Projects

The final project for this course involves writing a paper and giving a talk on a topic related to game theory. You can work in groups of 2 or 3 people, or you can work by yourself. The talk should be about 5 minutes + 5 minutes/person (for example, one person should give a 10 minute talk, and two people should give a 15 minute talk), and the paper should be about 1–3 pages + 2 pages/person (for example, one person should write a 3–5 page paper, and two people should write a 5–7 page paper). The presentations will occur on May 3, 5, 10, and 12. The paper will be due on Tuesday, May 17.

Your talk should focus on explaining the main ideas of your topic clearly, so that the audience (the other members of the class) can understand it. The more technical and complicated parts should go in the paper rather than the talk.

Deadlines:

• April 14: Tell me which topic you are doing, and who is in your group. (Each group needs to choose a different topic, so that all of the presentations are different.)
• April 26: Turn in a project proposal. This should be a half-page description of your project. What references do you plan to use? What topics do you plan to cover in your talk? What topics do you plan to cover in your paper?
• May 3, 5, 10, and 12: Presentations occur.
• May 17: Paper due.

There are three types of projects:

• Reading Topics: You can learn about a game theory topic that we haven't covered in class.
• Deterministic Games: You can analyze a deterministic game that we did not learn about in class.
• Applying Game Theory: You can apply the methods we have learned to a situation from real-life or fiction.

Reading Topics

For these topics, you should learn about a game theory topic that we haven't covered in class. Each of these topics lists one possible reference, but you should use more than one source (at least three sources). Wikipedia can be one of your sources, but the other sources should be from books or journals.

• Voting Theory: Voting Theory studies different voting systems, and how much power different people have under the voting systems. There are several possible topics on Voting Theory. The two main topics in voting theory are:
  • Different "yes or no" voting systems (ie, voting with only two possible outcomes). For this topic, I recommend starting by looking the book Mathematics and Politics by Alan D. Taylor (Chapters 3, 4, 8, and 9 are about "yes or no" voting systems).
  • Voting systems with multiple candidates. For this topic, I recommend starting by looking at Sections 3.1 and 3.2 of the book A Mathematical View of Our World by Parks, Musser, Trimpe, Maurer.
  For additional references on voting theory, you may want to look at Chaotic Elections! by Donald Saari, Chapter 15 of Games of Strategy by Dixit and Skeath, or parts of Game Theory and Strategy by Philip D. Straffin.
• Auction Theory: Auction Theory is about different ways to hold auctions and the optimal ways to bid in these auctions. I recommend starting by looking at Chapter 10 of The Art of Strategy by Dixit and Nalebuff (this book is not in the library — come see me if you want to borrow this book). You may also want to look at Chapter 16 of Games of Strategy by Dixit and Skeath.
• Brinkmanship and the Cuban Missile Crisis: We talked about the Cuban Missile Crisis in class, but there is much more to say. I recommend starting by looking at Chapter 14 of the book Games of Strategy by Dixit and Skeath.
• Evolutionary Game Theory: Game Theory has been applied to biology to study evolution. Some books to look at include Evolution and the Theory of Games by John Maynard Smith, Chapter 7 of Game Theory Evolving by Herbert Gintis, Chapter 13 of Games of Strategy by Dixit and Skeath, and Chapter 15 of Game Theory and Strategy by Philip D. Straffin.
• Poker and Game Theory: Game Theory can be used to analyze bluffing in poker. To start with I recommend looking at Chapter 10 of A Gentle Introduction to Game Theory by Saul Stahl. This chapter considers a simplified version of poker. (Note: This is not about analyzing the probabilities in poker, which is the topic below. It is just about analyzing the game theory involved in bluffing.)
• Games involving chance: We have only considered deterministic games, but there is also a lot that can be said involving games of chance. I recommend starting by looking at the book Mathematics in Games, Sports, and Gambling by Ronald J. Gould. There are many possible topics here, including:
  • You could choose a specific non-deterministic game such as Poker, Roulette, Backgammon, or Craps.
  • The Monty Hall Problem
  • Gambler's Ruin
• One-Player Games: We have only talked about games involving at least two players. A game involving only one person is more of a 'puzzle' than a game. I recommend starting by looking at the book Mathematics in Games, Sports, and Gambling by Ronald J. Gould. There are two possible topics:
  • Tower of Hanoi
  • Instant Insanity

Deterministic Games

For these topics, you should start by learning about the game. Then, you should try to analyze the game. Answer questions such as: Does the game have exactly one player with a winning strategy? Which player(s) have a winning strategy? What is the winning strategy? For some of these games, finding a winning strategy in general, can be very hard, so instead, you should find a winning strategy for small games (similar to how we found winning strategies for small Hex games).

• Look at Wikipedia for descriptions of the following games:
  • Sprouts
  • Hackenbush
  • Dots and Boxes
  • Domineering
• Tic-Tac-Toe on a Torus and Klein Bottle: Go the webpage http://www.math.ntnu.no/~dundas/75060/TorusGames/TorusGames.html. This page has games played on a torus and Klein bottle. Read the introduction to understand what a torus and a Klein bottle are. For this topic, try analyzing the Tic-Tac-Toe game on a torus and a Klein bottle.

Applying Game Theory

For this type of project, you should choose a situation (or possibly multiple situations) from real-life or fiction and model the situation as a strategic game. You could choose a situation from a historical event, from a current event, or from a situation in your life. Alternatively, you could choose an event from fiction, such as from a movie, television show, or book. For examples from fiction, look at the following webpage http://www.gametheory.net/popular/. A project of this type should involve the following:

1. Learn about the situation (for example, read about a historical event or watch a movie).
2. Model the situation with a strategic game (or several strategic games). You may want to use sequential games or simultaneous games (or both).
3. Analyize the strategic game — find Nash equilibria, prudential strategies, etc.
4. How does your analysis compare with what actually happened? You may want to go back to step 2 and create additional models