# Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games

## TL;DR

This paper studies a special type of 2×2×2 asymmetric evolutionary game involving three individuals and two strategies each.

## Contribution

The paper introduces new stability conditions for interior equilibrium points in asymmetric evolutionary games.

## Key findings

- Two solitary interior equilibrium points are analyzed for instability using the Jacobi matrix method.
- Interior equilibrium points distributed along a line are studied for stability using generalized Hamiltonian systems theory.
- Stable equilibrium points are surrounded by closed orbits, showing coexistence and fluctuation of strategies.

## Abstract

Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of 2×2×2 asymmetric games. In this paper, we explore a special class of 2×2×2 asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of 2×2×2 asymmetric games’ evolutionary dynamics.