A canonical game—75 years in the making—showing the equivalence of matrix games and linear programming

Abstract

According to Dantzig (Econometrica, 17, p.200, 1949), von Neumann was the first to observe that for any finite two-person zero-sum game, there is a feasible linear programming (LP) problem whose saddle points yield equilibria of the game, thus providing an immediate proof of the minimax theorem from the strong duality theorem. We provide an analogous construction going in the other direction. For any LP problem, we define a game and, with a brief and elementary proof, show that every equilibrium either yields a saddle point of the LP problem or certifies that one of the primal or dual programs is infeasible and the other is infeasible or unbounded. We thus obtain an immediate proof of the strong duality theorem from the minimax theorem. Taken together, von Neumann’s and our results provide a succinct and elementary demonstration that matrix games and linear programming are “equivalent” in a classical sense.