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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:

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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?

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Consider the subgame following 1's choice of $Y$. This is simply a simultaneous move Battle of Sexes game. To compute the SPNE, you first need to find the Nash equilibrium of this subgame.

It has 3 Nash equilibria: 2 pure and 1 mixed. The mixed strategy gives a value of $\frac{2}{3}$ to player 1 in this subgame. Now move up to the root node and compare whether player 1 should choose $Y$ or $N$. This would complete the computation of SPNE.

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