Look at the following payoff matrix, there are two stage Nash equlibria, and we consider a infinitely repeated game with discount factor $\delta\in(0,1)$, can players sustain an average payoff of (2,2) under any strategy? (I am confused because the individually rational(i.e. minmax) payoff is (3,3)) If not, why?
I think the result you are looking for is Lemma 2 (p.7) of these lecture notes by Johannes Hörner. Also see the papers he references in the end. There is a formal proof, but the idea is exactly what Herr K. wrote in his post. In your repeated game, all players must have an average payoff of at least 3 in Nash equilibrium.
I'm inclined to say $(2,2)$ is not sustainable, though I don't have a formal proof at the moment.
To achieve an average payoff of $(2,2)$, the players must play $(H,H)$ in some stages. But then a player can deviate to playing $L$ and avoid any subsequent punishments by ensuring that he can always get at least $3$, which is better than $2$.