Can't find the SPNE

Question

For a homework assignment, I need to find the subgame perfect equilibrium. The assignment asserts that there is only one subgame perfect equilibrium in this problem, but I am stuck between two possible options.

I found three Nash equilibria at (2,1), (3,2), and (1,2). The output of the left side is (2,1), but I am unsure if the output of the right side is definitively (3,2) or (1,2). Through backwards induction, P1 would choose (1,2) over (0,3), but I don't know if P2 would choose (3,2) or (1,2) since it's an equally good output for P2.

Michael Greinecker · Accepted Answer

Looking for the subgame perfect equilibrium here is hopeless; there really is more than one (in spite of what the assignment says).
When you use backward-induction, you have to use both. There is a subgame perfect equilibrium in which player $2$ chooses $d$ at the correspond decision node and a subgame perfect equilibrium in which player $2$ chooses $e$ at the same decision node. As a consequence, player $1$ makes different choices in their first decision node in these two subgame perfect equilibria.
By the way: An equilibrium specifies a strategy for each player, and a (pure) strategy specifies a decision at each information set. For example, player $1$ has four strategies:  $LW$, $LE$, $RW$, $RE$. If player $1$ moves left initially, their choice at the other decision node will have no effect on the resulting play. However, it is still important for determining whether a strategy profile is subgame perfect or not.

Can't find the SPNE

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