Minimax Theorem

"In a zero-sum game, optimal defense and optimal attack meet at the value."

Overview

The minimax theorem explains why mixed strategies can equalize worst-case guarantees in finite zero-sum games.

Game theory is the part of the curriculum that studies adaptive decision makers. It asks what happens when each model, user, attacker, defender, or agent optimizes while anticipating the choices of others.

This section is written in LaTeX Markdown. Inline mathematics uses $...$, and display equations use `

`. The notes emphasize strategy, payoff, best response, equilibrium, exploitability, and adversarial adaptation.

Prerequisites

• Convex Optimization
• Constrained Optimization
• Nash Equilibria
• Adversarial Robustness Preview

Companion Notebooks

Learning Objectives

After completing this section, you will be able to:

• Define finite two-player zero-sum matrix games and their mixed-strategy payoffs
• Compute maximin and minimax values for small payoff matrices
• Recognize pure saddle points and explain when mixed strategies are required
• Write the row-player and column-player linear programs for a zero-sum game
• State von Neumann's minimax theorem for finite games
• Explain the LP-duality proof idea behind equality of maximin and minimax values
• Use exploitability to diagnose approximate zero-sum equilibria
• Simulate no-regret dynamics as an approximation route to minimax play
• Translate minimax objectives into robust ML and adversarial training losses
• Distinguish finite zero-sum minimax from general-sum Nash analysis

Study Flow

1. Read the pages in order and pause after each page to restate the main definition or theorem.
2. Run theory.ipynb when you want to check the formulas numerically.
3. Use exercises.ipynb after the reading path, not before it.
4. Return to this overview page when you need the chapter-level navigation.

Runnable Companions

• Theory notebook
• Exercises notebook