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We study the homogenization properties in the small viscosity limit and inperiodic environments of the (viscous) backward-forward mean-field gamessystem. We consider separated Hamiltonians and provide results for systems with(i) "smoothing" coupling and general initial and terminal data, and (ii) with"local coupling" but well-prepared data.The limit is a first-orderforward-backward system. In the nonlocal coupling case, the averaged system isof mfg-type, which is well-posed in some cases. For the problems with localcoupling, the homogenization result is proved assuming that the formallyobtained limit system has smooth solutions with well prepared initial andterminal data. It is also shown, using a very general example (potential mfg),that the limit system is not necessarily of mfg-type.

Homogenization of the backward-forward mean-field games systems in periodic environments