  The prisoner's dilemma describes a conflict between a pair of players, in
which defection is a dominant strategy whereas cooperation is collectively
optimal. The iterated version of the dilemma has been extensively studied to
understand the emergence of cooperation. In the evolutionary context, the
iterated prisoner's dilemma is often combined with population dynamics, in
which a more successful strategy replicates itself with a higher growth rate.
Here, we investigate the replicator dynamics of three representative
strategies, i.e., unconditional cooperation, unconditional defection, and
tit-for-tat, which prescribes reciprocal cooperation by mimicking the
opponent's previous move. Our finding is that the dynamics is self-dual in the
sense that it remains invariant when we apply time reversal and exchange the
fractions of unconditional cooperators and defectors in the population. The
duality implies that the fractions can be equalized by tit-for-tat players,
although unconditional cooperation is still dominated by defection.
Furthermore, we find that mutation among the strategies breaks the exact
duality in such a way that cooperation is more favored than defection, as long
as the cost-to-benefit ratio of cooperation is small.
