  The finitely repeated Prisoners' Dilemma is a good illustration of the
discrepancy between the strategic behaviour suggested by a game-theoretic
analysis and the behaviour often observed among human players, where
cooperation is maintained through most of the game. A game-theoretic reasoning
based on backward induction eliminates strategies step by step until defection
from the first round is the only remaining choice, reflecting the Nash
equilibrium of the game. We investigate the Nash equilibrium solution for two
different sets of strategies in an evolutionary context, using
replicator-mutation dynamics. The first set consists of conditional
cooperators, up to a certain round, while the second set in addition to these
contains two strategy types that react differently on the first round action:
The "Convincer" strategies insist with two rounds of initial cooperation,
trying to establish more cooperative play in the game, while the "Follower"
strategies, although being first round defectors, have the capability to
respond to an invite in the first round. For both of these strategy sets,
iterated elimination of strategies shows that the only Nash equilibria are
given by defection from the first round. We show that the evolutionary dynamics
of the first set is always characterised by a stable fixed point, corresponding
to the Nash equilibrium, if the mutation rate is sufficiently small (but still
positive). The second strategy set is numerically investigated, and we find
that there are regions of parameter space where fixed points become unstable
and the dynamics exhibits cycles of different strategy compositions. The
results indicate that, even in the limit of very small mutation rate, the
replicator-mutation dynamics does not necessarily bring the system with
Convincers and Followers to the fixed point corresponding to the Nash
equilibrium of the game.
