Maximum-support Nash equilibria in zero-sum games Context: I would like to know how likely a player is to pick a specific action, provided that he plays optimally and the action is optimal. Phrased like this, the question is ill-defined. But is there some way around the problem?

Details: I have a zero-sum symmetric matrix game. Is there some good notion of a Nash equilibrium with a maximum support [edit: maximum among the NE strategies]? Like something along the lines of "take the center of the simplex of NE strategies"?

The question: Is there some canonical solution to the vague question above? Possibly something already written down in a textbook or a paper. (It would feel silly to reinvent the wheel.)
I am not sure about what exactly you mean by the maximum support NE but you may want to check the concept of Correlated Equilibrium. Any convex combination of NE payoff profiles is a Correlated Equilibrium (but not vice versa in general). So, what you wrote made me think about that. However, I don't think it makes much sense to use correlated equilibrium since, you know, there is no obvious advantage of correlation. Also, the NE payoff profile must be constant in your environment.

Another thing that comes to mind is restricting the attention to super-modular games. Then, you get a nice lattice of NE. So, I don't know if this makes sense either.

I would suggest playing around with some examples to understand what exactly it is that you want to characterize and asking a new one based on your examples.