# Advanced Game Theory

# Recommended Course Outline

This is a course outline that I use for my graduate-level game theory classes. It includes an ordered list of topics and lecture videos. It's important to watch the videos in the proper order because each concept builds on the ones introduced in previous videos. This will help ensure effective learning.

# Course Description and Objective

This is a course designed for graduate students which offers an introduction to game theory and strategic thinking. It mainly focuses on the theory of cooperative and non-cooperative games with an emphasis on economic applications. Game theory deals with multi-person decision problems where the actions of each decision maker or player may have an impact on the payoffs of others. In such situations, making optimal decisions requires strategic thinking. One must consider how their actions can influence the incentives of other players, and whether those players are aware of this interconnection.

Success in this course requires strong analytical and logical thinking and the habit of drawing conclusions based on qualitative information. The course requires a working knowledge of propositional logic, calculus (e.g., functions of one or several variables, derivatives), probability (e.g., random variables, probability distributions, conditional probabilities, expectations) and optimization.

Some of the topics covered in the class will be familiar to you from undergraduate or previous studies, but they will be treated in greater depth. At the end of the course, students should be able to

formulate any strategic interaction as a game form,

understand solution concepts in cooperative and non-cooperative games (whether it is in strategic or extensive form), and

develop analytical and problem-solving skills to analyze games.

# Suggested Textbooks

"Game Theory” by Drew Fudenberg and Jean Tirole.

“Microeconomic Theory” by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green.

“A Course in Game Theory” by Martin Osborne and Ariel Rubinstein.

“Strategy: An introduction to Game Theory” by Joel Watson.

# Course Outline

TOPIC 1: BASICS OF NON-COOPERATIVE GAME THEORY: STRATEGIC - FORM GAMES WITH COMPLETE INFORMATION

Dominant and dominated strategy, dominant strategy equilibrium

Solving Second Price Auction with Complete Information: An Equilibrium

Solving Second Price Auction with Complete Information: All Equilibrium

Solving Nash Demand Game with Outside Option: All Equilibrium

Proof of Nash Theorem

Never Best Response, IESDS, and IEWDS

Rationalizability

Correlated Equilibrium

TOPIC 2: BASICS OF NON-COOPERATIVE GAME THEORY: EXTENSIVE FORM GAMES WITH COMPLETE INFORMATION

Describing extensive form games: Part 2 (Player functions and information sets)

Games with Perfect Recall : Equivalence of Mixed and Behavioral Strategies

TOPIC 3: SOLVING EXTENSIVE FORM GAMES - APPLICATIONS

How to find Subgame Perfect Nash equilibrium of Simple Games: Backward Induction

How to find Subgame Perfect Nash equilibrium of Simple Games with Imperfect Information

Rubinstein' Alternating Offer Bargaining Game and its Solution for the Two-Period Version

Rubinstein's Alternating Offer Bargaining Game and its Solution for the Infinite Horizon Version

Finding Subgame Perfect Nash Equilibrium of a Finite Horizon Repeated Game: An Example

Infinitely Repeated Bertrand Competition: Collusion and Monopoly Pricing

Finding Nash Equilibrium of an Extensive Game with Imperfect Information: An Example

Robustness of Equilibrium Under Affine Payoff Transformations

Why is One-Shot Deviation Property Correct and the Key for Calculating SPNE

TOPIC 4: INTRODUCTION TO COOPERATIVE GAME THEORY AND ITS SOLUTION CONCEPTS

TOPIC 5: COOPERATIVE (NASH) BARGAINING

Pareto, Weak Pareto, and Individually Rational Sets of Bargaining Problems

Cooperative Bargaining Axioms: Pareto Optimality and Symmetry

Cooperative Bargaining Axioms: Independence of Irrelevant Alternatives (IIA)

Independence of Irrelevant Alternatives (IIA) and Nash Bargaining Rule

Solving for Kalai Smorodinsky Bargaining Outcome: Two Examples

Bargaining Axioms: Translation Invariance and Strong Monotonicity

Calculating Egalitarian Rule Bargaining Solution: A numerical Example

TOPIC 6: BANKRUPTCY (CLAIMS) PROBLEM

TOPIC 7: MATCHING THEORY

A Simple Theory of Matching: Gale and Shapley's One-to-One Matching

Strategy-Proof and Stable Mechanisms for One-to-One Matching

You Ask (Request) My House - I get Your Turn (YRMH-IGYT) Algorithm: A Numerical Example

TOPIC 8: VOTING

Voting Axioms: Independence of Irrelevant Alternatives (IIA)

Arrow's Impossibility Theorem and Condorcet Paradox (Cycles)

Strategic Voting: Gibbard-Satterthwaite Impossibility Theorem

TOPIC 9: BAYESIAN GAMES AND BAYESIAN NASH EQUILIBRIUM

An Example for Bayesian Nash Equilibrium: Public Good Provision

An Example for Bayesian Nash Equilibrium: First Price Auction

TOPIC 10: AUCTION THEORY

Solving Vickrey (Second Price) Auction: Weakly Dominant Strategy Equilibrium

Strategic and Revenue Equivalence between First, Second and English Auctions

Calculating Expected Revenue in Vickrey (Second Price) Auction

TOPIC 11: MECHANISM DESIGN

Mechanism Design Example: Cake Division Problem, Divide and Choose Rule

Direct Mechanisms, Dominant Strategy Incentive Compatibility, and Revelation Principle

Strategy Proof Mechanisms over Single Peaked Preferences: The Median Voting Rule

Vickrey–Clarke–Groves (VCG) Mechanism (a.k.a Pivotal Mechanism)

Budget Balancedness and Individual Rationality of VCG Mechanism

Bilateral Trade: VCG Mechanism Fails Budget Balancedness and Individual Rationality

Bilateral Trade: Proving Strategy-Proofness of VCG Mechanism

Bayesian Mechanism Design: Ex-post, Interim, and Ex-ante Individual rationality