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

Abstract

In biology, economics, sociology as well as other fields, there is often a 2 × 2 × 2 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 2 × 2 × 2 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.

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

“…In fact, the payoff matrix described in table 1 can be equitably represented by the following matrices [34]: Generalizing from Hofbauer's bi-matrix work [25], for 2 × 2 × 2 asymmetric games, there are six equations with three redundant equations. See appendix A and Song et al [34] for the detailed derivation process of equation (2.2). Let x ˙= 0, y ˙= 0, z ˙= 0, it follows that the equilibrium point of 2 × 2 × 2 asymmetric games evolution system is E = (x * , y * , z * ) ∈ [0, 1] 3 .…”

“…, where Π k ( ⋅ , ⋅ , ⋅ ) is a (0, 3) payoff tensor, and refer to Song et al [34] for detailed information,…”

“…In fact, the payoff matrix described in table 1 can be equitably represented by the following matrices [ 34 ]:…”

“…Here, this paper will probe into the simplest form of multi-party asymmetric games, namely asymmetric games involving three parties. The dynamic properties of face equilibrium points and edge equilibrium points in asymmetric games have been explored [ 34 ], and we will continue to discuss those of interior equilibrium points. Actually, as a mathematical analysis tool, asymmetric games have been widely used in practical problems, such as carbon emission reduction mechanisms, including carbon emission trading [ 35 – 37 ] and carbon-sink fishery [ 38 , 39 ], environmental mass incidents [ 40 ], supply chains [ 41 , 42 ], medical consortiums [ 43 ] and so on.…”

“…The fitness functions are then where is a payoff tensor, and refer to Song et al . [ 34 ] for detailed information, with and defined similarly. In this case, the dynamics are given by the compact form with and and …”