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.
In the second type of dynamics, however, evolutionarily stable strategy
is asymptotically stable for any size of a time delay.
Key Words : replicator dynamics, evolutionarily stable strategy,
asymptotic stability, time delay.