  We seek a route to the equilibrium where all the agents cooperate in the
iterated prisoner's dilemma game on a two-dimensional plane, focusing on the
role of tit-for-tat strategy. When a time horizon, within which a strategy can
recall the past, is one time step, an equilibrium can be achieved as
cooperating strategies dominate the whole population via proliferation of
tit-for-tat. Extending the time horizon, we filter out poor strategies by
simplified replicator dynamics and observe a similar evolutionary pattern to
reach the cooperating equilibrium. In particular, the rise of a modified
tit-for-tat strategy plays a central role, which implies how a robust strategy
is adopted when provided with an enhanced memory capacity.
