How are linear programming and game theory connected through von Neumann’s work?

John von Neumann’s work serves as a foundational bridge between linear programming and game theory. His contributions established the mathematical framework that allowed these two fields to be viewed as deeply interconnected.1

The connection is primarily rooted in the following developments:

• The Minimax Theorem: In 1928, von Neumann proved the minimax theorem, which asserts that every finite, two-person constant-sum game has a rational solution in pure or mixed strategies. This theorem provided the mathematical substance for game theory and established the concept of optimal strategies for competing adversaries.23
• Conjecture of Equivalence: In 1947, von Neumann conjectured that linear programming problems and matrix games are mathematically equivalent.1
• Duality: He introduced the critical concept of duality, which is central to the study of linear inequality systems and optimization.1

By demonstrating that finding optimal strategies in complex games can be formulated as a linear programming problem, von Neumann helped unify these areas of operations research, providing the tools necessary to solve practical problems in economics and defense planning.145