Existence of a subgame perfect Nash-equilibrium

Question

Existence of a subgame perfect Nash-equilibrium

Given is the following game

The game is repeated finitely many times and the total payoff is the sum of the payoff from each repetition.

If we assume that $T=2$, is it then possible to find a subgame perfect Nash-equilibrium such that the strategy $(B,L)$ is played in the first stage (first repetition)?

So far as I can count, it shouldn't be possible to construct any threats for $T=2$ such that the $(B,L)$ strategy is played in the first stage. This is due to the Nash-equilibrium laying in $(T,R)$ and it takes at least $T\geq 3$ to ensure that $(B,L)$ is played in the first stage.

Though, I'm not quite sure how to prove my statement. Can any of you confirm/deny my thoughts and help me with some arguments? Thanks.

Answer 1

The following proposition is well known:

If a stage game $G$ has a unique Nash equilibrium, then for any finite $T$, the repeated game $G(T)$ has a unique subgame perfect equilibrium outcome in which the Nash equilibrium of $G$ is played in every stage.

Since your stage game has a unique NE of $(T,R)$, this must be the outcome of the SPE of any finitely repeated game based on this stage game. Specifically regarding your question, it is impossible for $(B,L)$ to be played (in any stage including the first) in any SPE of the $T$-times repeated game.