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An Elementary Proof of a Minimax Theorem

Jeff Calder

TL;DR

This paper provides an elementary, self-contained proof of Fan's minimax theorem in a simplified setting suitable for undergraduates. It follows Nikaido's method by introducing the Phi functional and showing its minimum is zero, which yields a saddle point and the minimax equality. It also extends the result to certain unbounded y-domains when f is quadratic in y, providing a concrete unbounded-case proof. The contribution clarifies the role of convexity, duality, and saddle points in minimax problems and offers a teaching-friendly approach.

Abstract

Here, we give a self-contained and elementary proof of a minimax theorem due to Fan in a simplified setting that can be taught in an advanced undergraduate course. Our proof follows Nikaido's argument with some simplifications.

An Elementary Proof of a Minimax Theorem

TL;DR

This paper provides an elementary, self-contained proof of Fan's minimax theorem in a simplified setting suitable for undergraduates. It follows Nikaido's method by introducing the Phi functional and showing its minimum is zero, which yields a saddle point and the minimax equality. It also extends the result to certain unbounded y-domains when f is quadratic in y, providing a concrete unbounded-case proof. The contribution clarifies the role of convexity, duality, and saddle points in minimax problems and offers a teaching-friendly approach.

Abstract

Here, we give a self-contained and elementary proof of a minimax theorem due to Fan in a simplified setting that can be taught in an advanced undergraduate course. Our proof follows Nikaido's argument with some simplifications.

Paper Structure

This paper contains 3 sections, 4 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Minimax and Saddle Points
  3. A Minimax Theorem

Key Result

Lemma 1

Let $X\subseteq \mathbb{R}^d$, $Y\subseteq \mathbb{R}^k$, and $f:X\times Y \to \mathbb{R}$. Then

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Example 1
  • Definition 2
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 1 more