The Impact of Time Delay on Mutant Fixation in Evolutionary Games

Evolutionary game theory examines how strategies spread and endure in populations through reproduction and imitation based on their evolutionary fitness. Traditional models assume instantaneous population dynamics where the current population state solely determines fitness. However, some real-world evolutionary processes unfold over time, with outcomes emerging from historical events. This observation motivates incorporating time delays into evolutionary game models, where fitness relies on population history in the past. We studied the impact of time delay on the fixation probability of a mutant strategy in a Moran birth-death process with two strategies in a well-mixed population. At each time step of the process, an individual reproduces proportionally to fitness derived from the past population state. We model this as an absorbing Markov chain, allowing numerical computation of fixation probabilities and times. We focus on three important evolutionary games: Stag-Hunt, Snowdrift, and Prisoner’s Dilemma. We will show time delay reduces the fixation probability in the Stag-Hunt game but increases it in the Snowdrift game. The Prisoner’s Dilemma game shows a small decline in fixation probability. For the Stag-Hunt game, time delay increases the fixation time until a critical point, then reduces it. The Snowdrift game exhibits the opposite trend in fixation time. The Prisoner’s Dilemma game displays negligible change in fixation time.