Glossary

11.2 Evolutionary stable strategies and replicator dynamics

2 min read•july 25, 2024

Evolutionary game theory explores how strategies evolve in populations over time. It combines game theory with evolutionary biology, using concepts like evolutionary stable strategies () and to model and predict long-term behavioral outcomes.

The fundamentals of evolutionary game theory include ESS criteria, which determine if a strategy can resist invasion by alternatives. Replicator dynamics model strategy frequency changes based on fitness, while stability analysis helps identify equilibrium points and their stability in evolving populations.

Evolutionary Game Theory Fundamentals

Evolutionary stable strategies

Top images from around the web for Evolutionary stable strategies

  • Nash equilibrium - Simulace.info View original

    Is this image relevant?

  • Nash equilibrium - Wikipedia View original

    Is this image relevant?

  • Nash equilibrium - Simulace.info View original

    Is this image relevant?

1 of 3

Top images from around the web for Evolutionary stable strategies

  • Nash equilibrium - Simulace.info View original

    Is this image relevant?

  • Nash equilibrium - Wikipedia View original

    Is this image relevant?

  • Nash equilibrium - Simulace.info View original

    Is this image relevant?

1 of 3
  • ESS strategy adopted by population resists invasion by alternative strategies representing Nash equilibrium with evolutionary stability
  • ESS framework explains persistence of behaviors in populations predicts long-term evolutionary outcomes and dominance of certain traits
  • ESS characteristics include resistance to mutant strategies and self-reinforcement once established
  • ESS applications include animal behavior studies (mating rituals), human decision-making analysis (business competition), and evolution of cooperation investigations (prisoner's dilemma)

Criteria for ESS

  • Mathematical criteria for ESS using payoff function E(A,B)E(A,B) for strategy A against B
  • Strategy I is ESS if:
    1. E(I,I)≥E(J,I)E(I,I) \geq E(J,I) for all alternative J
    2. If E(I,I)=E(J,I)E(I,I) = E(J,I), then E(I,J)>E(J,J)E(I,J) > E(J,J)
  • Condition 1 means I performs at least as well against itself as alternatives
  • Condition 2 requires I to outperform J against J if J performs equally well against I
  • ESS determination methods include analysis best response functions and stability analysis

Replicator Dynamics and Stability Analysis

Replicator dynamics in evolution

  • Mathematical framework models strategy frequency changes in population over time based on fitness
  • Components include population state fitness function and replicator equation
  • Captures dynamic strategy evolution provides insights into long-term behavior
  • Assumes large population size asexual reproduction or genetic strategy inheritance and continuous time model
  • Replicator equation: dxidt=xi(fi(x)−fˉ(x))\frac{dx_i}{dt} = x_i(f_i(x) - \bar{f}(x)) where xix_i is strategy i frequency fi(x)f_i(x) is strategy i fitness and fˉ(x)\bar{f}(x) is average population fitness

Stability analysis of strategies

  • Fixed point analysis identifies equilibrium points where dxidt=0\frac{dx_i}{dt} = 0 for all strategies
  • Stability determined by analyzing Jacobian matrix eigenvalues
  • Phase diagrams graphically represent strategy dynamics with axes showing strategy frequencies
  • Phase diagram components include trajectories showing frequency changes over time and fixed points indicating equilibrium states
  • Stable equilibria act as attractors unstable equilibria as repellers and saddle points attract in some directions repel in others
  • ESS correspond to asymptotically stable fixed points in replicator dynamics but not all stable fixed points are necessarily ESS

Key Terms to Review (14)

Bishop's Theorem: Bishop's Theorem refers to a fundamental result in evolutionary game theory that characterizes evolutionary stable strategies (ESS). It provides a criterion for determining whether a strategy is an ESS by examining the fitness of a mutant strategy when introduced into a population predominantly composed of a resident strategy. The theorem is critical for understanding how certain strategies can persist in the face of competing alternatives through the lens of replicator dynamics.
Equilibrium Point: An equilibrium point is a state in a dynamic system where the variables remain constant over time, indicating that the forces acting on the system are balanced. This concept is crucial in understanding population dynamics, resource limitations, and evolutionary strategies, as it signifies stability in the face of environmental changes or interactions among species.
Ess: An evolutionary stable strategy (ess) is a strategy in a population that, if adopted by a majority of individuals, cannot be invaded by any alternative strategy. It represents a form of equilibrium where any small fraction of the population adopting a different strategy will not thrive or reproduce effectively, ensuring that the original strategy remains prevalent. This concept connects to broader ideas of stability and adaptation in populations, where strategies evolve over time through interactions among individuals.
Evolutionarily stable strategy: An evolutionarily stable strategy (ESS) is a strategy that, if adopted by a population, cannot be invaded by any alternative strategy that is initially rare. This concept helps explain how certain behaviors or traits can persist in a population over time, as they confer a selective advantage against any competing strategies. ESS is crucial in understanding the dynamics of cooperation, competition, and conflict within biological systems.
Fitness landscape: A fitness landscape is a conceptual model that represents the relationship between genotypes (or phenotypes) and their reproductive success, or 'fitness'. It visualizes how different genetic variations can lead to varying levels of fitness within a population, illustrating peaks (high fitness) and valleys (low fitness) that signify the adaptive potential of different traits in a given environment.
Frequency-dependent selection: Frequency-dependent selection is an evolutionary process where the fitness of a phenotype depends on its frequency relative to other phenotypes in a given population. This concept highlights how interactions between different species or individuals can change the success of a trait based on how common or rare it is, influencing competition, cooperation, and strategic interactions among organisms.
John Maynard Smith: John Maynard Smith was a British evolutionary biologist known for his contributions to the field of evolutionary game theory and the concept of evolutionary stable strategies (ESS). His work established a framework for understanding the dynamics of competition and cooperation among organisms, emphasizing how strategies can persist in populations over time. His insights helped bridge the gap between biology and mathematics, providing tools to analyze interactions in evolutionary contexts.
Kin selection: Kin selection is a type of natural selection that favors the reproductive success of an organism's relatives, even at a cost to the organism's own survival and reproduction. This phenomenon helps explain altruistic behaviors observed in nature, where individuals may sacrifice their own fitness for the benefit of their kin, ultimately enhancing the overall genetic success of shared relatives.
Natural selection: Natural selection is a fundamental process in evolutionary biology where organisms better adapted to their environment tend to survive and produce more offspring. This process leads to the gradual evolution of species over generations as advantageous traits become more common. It plays a crucial role in shaping the genetic makeup of populations and contributes to the development of evolutionary stable strategies within the framework of replicator dynamics.
Payoff matrix: A payoff matrix is a table that displays the potential outcomes of a strategic interaction between two or more players, representing their choices and the resulting payoffs for each combination of strategies. It serves as a critical tool in game theory, especially in biological contexts where individuals or species interact and make decisions based on their expected outcomes. Understanding the payoff matrix helps illustrate how various strategies can lead to different evolutionary outcomes, including the concept of evolutionary stable strategies.
Population dynamics: Population dynamics refers to the changes in population size, structure, and distribution over time, influenced by birth rates, death rates, immigration, and emigration. This concept helps in understanding how populations grow, shrink, or stabilize under various environmental pressures and interactions, such as competition and predation.
Replicator dynamics: Replicator dynamics is a mathematical framework used to model the evolution of strategies within populations, where the success of each strategy influences its frequency in future generations. This concept helps in understanding how different strategies compete for resources and how their prevalence changes over time, based on their relative fitness. It connects deeply with evolutionary stable strategies and provides insights into complex systems like cancer development and treatment optimization.
Resource Allocation: Resource allocation refers to the process of distributing available resources among various competing activities or strategies to maximize effectiveness and efficiency. In the context of evolutionary stable strategies and replicator dynamics, resource allocation is crucial for understanding how individuals or species make decisions about investing their energy, time, and resources into different behaviors or traits that can influence their survival and reproduction.
Robert Axelrod: Robert Axelrod is a prominent political scientist known for his work on cooperation in social systems, particularly through the lens of game theory and evolutionary dynamics. His research has provided insights into how cooperation can emerge and be sustained among self-interested agents, which ties into concepts like evolutionary stable strategies and replicator dynamics.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide