# Existence of Symmetric Pure Strategy Equilibrium

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?

• In general it does not even have a pure strategy equilibrium. I believe there are some results of existence of such equilibrium in supermodular games. Jun 3 '20 at 19:54

2. There is also no guarantee that symmetric games have pure-strategy equilibria. For example, the following game is symmetric and has no pure-strategy equilibria $$\begin{array}{cccc} & \mathrm{a} & \mathrm{b} & \mathrm{c}\\ \mathrm{a} & 1,1 & 0,2 & 5,0\\ \mathrm{b} & 2,0 & 3,3 & 0,4\\ \mathrm{c} & 0,5 & 4,0 & 1,1\\ \end{array}$$