Zero-Sum Games and Linear Programming Duality

TL;DR

This paper explores the connection between zero-sum games and linear programming duality, providing a more direct proof of the minimax theorem and extending Dantzig's game to identify optimal solutions or their absence.

Contribution

It offers a correct, more straightforward proof of the minimax theorem from linear programming duality and extends Dantzig's game for better solution detection.

Findings

A new, direct proof of the minimax theorem from linear programming duality.

Extension of Dantzig's game to identify optimal solutions or their non-existence.

Clarification of classical theorems related to solving linear equations with nonnegative variables.

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.