  We construct two models of discrete-time replicator dynamics with time delay.
In the social-type model, players imitate opponents taking into account average
payoffs of games played some units of time ago. In the biological-type model,
new players are born from parents who played in the past. We consider
two-player games with two strategies and a unique mixed evolutionarily stable
strategy. We show that in the first type of dynamics, it is asymptotically
stable for small time delays and becomes unstable for big ones when the
population oscillates around its stationary state. In the second type of
dynamics, however, evolutionarily stable strategy is asymptotically stable for
any size of a time delay.
