# Finding pure-strategy subgame-perfect Nash equilibria

I'm interested in finding the pure-strategy subgame-perfect Nash equilibria of the game below. What is confusing me is that after player A chooses between reducing and not reducing his end payoffs, the game is not sequential anymore, but simultaneous. How do I go about finding the subgame-perfect Nash equilibria?

I thought of first finding the Nash equilibria in the two simultaneous-move subgames and I got 4 equilibria in total (written as the outcomes):

$$(2,4)$$ and $$(4,2)$$ for the left subgame, and

$$(1,4)$$ and $$(3,2)$$ for the right subgame.

From this, I deduced what player A would choose in the first stage (don't reduce or reduce), based on possible payoffs, so I got the following two subgame-perfect Nash equilibria:

$$\{Don't\space reduce, a_2, b_2\}$$, and

$$\{Reduce,a_2,b_2\}$$.

I'm not sure if this is correct. After determining the Nash equilibria in the subgames, I get confused trying to determine what player A will do in the first stage. Am I doing this right and can someone explain how to approach solving this?

Because you are looking for subgame-perfect equilibria, it is the correct approach to solve this game backwards. There are 2 proper subgames, and you identified the correct Nash equilibria there. However, note that you specified the payoffs, not the strategies. The correct formulation would be $$(a_1,b_1)$$ instead of $$(2,4)$$.