The answer related to strict domination by mixed strategies. The following can be proved via Linear Programming duality:
Theorem: In a bimatrix game, the following two statements are equivalent:
Proof sketch: I am not going to write out the proof here, but an approach is to write down, in terms of linear equalities, what it means that row $i$ to get expected payoff $u$; next, that all other rows $k$ have smaller payoff than $u$, by introducing a variable that denotes the difference - that variable is then maximized in the linear program; and finally, that the variables describing the mixed strategy $q$ of player II are indeed a probability distribution. Then take the dual of the linear program, and use the fact that the dual has the same optimal objective function value as the primal (in particular when the primal has a positive optimal objective function value).