<p>Consider the following situation:</p>
<ul>
<li>Demand is given by <span class="math-container">$P^d(Q) = \max(12-Q, 0)$</span></li>
<li>There are two firms and each firm <span class="math-container">$i\in\{1,2\}$</span> has a cost function:
<span class="math-container">\begin{eqnarray*} c_i(q_i) =\begin{cases} 16 &amp; \text{if } q_i &gt; 0 \\ 0 &amp; \text{otherwise} \end{cases} \end{eqnarray*}</span></li>
<li>Firms choose quantities as in a Cournot Model.</li>
</ul>
<p>Firm <span class="math-container">$i$</span>'s profit maximisation problem is the following:
<span class="math-container">\begin{eqnarray*} \max_{q_i\geq 0} \max(12-q_i-q_j)q_i - c_i(q_i) \end{eqnarray*}</span> where <span class="math-container">$i, j \in \{1,2\}$</span> and <span class="math-container">$i\neq j$</span>.</p>
<p>Solving the above problem gives the following set of solutions (as a function of <span class="math-container">$q_j$</span>):
<span class="math-container">\begin{eqnarray*} \text{BR}_i(q_j) = \begin{cases} \{0\} &amp; \text{if } q_j &gt; 4 \\\{0, 4\} &amp; \text{if } q_j = 4 \\ \left\{\frac{12-q_j}{2}\right\} &amp; \text{if } q_j &lt; 4\end{cases} \end{eqnarray*}</span></p>
<p>Clearly there are three Cournot Nash equilibria:
<span class="math-container">\begin{eqnarray*} \left(q_1^{e_1}, q_2^{e_1}\right) = (6, 0) \\  \left(q_1^{e_2}, q_2^{e_2}\right) = (4,4) \\ \left(q_1^{e_3}, q_2^{e_3}\right) = (0,6)\end{eqnarray*}</span></p>
<p>This is a situation where we have two firms and there are multiple equilibria. Now consider the same demand as above and there are <span class="math-container">$n\geq 3$</span> firms with the same cost function as above, we'll see that again there will be multiple equilibria.</p>
<p>Firm <span class="math-container">$i$</span>'s profit maximisation problem is the following:
<span class="math-container">\begin{eqnarray*} \max_{q_i\geq 0} \max\left(12-q_i-\sum_{j\neq i}q_j\right)q_i - c_i(q_i) \end{eqnarray*}</span> where <span class="math-container">$i, j \in \{1,2,\ldots, n\}$</span> and <span class="math-container">$i\neq j$</span>.</p>
<p>Solving the above problem gives the following set of solutions (as a function of <span class="math-container">$q_{-i} = (q_1, q_2, \ldots, q_{i-1}, q_{i+1}, \ldots, q_n) $</span>):
<span class="math-container">\begin{eqnarray*} \text{BR}_i(q_{-i}) = \begin{cases} \{0\} &amp; \text{if } \sum_{j\neq i}q_j &gt; 4 \\\{0, 4\} &amp; \text{if } \sum_{j\neq i}q_j = 4 \\ \left\{\frac{12-\sum_{j\neq i}q_j}{2}\right\} &amp; \text{if } \sum_{j\neq i}q_j &lt; 4\end{cases} \end{eqnarray*}</span></p>
<p>Clearly there are two categories of Cournot Nash equilibria:</p>
<ul>
<li>There are <span class="math-container">$n$</span> equilibria of the type where exactly one of the firms produces 6 units, and rest of them produces <span class="math-container">$0$</span>.</li>
<li>There are <span class="math-container">${n\choose 2}$</span> equilibria of the type where exactly two out of <span class="math-container">$n$</span> firms produces 4 units each, and rest of them produces <span class="math-container">$0$</span>.</li>
</ul>
<p>To see how to find the best response functions (<span class="math-container">$\text{BR}_i$</span>), the following might be helpful: <a href="https://youtube.com/playlist?list=PLUJGfL_499TKJR9XDlmXLt9z3AMCfXNVC" rel="nofollow noreferrer">https://youtube.com/playlist?list=PLUJGfL_499TKJR9XDlmXLt9z3AMCfXNVC</a></p>

CC BY-SA 4.0
2022-05-12T09:45:47.817Z
2022-05-12T15:10:05.403Z
2022-05-12T15:10:05.403Z
Amit
<p>My approach for two cases was to draw reaction functions and successively eliminate what could not be played using rationality and common knowledge of rationality. I found indeed there is a unique NE when the two curves intersect. What would happen in more than two firms? I am having a hard time visualizing this. Would I need to set up an optimization problem to answer this (Which I can) but I don't know how I will follow the reasoning of iterated deletion to reason out a Nash prediction.</p>

CC BY-SA 4.0
2022-05-08T16:33:43.153Z
Do you have specific demand/profit/cost functions in mind? Do firms differ in some way?
Michael Greinecker
CC BY-SA 4.0
2022-05-08T18:55:46.977Z
How can you garantee that in the 2-firms case the reaction curves do not intersect several times? Even in the two firm case, multiple equilibria can occur. See for instance Fudenberg, D. and J. Tirole, 1991, Game theory, MIT Press.
Bertrand
CC BY-SA 4.0
2022-05-08T21:50:16.817Z
Excuse my comment, but I love that Bertrand comes to criticise a question on Cournot.
Walrasian Auctioneer
CC BY-SA 4.0
2022-05-09T05:03:14.540Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Giskard
CC BY-SA 4.0
2022-05-10T11:33:10.560Z
Why solve by iterated elimination? If you just want to know the Nash equilibrium, just intersecting the reaction functions does the job. Visualizing this with more than 2 firms is only possible for 3 firms: The graphs of the 3 reaction functions are 2-dimensional objects embedded in 3 dimensions (think of the surface of a landscape) and the Nash equilibria are where they intersect. Under standard assumptions this will be a single point. However, this NE is not guaranteed to be stable.
VARulle
<game-theory>
Does Cournot competition have unique Nash Equilibriium in more than 2 firms?
Jamal
# Does Cournot competition have unique Nash Equilibriium in more than 2 firms?

(Asked by: username_0 on 2022-05-08)

My approach for two cases was to draw reaction functions and successively eliminate what could not be played using rationality and common knowledge of rationality. I found indeed there is a unique NE when the two curves intersect. What would happen in more than two firms? I am having a hard time visualizing this. Would I need to set up an optimization problem to answer this (Which I can) but I don't know how I will follow the reasoning of iterated deletion to reason out a Nash prediction.

**Tags:** game-theory
> **Comments:**
> **username_1**: Do you have specific demand/profit/cost functions in mind? Do firms differ in some way?  
> **username_2**: How can you garantee that in the 2-firms case the reaction curves do not intersect several times? Even in the two firm case, multiple equilibria can occur. See for instance Fudenberg, D. and J. Tirole, 1991, Game theory, MIT Press.  
> **username_3**: Excuse my comment, but I love that Bertrand comes to criticise a question on Cournot.  
> **username_4**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  
> **username_5**: Why solve by iterated elimination? If you just want to know the Nash equilibrium, just intersecting the reaction functions does the job. Visualizing this with more than 2 firms is only possible for 3 firms: The graphs of the 3 reaction functions are 2-dimensional objects embedded in 3 dimensions (think of the surface of a landscape) and the Nash equilibria are where they intersect. Under standard assumptions this will be a single point. However, this NE is not guaranteed to be stable.  

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## 1 answers

### Answer by username_6 on 2022-05-12. Score: 4

Consider the following situation:

* Demand is given by $P^d(Q) = \max(12-Q, 0)$
* There are two firms and each firm $i\in\{1,2\}$ has a cost function:
  \begin{eqnarray\*} c\_i(q\_i) =\begin{cases} 16 & \text{if } q\_i > 0 \\ 0 & \text{otherwise} \end{cases} \end{eqnarray\*}
* Firms choose quantities as in a Cournot Model.

Firm $i$'s profit maximisation problem is the following:
\begin{eqnarray\*} \max\_{q\_i\geq 0} \max(12-q\_i-q\_j)q\_i - c\_i(q\_i) \end{eqnarray\*} where $i, j \in \{1,2\}$ and $i\neq j$.

Solving the above problem gives the following set of solutions (as a function of $q\_j$):
\begin{eqnarray\*} \text{BR}\_i(q\_j) = \begin{cases} \{0\} & \text{if } q\_j > 4 \\\{0, 4\} & \text{if } q\_j = 4 \\ \left\{\frac{12-q\_j}{2}\right\} & \text{if } q\_j < 4\end{cases} \end{eqnarray\*}

Clearly there are three Cournot Nash equilibria:
\begin{eqnarray\*} \left(q\_1^{e\_1}, q\_2^{e\_1}\right) = (6, 0) \\ \left(q\_1^{e\_2}, q\_2^{e\_2}\right) = (4,4) \\ \left(q\_1^{e\_3}, q\_2^{e\_3}\right) = (0,6)\end{eqnarray\*}

This is a situation where we have two firms and there are multiple equilibria. Now consider the same demand as above and there are $n\geq 3$ firms with the same cost function as above, we'll see that again there will be multiple equilibria.

Firm $i$'s profit maximisation problem is the following:
\begin{eqnarray\*} \max\_{q\_i\geq 0} \max\left(12-q\_i-\sum\_{j\neq i}q\_j\right)q\_i - c\_i(q\_i) \end{eqnarray\*} where $i, j \in \{1,2,\ldots, n\}$ and $i\neq j$.

Solving the above problem gives the following set of solutions (as a function of $q\_{-i} = (q\_1, q\_2, \ldots, q\_{i-1}, q\_{i+1}, \ldots, q\_n) $):
\begin{eqnarray\*} \text{BR}\_i(q\_{-i}) = \begin{cases} \{0\} & \text{if } \sum\_{j\neq i}q\_j > 4 \\\{0, 4\} & \text{if } \sum\_{j\neq i}q\_j = 4 \\ \left\{\frac{12-\sum\_{j\neq i}q\_j}{2}\right\} & \text{if } \sum\_{j\neq i}q\_j < 4\end{cases} \end{eqnarray\*}

Clearly there are two categories of Cournot Nash equilibria:

* There are $n$ equilibria of the type where exactly one of the firms produces 6 units, and rest of them produces $0$.
* There are ${n\choose 2}$ equilibria of the type where exactly two out of $n$ firms produces 4 units each, and rest of them produces $0$.

To see how to find the best response functions ($\text{BR}\_i$), the following might be helpful: <https://youtube.com/playlist?list=PLUJGfL_499TKJR9XDlmXLt9z3AMCfXNVC>


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