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Inertia in strategy switching transforms the strategy evolution

Physical Review E 2011年第6期84卷 066103-066103页

作者： Yanling Zhang Feng Fu Te Wu Guangming Xie Long Wang Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA

A recent experimental study [Traulsen et al., Proc. Natl. Acad. Sci. 107, 2962 (2010)] shows that human strategy updating involves both direct payoff comparison and the cost of switching strategy, which is equivalent to inertia. However, it remains largely unclear how such a predisposed inertia affects 2×2 games in a well-mixed population of finite size. To address this issue, the “inertia bonus” (strategy switching cost) is added to the learner payoff in the Fermi process. We find how inertia quantitatively shapes the stationary distribution and that stochastic stability under inertia exhibits three regimes, with each covering seven regions in the plane spanned by two inertia parameters. We also obtain the extended “1/3” rule with inertia and the speed criterion with inertia; these two findings hold for a population above two. We illustrate the above results in the framework of the Prisoner's Dilemma game. As inertia varies, two intriguing stationary distributions emerge: the probability of coexistence state is maximized, or those of two full states are simultaneously peaked. Our results may provide useful insights into how the inertia of changing status quo acts on the strategy evolution and, in particular, the evolution of cooperation.

关键词： INERTIA (Mechanics) PRISONER'S dilemma game PROBABILITY theory STOCHASTIC analysis PHYSICS research QUANTITATIVE research

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Effects of heterogeneous wealth distribution on public cooperation with collective risk

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Physical Review E 2010年第1期82卷 016102-016102页

作者： Jing Wang Feng Fu Long Wang Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University One Brattle Square Cambridge Massachusetts 02138 USA

The distribution of wealth among individuals in real society can be well described by the Pareto principle or “80-20 rule.” How does such heterogeneity in initial wealth distribution affect the emergence of public cooperation, when individuals, the rich and the poor, engage in a collective-risk enterprise, not to gain a profit but to avoid a potential loss? Here we address this issue by studying a simple but effective model based on threshold public goods games. We analyze the evolutionary dynamics for two distinct scenarios, respectively: one with fair sharers versus defectors and the other with altruists versus defectors. For both scenarios, particularly, we in detail study the dynamics of the population with dichotomic initial wealth—the rich versus the poor. Moreover, we demonstrate the possible steady compositions of the population and provide the conditions for stability of these steady states. We prove that in a population with heterogeneous wealth distribution, richer individuals are more likely to cooperate than poorer ones. Participants with lower initial wealth may choose to cooperate only if all players richer than them are cooperators. The emergence of pubic cooperation largely relies on rich individuals. Furthermore, whenever the wealth gap between the rich and the poor is sufficiently large, cooperation of a few rich individuals can substantially elevate the overall level of social cooperation, which is in line with the well-known Pareto principle. Our work may offer an insight into the emergence of cooperative behavior in real social situations where heterogeneous distribution of wealth among individual is omnipresent.

关键词： Pareto principle

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Partner switching stabilizes cooperation in coevolutionary prisoner’s dilemma

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Physical Review E 2009年第3期79卷 036101-036101页

作者： Feng Fu Te Wu Long Wang Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China

Previous studies suggest that cooperation prevails when individuals can switch their interaction partners quickly. However, it is still unclear how quickly individuals should switch adverse partners to maximize cooperation. To address this issue, we propose a simple model of coevolutionary prisoner’s dilemma in which individuals are allowed to either adjust their strategies or switch their defective partners. Interestingly, we find that, depending on the game parameter, there is an optimal tendency of switching adverse partnerships that maximizes the fraction of cooperators in the population. We confirm that the stabilization of cooperation by partner switching remains effective under some situations, where either normalized or accumulated payoff is used in strategy updating, and where either only cooperators or all individuals are privileged to sever disadvantageous partners. We also provide an extended pair approximation to study the coevolutionary dynamics. Our results may be helpful in understanding the role of partner switching in the stabilization of cooperation in the real world.

关键词： Stabilization

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Partner selections in public goods games with constant group size

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Physical Review E 2009年第2期80卷 026121-026121页

作者： Te Wu Feng Fu Long Wang Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA

Most of previous studies concerning the public goods game assume either participation is unconditional or the number of actual participants in a competitive group changes over time. How the fixed group size, prescribed by social institutions, affects the evolution of cooperation is still unclear. We propose a model where individuals with heterogeneous social ties might well engage in differing numbers of public goods games, yet with each public goods game being constant size during the course of evolution. To do this, we assume that each focal individual unidirectionally selects a constant number of interaction partners from his immediate neighbors with probabilities proportional to the degrees or the reputations of these neighbors, corresponding to degree-based partner selection or reputation-based partner selection, respectively. Because of the stochasticity the group formation is dynamical. In both selection regimes, monotonical dependence of the stationary density of cooperators on the group size was found, the latter over the whole range but the former over a restricted range of the renormalized enhancement factor. Moreover, the reputation-based regime can substantially improve cooperation. To interpret these differences, the microscopic characteristics of individuals are probed. We later extend the degree-based partner selection to general cases where focal individuals have preferences toward their neighbors of varying social ties to form groups. As a comparison, we as well investigate the situation where individuals locating on the degree regular graphs choose their coplayers at random. Our results may give some insights into better understanding the widespread teamwork and cooperation in the real world.

关键词： Condensed matter physics

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Emergence of social cooperation in threshold public goods games with collective risk

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Physical Review E 2009年第1期80卷 016101-016101页

作者： Jing Wang Feng Fu Te Wu Long Wang Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University One Brattle Square Cambridge Massachusetts 02138 USA

In real situations, people are often faced with the option of voluntary contribution to achieve a collective goal, for example, building a dam or a fence, in order to avoid an unfavorable loss. Those who do not donate, however, can free ride on others’ sacrifices. As a result, cooperation is difficult to maintain, leading to an enduring collective-risk social dilemma. To address this issue, here we propose a simple yet effective theoretical model of threshold public goods game with collective risk and focus on the effect of risk on the emergence of social cooperation. To do this, we consider the population dynamics represented by replicator equation for two simplifying scenarios, respectively: one with fair sharers, who contribute the minimum average amount versus defectors and the other with altruists contributing more than average versus defectors. For both cases, we find that the dilemma is relieved in high-risk situations where cooperation is likely to persist and dominate defection in the population. Large initial endowment to individuals also encourages the risk-averse action, which means that, as compared to poor players (with small initial endowment), wealthy individuals (with large initial endowment) are more likely to cooperate in order to protect their private accounts. In addition, we show that small donation amount and small threshold (collective target) can encourage and sustain cooperation. Furthermore, for other parameters fixed, the impacts of group size act differently on the two scenarios because of distinct mechanisms: in the former case where the cost of cooperation depends on the group size, large size of group readily results in defection, while easily maintains cooperation in the latter case where the cost of cooperation is fixed irrespective of the group size. Our theoretical results of the replicator dynamics are in excellent agreement with the individual based simulation results.

关键词： cooperation mechanisms simulation results Dilemma Simulation results Endowments Group Size zero fare public goods Threshold Replication Origin Emergence group cooperation

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Could Feedback-Based Self-Learning Help Solve Networked Prisoner's Dilemma?

Could Feedback-Based Self-Learning Help Solve Networked Pris...

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Joint 48th IEEE Conference on Decision and control and 28th Chinese control Conference

作者： Xiaojie Chen Feng Fu Long Wang State Key Laboratory for Turbulence and Complex Systems Center for Systems and ControlCollege of EngineeringPeking University Department of Zoology University of British Columbia Program for Evolutionary Dynamics Harvard University

We present a self-learning evolutionary Prisoner's Dilemma game model to study the evolution of cooperation in network-structured *** the evolutionary process,each agent updates its current strategy with a probability depending on the difference feedback between its actual score and score *** agent's score is a weighed mean of its payoff coming from its neighbors(social partners) and the payoff of its social partners obtaining from *** results show that the cooperation level in the structured populations increases with increasing the weight of partners' obtaining payoff in the *** interestingly,we find that very similar evolution of cooperation can respectively emerge in lattice,small-world and scale-free networks under the learningfeedback updating ***,we provide theoretical analysis and qualitative explanations for these numerical *** work may provide an effective way to solve the dilemma of cooperation for structured populations.

关键词： Could Feedback-Based Self-Learning Help Solve Networked Prisoner's Dilemma

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Social tolerance allows cooperation to prevail in an adaptive environment

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Physical Review E 2009年第5期80卷 051104-051104页

作者： Xiaojie Chen Feng Fu Long Wang Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Department of Zoology University of British Columbia Vancouver British Columbia Canada V6T 1Z4 Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA

In real situations, individuals often have moderate tolerance toward ambient cooperative environment in which they tend to avoid unfavorable interactions and search for favorable ones. How such social tolerance affects the evolution of cooperation and the resulting cooperative networks remains to be answered. To address this issue, here we present an effective model of co-evolutionary prisoner’s dilemma by introducing cooperative environment and social tolerance for networked players. An individual’s level of cooperative environment characterizes the cooperativity and sustainability of its interaction environment centered on itself. In our model, for paired individuals we assume that the one in better cooperative environment has a certain tolerance threshold to the opponent. If the opponent’s cooperative environment level is beyond the tolerance threshold, the one in better cooperative environment cuts unilaterally the link, and rewires to others. Otherwise, the link is not severed, and meanwhile an inhomogeneous strategy imitation process between them is considered. Moreover, a player’s cooperative environment is adjusted in response to the strategy choices in the neighborhood. Interestingly, we find that there exists a moderate tolerance threshold warranting the best promotion of cooperation. We explain the nontrivial results by investigating the time ratio of strategy (network) updating during the whole process and properties in emerging networks. Furthermore, we investigate the effect of memory-dependent discounting of individuals’ cooperative environment on the evolution of cooperation. We also demonstrate the robustness of our results by considering two other modified co-evolutionary rules. Our results highlight the importance of appropriate tolerance threshold for the evolution of cooperation during the entangled co-evolution of strategy and structure.

关键词： Sustainable development

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Evolutionary dynamics on graphs: Efficient method for weak selection

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Physical Review E 2009年第4期79卷 046707-046707页

作者： Feng Fu Long Wang Martin A. Nowak Christoph Hauert Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Department of Organismic and Evolutionary Biology and Department of Mathematics Harvard University Cambridge Massachusetts 02138 USA Department of Mathematics University of British Columbia Vancouver British Columbia Canada V6T 1Z2

Investigating the evolutionary dynamics of game theoretical interactions in populations where individuals are arranged on a graph can be challenging in terms of computation time. Here, we propose an efficient method to study any type of game on arbitrary graph structures for weak selection. In this limit, evolutionary game dynamics represents a first-order correction to neutral evolution. Spatial correlations can be empirically determined under neutral evolution and provide the basis for formulating the game dynamics as a discrete Markov process by incorporating a detailed description of the microscopic dynamics based on the neutral correlations. This framework is then applied to one of the most intriguing questions in evolutionary biology: the evolution of cooperation. We demonstrate that the degree heterogeneity of a graph impedes cooperation and that the success of tit for tat depends not only on the number of rounds but also on the degree of the graph. Moreover, considering the mutation-selection equilibrium shows that the symmetry of the stationary distribution of states under weak selection is skewed in favor of defectors for larger selection strengths. In particular, degree heterogeneity—a prominent feature of scale-free networks—generally results in a more pronounced increase in the critical benefit-to-cost ratio required for evolution to favor cooperation as compared to regular graphs. This conclusion is corroborated by an analysis of the effects of population structures on the fixation probabilities of strategies in general 2×2 games for different types of graphs. Computer simulations confirm the predictive power of our method and illustrate the improved accuracy as compared to previous studies.

关键词： dynamics

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Interaction stochasticity supports cooperation in spatial Prisoner’s dilemma

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Physical Review E 2008年第5期78卷 051120-051120页

作者： Xiaojie Chen Feng Fu Long Wang State Key Laboratory for Turbulence and Complex Systems Center for Systems and Control College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University Cambridge Massachusetts 02138 USA

Previous studies mostly assume deterministic interactions among neighboring individuals for games on graphs. In this paper, we relax this assumption by introducing stochastic interactions into the spatial Prisoner’s dilemma game, and study the effects of interaction stochasticity on the evolution of cooperation. Interestingly, simulation results show that there exists an optimal region of the intensity of interaction resulting in a maximum cooperation level. Moreover, we find good agreement between simulation results and theoretical predictions obtained from an extended pair-approximation method. We also show some typical snapshots of the system and investigate the mean payoffs for cooperators and defectors. Our results may provide some insight into understanding the emergence of cooperation in the real world where the interactions between individuals take place in an intermittent manner.

关键词： Game theory

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Consensus of population systems with community structures

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Physical Review E 2008年第5期78卷 051923-051923页

作者： Jing Wang Bin Wu Long Wang Feng Fu Center for Systems and Control State Key Laboratory for Turbulence and Complex Systems College of Engineering Peking University Beijing 100871 China Program for Evolutionary Dynamics Harvard University One Brattle Square Cambridge Massachusetts 02138 USA

Multicommunity population systems may reach a consensus state where the fractions of each species in different communities agree on a common value. In this paper, by analyzing the evolutionary dynamics based on an extended replicator equation incorporating community effects, the consensus problem of population systems with n communities is studied. In particular, the simple case of two communities is investigated in detail. In general, for n communities, a sufficient and necessary condition for population systems to reach a consensus of coexistent state is provided. Regarding the population dynamics for the four different types of games, whether the population systems can achieve consensus is determined. The dynamics of community-structured populations shows richer features than nonstructured populations, and some nontrivial phenomena arising from different community-structured population systems are illustrated with concrete numerical examples.

关键词： Population dynamics

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