  We solve the dynamics of the on-line minority game, with general types of
decision noise, using generating functional techniques a la De Dominicis and
the temporal regularization procedure of Bedeaux et al. The result is a
macroscopic dynamical theory in the form of closed equations for correlation-
and response functions defined via an effective continuous-time single-trader
process, which are exact in both the ergodic and in the non-ergodic regime of
the minority game. Our solution also explains why, although one cannot formally
truncate the Kramers-Moyal expansion of the process after the Fokker-Planck
term, upon doing so one still finds the correct solution, that the previously
proposed diffusion matrices for the Fokker-Planck term are incomplete, and how
previously proposed approximations of the market volatility can be traced back
to ergodicity assumptions.
