Does the value of a pure strategy Nash equilibrium(if exists) equal the value of the mix strategy Nash equilibrium in two-person zero-sum game? Given a two-person zero-sum game, a mixed strategy Nash equilibrium always exists and all such equilibria have the same value. A pure strategy Nash equilibrium, however, may not exist.

My question is: suppose that a pure strategy Nash equilibrium exists, does the value of it equal the value of the mixed strategy Nash equilibrium?
Pure strategy Nash equilibria are a subset of mixed strategy Nash equilibria, so as long as your statement 


  a mixed strategy Nash equilibrium always exists and all such equilibria have the same value


is true, all pure strategy Nash equilibria will have the same value as well.