Zero-Sum Games and Linear Programming Duality
Bernhard von Stengel
TL;DR
The paper addresses the long-standing question of deriving LP duality from the minimax theorem, clarifying that Dantzig's classical proof relies on a strong complementarity assumption. It delivers two core contributions: a self-contained proof of LP duality from the minimax theorem using Gordan's and Tucker's theorems (Theorem t-gotu), and a strengthened extension of Dantzig's zero-sum game with an extra row that yields either an optimal primal–dual pair or a certificate of infeasibility (Theorem t-BM), with a polynomial-size encoding bound derived from Carathéodory's theorem. It also shows how Tucker's Theorem can be derived from Gordan's Theorem (via a variable-elimination argument) and discusses a constructive reduction framework that relates LP solving to zero-sum games, including comparisons to related work by Adler and Brooks & Reny. The discussion culminates in a concise treatment of minimally infeasible systems of inequalities and their equivalence to minimally infeasible equalities, tying back to the foundational Lemmas of Farkas and related theorems. Collectively, the results provide a rigorous, self-contained, and constructive bridge between minimax theory, classical linear-algebraic alternative theorems, and linear programming duality, with explicit certificates and polynomial-size encodings for practical reductions.
Abstract
The minimax theorem for zero-sum games is easily proved from the strong duality theorem of linear programming. For the converse direction, the standard proof by Dantzig (1951) is known to be incomplete. We explain and combine classical theorems about solving linear equations with nonnegative variables to give a correct alternative proof, more directly than Adler (2013). We also extend Dantzig's game so that any max-min strategy gives either an optimal LP solution or shows that none exists.