I am trying to understand the proof for the folk theorem for infinitely repeated games without discounting (average reward).
The proof says that given that a rewarding payoff is enforceable and feasible, then the state must be a Nash equilibrium.
It also says that if a state is a Nash equilibrium, then the payoff should be enforceable, but that's easy to prove.
For the first statement, they do a proof by construction, which can be found here. I understood till the part where the reward payoff of the player $i$ is $r_i$.
Now as a part of the construction, the state that the players will use a grim strategy, and if anyone diverges from the agreed upon strategy, then all the other players will play the minimax strategy for the player who diverged. So the payoff of the person will be less or equal to $r_i$, because earlier, the payoff was enforceable.
Understood.
But what I don't understand is, how can they assume that a grim strategy would be played by the other players? What if they don't enforce this strategy? What if, the other players just didn't react to what the diverger did, when the game earlier was enforceable and was feasible?
Since this condition is missing from the proof, I don't totally understand how this works. Any pointers appreciated.