<p><a href="https://i.sstatic.net/HqRAf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HqRAf.png" alt="enter image description here"></a></p>

<p>I'm trying to find the Mixed-Strategy Subgame-Perfect Equilibrium of the sequential-move Battle of the Sexes game.</p>

<p>I know how to find the mixed strategy Nash equilibrium of the battle of the sexes:</p>

<p>Let player 1 play <span class="math-container">$O$</span> with probability <span class="math-container">$p$</span> and <span class="math-container">$F$</span> with <span class="math-container">$1-p$</span>. (and symmetric for player <span class="math-container">$2$</span>).</p>

<p>So, finding the best response correspondences for each player:</p>

<p><span class="math-container">$\partial\pi_1/\partial p=\partial(2pq+(1-p)(1-q))/\partial p=3q-1$</span></p>

<p>Symmetrical for player 2 is <span class="math-container">$3p-1$</span>. </p>

<p>So I graph the best response correcspondence and find the Nash Equilibrium:</p>

<p><a href="https://i.sstatic.net/c9ihJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/c9ihJ.png" alt="enter image description here"></a></p>

<p>So, now that I found the mixed strategy Nash Equilibrium is <span class="math-container">$(p,q)=(2/3,1/3)$</span>, what do I do next? What is the Subgame-Perfect Equilibrium?</p>

<game-theory>
Finding Mixed-Strategy Subgame-Perfect Equilibrium
Slim Shady
# Finding Mixed-Strategy Subgame-Perfect Equilibrium

(Asked by: username_0 on 2020-06-12)

[![enter image description here](https://i.sstatic.net/HqRAf.png)](https://i.sstatic.net/HqRAf.png)

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:

[![enter image description here](https://i.sstatic.net/c9ihJ.png)](https://i.sstatic.net/c9ihJ.png)

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?

**Tags:** game-theory

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## 0 answers

