The Stability of Imitation Dynamics with Continuously Distributed Delays

This paper investigates imitation dynamics with continuously distributed delay. In realistic technological, economic, and social environments, individuals are involved in strategic interactions simultaneously while the influences of their decision-making may not be observable instantaneously. It shows that there exists a time delay effect. Different distributions of delay are further considered to efficiently lucubrate the stability of interior equilibrium in the imitation dynamics with continuous distributions of delay in the two-strategy game contexts. Precisely, when the delay follows the uniform distributions and Gamma distributions, the authors present that interior equilibrium can be asymptotically stable. Furthermore, when the probability density of the delay is general density, the authors also determine a sufficient condition for stability derived from the expected delay. Last but not least, the interested but uncomplicated Snowdrift game is utilized to demonstrate our theoretical results.