# 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?

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?

• The PBE is that 1 picks $Y$, then 1 plays the mixed strategy, 2's beliefs correspond to 1's mix, and 2 plays his part of the mix. That's the eqm. What you wrote looks like you're going back to talking about pure strategies, with the $(YO,o)$. – user762914 Jun 12 '20 at 16:09