Linear Systems Equivalent to Two-Person Zero Sum Games

Abstract

We provide two characterizations of the set of equilibria of a two-person zero-sum “matrix” (TPZS) game. The first is a lemma, which says that a strategy profile (pair of randomizations over pure strategies) is an equilibrium if an only if along with another real number, it satisfies a specific system of linear inequalities. The second is a proposition, which says that a strategy profile is an equilibrium if an only if along with two real numbers it solves a certain linear programming problem. The proposition is a special case of proposition 2 in Lahiri (2025) which in turn is a special case of the “Equivalence Theorem” in Mangasarian and Stone (1964).

A strategy profile is said to be completely mixed if it assigns positive probabilities to all pure strategies for both players and a TPZS game is said to be completely mixed if all equilibria of the game are completely mixed. We show that a completely mixed TPZS game has a unique equilibrium strategy. In the proof of this result, the lemma we referred to earlier plays an important role. We then apply this result for the case of a matrix corresponding to a completely mixed TPZS game with value zero, to show (i) that a non-zero linear combination of the columns of the matrix is zero, if and only if the vector of coefficients used for the linear combination is a scalar multiple of the column player’s strategy in the unique equilibrium, (ii) that a non-zero linear combination of the rows of the matrix is zero, if and only if the vector of coefficients used for the linear combination is a scalar multiple of the row player’s strategy in the unique equilibrium. We apply a corollary of this proposition to obtain a complete characterization of all completely mixed TPZS games that have value zero. This characterization is different from the characterization available in Kaplansky (1945).