  We study the role of strategy correlations and timing of adaptation for the
dynamics of Minority Games, both simulationally and analytically. Using the
exact generating functional approach a la De Dominicis we compute the phase
diagram and the behaviour of batch and on-line games with correlated
strategies, complementing exisiting replica studies of their statics. It is
shown that the timing of adaptation can be relevant; while conventional games
with uncorrelated strategies are nearly insensitive to the choice of on-line
versus batch learning, we find qualitative differences when anti-correlations
are present in the strategy assignments. The available standard approximations
for the volatility in terms of persistent order parameters in the stationary
ergodic states become unreliable in batch games under such circumstances. We
then comment on the role of oscillations and the relation to the breakdown of
ergodicity. Finally, it is discussed how the generating functional formalism
can be used to study mixed populations of so-called `producers' and
`speculators' in the context of the batch minority games.
