Why is the tat-for-tat strategy a Nash equilibrium in infinitely repeated games? Why is the tat-for-tat strategy a Nash equilibrium in infinitely repeated games, but not a Nash equilibrium in a finite scenario? Specifically for this matrix:



Assume higher payoffs reflect higher utility. It's a prisoner's dilemma situation.

Since tit-for-tat assumes we start at (Honor, Honor), and play the strategy that the other player last played in future rounds, I don't really see why it's a Nash equilibrium in an infinite scenario and not a finite scenario.

In a finite scenario (e.g. one round), wouldn't the players end up at the NE (Cheat, Cheat) because they follow their self-interest? And in an infinite scenario, wouldn't they end up at (Honor, Honor) (not a NE) assuming the discount factor is high enough?

Any clarity appreciated!
(i) In the 1 round case, tit-for-tat is not a NE. To see this notice that the tit-for-tat strategy, as you describe, dictates that the players play $(H,H)$ in the first (and only) round---as you point out, this is clearly not a NE, since either player can increase her payoff by changing her strategy from $H$ to $C$. Perhaps what you missed is that the strategy includes proscription in first round, which does not cohere from the 1-shot Nash behavior.

(ii) In the $N$-round case, again tit-for-tat is not a NE. This is a little more subtle that point (i) but not much. If both players play according to the strategy, then they will make it to the $N-1$ round playing $H$. At this point, the strategy dictates they play $H$ in the $N^{th}$, and final, round. For the same reason as in (i) this is not a best response, hence not an equilibrium (note that forward looking agents would anticipate this, so the strategy would break down immediately, but analyzing the last round is sufficient to see it is not an equilibrium).

(iii) In the infinitely-repeated case, tit-for-tat can persist, but it depends on how agents discount future utility in comparison with current utility. The general logic is that since there is no final period where things begin to unravel, players are always willing to forgo current utility to maintain good standing (the tit part of the tit-for-tat?) and therefore a higher future (continuation) payoff. Of course, if the players care about today's payoff far more than tomorrow's, they will defect and play $C$. See here for more about NE in infinitely-repeated games.