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In recent and past works, convexity is usually assumed on each individualpart of the action functional in order to demonstrate the existence anduniqueness of a Nash equilibrium on some interval [0, T] (this meant that eachhessian was assumed to be nonnegative). Particularly, a certain assumption wasimposed in order to quantify the smallness of T. The contribution of thisproject is to expand on this with the key insight being that one does not needthe convexity of each part of the action, but rather just an appropriatecombination of them, which will essentially "compensate" for the other twoterms to yield convexity in the action.  This is meaningful in both the pure and applied settings as it generalizesthe existence and uniqueness of a Nash equilibrium slightly more, but maybemore importantly matches real world application slightly closer, as in realitythere are many settings in which not each part of the action have convexity.Thus, it is more accurate for modern application of Mean Field Game Theory.

The Existence and Uniqueness of a Nash Equilibrium in Mean Field Game Theory