Subgame Perfect Nash equilibrium (Mixed strategy)

Question

The following extensive form game is given:

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

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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.

Thanks in advance.

Answer 1

There are only mixed strategies if the payoffs at the terminal nodes are not unique.

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 $J$.