Nash Equilibria

"Equilibrium is not harmony; it is the absence of profitable unilateral change."

Overview

Nash equilibrium formalizes strategic stability: each agent's policy is optimal against the policies of the others.

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

• Optimization Landscape
• Probability Measure Spaces Preview
• Causal Discovery
• Generative Models

Companion Notebooks

Learning Objectives

After completing this section, you will be able to:

• Define finite normal-form games with players, actions, payoffs, and information assumptions
• Compute best-response correspondences from a payoff table
• Identify pure Nash equilibria and distinguish them from Pareto-optimal outcomes
• Derive mixed equilibria in two-action games using the indifference principle
• Explain why randomization changes strategic stability in games with no pure equilibrium
• State the finite-game Nash existence theorem and the fixed-point idea behind it
• Recognize when support enumeration or Lemke-Howson style methods are appropriate
• Connect Nash reasoning to GANs, self-play, routing markets, and LLM tool agents
• Measure equilibrium failure using unilateral deviations and exploitability
• Separate strategic stability from social welfare, fairness, and safety

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