# Finding Mixed-Strategy Subgame-Perfect Equilibrium

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?

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.