  We use analytical techniques based on an expansion in the inverse system size
to study the stochastic evolutionary dynamics of finite populations of players
interacting in a repeated prisoner's dilemma game. We show that a mechanism of
amplification of demographic noise can give rise to coherent oscillations in
parameter regimes where deterministic descriptions converge to fixed points
with complex eigenvalues. These quasi-cycles between co-operation and defection
have previously been observed in computer simulations; here we provide a
systematic and comprehensive analytical characterization of their properties.
We are able to predict their power spectra as a function of the mutation rate
and other model parameters, and to compare the relative magnitude of the cycles
induced by different types of underlying microscopic dynamics. We also extend
our analysis to the iterated prisoner's dilemma game with a win-stay lose-shift
strategy, appropriate in situations where players are subject to errors of the
trembling-hand type.
