# Solving a two stage game by backward induction: which is the equilibrium notion?

Take a two-stage game with complete information and simultaneous actions in each state:

(1) Player 1 and 2 simultaneously choose action $$a_1\in A_1$$ and $$a_2\in A_2$$ respectively.

(2) Player 1 and 2 observe the outcome of the 1st stage $$(a_1, a_2)$$, then simultaneously choose action $$a_3\in A_3$$ and $$a_4\in A_4$$ respectively.

Payoffs are $$u_i(a_1, a_2, a_3, a_4)$$ for $$i = 1,2$$.

Suppose that I do the following (typically called backward induction):

(A) find the functions $$a^*_3(a_1,a_2)$$ and $$a^*_4(a_1,a_2)$$ such that

$$\begin{cases} a_3^*(a_1, a_2)\in argmax_{a_3(\cdot)}u_1(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\ a_4^*(a_1, a_2)\in argmax_{a_4(\cdot)}u_2(a_1, a_2, a_3(a_1, a_2), a_4(a_1, a_2))\\ \end{cases}$$

(B) find $$a_1^*, a_2^*$$ such that $$\begin{cases} a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ \end{cases}$$

Which equilibrium notion I'm I applying?

Basically, in (A) I find the pure strategy Nash Equilibrium of stage 2, for each possible action played in stage 1; in (B) I find the pure strategy Nash Equilibrium of stage 1, given the solution from (A).

How is $$\{a_1^*, a_2^*, a^*_3(a_1, a_2), a^*_4(a_1, a_2)\}$$ called?