So I have this little game between two players, played T times without any discouting factor with the following payoff table:
now if T is 2, is there a SPNE where (B,L) is being played first round?
I'm thinking no, since (T,R) is the unique NE and no credible threat can be made when there's only 2 rounds. Is this reasoning sound?
What happens if $2<T<\infty?$
Because the stage game has a unique NE, in any finitely repeated game of the stage game, there is a unique SPNE. In this SPNE, the stage NE is played after every history. This can be easily shown by backward induction as follows.
Suppose horizon is $T$. Let's use $t=0,1,\cdots, T-1$ to denote each period. In period $T-1$, given any history $h^{T-1}$, SPNE requires the stage NE. In period $T-2$, because after any history in $h^{T-1}$, the same NE is played, there is no intertemporal incentives. Hence after any history $h^{T-2}$, the unique NE must be played. The same logic follows.
This is true because the stage game has a unique stage NE. If the stage game has multiple NE, then non trivial intertemporal incentives can be provided.