopting the prisoner's dilemma game and the snowdrift game as metaphors, we find that cooperation can be maintained and even enhanced for low velocities and small payoff parameters, when compared with the case that all agents do not move. But such enhancement of cooperation is largely determined by the value of $R$, and for modest values of $R$, there is an optimal value of velocity to induce the…
Game theory is fundamental to understanding cooperation between agents. Mainly, the Prisoner's Dilemma is a well-known model that has been extensively studied in complex networks. However, although the emergence of cooperation has been investigated before, the influence of memory in its evolution is not well understood. This paper presents a detailed study of cooperation dynamics in which agents have memory. We simulate the evolutionary Prisoner's dilemma game on random, scale-free and networks presenting degree-degree correlation. Through extensive simulations, we show that assortativity can …
Cooperation is a major factor in the evolution of human societies. The structure of human social networks, which affects the dynamics of cooperation and other interpersonal phenomena, have common structural signatures. One of these signatures is the tendency to organize as groups. Among the generative models that network theorists use to emulate this feature is the Stochastic Block Model (SBM). In this paper, we study evolutionary game dynamics on SBM networks. Using a recently-discovered duality between evolutionary games and coalescing random walks, we obtain analytical conditions such that …
The evolution of cooperation among unrelated individuals in human and animal societies remains a challenging issue across disciplines. It is an important subject also in the evolutionary game theory to research how cooperation arises. The subject has been extensively studied especially in Prisonars' dilemma game(PD) and the emergence of cooperation is important subject also in public goods game(PGG). In this article, we consider evolutionary PGG on complex networks where the topology of the networks varies under the infulence of game dynamics. Then we study what effects on the evolution of pla…
In this paper, the total payoff of each agent is regulated to reduce the heterogeneity of the distribution of the total payoffs. It is found there is an optimal regulation strength where the fraction of cooperation is prominently promoted, too weak or too strong of the strength will have little effects or result in the disappearance of the cooperators. It is also found that most of the cooperators are not distributed in isolation but form the cooperator clusters, and to promote the cooperation the only way is to enlarge the size of the cooperator clusters. Finally, we try to explain the emerge…
We study the evolutionary Prisoner's Dilemma on two social networks obtained from actual relational data. We find very different cooperation levels on each of them that can not be easily understood in terms of global statistical properties of both networks. We claim that the result can be understood at the mesoscopic scale, by studying the community structure of the networks. We explain the dependence of the cooperation level on the temptation parameter in terms of the internal structure of the communities and their interconnections. We then test our results on community-structured, specifical…
Most real life systems have a random component: the multitude of endogenous and exogenous factors influencing them result in stochastic fluctuations of the parameters determining their dynamics. These empirical systems are in many cases subject to noise of multiplicative nature. The special properties of multiplicative noise as opposed to additive noise have been noticed for a long while. Even though apparently and formally the difference between free additive vs. multiplicative random walks consists in just a move from normal to log-normal distributions, in practice the implications are much …
We study the self-assembly of a complex network of collaborations among self-interested agents. The agents can maintain different levels of cooperation with different partners. Further, they continuously, selectively, and independently adapt the amount of resources allocated to each of their collaborations in order to maximize the obtained payoff. We show analytically that the system approaches a state in which the agents make identical investments, and links produce identical benefits. Despite this high degree of social coordination some agents manage to secure privileged topological position…
Social structure affects the emergence and maintenance of cooperation. Here we study the evolutionary dynamics of cooperation in fragmented societies, and show that conjoining segregated cooperation-inhibiting groups, if done properly, rescues the fate of collective cooperation. We highlight the essential role of inter-group ties, that sew the patches of the social network together and facilitate cooperation. We point out several examples of this phenomenon in actual settings. We explore random and non-random graphs, as well as empirical networks. In many cases we find a marked reduction of th…
Social networks have a scale-free property and community structure, and many problems in life have the characteristic of public goods, such as resource shortage. Due to different preferences of individuals, there exist individuals who adopt heterogeneous strategies updating rules in the network. We investigate the evolution of cooperation in the scale-free community network with public goods games and the influence of multiple strategy updating rules. Here, two types of strategy updating rules are considered which are pairwise comparison rules and aspiration-driven rules. Numerical simulations…
We outline a model to study the evolution of cooperation in a population of agents playing the prisoner's dilemma in signed networks. We highlight that if only dyadic interactions are taken into account, cooperation never evolves. However, when triadic considerations are introduced, a window of opportunity for emergence of cooperation as a stable behaviour emerges.
The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that cooperator clusters in spatial games are effective to protect cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that assures the prevalence of cooperation in a system. Here, we study the dynamical organization of cooperator clusters in spatial prisoner's dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized b…
We study the problem of the emergence of cooperation in dynamic signed networks where agent strategies coevolve with relational signs and network topology. Running simulations based on an agent-based model, we compare results obtained in a regular lattice initialization with those obtained on a comparable random network initialization. We show that the increased degree heterogeneity at the outset enlarges the parametric conditions in which cooperation survives in the long run. Furthermore, we show how the presence of sign-dependent emotional strategies catalyze the evolution of cooperation wit…
Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini. They showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighb…
The emergence of cooperation among self-interested agents has been a key concern of the multi-agent systems community for decades. With the increased importance of network-mediated interaction, researchers have shifted the attention on the impact of social networks and their dynamics in promoting or hindering cooperation, drawing various context-dependent conclusions. For example, some lines of research, theoretical and experimental, suggest the existence of a threshold effect in the ratio of timescales of network evolution, after which cooperation will emerge, whereas other lines dispute this…
