Stability analysis of evolutionary dynamics of 2 x 2 x 2 asymmetric games

In biology, economics, sociology as well as other fields, there is often a 2x2x2 asymmetric evolutionary game problem in which each party has a set of strategies, and different strategy combinations correspond to the specific pay-offs of each party. Since each participant dynamically adjusts the strategy for maximizing their own interests, the pay-off matrix plays an important role in the evolution of the game system. Based on the pay-off matrix, we probe into the resulting state of 2x2x2 asymmetric evolutionary games. The results show that from the information of the pay-off matrix, the judgement conditions for the system to evolve into three pure strategies, two pure strategies and one pure strategy can be determined directly. What is more, under a certain type of fixed pay-off matrix, the strategy combinations observed at different evolution times is always varying. Here, we explore the connection between the pay-off matrix and the evolution of behaviours through stability theory, and results obtained are conducive to deeply understand and predict the dynamic evolution of behaviour in game systems.