Firms 1 and 2 are competing in the same market. Each firm $i$ must choose a quantity $q_{i}$ to supply, and the market price $p$ will depend on their choices according to the inverse demand formula $p(q_{1},q_{2}) = \text{max}[A-(q_{1} + q_{2}), 0]$.

The total cost of production for each firm $i$ is $q_{i}^2$, and so the total profit for firm ${i}$ will be $u_{i}(q_{1},q_{2}) = p(q_{1},q_{2}) \cdot q_{i} - (q_{i})^{2}$

Suppose that firm 1 chooses $q_{1}$ first, and then firm 2 chooses its $q_{2}$ after observing $q_{1}$.

Find : The subgame-perfect equilibrium of this game with perfect information, and compute each firm's expected profit in this equilibrium.

(What I have tried:)

For any given $q_{2}$, firm 1's best response $q_{1}$ to maximize $u_{1}$ is:

\begin{equation} q_{1}(q_{2}) = \begin{cases} \frac{A-q_{2}}{4} & \text{if } q_{2} < A \\ 0 & \text{if } q_{2} > A \end{cases} \end{equation}

Similarly, for any given $q_{1}$, firm 2's best response $q_{2}$ to maximize $u_{2}$ is:

\begin{equation} q_{2}(q_{1}) = \begin{cases} \frac{A-q_{1}}{4} & \text{if } q_{1} < A \\ 0 & \text{if } q_{1} > A \end{cases} \end{equation}

Therefore, the Nash equilibrium when the two firms choose $q_{1}$ and $q_{2}$ simultaneously and independently are:

$$(q_{1}, q_{2}) = (\frac{A}{5}, \frac{A}{5})$$

With that Nash equilibrium, the expected utilities of firm 1 and 2 are:

$$EU_{1} = EU_{2} = \frac{2A^2}{25}$$

(Question):

For the subgame-perfect equilibrium, though, how would one use backward induction to find it in this game?

The answer key notes that the subgame-perfect equilibrium is

$$(q_{1}^{s}, q_{2}^{s}) = (\frac{3}{14}A, \frac{11}{56}A)$$

,but how does one arrive at that answer?