# Finding the subgame-perfect equilibrium in sequential games with infinite action spaces

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:

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

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

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

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?

• What have you tried?
– smcc
Commented May 13 at 18:31

In the sequential game, since firm 2 chooses its $$q_2$$ after observing firm $$1$$'s $$q_1$$, firm 2's strategy will be its best response function which is given by:

$$q_2(q_1)=\max\left(\dfrac{A-q_1}{4},0\right)$$

Given firm $$2$$'s strategy above, firm $$1$$ will choose its action (or strategy) $$q_1$$ by solving the following problem:

$$\begin{eqnarray*} \max_{q_1\geq 0} & \max(A-q_1-q_2,0)q_1-q_1^2 \\ \text{s.t. } & q_2=\max\left(\dfrac{A-q_1}{4},0\right)\end{eqnarray*}$$

which is equivalent to solving: $$\begin{eqnarray*} \max_{0\leq q_1\leq A} & \dfrac{3(A-q_1)}{4}q_1-q_1^2 \end{eqnarray*}$$ Solving it, we get $$q_1=\dfrac{3A}{14}$$. And therefore, $$q_2=\max\left(\dfrac{A-q_1}{4},0\right)=\dfrac{11A}{56}$$

• Have you by any chance seen how ChatGPT struggles to solve altered versions of the riddle where "son has accident, father is notified that son is taken to the hospital and he will come two hours later, but at the hospital the doctor says they cannot operate on son because that is their son, how is this possible"? Commented May 21 at 13:40
• No, I am not aware of it.
– Amit
Commented May 22 at 4:34
• The solution to the original puzzle is of course that the doctor is the mother of the son. This puzzle seems to be somewhat well-known. People have noticed that since chatGPT has not understanding of logic, it just memorizes frequent patterns, if they alter the puzzle slightly: instead of the father they say that the mother is notified and she will come to the hospital two hours later, chatGPT gets the solution wrong: it still says that the doctor is the mother, as that was the original answer to the recognized pattern. Commented May 22 at 6:17
• I have compiled and graded many exams, and I think when people teach through exercises instead of explaining and reexplaining the theory, we tend to get phenomena similar to chatGPT's "false understanding". Exercises are still useful, but only after at least a medium understanding of the theory, otherwise what one learns is patterns in the sample. E.g. OP said that backward induction cannot be used in this case, because they seem to have understood it as underlining some payoffs in a finite game tree, instead of the actual logic. Commented May 22 at 6:21
• Of course, not everyone wants to understand, grades and passing are important in themselves; and since few professors bother with truly original exams, learning the patterns in the sample exercises can be very useful. I think this is similar to the story of another AI critique, Searle's Chinese Room problem. Commented May 22 at 6:23

You should still use backward induction. Since the game is not finite, you should not look at the penultimate nodes corresponding to different $$q_1$$ quantities one by one, but rather figure out what the second mover does in response to a general $$q_1$$ quantity. (Treat $$q_1$$ as a parameter in the second player's decision problem.) Once you have done this, just like in finite backward induction, player 1 can 'foresee' the action of player 2, and optimize their own choice of action accordingly.

• Hi! This is an answer, not a comment. Not sure why you think backward induction would lead to a non subgame perfect equilibrium. Commented May 18 at 13:45
• Could you elaborate on what you did, include your calculations? Commented May 20 at 12:33
• "tried to find player 1's optimum decision in response to P2's decision" Perhaps here you treated P2's decision as a parameter $q_2$, but that is not the way to go? Commented May 20 at 12:44