I have this problem about Cournot Duopoly game. Actually I don't know if I have understood the "real sense" of the problem. I consider CD game described by the following payoff fucntions: $$ u_{1}(x_1,x_2)=x_1\Big{(}1-\frac{x_1+x_2}{2}\Big{)} $$ $$ u_{2}(x_1,x_2)=x_2\Big{(}1-\frac{x_1+x_2}{2}\Big{)} $$ At this point computing the partial derivatives and set them equal to zero I obtain as NE $x_{1}=x_{2}=\frac{2}{3}$ and so the payoffs at NE are both equal to $\frac{2}{9}$.
Now it has been considered the total payoff function $u(x_1,x_2)=u_1(x_1,x_2)+u_2(x_1,x_2)$ and I would like to have $$\frac{\partial{u}}{\partial{x_1}}=\frac{\partial{u}}{\partial{x_2}}=0$$ (and they called this "cooperative solution").
Solving with this new framework I obtain $x_1=x_2=\frac{1}{2}$ and the payoffs are both $\frac{1}{4}$. So in this case with respect the NE that I have found at the beginning I can say that the production in smaller but the profit is bigger.
Now, my question is: How to prove that $ \left(\frac{1}{2},\frac{1}{2} \right)$ is not a Nash equilibrium? And what is the best response to $\frac{1}{2}$? I hope that someone could give me a hint because I'm new to game theory.
Thank you very much!
If player 2 chooses $x_2=1/2$, then, plugging this into player 1's payoff, we get for player 1's payoff: $$x_1\left(1-\frac{x_1+0.5}{2}\right)=x_1\left(\frac{3}{4}-\frac{x_1}{2}\right).$$ Maximizing this using calculus, we find that the best choice of player 1 is: $$x_1=\frac{3}{4}.$$ So, the best reply of player 1, if the other player chooses 1/2, is to choose 3/4. The strategy pair (1/2,1/2) is not a Nash equilibrium because player 1 is not playing his best reply. Of course, by the same argument, player 2 is not playing his best reply either.