# Mean field games and approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.

I have a question about a step in theorem 3.8 on page 17. Let me give the set-up. Each player $$i$$ has the payoff function $$\mathcal{J^i}(\alpha^1, \ldots \alpha^N) = \mathbb{E}\left[\int_0^T \frac12|\alpha_s^i|^2 + F\left(X_s^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right) \ ds + G\left(X_T^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_T^j}\right)\right].$$

The dynamics of the player are given by $$dX_t^i = \alpha_t^i \ dt + \sqrt{2}\ dB_t^i$$, as usual. $$F$$ and $$G$$ are Lipschitz in both variables.

The $$X_t^i$$ are i.i.d. with law $$m(t)$$. The crucial step that I fail to understand is the following: there is $$N_0$$ large such that for $$N \ge N_0$$, $$\mathbb{E}\left[\sup_{|y|\le 1/\sqrt{\epsilon}} \left|F(y, m(t)) - F\left(y, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right)\right|\right]\le \epsilon$$ for any $$t \in [0,T]$$. Note that the subscript on the $$X^i$$ is $$s$$, not $$t$$.

He says this is an application of Hewitt-Savage. However, I fail to see how this works at all. I'm not sure where to introduce my integrals, and I definitely don't see what role the restriction $$|y|\le 1/\sqrt{\epsilon}$$ is playing here. $$F$$ is uniformly Lipschitz in the $$m$$ variable, independent of the parameter $$y$$. Because this is the case, clearly there has to be something more do just passing to the Lipschitz bound on that difference, because we'd be dropping anything to do with $$y$$ (and then I really can't see how Hewitt-Savage would help). Can someone help shed light on this step?

From the reference, the limit under $$n \rightarrow + \infty$$ in the result of corollary 5.13 (https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf) implies that there exists $$1 \leq N_0$$ and $$0 \leq \epsilon_1$$ such that for all $$N_0 \leq N$$, $$\mathbb{E}\left[ \underset{y}{\sup} \left| F \left( y, \frac{1}{N - 1} \underset{n \geq 2}{\sum} \delta_{\bar{X}^j_s} \right) - F(y, m(s)) \right| \right] \leq \epsilon_1$$ where the expectation indicates an integral over the joint probability distribution of the states $$(\bar{X}^j_s)$$. The Lipschtiz condition w.r.t. $$m(\cdot)$$ implies that there exists $$0 < \epsilon_2$$ such that for given $$0 \leq s, t$$, $$\mathbb{E}\left[ \underset{y}{\sup} \left| F \left(y,m(t) \right) - F(y, m(s)) \right| \right] \leq \epsilon_2.$$ Applying the triangle inequality and choosing $$\epsilon = \epsilon_1 + \epsilon_2$$ we can obtain the inequality which takes the $$\sup$$ over all $$y$$, so that it is also true for $$|y| \leq 1/\sqrt{\epsilon}$$. The inequality constraint referred to in the question is related to the construction of the proof and not directly related to showing this particular expectation-inequality per se.