Evolutionary Stability and Neutrality in Population Games

In this article, we delve into the concept of population games, which are used to model evolutionary dynamics in a wide range of fields, including biology, economics, and social sciences. The main idea is that a population consists of individuals or groups that make decisions based on their preferences, leading to changes in the overall distribution of strategies within the population over time.
At its core, a population game is defined by a strategy set S, which represents the set of available strategies for each individual in the population, and a population state space P(S), which encodes how these strategies are being employed across the population. The mean payoff function F : P(S) → C(S) assigns a numerical value to each possible distribution of strategies in the population, based on how well they perform on average.
One key concept in population games is the notion of Nash equilibria, which describe situations where no individual can improve their payoff by unilaterally changing their strategy, assuming all others keep their strategies unchanged. However, evolutionary stability is a more complex and important idea, as it captures the ability of a population to resist invading mutants that might offer higher payoffs.
In this context, an invasion barrier ϵ(ν) represents the threshold level of payoff improvement required for a mutant population to invade a given population. If such an invasion barrier exists, then the original population is considered evolutionarily stable. Intuitively, if a population is not evolving towards any mutant population under small fluctuations, it’s because there’s no incentive to do so, and it’s evolutionarily stable.
Monotonicity and dissipativity are additional concepts that play important roles in the study of population games. Monotonicity refers to the idea that a population is moving towards more cooperative or more competitive strategies over time, while dissipativity ensures that any deviation from the current state will lead to a decrease in payoffs over time.
In summary, population games provide a useful framework for understanding how evolutionary dynamics can shape the behavior of individuals within a population, leading to emergent patterns and outcomes. By studying these dynamics through concepts like Nash equilibria, evolutionary stability, invasion barriers, monotonicity, and dissipativity, we can gain insights into a wide range of phenomena in various fields.