Economics - Game Theory Question

[tabako]

I am having a hard time intuitively understanding the mixed Nash equilibrium in certain games. I can work out the numbers well enough, but I can't seem to grasp why they are Nash equilibrium.

Take the coordination game:


I can easily identify the 3 Nash equilibrium, but the mixed strategy one does not really make sense to me.

The way I see it, is if I know that the other player is going to random his choices (50%,50%), then any mixed strategy of mine will have the same expected utility. I make the case that any strategy other than (50%,50%) is optimal because then the other player would adjust to a pure strategy, and we would be able to shift to a pure Nash equilibrium, both of which have higher expected utilities than the mixed strategy Nash equilibrium (50% of the time in the mixed equilibrium, we choose opposing strategies and both receive the payoff of 0).

Maybe my problem is that I am thinking of this in terms of playing the game multiple times (am I supposed to? I don't really know). If anybody can add some insight into this, it would be greatly appreciated. Thanks.