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. We also perform a detailed analysis of how
the evolutionary behaviour depends on payoffs, game length, and mutation rate.

Reference:

Lindgren K, Verendel V. Evolutionary Exploration of the Finitely Repeated Prisoners Dilemma – The Effect of Out-of-
Equilibrium Play. Games, 4(1):1-20, 2013.