Generating functional analysis of minority games with real market histories

It is shown how the generating functional method of De Dominicis can be used to solve the dynamics of the original version of the minority game (MG), in which agents observe real as opposed to fake market histories. Here one again finds exact closed equations for correlation and response functions, but now these are defined in terms of two connected effective non-Markovian stochastic processes: a single effective agent equation similar to that of the ‘fake’ history models, and a second effective equation for the overall market bid itself (the latter is absent in ‘fake’ history models). The result is an exact theory, from which one can calculate from first principles both the persistent observables in the MG and the distribution of history frequencies. PACS numbers: 02.50.Le, 87.23.Ge, 05.70.Ln, 64.60.Ht