Evolutionary Games on Star Graphs Under Various Updating Rules

Evolutionary game dynamics have been traditionally studied in well-mixed populations where each individual is equally likely to interact with every other individual. Recent studies have shown that the outcome of the evolutionary process might be significantly affected if the population has a non-homogeneous structure. In this paper we study analytically an evolutionary game between two strategies interacting on an extreme heterogeneous graph, the star graph. We find explicit expressions for the fixation probability of mutants, and the time to absorption (elimination or fixation of mutants) and fixation (absorption conditional on fixation occurring). We investigate the evolutionary process considering four important update rules. For each of the update rules, we find appropriate conditions under which one strategy is favoured over the other. The process is considered in four different scenarios: the fixed fitness case, the Hawk-Dove game, the Prisoner's dilemma and a coordination game. It is shown that in contrast with homogeneous populations, the choice of the update rule might be crucial for the evolution of a non-homogeneous population.