I'm trying to find the Mixed-Strategy Subgame-Perfect Equilibrium of the sequential-move Battle of the Sexes game.
I know how to find the mixed strategy Nash equilibrium of the battle of the sexes:
Let player 1 play $O$ with probability $p$ and $F$ with $1-p$. (and symmetric for player $2$).
So, finding the best response correspondences for each player:
$\partial\pi_1/\partial p=\partial(2pq+(1-p)(1-q))/\partial p=3q-1$
Symmetrical for player 2 is $3p-1$.
So I graph the best response correcspondence and find the Nash Equilibrium:
So, now that I found the mixed strategy Nash Equilibrium is $(p,q)=(2/3,1/3)$, what do I do next? What is the Subgame-Perfect Equilibrium?
So, by the answer from Renard:
The expected payoff from $N$ for player $1$ is 1.5.
If he chooses $Y$, then the expected payoff is $(2/3)2+(1/3)1=5/3\gt 1.5$.
So for player 1 the optimal strategy would to go $Y$ and then $O$. And for player 2 would be to go $o$ if player $1$ goes $O$ and $f$ if player $1$ goes $f$.
Hence, the following the the Mixed-Strategy Subgame-Perfect Equilibrium: $(YO,o)$. Right?