Evolution, games theory and polyhedra

Area of Science:

• Evolutionary Game Theory
• Mathematical Biology
• Animal Behavior

Background:

• Evolutionary Stable Strategy (ESS) is crucial for understanding animal species' behavioral dynamics.
• ESS represents a subset of equilibrium solutions in non-zero sum games.
• Existing methods for identifying ESS can be computationally intensive.

Purpose of the Study:

• To define the problem of finding an Evolutionary Stable Strategy (ESS) for animal species.
• To demonstrate that ESS are a subset of equilibrium solutions in specific non-zero sum games.
• To present a novel method for identifying ESS through game-theoretic equilibria.

Main Methods:

• Defining ESS within the framework of non-zero sum game theory.
• Characterizing equilibrium solutions as vertices of a convex polyhedron.
• Developing a vertex-based algorithm to find equilibrium and subsequently ESS solutions.
• Illustrating the method with numerical examples from existing literature.
• Discussing an alternative approach using Linear Complementarity Problems (LCP).

Main Results:

• Equilibrium solutions for the defined non-zero sum game arise from the vertices of a specific convex polyhedron.
• A practical method is provided to identify ESS by first finding these vertex-based equilibrium solutions.
• The method's efficacy is validated through multiple numerical examples.

Conclusions:

• The geometric approach provides a structured way to find ESS.
• The connection between game theory equilibria and convex polyhedra offers new insights.
• Linear Complementarity Problems present a viable alternative computational strategy for ESS determination.