14.2 Replicator dynamics and population games
3 min read•august 7, 2024
model how strategy frequencies change in populations over time based on fitness. This powerful tool in evolutionary game theory helps us understand how successful strategies spread and less successful ones decline.
Fixed points, where strategy frequencies remain constant, are crucial in replicator dynamics. attract nearby states, while unstable ones repel them. This concept is key to predicting long-term evolutionary outcomes in populations.
Replicator Dynamics
Replicator Equation and Population State
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- describes how the frequency of strategies in a population changes over time based on their relative fitness
- represents the distribution of strategies within a population at a given time
- Changes in population state are driven by differences in fitness between strategies
- Strategies with higher fitness tend to increase in frequency while those with lower fitness decrease
Fitness Landscape and Selection Pressure
- maps the relationship between strategies and their corresponding fitness values
- Shape of the fitness landscape determines the direction and magnitude of
- Selection pressure drives the population towards strategies with higher fitness
- Peaks in the fitness landscape represent locally optimal strategies that are resistant to invasion by other strategies
- Valleys in the fitness landscape indicate strategies with lower fitness that are vulnerable to being outcompeted
Fixed Points and Stability
Stable and Unstable Fixed Points
- Fixed points are population states where the frequencies of strategies remain constant over time
- Stable fixed points are attractors in the replicator dynamics where nearby population states converge towards them
- are repellers where nearby population states diverge away from them
- Stability of fixed points depends on the local shape of the fitness landscape around them
- Stable fixed points occur at local peaks in the fitness landscape (evolutionary stable strategies)
- Unstable fixed points occur at local valleys or saddle points in the fitness landscape
Frequency-Dependent Selection
- occurs when the fitness of a strategy depends on its frequency in the population
- means a strategy's fitness increases as it becomes more common (network effects, economies of scale)
- means a strategy's fitness decreases as it becomes more common (resource competition, predator-prey dynamics)
- Frequency-dependent selection can lead to complex dynamics and multiple stable equilibria
- is an example of cyclical frequency-dependent selection where no single strategy dominates
Evolutionary Processes
Evolutionary Drift and Coexistence
- refers to random changes in strategy frequencies due to finite population size and stochastic effects
- Drift can cause populations to move between basins of attraction for different fixed points
- In the absence of selection, drift leads to the eventual fixation of a single strategy in the population
- of multiple strategies can occur when there are multiple stable fixed points in the replicator dynamics
- Coexistence requires negative frequency-dependent selection or other mechanisms that maintain diversity
- are population states where multiple strategies coexist at stable frequencies
- is an example where two strategies can coexist in a mixed equilibrium
Key Terms to Review (16)
Coexistence: Coexistence refers to the situation where multiple strategies or types can exist together in a population without one dominating the others. In this context, it highlights how different strategies can maintain their presence and function effectively even in competitive environments. This phenomenon is crucial for understanding the dynamics of strategy interactions and the balance of populations over time.
Evolutionary drift: Evolutionary drift refers to the random changes in the frequency of alleles (gene variants) in a population over time, which can lead to significant differences in traits within that population. This concept highlights how genetic variation can occur due to chance events, rather than being driven solely by natural selection. Understanding evolutionary drift is crucial for analyzing how populations change and adapt in uncertain environments, especially when considering the dynamics of strategies and behaviors among competing agents.
Evolutionary stable strategy: An evolutionary 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 connects to the idea of stability in evolutionary biology and game theory, highlighting how certain behaviors or strategies can become predominant within a population due to their ability to withstand challenges from competing strategies. The dynamic nature of populations and interactions emphasizes how these strategies evolve over time through replicator dynamics and various population games.
Fitness Landscape: A fitness landscape is a conceptual model used to visualize the relationship between genotypes or strategies and their reproductive success or fitness in a given environment. This landscape can be represented as a multi-dimensional surface where peaks represent high fitness and valleys represent low fitness, illustrating how populations evolve and adapt over time based on the dynamics of selection. In this model, the structure of the landscape can influence evolutionary trajectories, adaptation rates, and stability within populations.
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 population. This type of selection can lead to diverse strategies in populations, as the success of one phenotype can change depending on how common or rare it is. As a result, this dynamic can drive the evolution of traits and behaviors in various species, often seen in population games where interactions between individuals play a crucial role.
Matching pennies game: The matching pennies game is a simple two-player game where each player simultaneously chooses either heads or tails for a coin. The goal is for one player to match the other player's choice, while the other player aims to choose the opposite. This game illustrates key concepts in game theory, particularly with regard to mixed strategies and the concept of Nash equilibrium, as players must randomize their choices to avoid predictability.
Negative frequency-dependence: Negative frequency-dependence refers to a phenomenon in evolutionary biology where the fitness of a phenotype decreases as it becomes more common in a population. This concept is important in understanding how different strategies can coexist within a population, as it creates a dynamic balance where rarer strategies gain a fitness advantage, promoting diversity among phenotypes.
Polymorphic equilibria: Polymorphic equilibria refer to a situation in evolutionary game theory where multiple strategies coexist stably within a population, each having a positive frequency. This concept highlights the diversity of strategies that can be maintained in a population through mechanisms such as natural selection and frequency-dependent payoffs, allowing for the persistence of different traits or behaviors over time.
Population State: A population state refers to a specific configuration of individuals in a population characterized by their strategies and proportions at a given time. This concept is crucial in understanding how different strategies evolve over time within the context of replicator dynamics, as it provides a snapshot of the interactions among various strategies in a population game, determining which strategies may flourish or decline based on their relative fitness.
Positive Frequency-Dependence: Positive frequency-dependence refers to a scenario in which the fitness of a phenotype increases as it becomes more common in the population. This concept suggests that individuals with a more frequent trait may have advantages due to increased recognition or social preferences, thereby reinforcing the prevalence of that trait over time. In the context of replicator dynamics and population games, this idea is critical for understanding how strategies evolve based on their frequency and the interactions within a population.
Replicator Dynamics: Replicator dynamics is a mathematical framework used to model the evolution of strategies in population games, focusing on how certain strategies become more or less prevalent over time based on their relative success. This concept helps explain how populations adapt and evolve, capturing the idea that successful strategies tend to replicate more frequently, leading to changes in the overall composition of the population. It connects deeply to evolutionary biology and social behavior by providing insights into how competing strategies can evolve through interactions among individuals in a population.
Replicator equation: The replicator equation is a mathematical model used to describe the dynamics of strategies in evolutionary game theory, where the change in the frequency of a strategy over time is proportional to its fitness relative to the average fitness of the population. This concept is crucial for understanding how strategies evolve and spread within populations, particularly in contexts like population games. The equation captures how successful strategies become more common, driving the evolutionary process in competitive environments.
Rock-paper-scissors game: The rock-paper-scissors game is a simple hand game commonly used as a decision-making tool between two players, where each player simultaneously chooses one of three options: rock, paper, or scissors. The game has a cyclical structure where rock crushes scissors, scissors cuts paper, and paper covers rock, which makes it a classic example of a zero-sum game and a basic model for understanding strategic interactions in replicator dynamics and population games.
Selection Pressure: Selection pressure refers to the environmental factors that influence the survival and reproduction of individuals within a population, driving the process of natural selection. It determines which traits become more or less common in a population over time by favoring individuals with certain advantageous characteristics. In the context of replicator dynamics and population games, selection pressure plays a crucial role in shaping the strategies that individuals adopt, leading to the evolution of optimal behaviors over generations.
Stable fixed points: Stable fixed points are specific states in a dynamical system where the system tends to return if perturbed, indicating a balance in the underlying dynamics. In the context of replicator dynamics and population games, these points reflect strategies that remain consistently optimal, as they are resistant to small changes in the population's composition or strategy distribution. This stability can lead to equilibrium in populations, influencing evolutionary processes and strategic interactions among individuals.
Unstable fixed points: Unstable fixed points are specific solutions in dynamical systems where a small perturbation can lead to significant changes in the system's behavior, causing trajectories to diverge away from these points. In the context of replicator dynamics and population games, these fixed points can indicate strategies that are not robust against small deviations, making them critical for understanding evolutionary stability and strategy convergence in populations.