<p>In the sequential game, since firm 2 chooses its <span class="math-container">$q_2$</span> after observing firm <span class="math-container">$1$</span>'s <span class="math-container">$q_1$</span>, firm 2's strategy will be its best response function which is given by:</p>
<p><span class="math-container">$q_2(q_1)=\max\left(\dfrac{A-q_1}{4},0\right)$</span></p>
<p>Given firm <span class="math-container">$2$</span>'s strategy above, firm <span class="math-container">$1$</span> will choose its action (or strategy) <span class="math-container">$q_1$</span> by solving the following problem:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{q_1\geq 0} &amp; \max(A-q_1-q_2,0)q_1-q_1^2 \\ \text{s.t. } &amp; q_2=\max\left(\dfrac{A-q_1}{4},0\right)\end{eqnarray*}</span></p>
<p>which is equivalent to solving:
<span class="math-container">\begin{eqnarray*} \max_{0\leq q_1\leq A} &amp; \dfrac{3(A-q_1)}{4}q_1-q_1^2 \end{eqnarray*}</span>
Solving it, we get <span class="math-container">$q_1=\dfrac{3A}{14}$</span>. And therefore, <span class="math-container">$q_2=\max\left(\dfrac{A-q_1}{4},0\right)=\dfrac{11A}{56}$</span></p>

CC BY-SA 4.0
2024-05-21T13:40:38.413Z
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"?
Giskard
CC BY-SA 4.0
2024-05-22T04:34:29.003Z
No, I am not aware of it.
Amit
CC BY-SA 4.0
2024-05-22T06:17:05.147Z
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.
Giskard
CC BY-SA 4.0
2024-05-22T06:21:20.670Z
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.
Giskard
CC BY-SA 4.0
2024-05-22T06:23:14.473Z
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](https://en.wikipedia.org/wiki/Chinese_room#Chinese_room_thought_experiment).
Giskard
CC BY-SA 4.0
2024-05-22T06:32:22.870Z
I think this is also the reason why [your answer here](https://economics.stackexchange.com/a/58460/1601) may be more useful for the normal student, [Adam Bailey's answer](https://economics.stackexchange.com/a/58458/1601) is more useful to the student who wants to understand what they are doing.
Giskard
CC BY-SA 4.0
2024-05-22T14:58:02.277Z
I have posted the answer. I just hope it would be useful for someone.
Amit
CC BY-SA 4.0
2024-05-22T15:14:46.007Z
Well yes, but useful how.
Giskard
CC BY-SA 4.0
2024-05-22T17:04:55.197Z
As I have said, I hope it is useful for someone (not necessarily everyone). If there is such a 'someone,' they would be able to explain how.
Amit
CC BY-SA 4.0
2024-05-20T16:25:21.433Z
2024-05-20T16:25:21.433Z
Amit
<p>You should still use backward induction. Since the game is not finite, you should not look at the penultimate nodes corresponding to different <span class="math-container">$q_1$</span> quantities one by one, but rather figure out what the second mover does in response to a general <span class="math-container">$q_1$</span> quantity. (Treat <span class="math-container">$q_1$</span> 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.</p>

CC BY-SA 4.0
2024-05-18T13:45:43.047Z
Hi! This is an answer, not a comment. Not sure why you think backward induction would lead to a non subgame perfect equilibrium.
Giskard
CC BY-SA 4.0
2024-05-20T12:33:16.183Z
Could *you* elaborate on what you did, include your calculations?
Giskard
CC BY-SA 4.0
2024-05-20T12:44:31.837Z
"*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?
Giskard
CC BY-SA 4.0
2024-05-14T04:32:15.417Z
2024-05-18T18:32:30.253Z
2024-05-18T18:32:30.253Z
Giskard
<p>Firms 1 and 2 are competing in the same market. Each firm <span class="math-container">$i$</span> must choose a quantity <span class="math-container">$q_{i}$</span> to supply, and the market price <span class="math-container">$p$</span> will depend on their choices according to the inverse demand formula <span class="math-container">$p(q_{1},q_{2}) = \text{max}[A-(q_{1} + q_{2}), 0]$</span>.</p>
<p>The total cost of production for each firm <span class="math-container">$i$</span> is <span class="math-container">$q_{i}^2$</span>, and so the total profit for firm <span class="math-container">${i}$</span> will be <span class="math-container">$u_{i}(q_{1},q_{2}) = p(q_{1},q_{2}) \cdot q_{i} - (q_{i})^{2}$</span></p>
<p>Suppose that firm 1 chooses <span class="math-container">$q_{1}$</span> first, and then firm 2 chooses its <span class="math-container">$q_{2}$</span> after observing <span class="math-container">$q_{1}$</span>.</p>
<p><strong>Find</strong> : The <em>subgame-perfect equilibrium</em> of this game with perfect information, and compute each firm's expected profit in this equilibrium.</p>
<hr />
<p>(<em>What I have tried:</em>)</p>
<p>For any given <span class="math-container">$q_{2}$</span>, firm 1's best response <span class="math-container">$q_{1}$</span> to maximize <span class="math-container">$u_{1}$</span> is:</p>
<p><span class="math-container">\begin{equation}
q_{1}(q_{2}) = 
\begin{cases} 
\frac{A-q_{2}}{4} &amp; \text{if } q_{2} &lt; A \\
0 &amp; \text{if } q_{2} &gt; A
\end{cases}
\end{equation}</span></p>
<p>Similarly, for any given <span class="math-container">$q_{1}$</span>, firm 2's best response <span class="math-container">$q_{2}$</span> to maximize <span class="math-container">$u_{2}$</span> is:</p>
<p><span class="math-container">\begin{equation}
q_{2}(q_{1}) = 
\begin{cases} 
\frac{A-q_{1}}{4} &amp; \text{if } q_{1} &lt; A \\
0 &amp; \text{if } q_{1} &gt; A
\end{cases}
\end{equation}</span></p>
<p>Therefore, the <em>Nash equilibrium</em> when the two firms choose <span class="math-container">$q_{1}$</span> and <span class="math-container">$q_{2}$</span> simultaneously and independently are:</p>
<p><span class="math-container">$$(q_{1}, q_{2}) = (\frac{A}{5}, \frac{A}{5})$$</span></p>
<p>With that Nash equilibrium, the expected utilities of firm 1 and 2 are:</p>
<p><span class="math-container">$$EU_{1} = EU_{2} = \frac{2A^2}{25}$$</span></p>
<hr />
<p><em>(Question):</em></p>
<p>For the <em>subgame-perfect equilibrium</em>, though, how would one use backward induction to find it in this game?</p>
<p>The answer key notes that the subgame-perfect equilibrium is</p>
<p><span class="math-container">$$(q_{1}^{s}, q_{2}^{s}) = (\frac{3}{14}A, \frac{11}{56}A)$$</span></p>
<p>,but how does one arrive at that answer?</p>

CC BY-SA 4.0
2024-05-13T18:31:15.703Z
What have you tried?
smcc
<game-theory><dynamic-games>
Finding the subgame-perfect equilibrium in sequential games with infinite action spaces
user9487
# Finding the subgame-perfect equilibrium in sequential games with infinite action spaces

(Asked by: username_0 on 2024-05-13)

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?

**Tags:** game-theory, dynamic-games

> **Comments:**
> **username_1**: What have you tried?  

---

## 2 answers

### Answer by username_2 on 2024-05-20. Score: 2 | Accepted answer

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, $q\_2=\max\left(\dfrac{A-q\_1}{4},0\right)=\dfrac{11A}{56}$

> **Comments:**
> **username_3**: 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"?  
> **username_2**: No, I am not aware of it.  
> **username_3**: 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.  
> **username_3**: 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.  
> **username_3**: 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](https://en.wikipedia.org/wiki/Chinese_room#Chinese_room_thought_experiment).  
> **username_3**: I think this is also the reason why [your answer here](https://economics.stackexchange.com/a/58460/1601) may be more useful for the normal student, [Adam Bailey's answer](https://economics.stackexchange.com/a/58458/1601) is more useful to the student who wants to understand what they are doing.  
> **username_2**: I have posted the answer. I just hope it would be useful for someone.  
> **username_3**: Well yes, but useful how.  
> **username_2**: As I have said, I hope it is useful for someone (not necessarily everyone). If there is such a 'someone,' they would be able to explain how.  

---

### Answer by username_3 on 2024-05-14. Score: 1

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.

> **Comments:**
> **username_3**: Hi! This is an answer, not a comment. Not sure why you think backward induction would lead to a non subgame perfect equilibrium.  
> **username_3**: Could *you* elaborate on what you did, include your calculations?  
> **username_3**: "*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?  

---

