As a simple example, suppose two players A and B play a game wherein each picks a positive integer, and if they both pick the same integer $N$ then B pays $f(N)$ dollars to A, for some given payoff function $f$. This game should be particularly simple since the payoffs are symmetric, i.e., if A and B choose different strategies then no money is exchanged at all. What general conditions can one place on $f$ to ensure this game has a strict Nash equilibrium in mixed strategies?
For instance, we might suppose $f$ to assume nonnegative integer values with a unique minimum of $0$ at some integer, say $M$. Clearly then B should always play $M$, and then it doesn't matter what A plays. This isn't a strict equilibrium, though, because A has no preference at all for what number to pick given that B picks $M$.