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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?

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?

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You've solved for the strategies if 1 chooses Y. Now compute the expected utility for 1 for that subgame, where 1 chooses Y and then the two players randomize. Then compare it to the payoff from N to determine whether 1 chooses Y or N. That will give you the perfect Bayesian equilibrium of the game (where beliefs are given by the mixed strategies).

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  • $\begingroup$ thank you, could you see my edit $\endgroup$ – The Poor Jew Jun 12 '20 at 15:50
  • $\begingroup$ 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 $(YO,o)$. $\endgroup$ – user762914 Jun 12 '20 at 16:09

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