<p>The probabilities are obtained using Bayes updating.</p>
<p>Let <span class="math-container">$f_i = L$</span> be the event that firm <span class="math-container">$i$</span> is low and let <span class="math-container">$f_i = H$</span> be the event that firm <span class="math-container">$i$</span> is a high type.</p>
<p>Assume that firm 1 knows she herself is a high type then:
<span class="math-container">$$
\begin{align*}
\Pr(f_2 = H|f_1 = H) &amp;= \frac{\Pr(f_2 = H \text{ and } f_1 = H)}{\Pr(f_1 = H)},\\
&amp;= \frac{1/3}{1/3 + 1/6},\\
&amp;= \frac{1/3}{1/2} = \frac{2}{3}
\end{align*}
$$</span>
Then also <span class="math-container">$\Pr(f_2 = L|f_1 = H) = 1- \dfrac{2}{3} =  \dfrac{1}{3}$</span>.</p>
<p>The other probabilities can be computed in a similar way.</p>
<p>Also notice that for the expected profits that in general, maximising <strong>expected profits</strong> is not the same as maximising the <strong>profits of playing against a firm with the expected output</strong>. In case of linear inverse demand, it probably does not make a difference, but it might make a difference in general (e.g. if the inverse demand is non-linear).  It seems like the solution manual makes somewhat of an error here.</p>
<p>The expected profits of the solution manual seems strange. They should not be pre-multipied by <span class="math-container">$q_1^L$</span> (<span class="math-container">$q_1^H$</span> etc.)</p>

CC BY-SA 4.0
2021-05-30T18:22:38.277Z
2021-05-30T18:22:38.277Z
tdm
<p>Consider two risk-neutral firms that compete in quantities (Cournot). The aggregate inverse demand is given by <span class="math-container">$P(Q) = 3-Q$</span>. Each firm can only observe its own cost. Find a symmetric BNE.</p>
<p>The constant marginal costs are distributed as follows:</p>
<p><a href="https://i.sstatic.net/jU8i6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jU8i6.png" alt="enter image description here" /></a></p>
<p><strong>My Solution:</strong></p>
<p>There are two types: MC is 1 (low type) and MC is 2 (high type).</p>
<p>First, let's define types and quantities</p>
<ol>
<li><p>When Firm 1 is high type, it knows <span class="math-container">$q_1 = q_1^H$</span>, but it believes <span class="math-container">$q_2 = 1/6q_2^L + 1/3 q_2^H $</span></p>
</li>
<li><p>When Firm 1 is low type, it knows <span class="math-container">$q_1 = q_1^L$</span>, but it believes <span class="math-container">$q_2 = 1/6q_2^H + 1/3 q_2^L $</span></p>
</li>
<li><p>When Firm 2 is high type, it knows <span class="math-container">$q_2 = q_2^H$</span>, but it believes <span class="math-container">$q_1 = 1/6q_1^L + 1/3 q_1^H $</span></p>
</li>
<li><p>When Firm 2 is low type, it knows <span class="math-container">$q_2 = q_2^L$</span>, but it believes <span class="math-container">$q_1 = 1/3q_1^L + 1/6 q_1^H $</span></p>
</li>
</ol>
<p>Second, calculate the expected payoffs:</p>
<p><span class="math-container">$EU_1^H(q_1^H,q_2) = (3-q_1^H-q_2)q_1^H -2q_1^H$</span></p>
<blockquote>
<p><span class="math-container">$EU_1^H(q_1^H,q_2) = (3-q_1^H- [1/6q_2^L + 1/3 q_2^H])q_1^H -2q_1^H$</span></p>
</blockquote>
<p><span class="math-container">$EU_1^L(q_1^L,q_2) = (3-q_1^L-q_2)q_1^L -q_1^L$</span></p>
<blockquote>
<p><span class="math-container">$EU_1^L(q_1^L,q_2) = (3-q_1^L- [1/3q_2^L + 1/6 q_2^H])q_1^L -q_1^L$</span></p>
</blockquote>
<p><span class="math-container">$EU_2^H(q_1,q_2^H) = (3-q_1-q_2^H)q_2^H -2q_2^H$</span></p>
<blockquote>
<p><span class="math-container">$EU_2^H(q_1,q_2^H) = (3-[1/6q_1^L + 1/3 q_1^H  ]-q_2^H)q_2^H -2q_2^H$</span></p>
</blockquote>
<p><span class="math-container">$EU_2^L(q_1,q_2^L) = (3-q_1-q_2^L)q_2^L -q_2^L$</span></p>
<blockquote>
<p><span class="math-container">$EU_2^L(q_1,q_2^L) = (3-[1/3q_1^L + 1/6 q_1^H ]-q_2^L)q_2^L -q_2^L$</span></p>
</blockquote>
<p>Third, Find FOCs:</p>
<p><span class="math-container">$$q_1^H = \frac{1-1/6 q_2^L-1/3q_2^H}{2}$$</span>
<span class="math-container">$$q_1^L = \frac{2-1/3 q_2^L-1/6q_2^H}{2}$$</span>
<span class="math-container">$$q_2^H = \frac{1-1/6 q_1^L-1/3q_1^H}{2}$$</span>
<span class="math-container">$$q_2^L = \frac{2-1/3 q_1^L-1/6q_1^H}{2}$$</span></p>
<p>Finally, at equilibrium, <span class="math-container">$q_1^H = q_1^L = 2/5$</span> and <span class="math-container">$q_2^H = q_2^L = 2/5$</span></p>
<p>HOWEVER, IN SOLUTION MANUAL, its solution is very different. WHAT is my mistake? Why is it different?</p>
<p><strong>SOLUTION MANUAL</strong> says:</p>
<p>Expected utility of low and high type firms are follow:</p>
<p><span class="math-container">$EU_1^L (q_1, q_2')=q_1^L (2-q_1^L-q_2')q_1^L$</span></p>
<p><span class="math-container">$EU_1^H (q_1, q_2^{''})=q_1^H (2-q_1^H-q_2^{''})q_1^H$</span></p>
<p><span class="math-container">$EU_2^L (q_1', q_2)=q_2^L (2-q_1'-q_2^L)q_2^L$</span></p>
<p><span class="math-container">$EU_2^L (q_1^{''}, q_2)=q_2^H (2-q_1^{''}-q_2^H)q_2^H$</span></p>
<p>FOCs are</p>
<p><span class="math-container">$q_1^L = (2-q_2')/2$</span></p>
<p><span class="math-container">$q_1^H = (1-q_2^{''})/2$</span></p>
<p><span class="math-container">$q_2^L = (2-q_1')/2$</span></p>
<p><span class="math-container">$q_2^H = (1-q_1^{''})/2$</span></p>
<p>where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,</p>
<p><span class="math-container">$ q_2' = 2/3 q_2^L + 1/3 q_2^H $</span></p>
<p><span class="math-container">$ q_2^{''} = 1/3 q_2^L + 2/3 q_2^H $</span></p>
<p><span class="math-container">$ q_1' = 2/3 q_1^L + 1/3 q_1^H $</span></p>
<p><span class="math-container">$ q_1^{''} = 1/3 q_1^L + 2/3 q_1^H $</span></p>
<p>At equilibrium, <span class="math-container">$q_1^H = q_1^L = 5/7$</span> and <span class="math-container">$q_2^H = q_2^L = 2/7$</span></p>
<hr />
<p>In the solution manual, I especially don't understand the probability part and last part (the following part:)</p>
<blockquote>
<blockquote>
<p>where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,</p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$ q_2' = 2/3 q_2^L + 1/3 q_2^H $</span></p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$ q_2^{''} = 1/3 q_2^L + 2/3 q_2^H $</span></p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$ q_1' = 2/3 q_1^L + 1/3 q_1^H $</span></p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$ q_1^{''} = 1/3 q_1^L + 2/3 q_1^H $</span></p>
</blockquote>
</blockquote>
<p>Please explain the solution manual's solution or please share your more understandable clear solution way with me. Thank you a lot.</p>

<game-theory><self-study><theory-of-the-firm>
BNE: Incomplete Information Cournot
studentp
# BNE: Incomplete Information Cournot

(Asked by: username_0 on 2021-05-30)

Consider two risk-neutral firms that compete in quantities (Cournot). The aggregate inverse demand is given by $P(Q) = 3-Q$. Each firm can only observe its own cost. Find a symmetric BNE.

The constant marginal costs are distributed as follows:

[![enter image description here](https://i.sstatic.net/jU8i6.png)](https://i.sstatic.net/jU8i6.png)

**My Solution:**

There are two types: MC is 1 (low type) and MC is 2 (high type).

First, let's define types and quantities

1. When Firm 1 is high type, it knows $q\_1 = q\_1^H$, but it believes $q\_2 = 1/6q\_2^L + 1/3 q\_2^H $
2. When Firm 1 is low type, it knows $q\_1 = q\_1^L$, but it believes $q\_2 = 1/6q\_2^H + 1/3 q\_2^L $
3. When Firm 2 is high type, it knows $q\_2 = q\_2^H$, but it believes $q\_1 = 1/6q\_1^L + 1/3 q\_1^H $
4. When Firm 2 is low type, it knows $q\_2 = q\_2^L$, but it believes $q\_1 = 1/3q\_1^L + 1/6 q\_1^H $

Second, calculate the expected payoffs:

$EU\_1^H(q\_1^H,q\_2) = (3-q\_1^H-q\_2)q\_1^H -2q\_1^H$

> $EU\_1^H(q\_1^H,q\_2) = (3-q\_1^H- [1/6q\_2^L + 1/3 q\_2^H])q\_1^H -2q\_1^H$

$EU\_1^L(q\_1^L,q\_2) = (3-q\_1^L-q\_2)q\_1^L -q\_1^L$

> $EU\_1^L(q\_1^L,q\_2) = (3-q\_1^L- [1/3q\_2^L + 1/6 q\_2^H])q\_1^L -q\_1^L$

$EU\_2^H(q\_1,q\_2^H) = (3-q\_1-q\_2^H)q\_2^H -2q\_2^H$

> $EU\_2^H(q\_1,q\_2^H) = (3-[1/6q\_1^L + 1/3 q\_1^H ]-q\_2^H)q\_2^H -2q\_2^H$

$EU\_2^L(q\_1,q\_2^L) = (3-q\_1-q\_2^L)q\_2^L -q\_2^L$

> $EU\_2^L(q\_1,q\_2^L) = (3-[1/3q\_1^L + 1/6 q\_1^H ]-q\_2^L)q\_2^L -q\_2^L$

Third, Find FOCs:

$$q\_1^H = \frac{1-1/6 q\_2^L-1/3q\_2^H}{2}$$
$$q\_1^L = \frac{2-1/3 q\_2^L-1/6q\_2^H}{2}$$
$$q\_2^H = \frac{1-1/6 q\_1^L-1/3q\_1^H}{2}$$
$$q\_2^L = \frac{2-1/3 q\_1^L-1/6q\_1^H}{2}$$

Finally, at equilibrium, $q\_1^H = q\_1^L = 2/5$ and $q\_2^H = q\_2^L = 2/5$

HOWEVER, IN SOLUTION MANUAL, its solution is very different. WHAT is my mistake? Why is it different?

**SOLUTION MANUAL** says:

Expected utility of low and high type firms are follow:

$EU\_1^L (q\_1, q\_2')=q\_1^L (2-q\_1^L-q\_2')q\_1^L$

$EU\_1^H (q\_1, q\_2^{''})=q\_1^H (2-q\_1^H-q\_2^{''})q\_1^H$

$EU\_2^L (q\_1', q\_2)=q\_2^L (2-q\_1'-q\_2^L)q\_2^L$

$EU\_2^L (q\_1^{''}, q\_2)=q\_2^H (2-q\_1^{''}-q\_2^H)q\_2^H$

FOCs are

$q\_1^L = (2-q\_2')/2$

$q\_1^H = (1-q\_2^{''})/2$

$q\_2^L = (2-q\_1')/2$

$q\_2^H = (1-q\_1^{''})/2$

where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,

$ q\_2' = 2/3 q\_2^L + 1/3 q\_2^H $

$ q\_2^{''} = 1/3 q\_2^L + 2/3 q\_2^H $

$ q\_1' = 2/3 q\_1^L + 1/3 q\_1^H $

$ q\_1^{''} = 1/3 q\_1^L + 2/3 q\_1^H $

At equilibrium, $q\_1^H = q\_1^L = 5/7$ and $q\_2^H = q\_2^L = 2/7$

---

In the solution manual, I especially don't understand the probability part and last part (the following part:)

> > where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,

> > $ q\_2' = 2/3 q\_2^L + 1/3 q\_2^H $

> > $ q\_2^{''} = 1/3 q\_2^L + 2/3 q\_2^H $

> > $ q\_1' = 2/3 q\_1^L + 1/3 q\_1^H $

> > $ q\_1^{''} = 1/3 q\_1^L + 2/3 q\_1^H $

Please explain the solution manual's solution or please share your more understandable clear solution way with me. Thank you a lot.

**Tags:** game-theory, self-study, theory-of-the-firm

---

## 1 answers

### Answer by username_1 on 2021-05-30. Score: 2 | Accepted answer

The probabilities are obtained using Bayes updating.

Let $f\_i = L$ be the event that firm $i$ is low and let $f\_i = H$ be the event that firm $i$ is a high type.

Assume that firm 1 knows she herself is a high type then:
$$
\begin{align\*}
\Pr(f\_2 = H|f\_1 = H) &= \frac{\Pr(f\_2 = H \text{ and } f\_1 = H)}{\Pr(f\_1 = H)},\\
&= \frac{1/3}{1/3 + 1/6},\\
&= \frac{1/3}{1/2} = \frac{2}{3}
\end{align\*}
$$
Then also $\Pr(f\_2 = L|f\_1 = H) = 1- \dfrac{2}{3} = \dfrac{1}{3}$.

The other probabilities can be computed in a similar way.

Also notice that for the expected profits that in general, maximising **expected profits** is not the same as maximising the **profits of playing against a firm with the expected output**. In case of linear inverse demand, it probably does not make a difference, but it might make a difference in general (e.g. if the inverse demand is non-linear). It seems like the solution manual makes somewhat of an error here.

The expected profits of the solution manual seems strange. They should not be pre-multipied by $q\_1^L$ ($q\_1^H$ etc.)


---

