Suppose the game consists of only $2$ players, player $1$ and player $2$, and each of them has only $2$ strategies to choose between. This gives a $2$ by $2$ payoff matrix. Player $2$ has no preference when choosing one of his strategies, while player $1$ chooses the strictly dominant strategy for himself.
In this case, is player $2$ going to mix her strategies? Given that whatever proportion player $2$ mixes her strategies, player $1$ would definitely gain positive earning.
In this case, is there any mixed Nash equilibrium?
Thanks for help!