Consider the following matrix game:
\begin{matrix} & L &M&R \\ T&8,8 & 0,9 & 0,0 \\ C&9,0 & 0,0 & 3,1 \\ B&0,0 & 1,3 & 3,3 \end{matrix}
For the twice repeated version of this game, describe a subgame perfect equilibrium in which $(T,L)$ is played in the first round.
So suppose $(T,L)$ is played in the first round. My book gives the following matrix for second round play:
\begin{matrix} & L &M&R \\ T&B,R & C,R & C,R \\ C&B,M & B,R & B,R \\ B&B,M & B,R & B,R \end{matrix}
How am I supposed to interpret this second matrix? I was thinking of something like this, Suppose $(T,M)$ is played in round 1. Both players observe the other player's move. Player 1 sees Player 2's choice of $M$ and realizes that if he were to switch to playing $C$ than he could potentially increase his payoff to 9 if Player 2 plays $L$ in the second round , if Player 2 does not play L, then Player 1 is either getting a payoff of 0 or 3, both which are not worse than his previous result. Therefore Player 1 should switch to $C$. Similarly, Player 2 should switch to $R$, since all three scenarios are either better or not worse for him. Is this interpretation correct? We could also argue that $L$ is a dominated strategy which might make things easier.
At the request of a commenter, I'll post my comment (slightly edited) as an answer. Note that I made up the term "response matrix".
Given the submitted second-round response matrix being known to both players in advance, it is optimal for them to play $(T,L)$ in the first round. Also the submitted second-round response matrix only contains Nash equilibria. So, I guess that means $(T,L)$ and the second matrix together form a subgame-perfect equilibrium. This needn't be unique, but the question asks for "a", not "the", subgame-perfect equilibrium.