Existence of Symmetric Pure Strategy Equilibrium

Question

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?

Answer 1

1. There is no guarantee that symmetric games have symmetric equilibria. See this paper for concrete examples.

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 $$ \begin{array}{cccc} & \mathrm{a} & \mathrm{b} & \mathrm{c}\\ \mathrm{a} & 1,1 & 0,2 & 5,0\\ \mathrm{b} & 2,0 & 3,3 & 0,4\\ \mathrm{c} & 0,5 & 4,0 & 1,1\\ \end{array} $$

3. There are two special kinds of games that are known to always have pure strategy equilibria: supermodular games and potential games.