A Canonical Game -- Nearly 75 Years in the Making -- Showing the Equivalence of Matrix Games and Linear Programming

Abstract

Around 1947, von Neumann showed that for any finite two-person zero-sum game, there is a feasible linear programming (LP) problem consisting of a primal-dual pair of linear programs whose saddle points yield equilibria of the game, thus providing an immediate proof of the minimax theorem from the strong duality theorem. But going in the other direction has since remained an unsolved problem. For any LP problem, we define a canonical game and, with an elementary proof, show that every equilibrium either yields a saddle point of the LP problem or certifies that one of the primal or dual problems is infeasible and the other has no optimal solution. We thus obtain an immediate proof of the strong duality theorem from the minimax theorem. Taken together, von Neumann's classical result and the present result provide a definitive formalization of the very old idea that matrix games and linear programming are ``equivalent.''