<p>A function <span class="math-container">$P(q_1,q_2)$</span> is said to be the potential function of a game if its derivatives, coincide with the marginal payoff of each player. </p>

<p>So for example, in your case, if you have that the  payoff of a firm is its profit. Let me call it <span class="math-container">$\pi_1(q_1,q_2)=TR-TC=p\cdot q_1-c\cdot q_1$</span> for firm 1 and <span class="math-container">$\pi_2(q_1,q_2)=TR-TC=p\cdot q_2-c\cdot q_2$</span> for firm 2, then to check that P is the potential of this game you want to check that <span class="math-container">$\frac{\partial P(q_1,q_2)}{\partial q_1}=\frac{\partial \pi_1(q_1,q_2)}{\partial q_1}$</span> and that <span class="math-container">$\frac{\partial P(q_1,q_2)}{\partial q_2}=\frac{\partial \pi_2(q_1,q_2)}{\partial q_2}$</span>. </p>

<p>The problem is not asking you to solve for the optimal quantity just yet, so <span class="math-container">$MR=MC$</span> will not quite help you (at least not in that specific question).</p>

<p>To see more aboutthe definition of a "potential function" and a "potential game" see <a href="https://en.wikipedia.org/wiki/Potential_game" rel="nofollow noreferrer">here</a>.</p>

CC BY-SA 4.0
2019-06-27T20:15:01.720Z
2019-06-27T20:15:01.720Z
Regio
<p>So i have this question</p>

<p><a href="https://i.sstatic.net/3dbRU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3dbRU.png" alt="enter image description here"></a></p>

<p>And i have no clue on how to get to verifying that that function on the answer sheet is a potential Cournot game.</p>

<p>I started answering the question like this, but then i got lost when calculating the MC since i only have <em>c</em> as a unit cost and not total cost.</p>

<p>But probably i was answering the question in a wrong way.</p>

<p>Please tell me if i am.</p>

<p><a href="https://i.sstatic.net/OpfbD.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OpfbD.jpg" alt="enter image description here"></a></p>

<p>Could you show me the steps on how to get to the function which is a potential for the Cournot game ? </p>

<p>I have learning disabilities and I have difficulties grasping on to hard concepts like these. If you could show me clear and simple steps to get to the answer it would be greatly appreciated ! Thanks!</p>

<game-theory><potential-game>
How to solve for the potential of a cournot duopoly problem?
Sara Saletti
# How to solve for the potential of a cournot duopoly problem?

(Asked by: username_0 on 2019-06-27)

So i have this question

[![enter image description here](https://i.sstatic.net/3dbRU.png)](https://i.sstatic.net/3dbRU.png)

And i have no clue on how to get to verifying that that function on the answer sheet is a potential Cournot game.

I started answering the question like this, but then i got lost when calculating the MC since i only have *c* as a unit cost and not total cost.

But probably i was answering the question in a wrong way.

Please tell me if i am.

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

Could you show me the steps on how to get to the function which is a potential for the Cournot game ?

I have learning disabilities and I have difficulties grasping on to hard concepts like these. If you could show me clear and simple steps to get to the answer it would be greatly appreciated ! Thanks!

**Tags:** game-theory, potential-game

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## 1 answers

### Answer by username_1 on 2019-06-27. Score: 3

A function $P(q\_1,q\_2)$ is said to be the potential function of a game if its derivatives, coincide with the marginal payoff of each player.

So for example, in your case, if you have that the payoff of a firm is its profit. Let me call it $\pi\_1(q\_1,q\_2)=TR-TC=p\cdot q\_1-c\cdot q\_1$ for firm 1 and $\pi\_2(q\_1,q\_2)=TR-TC=p\cdot q\_2-c\cdot q\_2$ for firm 2, then to check that P is the potential of this game you want to check that $\frac{\partial P(q\_1,q\_2)}{\partial q\_1}=\frac{\partial \pi\_1(q\_1,q\_2)}{\partial q\_1}$ and that $\frac{\partial P(q\_1,q\_2)}{\partial q\_2}=\frac{\partial \pi\_2(q\_1,q\_2)}{\partial q\_2}$.

The problem is not asking you to solve for the optimal quantity just yet, so $MR=MC$ will not quite help you (at least not in that specific question).

To see more aboutthe definition of a "potential function" and a "potential game" see [here](https://en.wikipedia.org/wiki/Potential_game).


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