<p>Given the demand function:
<span class="math-container">\begin{eqnarray*}P(Q) = \max(60-Q,0) \end{eqnarray*}</span>
where <span class="math-container">$Q=q_1+q_2$</span>.</p>
<p>Cost function of company 1 is
<span class="math-container">\begin{eqnarray*}c_1(q_1)= \begin{cases} 18q_1 &amp; \text{with prob. } \frac{1}{3} \\ 24q_1 &amp; \text{with prob. } \frac{2}{3} \end{cases} \end{eqnarray*}</span>
Cost function of company 2 is
<span class="math-container">\begin{eqnarray*}c_2(q_2)= \begin{cases} 12q_2 &amp; \text{with prob. } \frac{1}{4} \\ 24q_2 &amp; \text{with prob. } \frac{3}{4} \end{cases} \end{eqnarray*}</span>
Let <span class="math-container">$(q_1^L, q_1^H)$</span> denotes the strategy of company 1, where <span class="math-container">$q_1^L$</span> is the quantity it produces when it faces low costs i.e. <span class="math-container">$18q_1$</span> (Type L) and <span class="math-container">$q_1^H$</span> is the quantity it produces when it faces high costs i.e. <span class="math-container">$24q_1$</span> (Type H). Likewise, let <span class="math-container">$(q_2^L, q_2^H)$</span> denotes the strategy of company 2, where <span class="math-container">$q_2^L, q_2^H$</span> are the quantities chosen by company 2 when it faces costs of <span class="math-container">$12q_2$</span> (Type L) and <span class="math-container">$24q_2$</span> (Type H) respectively.</p>
<p>To determine the Bayesian Nash equilibrium, we'll first find the best response functions of both types of two companies:</p>
<p>Type L of company 1's maximisation problem:
<span class="math-container">\begin{eqnarray*} \max_{q^L_1\geq 0}\frac{1}{4}(60-q_2^L-q_1^L)q^L_1+ \frac{3}{4}(60-q_2^H-q_1^L)q^L_1-18q^L_1 \end{eqnarray*}</span>
and we get the best response function as:
<span class="math-container">\begin{eqnarray*} q_1^L=\frac{42 -\left(\frac{1}{4}q_2^L+\frac{3}{4}q_2^H\right)}{2} \end{eqnarray*}</span>
Likewise, Type H of company 1's maximisation problem:
<span class="math-container">\begin{eqnarray*} \max_{q^H_1\geq 0}\frac{1}{4}(60-q_2^L-q_1^H)q^H_1+ \frac{3}{4}(60-q_2^H-q_1^H)q^H_1-24q^H_1 \end{eqnarray*}</span>
and we get the best response function as:
<span class="math-container">\begin{eqnarray*} q_1^H=\frac{36 -\left(\frac{1}{4}q_2^L+\frac{3}{4}q_2^H\right)}{2} \end{eqnarray*}</span></p>
<p>Similarly, we can derive the best response of two types of company 2 and get
<span class="math-container">\begin{eqnarray*} q_2^L=\frac{48 -\left(\frac{1}{3}q_1^L+\frac{2}{3}q_1^H\right)}{2} \\ q_2^H=\frac{36 -\left(\frac{1}{3}q_1^L+\frac{2}{3}q_1^H\right)}{2}  \end{eqnarray*}</span></p>
<p>Solving the system of four best response functions we get the following as strategies of company 1 and 2 in the Bayesian Nash equilibrium.</p>
<ul>
<li>Strategy of firm 1: <span class="math-container">$q_1^L = \frac{43}{3}$</span>, <span class="math-container">$q_1^H=\frac{34}{3}$</span></li>
<li>Strategy of firm 2: <span class="math-container">$q_2^L = \frac{107}{6}$</span>, <span class="math-container">$q_2^H=\frac{71}{6}$</span></li>
</ul>

CC BY-SA 4.0
2024-05-30T15:42:24.990Z
Thanks man, but i solved it about tow weeks ago
morteza bigdeli
CC BY-SA 4.0
2024-05-28T12:22:47.987Z
2024-05-28T12:29:02.063Z
2024-05-28T12:29:02.063Z
Amit
<p>Consider the Cournot game with two companies, whose function (Q) P is in the form of a relation:</p>
<p>P (Q) = 60 − Q if Q ≤ 60
0 if Q ≥ 60</p>
<p>where q2 + q1 = Q is the sum of outputs in the market. The cost of company 2 is 12q2 = c2(q2) with probability 1/4 and 24q2 = c2(q2) with probability 3/4, and the cost of company 1 is 18q1 = c1(q1) with probability 1/3 and 24q1 = c1(q1) with probability 2/3. it is The companies are different from the situation: They know which of the two states they are in, but the opposite company only knows about the types and their probabilities.
A) Show the given information in the form of a Bayesian game, and for this, the set of players, the set of each type, the set of strategiesand identify the profit function.
B)Find Bayesian nash equilibrium and expected payoffs</p>

CC BY-SA 4.0
2024-05-11T15:51:43.887Z
Cool but what is your question, where do you get stuck with this simple exercise? Do you not know some of the definitions?
Giskard
CC BY-SA 4.0
2024-06-04T14:10:23.523Z
there are four types in this problem and each player only sees the probablities of the other one not anything else, the solution which is provided by someone here is not compeletly true to my problem ayway, if you wanted the whole question, i can send it to you, regards
morteza bigdeli
<microeconomics><bayesian-game><cournot>
bayesian nash equilibrium cournot game
morteza bigdeli
# bayesian nash equilibrium cournot game

(Asked by: username_0 on 2024-05-11)

Consider the Cournot game with two companies, whose function (Q) P is in the form of a relation:

P (Q) = 60 − Q if Q ≤ 60
0 if Q ≥ 60

where q2 + q1 = Q is the sum of outputs in the market. The cost of company 2 is 12q2 = c2(q2) with probability 1/4 and 24q2 = c2(q2) with probability 3/4, and the cost of company 1 is 18q1 = c1(q1) with probability 1/3 and 24q1 = c1(q1) with probability 2/3. it is The companies are different from the situation: They know which of the two states they are in, but the opposite company only knows about the types and their probabilities.
A) Show the given information in the form of a Bayesian game, and for this, the set of players, the set of each type, the set of strategiesand identify the profit function.
B)Find Bayesian nash equilibrium and expected payoffs

**Tags:** microeconomics, bayesian-game, cournot

> **Comments:**
> **username_1**: Cool but what is your question, where do you get stuck with this simple exercise? Do you not know some of the definitions?  
> **username_0**: there are four types in this problem and each player only sees the probablities of the other one not anything else, the solution which is provided by someone here is not compeletly true to my problem ayway, if you wanted the whole question, i can send it to you, regards  

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## 1 answers

### Answer by username_2 on 2024-05-28. Score: 3

Given the demand function:
\begin{eqnarray\*}P(Q) = \max(60-Q,0) \end{eqnarray\*}
where $Q=q\_1+q\_2$.

Cost function of company 1 is
\begin{eqnarray\*}c\_1(q\_1)= \begin{cases} 18q\_1 & \text{with prob. } \frac{1}{3} \\ 24q\_1 & \text{with prob. } \frac{2}{3} \end{cases} \end{eqnarray\*}
Cost function of company 2 is
\begin{eqnarray\*}c\_2(q\_2)= \begin{cases} 12q\_2 & \text{with prob. } \frac{1}{4} \\ 24q\_2 & \text{with prob. } \frac{3}{4} \end{cases} \end{eqnarray\*}
Let $(q\_1^L, q\_1^H)$ denotes the strategy of company 1, where $q\_1^L$ is the quantity it produces when it faces low costs i.e. $18q\_1$ (Type L) and $q\_1^H$ is the quantity it produces when it faces high costs i.e. $24q\_1$ (Type H). Likewise, let $(q\_2^L, q\_2^H)$ denotes the strategy of company 2, where $q\_2^L, q\_2^H$ are the quantities chosen by company 2 when it faces costs of $12q\_2$ (Type L) and $24q\_2$ (Type H) respectively.

To determine the Bayesian Nash equilibrium, we'll first find the best response functions of both types of two companies:

Type L of company 1's maximisation problem:
\begin{eqnarray\*} \max\_{q^L\_1\geq 0}\frac{1}{4}(60-q\_2^L-q\_1^L)q^L\_1+ \frac{3}{4}(60-q\_2^H-q\_1^L)q^L\_1-18q^L\_1 \end{eqnarray\*}
and we get the best response function as:
\begin{eqnarray\*} q\_1^L=\frac{42 -\left(\frac{1}{4}q\_2^L+\frac{3}{4}q\_2^H\right)}{2} \end{eqnarray\*}
Likewise, Type H of company 1's maximisation problem:
\begin{eqnarray\*} \max\_{q^H\_1\geq 0}\frac{1}{4}(60-q\_2^L-q\_1^H)q^H\_1+ \frac{3}{4}(60-q\_2^H-q\_1^H)q^H\_1-24q^H\_1 \end{eqnarray\*}
and we get the best response function as:
\begin{eqnarray\*} q\_1^H=\frac{36 -\left(\frac{1}{4}q\_2^L+\frac{3}{4}q\_2^H\right)}{2} \end{eqnarray\*}

Similarly, we can derive the best response of two types of company 2 and get
\begin{eqnarray\*} q\_2^L=\frac{48 -\left(\frac{1}{3}q\_1^L+\frac{2}{3}q\_1^H\right)}{2} \\ q\_2^H=\frac{36 -\left(\frac{1}{3}q\_1^L+\frac{2}{3}q\_1^H\right)}{2} \end{eqnarray\*}

Solving the system of four best response functions we get the following as strategies of company 1 and 2 in the Bayesian Nash equilibrium.

* Strategy of firm 1: $q\_1^L = \frac{43}{3}$, $q\_1^H=\frac{34}{3}$
* Strategy of firm 2: $q\_2^L = \frac{107}{6}$, $q\_2^H=\frac{71}{6}$

> **Comments:**
> **username_0**: Thanks man, but i solved it about tow weeks ago  

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