Can a game with a pure strategy Nash equilibrium also have mixed strategy equilibria?

Question

I have questions:

1. If zero-sum game has pure strategy Nash equilibrium (saddle point), can it have also mixed strategy equilibria?

2. What if game is not zero-sum?

Answer 1

I am not sure about the first question, but for the second question it definitely exists: consider a game with two players who can both choose left $(L,l)$ and right $(R,r)$ and the following payoff matrix:

Clearly, there are two pure strategy Nash equilibria: $(L,l)$ and $(R,r)$. However, there is also a mixed strategy Nash equilibrium where Player 1 chooses $L$ with probability $\frac{2}{3}$ and Player 2 chooses $l$ with probability $\frac{1}{3}$.