  We analyze grand-canonical minority games with infinite and finite score
memory and different updating timescales (from `on-line' games to `batch'
games) in detail with various complementary methods, both analytical and
numerical. We focus on the emergence of `stylized facts' and on the production
of exploitable information, as well as on the dynamic behaviour of the models.
We find that with finite score memory no agent can be frozen, and that all the
current analytical methods fail to provide satisfactory explanation of the
observed behaviours.
