Lecture 13: Game Theory & Nash Equilibrium

Strategic Interactions in Multi-Agent Environments

When rational agents compete and cooperate
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Building on Lecture 12

In Lecture 12, we learned how to make optimal decisions under uncertainty using MDPs. We could answer: "What action maximizes my expected utility?"

But what happens when other rational agents are also making decisions that affect our outcomes? Game Theory extends decision theory to multi-agent settings where strategic interaction matters: "What should I do, knowing that others are also optimizing?"

Lecture Overview

Game Theory provides a mathematical framework for analyzing strategic interactions between rational agents. It finds applications in economics, AI, networking, biology, and anywhere multiple decision-makers interact.

Key Concepts: Players, Strategies, Payoffs, Nash Equilibrium, Dominant Strategies, Mixed Strategies
Applications: Multi-Agent AI, Auctions, Network Security, Autonomous Vehicles, Market Competition

The Central Question

In a multi-agent environment, an intelligent agent faces a new challenge:

How should I act when my outcome depends on what others do?
The Challenge:
  • Other agents are also rational optimizers
  • My best action depends on their actions
  • Their best actions depend on mine
  • Circular reasoning problem!
Game Theory Solution:
  • Model players, strategies, and payoffs
  • Find Nash Equilibria: stable strategy profiles
  • No player can improve by changing alone
  • Rational, principled, mathematically sound

Main Topics

1. Introduction to Game Theory

What is game theory and why it matters for AI. Historical foundations from Von Neumann and Nash. Types of games: cooperative vs non-cooperative, zero-sum vs non-zero-sum.

History Types of Games Strategic Thinking Real-World Examples
2. System Model: Game Components

Formal definition of multi-player games. Players, actions, strategies, payoff functions, and information structure. Key assumptions: rationality, common knowledge.

Players (N) Actions (Sᵢ) Payoffs (uᵢ) Rationality
3. Dominant Strategies

When one strategy is always best regardless of opponents. Strictly and weakly dominant strategies. Iterated elimination of dominated strategies (IESDS). The Prisoner's Dilemma.

Dominance IESDS Prisoner's Dilemma Best Response
4. Nash Equilibrium

The most important solution concept in game theory. Definition, finding pure strategy Nash equilibria, best response analysis, and coordination games.

Nash Equilibrium Best Response Multiple Equilibria Coordination
5. Mixed Strategies

When randomization is optimal. Mixed strategy Nash equilibria, computing probabilities, and the Rock-Paper-Scissors example.

Mixed Strategies Randomization Expected Payoffs RPS Analysis
6. Applications & Summary

Real-world applications in AI, economics, and networking. Multi-agent reinforcement learning, auctions, and mechanism design. Course summary and connections.

Multi-Agent AI Auctions Mechanism Design Summary

Interactive Demonstrations

Explore game theory through interactive simulations and strategy analyzers.

Prisoner's Dilemma
Payoff Matrix Builder
Nash Equilibrium Finder
Rock-Paper-Scissors
Coordination Games
Mixed Strategy Calculator

Practical Exercises

Master game theory through hands-on problem-solving.

Game Modeling Problems

Model real-world scenarios as strategic games.

Nash Equilibrium Practice

Find pure and mixed strategy Nash equilibria.

Dominance Analysis

Identify dominant strategies and apply IESDS.

Strategic Reasoning

Apply game theory to AI and real-world decisions.

Key Concepts Summary

Game Theory Framework:
  • Players (N): rational decision-makers
  • Strategies (Sᵢ): available actions for each player
  • Payoffs (uᵢ): utility function mapping outcomes
  • Information: what players know
Nash Equilibrium:
  • Strategy profile where no player can improve alone
  • s* is NE if uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*) ∀ sᵢ
  • May be pure or mixed strategies
  • Every finite game has at least one NE
Dominant Strategies:
  • Strategy that's best regardless of opponents
  • IESDS: iteratively remove dominated strategies
  • Dominant strategy equilibrium (if exists)
  • Prisoner's Dilemma: (Defect, Defect)
Mixed Strategies:
  • Probability distribution over pure strategies
  • Optimal when pure NE doesn't exist
  • Rock-Paper-Scissors: (⅓, ⅓, ⅓)
  • Nash's theorem: every finite game has mixed NE
SE444: Artificial Intelligence | Lecture 13: Game Theory & Nash Equilibrium