Last modified: 2014-06-09

#### Abstract

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.