  In a finite two player game consider the matrix of one player's payoff
difference between any two consecutive pure strategies. Define the half space
induced by a column vector of this matrix as the set of vectors that form an
obtuse angle with this column vector. We use Farkas' lemma to show that this
player can be made indifferent between all pure strategies if and only if the
union of all these half spaces covers the whole vector space. This result leads
to a necessary (and almost sufficient) condition for a game to have a
completely mixed Nash equilibrium. We demonstrate its usefulness by providing
the class of all symmetric two player three strategy games that have a unique
and completely mixed symmetric Nash equilibrium.
