Farkas' Lemma and Complete Indifference

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.