# Existence of a subgame perfect Nash-equilibrium

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.

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.