# How to solve for the potential of a cournot duopoly problem?

So i have this question

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.

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!

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).