How does an evolutionary stable strategy impact competition in a market?

An evolutionary stable strategy impacts market competition by promoting strategy persistence, as any deviation becomes disadvantageous. It leads to the prevalence of behaviors that maximize individual benefits, ensuring market participants adopt robust strategies that can resist invasion by alternative, less successful strategies, thereby stabilizing competition dynamics.

How can evolutionary stable strategies be applied in real-world economic scenarios?

Evolutionary stable strategies (ESS) can be applied in real-world economic scenarios to analyze behaviors in competitive markets, such as pricing strategies, product positioning, and resource allocation. They help predict which strategies will persist and be resilient to changes, aiding firms in understanding competitive dynamics and making strategic decisions.

Can an evolutionary stable strategy change over time?

Yes, an evolutionary stable strategy (ESS) can change over time if the population's environment or conditions change. As these factors evolve, the strategies that provide the highest fitness can shift, potentially leading to a new ESS that better adapts to the current situation.

What defines an Evolutionary Stable Strategy (ESS)?

A strategy that maximizes short-term gains regardless of stability.

In the Forager-Warrior example, what condition balances the strategies?

The solution for \( p \) such that expected payoffs are equal.

How are Pure ESS evaluated in a population?

By observing random deviations in strategies over time.

What is a Mixed Evolutionary Stable Strategy (ESS)?

A mixed ESS only applies in predictable environments.

What is the condition for a strategy A to be considered an ESS?

\[ E(A, A) > E(B, A) \] and if \[ E(A, A) = E(B, A) \], then \[ E(A, B) > E(B, B) \].

What defines a Pure Evolutionary Stable Strategy (ESS)?

Strategies that change dynamically to improve population fitness.

Evolutionary Stable Strategy: Game Theory

Q: What is the difference between an evolutionary stable strategy and a Nash equilibrium?

An evolutionary stable strategy (ESS) is a refinement of Nash equilibrium, specific to games in which strategy frequencies evolve over time. ESS is a strategy that, if adopted by a population, cannot be invaded by any alternative strategy. A Nash equilibrium, on the other hand, is broader and refers to a set of strategies where no player can benefit by unilaterally changing their strategy.

Q: How is an evolutionary stable strategy identified in a population model?

An evolutionary stable strategy (ESS) in a population model is identified when a strategy, once it is adopted by a majority of the population, cannot be invaded by any alternative strategy. This requires that the ESS yields a higher or equal fitness compared to the alternative when prevalent and a higher fitness when rare.

evolutionary stable strategy

An Evolutionary Stable Strategy (ESS) is a concept from evolutionary game theory defining a strategy that, if adopted by a population, cannot be invaded by an alternative strategy due to the decreased fitness of invaders. ESS relies on the principles of natural selection and is pivotal in understanding the development of behaviors in biological ecosystems, ensuring that certain strategies persist over time as they yield higher survival or reproductive success. Recognizing ESS helps in grasping the foundational mechanisms of how species evolve and adapt, staying "stable" despite potential evolutionary changes.

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Evolutionary Stable Strategy Definition

A fundamental concept in game theory, an Evolutionary Stable Strategy (ESS) represents a strategy that, if adopted by a population, cannot be invaded by any alternative strategy. This occurs because an ESS provides a higher fitness level when compared to mutant strategies. Utilizing the framework of evolutionary game theory, ESS helps explain the persistence of certain behavioral strategies within populations over time.

An ESS is particularly important in scenarios where individuals' interactions impact their relative success or survival. It provides a formal model for understanding how strategies evolve through natural selection. The mathematical formulation of an ESS is built upon the foundational principles of game theory explored by John Maynard Smith and George Robert Price in the early 1970s.

An Evolutionary Stable Strategy (ESS) is a strategy which, if adopted by a population, cannot be overtaken by any small number of individuals with a different strategy because it leads to the highest fitness.

Mathematical Representation of ESS

The concept of an ESS can be better understood through its mathematical representation. Given two strategies, say A and B, within a population, the fitness outcomes of individuals using these strategies can be evaluated. Suppose individuals using a strategy A play against others either using strategy A or strategy B. The fitness outcomes can be represented as follows:

The payoff when A meets A: \[ E(A, A) \]
The payoff when A meets B: \[ E(A, B) \]
The payoff when B meets A: \[ E(B, A) \]
The payoff when B meets B: \[ E(B, B) \]

For a strategy to be evolutionarily stable, the following conditions must be satisfied:

Condition 1: \[ E(A, A) > E(B, A) \] (Strategy A must do better against itself than strategy B does against A)
Condition 2: If \[ E(A, A) = E(B, A) \], then \[ E(A, B) > E(B, B) \] (This condition applies when strategy A's payoff against itself equals that of strategy B's against A; A must still do better against B than B would against itself)

What is an Evolutionary Stable Strategy in Game Theory?

An Evolutionary Stable Strategy (ESS) is a foundational concept in game theory and evolutionary biology. It is a strategy that, once it becomes prevalent within a population, cannot easily be invaded or replaced by an alternative strategy. The ESS model explains why certain behaviors persist in populations over evolutionary time.

These strategies are intimately tied with the notion of natural selection, where the fitness, or success, of an individual is influenced by the strategy it employs in interactions with others. When a strategy is evolutionarily stable, it confers an advantage such that any small number of mutants using a different strategy do not fare better.

The concept of ESS was introduced by John Maynard Smith and George Robert Price, who applied game theory to biological scenarios. This was a significant development, allowing the modeling of evolution not merely as a straightforward competition for resources, but as a series of strategic interactions.

ESS has applications beyond biology in psychology, economics, and social sciences, helping explain phenomena such as cooperation and altruism, which might seem counterintuitive under simple Darwinian competition models.

Conditions for Evolutionary Stability

To understand when a strategy is evolutionarily stable, consider a scenario where two strategies, A and B, exist. The fitness or payoff matrix for these interactions is expressed as:

	A	B
A	\[ E(A, A) \]	\[ E(A, B) \]
B	\[ E(B, A) \]	\[ E(B, B) \]

Two key conditions define when A is an evolutionary stable strategy:

Primary condition: \[ E(A, A) > E(B, A) \], meaning strategy A must perform better against itself than strategy B does against A.
Secondary condition: If \[ E(A, A) = E(B, A) \], then \[ E(A, B) > E(B, B) \] must hold true. This means that even if both strategies perform equally well against A, strategy A must still perform better against B than B does against itself.

An Evolutionary Stable Strategy (ESS) is defined as a strategy that, if prevalent in a population, cannot be invaded by any small group using a different strategy due to its higher payoff.

Consider the classic example of the Hawk-Dove game to illustrate the concept of an ESS. In this game, two strategies are possible: Hawk (aggressive) and Dove (peaceful).

The payoff matrix is:

	Hawk	Dove
Hawk	\[(V-C)/2\]	\[V\]
Dove	\[0\]	\[V/2\]

Where V represents the value of the resource, and C the cost of fighting. If the cost is high compared to the value, a mixture of Hawks and Doves becomes an ESS, where neither strategy completely dominates. This balance is stable because if there are too many Hawks, the high fighting cost makes Doves more attractive, and vice-versa.

Pure Evolutionary Stable Strategy

A Pure Evolutionary Stable Strategy (ESS) refers to a scenario where individuals in a population adopt a single strategy exclusively, without deviation. In this context, 'pure' implies that the strategy is not a mixed one, but rather a strictly defined action consistently applied by each individual. This contrasts with mixed strategies, where individuals may randomly choose between strategies based on a probability distribution.

Pure strategies are essential to understanding simple and direct evolutionary outcomes in a game-theoretic context. They help in predicting behaviors within populations under specific evolutionary pressures, especially when strategies are discrete and distinguishable.

A Pure Evolutionary Stable Strategy (ESS) is a strategy that, when adopted uniformly by a population, cannot be successfully invaded by a small number of individuals implementing an alternative strategy due to it yielding a higher fitness.

Understanding Pure ESS through Game Theory

To analyze and comprehend how a pure ESS functions within a population, it is crucial to employ mathematical formulations from game theory. These formulations typically involve payoffs that determine the fitness of competing strategies.

Consider a game where Strategy X and Strategy Y are the players in a competitive evolutionary scenario. The expected payoffs in a game matrix would relate these strategies as follows:

Let's take an example with two competing strategies: Strategy X and Strategy Y. The payoff matrix might look like this:

	Strategy X	Strategy Y
Strategy X	\[ E(X, X) \]	\[ E(X, Y) \]
Strategy Y	\[ E(Y, X) \]	\[ E(Y, Y) \]

For Strategy X to be a pure ESS, it must satisfy:

Condition 1: \[ E(X, X) > E(Y, X) \]
Condition 2: If \[ E(X, X) = E(Y, X) \], then \[ E(X, Y) > E(Y, Y) \]

Mixed Evolutionary Stable Strategy

A Mixed Evolutionary Stable Strategy (ESS) extends the concept of evolutionary stability to situations where individuals can adopt a range of strategies in a probabilistic manner. Unlike pure strategies, where a single action is consistently applied, mixed strategies incorporate a probability distribution over diverse strategies. This results in a more flexible strategy that provides stability in environments where variability and unpredictability are high.

Mixed strategies are crucial in ecologically diverse systems where discrete strategies may not suffice due to frequent environmental or contextual shifts. By balancing different strategies, populations can achieve stability even when facing a variety of challenges.

A Mixed Evolutionary Stable Strategy is an equilibrium strategy that combines several potential strategies, each chosen based on a probability distribution, making it resistant to invasions by mutant strategies.

How to Find an Evolutionary Stable Strategy

To determine a mixed ESS, it's essential to leverage the mathematical frameworks of game theory. Consider a two-strategy game, with competing strategies A and B. These strategies are chosen based on certain probabilities, with p representing the probability of choosing A and (1-p) the probability of choosing B. The respective payoffs are calculated as follows:

	A	B
A	\[ E(A, A) \]	\[ E(A, B) \]
B	\[ E(B, A) \]	\[ E(B, B) \]

The expected payoff for Strategy A in the population can be expressed as:

\[ p \times E(A, A) + (1-p) \times E(A, B) \]

Similarly, for Strategy B, the expected payoff is:

\[ p \times E(B, A) + (1-p) \times E(B, B) \]

For a strategy to be mixed ESS, the conditions:E(p) for the strategy mixes must satisfy:

Balance equation: The marginal benefit of switching strategies must equal the marginal benefit of staying with the current strategy.
Equilibrium: Both strategies should provide equal average payoffs when employed using respective probabilities.

Consider a scenario with two strategies: Forager and Warrior within an animal population. The payoff matrix is given by:

	Forager	Warrior
Forager	3	1
Warrior	2	2

If the population mixes these two strategies, the equilibrium probability for adopting Forager strategy (say p) is calculated such that both strategies offer equal payoffs:

\[ p \times 3 + (1-p) \times 1 = p \times 2 + (1-p) \times 2 \]

This simplifies to finding the solution for p which balances the expected payoffs for the mixed strategy equilibrium.

When dealing with mixed strategies, focus on calculating the expected values of the payoff using probabilities for a more flexible evolutionary context.

evolutionary stable strategy - Key takeaways

Evolutionary Stable Strategy (ESS) is a game theory concept representing a strategy that cannot be overtaken by an alternative if prevalent in a population due to its higher fitness.
ESS was introduced in the 1970s by John Maynard Smith and George Robert Price and helps explain persistence of behavioral strategies in populations.
Pure Evolutionary Stable Strategy involves a population adopting a single strategy consistently, providing higher fitness than any small alternative strategy.
Mixed Evolutionary Stable Strategy allows for adopting multiple strategies probabilistically, maintaining equilibrium by balancing payoffs across strategies.
ESS conditions include that a strategy must do better when interacting with itself than an alternative, and if equal, it must outperform when interacting with other strategies.
Mathematical representation helps determine ESS by comparing fitness or payoffs for different strategies within a game matrix.

evolutionary stable strategy

StudySmarter Editorial Team