two-person continuous game

Question

Can anyone help me solve this question? Thanks in advance! In a two-person continuous game the strategies are between 0 and 1, and the two payoff functions are f1 = x+y-(x+y)^2 and f2 = xy - y^2 + 2x. Find the equilibrium.

Answer 1

Assuming x is for player 1 and y is for player 2, we can proceed as

$\frac{\partial f_1}{\partial x}=1-2(x+y)=0 \Rightarrow x+y=\frac{1}{2}$

$\frac{\partial f_2}{\partial y}=x-2y=0$

On solving, $x=\frac{1}{3}$ and $y=\frac{1}{6}$

So, the Nash equilibrium is ($\frac{1}{3}$,$\frac{1}{6}$)