I have 2 symmetric players $A$ and $B$.
Each of them has 2 decision variables $x_i\in[0, \beta]$ and $y_i\in[0,1]$, where $i\in\{A,B\}$.
Their payoff functions are symmetric, i.e., if you swap the label $A$ and $B$, you get the other player's payoff function. And the payoff function is continuous in both $x$ and $y$.
My question is: for this type of game, does it always have symmetric pure strategy equilibrium? In general, what are the conditions to guarantee the existence of symmetric pure strategy equilibrium?