Math 5750-1. Topics in Applied Mathematics: Game Theory. Spring 2013.

Time and place: 10:45--11:35 MWF in JWB 308.

Instructor: Stewart Ethier (Prof.), JWB 119, 581-6148, ethier@math.utah.edu. Office hours 2:00--2:50 MWF.

Text: Game Theory by Thomas Ferguson. Free download at http://www.math.ucla.edu/~tom/Game_Theory/Contents.html.

Prerequisite: Math 2270, Linear Algebra.

Topics covered, if time permits: Impartial Combinatorial Games (Take-Away Games, The Game of Nim, Graph Games, Sums of Combinatorial Games), Two-Person Zero-Sum Games (The Strategic Form of a Game, Matrix Games, Domination, The Principle of Indifference, Solving Finite Games, The Extensive Form of a Game), Two-Person General-Sum Games (Bimatrix Games -- Safety Levels, Noncooperative Games -- Equilibria, Models of Duopoly, Cooperative Games), and n-Person Games in Coalitional Form (Many-Person TU Games, Imputations and the Core, The Shapley Value, The Nucleolus).

Grades: Grades will be based on weekly homework assignments (20%), a midterm exam (25%), a term project (20%), and a final exam (35%). The final exam will occur at the scheduled time.

Assignments: Assignments on the week's material will be posted Fridays on this page. They will be due the following Friday. Late assignments will not be accepted. (Do not use paper torn from a spiral notebook; staple pages together; reduced credit for illegible handwriting; extra credit for typed assignments.)

Project: For the project there is some flexibility. It could be a report on an application of game theory or it could be an analysis of a game we didn't cover. It could be applied or theoretical. Grades will be based on how interesting it is and on how well you appear to understand it. It should not exceed 10 pages (it is not a thesis). The project is due at last class period. How should you find a topic? Do a literature search based on your interests. If you are on campus, you can use JSTOR (journal storage) or MathSciNet. The former is better because you can download the article, whereas the latter may be more complete but you may have to get the article from the library. Google Scholar may also be useful, and you don't have to be on campus. Important: If you use a published source, please submit a photocopy of it with your project. It is OK to use someone else's ideas as long as proper credit is given. And you can use someone else's words if they appear as properly attributed quotations. Example: The link below to Garrison Hansen's combinatorial games was created as a project for this class in 2011. It received the highest possible score.

Expected learning outcomes: The student who completes this course successfully will have a working knowledge of several areas of game theory, specifically Impartial Combinatorial Games, Two-Person Zero-Sum Games, Two-Person General-Sum Games, and n-Person Games in Coalitional Form. This knowledge should be sufficient to apply game theory to your own area of interest.

Some useful links:
Games you can play. Includes Chomp!, Fibonacci Nim, Moore's Nim, Dawson's Chess, Dots and Boxes, and Dominotion.
Garrison Hansen's combinatorial games. (A project for Math 5750, Spring 2011.)
Matrix game solver (five decimal places).
Bimatrix game solver (four decimal places, and exact).

Week 1 (Jan. 7, 9, 11). We gave an overview of the subject and covered Chapter 1 of Part I (Take-Away Games). Assignment for next week: page I-6, Exercises 2, 3, 4.
Week 2 (Jan. 14, 16, 18). We covered Chapter 2 (Nim) and Chapter 3 (Graph Games), omitting the material on more general graphs. Assignment for next week: page I-11, Exercises 2, 5; page I-19, Exercise 5.
Week 3 (Jan. 23, 25). We covered Chapter 4 (Sprague-Grundy theorem). Assignment for next week: page I-26, Exercises 3, 5, 10.
Week 4 (Jan. 28, 30, Feb. 1). We started Part II, getting through Section 2.2. Assignment for next week: page II-7, 2, 4; page II-14, 2. In Exercise 4, to avoid possible misinterpretation, note that the phrase, "I let you off with a payment of a dime," means that Olaf pays Alex 10 cents.
Week 5 (Feb. 4, 6, 8). We finished Chapter 2 of Part II (Domination) and started Chapter 3. Assignment for next week: page II-14, Exercises 4, 5, 11. Information about the game of Le Her. Information about the game of baccarat chemin de fer. In the latter file, there is a bad typo. The numbers in the two columns in Table 5.5 should be reversed, keeping the column labels the same. Table 5.6 is correct, however. Also, here is the plot needed to solve baccarat chemin de fer.
Week 6 (Feb. 11, 13, 15). We finished Chapter 3 of Part II (Indifference Principle). (We'll skip Section 3.6.) Assignment for next week: page II-29, Exercises 1, 2, 5.
Week 7 (Feb. 20, 22). We proved the minimax theorem using linear algebra, not using the linear programming approach in the notes. No assignment for next week except to prepare for the midterm exam on March 1. Sample exam, given two years ago. It is best to work out your own solutions first, before looking at my solutions.
Week 8 (Feb. 25, 27, Mar. 1). We nearly finished Chapter 5 of Part II (Extensive Form) and had the Midterm Exam. Solutions here. Assignment for next week: page II-55, Exercises 3, 5, 7.
Week 9 (Mar. 4, 6, 8). We started Chapter 2 of Part III (Noncooperative games). Assignment for March 22: page III-6, Exercises 1, 2, 3.
Week 10 (Mar. 18, 20, 22). We covered Section 2 (Noncooperative games) and part of Section 3 (Cournot's model), and we started Section 4 (Cooperative games). Assignment for next week: page III-13, Exercises 3, 4, 6.
Week 11 (Mar. 25, 27, 29). We finished TU games and began Part III, Section 4.3 (NTU games), getting through Nash's approach on page III-34. Assignment for next week: page III-39, Exercises 2*, 3(b), 4. *There are two problems labeled 2, and it is the second 2 that is intended.