Stochastic dynamics of invasion and fixation.

We study evolutionary game dynamics in finite populations. We analyze an evolutionary process, which we call pairwise comparison, for which we adopt the ubiquitous Fermi distribution function from statistical mechanics. The inverse temperature in this process controls the intensity of selection, leading to a unified framework for evolutionary dynamics at all intensities of selection, from random drift to imitation dynamics. We derive a simple closed formula that determines the feasibility of cooperation in finite populations, whenever cooperation is modeled in terms of any symmetric two-person game. In contrast with previous results, the present formula is valid at all intensities of selection and for any initial condition. We investigate the evolutionary dynamics of cooperators in finite populations, and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations, as a function of a given initial number of cooperators, showing how this interplay strongly affects the approach to fixation of a given trait in finite populations, leading to counterintuitive results at different intensities of selection.

• Publication year

  2006

• Venue

  Physical review. E, Statistical, nonlinear, and soft matter physics

• Publication date

  2006-07-17

• Fields of study

  Biology, Mathematics, Physics, Medicine

• Identifiers
  DOI 10.1103/PhysRevE.74.011909arXiv q-bio/0609020PMID 16907129PMCID PMC2904085
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

• Biased social learning

  2009cited by this paper
• Pairwise comparison and selection temperature in evolutionary game dynamics.

  2007cited by this paper
• Evolutionary dynamics of social dilemmas in structured heterogeneous populations.

  2006cited by this paper
• Synergy and discounting of cooperation in social dilemmas.

  2006cited by this paper
• J. Math. Biol. (2006) Mathematical Biology Digital Object Identifier (DOI):

  2005cited by this paper
• Game theory and physics

  2005cited by this paper
• Scale-free networks provide a unifying framework for the emergence of cooperation.

  2005cited by this paper
• Stochastic Processes in Physics and Chemistry

  2005cited by this paper
• IL-13受体α2降低血吸虫病肉芽肿的炎症反应并延长宿主存活时间[英]/Mentink-Kane MM,Cheever AW，Thompson RW，et al//Proc Natl Acad Sci U S A

  2005cited by this paper
• Evolutionary Dynamics of Biological Games

  2004cited by this paper
• Coevolutionary dynamics: from finite to infinite populations.

  2004cited by this paper
• Evolutionary game dynamics in finite populations

  2004cited by this paper
• Emergence of cooperation and evolutionary stability in finite populations

  2004cited by this paper
• Spatial structure often inhibits the evolution of cooperation in the snowdrift game

  2004cited by this paper
• Evolutionary prisoner's dilemma game on hierarchical lattices.

  2004cited by this paper
• Cooperation for volunteering and partially random partnerships.

  2004cited by this paper
• Effects of neighbourhood size and connectivity on the spatial Continuous Prisoner's Dilemma.

  2004cited by this paper
• Prisoner's dilemma and public goods games in different geometries: Compulsory versus voluntary interactions

  2003cited by this paper
• Near-neutrality in evolution of genes and gene regulation

  2002cited by this paper
• Learning dynamics in social dilemmas

  2002cited by this paper
• Evolutionary Games and Population Dynamics

  1998cited by this paper
• Evolutionary prisoner's dilemma game on a square lattice

  1997cited by this paper
• Spatial evolutionary game theory: Hawks and Doves revisited

  1996cited by this paper
• Stochastic Processes in Physics and Chemistry

  1995cited by this paper
• Spatial games and the maintenance of cooperation.

  1994cited by this paper
• The Statistical Mechanics of Strategic Interaction

  1993cited by this paper
• Evolutionary games and spatial chaos

  1992cited by this paper
• The neutral theory of molecular evolution

  1985cited by this paper
• The neutral theory of molecular evolution.

  1983cited by this paper
• Handbook of Stochastic Methods

  1983cited by this paper
• Mathematical population genetics

  1981cited by this paper
• Evolutionarily Stable Strategies and Game Dynamics

  1978cited by this paper
• Evolution and the theory of games

  1976cited by this paper
• Nature (London

  1975cited by this paper
• The Logic of Animal Conflict

  1973cited by this paper
• A First Course on Stochastic Processes

  1968cited by this paper
• A First Course in Stochastic Processes

  1967cited by this paper
• Evolution and the theory of games.

  1961cited by this paper
• Journal of physics

  1939cited by this paper

• Methanol production technology development pathways under multi-agent stochastic evolutionary game: a complex network perspective

  2026cites this paper
• Research on the diffusion of low-carbon technologies through trilateral evolutionary game based on incentive mechanisms

  2026cites this paper
• Cooperation in collective dilemmas under opinion-based risk perceptions

  2026cites this paper
• Fairness Dynamics in Digital Economy Platforms with Biased Ratings

  2026cites this paper
• Evolutionary dynamics of reactive partner strategy in stochastic games

  2026cites this paper
• Non-participant externalities reshape the evolution of altruistic punishment.

  2026cites this paper
• Cooperative or competitive? Resolving social dilemmas in autonomous vehicles through evolutionary game theory.

  2026cites this paper
• Disentangling trust from cooperation: Evolution of trust as reduced monitoring in social dilemmas

  2026cites this paper
• The dynamics of cooperation in asymmetric public goods games

  2026cites this paper
• Structure-aware imitation dynamics on higher-order networks

  2026cites this paper
• Impact of uncertain payoffs on evolution of cooperation in TFT-AllD game based on imitation dynamics

  2026cites this paper
• Evolution of fairness in hybrid populations with specialised AI agents

  2026influential citation
• Disentangling trust from cooperation: Trust as reduced monitoring across social dilemmas

  2025influential citation
• Environmental feedback promotes cooperation in a spatial prisoner's dilemma game with preferential selection

  2025cites this paper
• Exploring cooperation and defection in multi-agent systems via Ising model Hamiltonian

  2025cites this paper
• Evolution of Coordination Through Institutional Incentives: An Evolutionary Game Theory Approach

  2025influential citation
• A multi-agent reinforcement learning framework for exploring dominant strategies in iterated and evolutionary games

  2025cites this paper
• A new electoral bottom-up model of institutional governance

  2025cites this paper
• Reporting is a double-edged sword for promoting cooperation in the public goods game.

  2025cites this paper
• Effects of tag mediation and structural balance on the evolution of cooperation on signed networks.

  2025cites this paper
• Evolution of boundedly rational learning in games

  2025cites this paper
• Reconstruction of micro-dynamics characterizing human decision-making behavior in repeated social dilemmas

  2025cites this paper
• Heterogeneous fragmentation of empty sites promotes cooperation in phenotypically diverse populations with tag-mediated interactions

  2025cites this paper
• Non-participant externalities reshape the evolution of altruistic punishment

  2025cites this paper
• Dynamics of cooperation in concurrent games

  2025cites this paper
• Can Media Act as a Soft Regulator of Safe AI Development? A Game Theoretical Analysis

  2025cites this paper
• Bayesian tit-for-tat fosters cooperation in evolutionary stochastic games.

  2025cites this paper
• The Evolution of Trust under Institutional Moral Hazard

  2025cites this paper
• Social imitation dynamics of vaccination driven by vaccine effectiveness and beliefs

  2025influential citation
• Memory-based prisoner’s dilemma game with payoff-driven preferential selection

  2025cites this paper
• Systemic financial risk analysis of the U.S. based on the complex network

  2025cites this paper
• Prejudice-driven spite: A discontinuous phase transition in the ultimatum game.

  2025cites this paper
• Breaking coexistence: Zealotry vs nonlinear social impact.

  2025cites this paper
• Public goods games with environmental feedbacks in well-mixed and structured populations

  2025cites this paper
• Words are not wind - how public joint commitment and reputation solve the Prisoner's Dilemma

  2025cites this paper
• Emergence of Cooperation and Commitment in Optional Prisoner's Dilemma

  2025cites this paper
• Impact of multiple delays on the governance of risky commons.

  2025cites this paper
• Evolutionary game dynamics of index insurance with refund or non-refund risk sharing mechanism.

  2025cites this paper
• Media and responsible AI governance: a game-theoretic and LLM analysis

  2025influential citation
• Chaos and noise in evolutionary game dynamics

  2025cites this paper
• How large language models judge and influence human cooperation

  2025cites this paper
• The Dominant Role of Exit Familiarity over Crowd Interactions and Spatial Layout in Pedestrian Evacuation Efficiency

  2025cites this paper
• Drivers of cooperation in social dilemmas on higher‐order networks

  2025cites this paper
• Network reciprocity turns cheap talk into a force for cooperation.

  2025cites this paper
• Cooperation versus social welfare.

  2025cites this paper
• Coevolution of individual perception and cooperative behavior in the norm compliance dilemma.

  2025cites this paper
• Social welfare optimisation in well-mixed and structured populations

  2025cites this paper
• Public goods cooperation via group interaction with opponent selection

  2025cites this paper
• Higher-order evolutionary dynamics with game transitions.

  2025cites this paper
• Environmental stochastic fluctuations lead to different evolutionary directions of strategies

  2025cites this paper
• Evolution favours positively biased reasoning in sequential interactions with high future gains

  2025cites this paper
• Co-evolution of risk and cooperation in climate policies under wealth inequality

  2024cites this paper
• Memory-based spatial evolutionary prisoner’s dilemma

  2024cites this paper
• The impact of time delays on mutant fixation in pairwise social dilemma games

  2024cites this paper
• Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level Competition

  2024cites this paper
• Agent-based evolutionary game dynamics uncover the dual role of resource heterogeneity in the evolution of cooperation.

  2024cites this paper
• A study of factors in the formation of population game cooperation based on mixed learning rules

  2024cites this paper
• Emergence of cooperation in the one-shot Prisoner’s dilemma through Discriminatory and Samaritan AIs

  2024influential citation
• Green technology diffusion mechanism in China’s aviation industry cluster based on complex network game model

  2024cites this paper
• Evolution of cooperation with early social influence for explaining collective action.

  2024cites this paper
• The effect of multi-tasks mechanism on cooperation in evolutionary game.

  2024cites this paper
• A social monitoring mechanism for third-party judges promotes cooperation in evolutionary games

  2024cites this paper
• Evolutionary dynamics in stochastic nonlinear public goods games

  2024cites this paper
• q-deformed evolutionary dynamics in simple matrix games.

  2024cites this paper
• To Promote Full Cooperation in Social Dilemmas, Agents Need to Unlearn Loyalty

  2024cites this paper
• Coevolutionary game dynamics with localized environmental resource feedback.

  2024cites this paper
• Social learning with complex contagion

  2024cites this paper
• The nested structures of higher-order interactions promote the cooperation in complex social networks

  2024cites this paper
• Computational evolution of social norms in well-mixed and group-structured populations

  2024cites this paper
• The evolution of cooperation affected by unidirectional acceptability mechanism on interdependent networks

  2024cites this paper
• Social dilemma of nonpharmaceutical interventions: Determinants of dynamic compliance and behavioral shifts

  2024cites this paper
• Network reciprocity and inequality: The role of additional mixing links among social groups

  2024cites this paper
• Opinion dynamics on biased dynamical networks: Beyond rare opinion updating.

  2024cites this paper
• Trust AI Regulation? Discerning users are vital to build trust and effective AI regulation

  2024cites this paper
• The whale optimization algorithm and the evolution of cooperation in the spatial public goods game

  2024cites this paper
• The Impact of Time Delay on Mutant Fixation in Evolutionary Games

  2024cites this paper
• The implications of deep cooperation strategy for the evolution of cooperation in social dilemmas

  2024cites this paper
• Success-driven opinion formation determines social tensions

  2024cites this paper
• On some dynamical features of the complete Moran model for neutral evolution in the presence of mutations

  2024cites this paper
• How norms shape the evolution of prosocial behaviour–compassion, universalizability, reciprocity and equity: a C.U.R.E for social dilemmas

  2024cites this paper
• Evolution of Cooperation Among Unequals with Game-Environment Feedback

  2024cites this paper
• Impact of environmental stochastic fluctuations on the evolutionary stability of imitation dynamics.

  2024cites this paper
• Both eyes open: Vigilant Incentives help auditors improve AI safety

  2024cites this paper
• Evolution of a theory of mind

  2024cites this paper
• The distorting effects of producer strategies: Why engagement does not reveal consumer preferences for misinformation

  2024cites this paper
• Impact of individual sensitivity to payoff difference between individuals on a discrete-time imitation dynamics

  2023cites this paper
• Adaptations to infer fitness interdependence promote the evolution of cooperation

  2023cites this paper
• A robust way to speed up consensus via adaptive social networks

  2023cites this paper
• Imitation dynamics on networks with incomplete information

  2023cites this paper
• A geometric process of evolutionary game dynamics

  2023cites this paper
• EGTtools: Evolutionary game dynamics in Python.

  2023cites this paper
• Cooperation dynamics in public goods games with evolving cognitive bias

  2023cites this paper
• Task Allocation Game with the Loss Function

  2023cites this paper
• Is cooperation sustained under increased mixing in evolutionary public goods games on networks?

  2023cites this paper
• Rewarding policies in an asymmetric game for sustainable tourism

  2023cites this paper
• Exit options sustain altruistic punishment and decrease the second-order free-riders, but it is not a panacea

  2023cites this paper
• Minimum-time strategy optimization for networked evolutionary games with bankruptcy mechanism

  2023cites this paper
• Indirect reciprocity in the public goods game with collective reputations

  2023cites this paper
• The evolution and social cost of herding mentality promote cooperation

  2023cites this paper
• Seeding Leading Cooperators and Institutions in Networked Climate Dilemmas

  2023cites this paper