The following extensive form game is given:

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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.

Thanks in advance.


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 $J$.

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