Definition of subgame perfect Nash equilibrium

Question

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 $i = 1,2$.

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As equilibrium concept I use subgame perfect Nash equilibrium. I find it by backward induction:

(A) find the functions $a^*_3(\cdot)$ and $a^*_4(\cdot)$ such that $\forall (a_1,a_2)\in A_1\times A_2$

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

(B) find $(a_1^*, a_2^*)\in A_1\times A_2$ such that $$ \begin{cases} a_1^*\in argmax_{a_1}u_1(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ a_2^*\in argmax_{a_2}u_2(a_1, a_2, a^*_3(a_1, a_2), a^*_4(a_1, a_2))\\ \end{cases} $$

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Question: a subgame perfect Nash equilibrium is $$ \{a_1^*, a_2^*, \underbrace{a^*_3(\cdot), a^*_4(\cdot)}_{\text{Functions}}\} $$ or $$ \{a_1^*, a_2^*, \underbrace{a^*_3(a_1^*), a^*_4(a_2^*)}_{\text{Point in the image set of the functions $a^*_3(\cdot), a^*_4(\cdot)$}}\} $$ ?

Answer 1

An equilibrium consists of a profile of strategies, which specifies an action for every player at each possible contingency. Since each action profile $(a_1,a_2)$ is a contingency, the SPE must include functions $a^*_3(a_1,a_2), a^*_4(a_1,a_2)$ that specifies what to do at those contingencies.