  We use generating functional methods to solve the so-called inner product
versions of the minority game (MG), with fake and/or real market histories, by
generalizing the theory developed recently for look-up table MGs with real
histories. The phase diagrams of the lookup table and inner product MG versions
are generally found to be identical, with the exception of inner product MGs
where histories are sampled linearly, which are found to be structurally
critical. However, we encounter interesting differences both in the theory
(where the role of the history frequency distribution in lookup table MGs is
taken over by the eigenvalue spectrum of a history covariance matrix in inner
product MGs) and in the static and dynamic phenomenology of the models. Our
theoretical predictions are supported by numerical simulations.

Physics and Society