# Subgame Perfect Nash equilibrium (Mixed strategy)

The following extensive form game is given:

Find a Subgame Perfect Nash equilibrium of the game featuring one player using a mixed strategy.

I know that in order to find a SPNE (Subgame Perfect Nash Equilibrium), we can use backward induction procedure and I am familiar with this procedure. In fact, I can solve this game for SPNE in pure strategies, but I don't know know how to solve it using a mixed strategy. I also know how to find a mixed strategy Nash equilibrium in static games, but I don't know how to do it in dynamic games, i.e. combine it with backward induction. I tried to represent some subgames in a payoff matrix and to solve for indifference condition for both players like in static games, but I obtained negative probability values, which is, of course, wrong.

Any help is appreciated.

So in the final subgame on the left, 1 prefers $$G$$ to $$H$$, so the continuation payoffs at that node are $$(-5,2)$$, and at the final subgame on the right, 1 prefers $$J$$ to $$I$$, so the continuation payoffs at that node are $$(5,-1)$$.
On the right, 2 then prefers $$e$$ and a payoff of 5 to $$f$$ and a payoff of -1. The only mixing can occur on the left, where 2 is indifferent between $$c$$ and a payoff of 2 after $$G$$ is played, or ending the game by playing $$d$$ and getting $$2$$. Because he is indifferent, any mix is part of an equilibrium strategy, but that affects 1's incentives to choose $$A$$ or $$B$$.
If 1 chooses $$A$$, the payoff is $$p(-5) +(1-p)1$$, while $$B$$ gives a payoff of -12.
So all of the SPNE are of the form: 1 chooses $$A$$; 2 mixes over $$c$$ and $$d$$ with any $$p \in [0,1]$$, 2 chooses $$e$$; 1 chooses $$G$$, 1 chooses $$J$$.