We study in this paper some evolutionary games where competition between individuals from a large population occurs through many local interactions between randomly selected individuals. We focus on games that have the property of possessing a single interior evolutionarily stable strategy (ESS). We study in particular the effect of the time delays on the convergence of evolutionary dynamics to the ESS in an evolutionary game in which each pure strategy is associated with its own delay. In particular, we study a multiple access game as well as a Hawk and Dove game. We study the properties of the ESS in these games and also the effect of time delays on the convergence of various bio-inspired evolutionary game dynamics to the ESS.