Dynamics and stability of evolutionary games with time-invariant delay in strategies

This paper investigates the modeling and stability of finite evolutionary games (EGs) with time-invariant delay in strategies. Unlike EGs without delay, the evolutionary dynamics of a sequence of strategy profiles, named as the profile trajectory, is proposed to describe the strategy updating process of the delayed EGs. Using the semi-tensor product of matrices, the evolutionary dynamics of the delayed EGs is expressed into an algebraic model, and then some sufficient conditions are proposed to assure the convergence of profile trajectory to a pure Nash equilibrium. Finally, we apply our model to the networked evolutionary games and propose a new strategy updating rule, called the distributed sequential Myopic Best Response Adjustment Rule (MBRAR), and prove that under the distributed sequential MBRAR, a delayed networked evolutionary game will also converge to a pure Nash equilibrium. Some examples are given to illustrate the theoretical results.