On Fortification of General Games

A recent result of Moshkovitz [Mos14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Mos14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both `1 and `2 guarantees on induced distributions from large subsets. We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update [Mos15] of [Mos14] with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular `2 guarantees) is necessary for obtaining the robustness required for fortification. Furthermore, we show that this can yield a similar parallel repetition theorem for robust general games and not just robust projection games on bi-regular graphs.