# Algorithmic Game Theory

Spring 2012
Instructor: Aaron Roth
Time: Tuesday/Thursday 1:30-3:00 pm
Room: Towne 303

Overview: In this course, we will take an algorithmic perspective on problems in game theory. We will consider questions such as: how should an auction for scarce goods be structured if the seller wishes to maximize his revenue? How badly will traffic be snarled if drivers each selfishly try to minimize their commute time, compared to if a benevolent dictator directed traffic? How can couples be paired so that no two couples wish to swap partners in hindsight? How can you be as successful at betting on horse races as the best horse racing expert, without knowing anything about horse racing?

Prerequisites: This will be a mathematically rigorous theory course for advanced undergraduates. Students should have taken, or be taking concurrently a course in algorithms (such as CIS 320), be mathematically mature, and be familiar with big-O notation. Prior coursework in game theory is helpful, but not necessary. Everything will be presented from first principles.

Goals and Grading: The goal of this course is to give students a rigorous introduction to game theory from a computer science perspective, and to prepare students to think about economic and algorithmic interactions from the perspective of incentives. Grading will be based on participation, problem sets, a midterm, and a final exam.

Textbook: A recommended textbook is Algorithmic Game Theory, which is also available for free on the web here (username=agt1user, password=camb2agt).

Office Hours and Discussion: By appointment in Levine 603.
We will be using Piazza to discuss class material and answer questions. The Piazza page for CIS 399 is piazza.com/upenn/spring2012/cis399. Students are encouraged to ask questions about the material on Piazza so that everyone can benefit and contribute to their answers.

Topics Covered:
1. Part 1: Game Theory and Game Dynamics
1. Quick introduction to game theory: Zero sum and general sum games, repeated games, Minmax strategies, Nash equilibrium, correlated equilibrium
2. Game Dynamics: Sequential best response, weighted majority algorithm, fictitious play, perturbed follow the leader
3. Game Dynamics converging to Nash equilibrium in zero sum games; Game dynamics converging to correlated equilibrium in general sum games
4. Price of anarchy: Definition, routing games, hoteling games
5. Smooth games and "Price of Total Anarchy"
6. More if time allows...
2. Part 2: Auctions and Mechanism Design
1. Auction basics: First price auctions, second price auctions, truthfulness
2. Maximizing welfare: The VCG Mechanism
4. Maximizing revenue: Bayesian optimal auctions: How to set a reserve price
5. Maximizing revenue: Prior Free Mechanism Design
6. Online auctions for digital goods
7. Stable Marriages and the Deferred Acceptance Algorithm
8. More if time allows...

Problem Sets and Exams:
1. Homework 1: Due Thursday, February 9 in class.
2. Homework 2: Due Thursday, March 1 in class.
3. Practice Midterm: Real Midterm will be February 23 in class.
4. Homework 3: Due Tuesday, March 20 in class.
5. Homework 4: Due Thursday, April 12 in class.
6. Practice Final: Real Midterm will be April 24 in class.

Lectures:
• Lecture 1: Overview (Some slides stolen from Costis Daskalakis)
• Lecture 2: Basic definitions. Normal form game, dominant strategies, iterated elimination of dominated strategies, nash equilibrium in pure and mixed strategies, the 2 player parables of game theory (prisoners dilemma, battle of the sexes, matching pennies). See AGT Chapter 1.
• Lecture 3: Congestion games and best response dynamics. Existence of and convergence to pure strategy nash equilibrium. See Yishay Mansour's notes
• Lecture 4: Fast convergence to approximate Nash equilibrium in congestion games. Weighted congestion games do -not- have pure strategy Nash equilibria in general, and load balancing games on identical machines: a special case of weighted congestion games that -do- have pure strategy nash equilibria. Can compute a Nash equilibrium by minimizing the potential function, and in symmetric network routing games we can do this in polynomial time.
• Lecture 5: Last lecture on Best Response Dynamics. Proved that best response dynamics converge if and only if the game has an -ordinal potential function-.
• Lecture 6: Doing as well as the best expert with the polynomial weights algorithm. See AGT Chapter 4.
• Lecture 7: 2-Player zero sum games, and a proof of the min-max theorem via the polynomial weights algorithm. See AGT Chapter 4
• Lecture 8: n-player Separable Graphical zero sum games, and convergence to Nash equilibrium of no-regret algorithms.
• Lecture 9: Correlated equilibria and coarse correlated equilibria. See AGT Chapters 1, 4.
• Lecture 10: An algorithm guaranteeing no swap-regret, converging to correlated equilibrium in general games. See end of AGT Chapter 4. Last lecture on game dynamics!
• Lecture 11: Begin price of anarchy. See AGT Chapter 17.
• Lecture 12: Price of anarchy and stability of the fair cost sharing game, and the Hotelling game. See AGT Chapters 17 and 19.
• Lecture 13: Smooth games and the robust price of anarchy. Matching lower bound for linear congestion games. Finished part 1 of the class!
• Lecture 14: Stable matchings and the deferred acceptance algorithm. See Parkes and Seuken's notes
• Lecture 15: Male proposing deferred acceptance algorithm is incentive compatible for men, but not for women. See Shoham and Leyton-Brown Chapter 10
• Lecture 16: Auction design -- The Groves Mechanism. See Parkes and Seuken's notes
• Lecture 17: Auction design -- The VCG Mechanism.
• Lecture 18: Single parameter domains and monotone allocation rules (Myerson's Lemma). See Hartline Chapter 2
• Lecture 19: Monotone approximation algorithms and approximate mechanism design: Knapsack Auctions.
• Lecture 20: Knapsack Auctions continued...
• Lecture 21: Revenue maximization: Profit extractors. See Hartline Chapter 6
• Lecture 22: Revenue maximization: Random Sampling Auctions. See Hartline Chapter 6
• Lecture 23: Revenue maximization: Online auctions for digital goods.

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