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?
