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

Time and place: 2:00--2:50 MWF in LCB 225.

Instructor: Stewart Ethier (Prof.), JWB 119, 581-6148, ethier@math.utah.edu. Office hours are 10--11 MWF unless announced otherwise. Other times are available by appointment. I will be out of town on March 23.

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

Prerequisite: Math 2270, Linear Algebra.

Topics covered: 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 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%).

Assignments: Assignments on the week's material will be posted Fridays on this page. They will be due the following Friday. Because of the size of the class, I may grade only a subset of the assigned problems.

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 April 22 (a week before the end of classes). 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'll 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.

Some useful links:
Games you can play. Includes Chomp!, Fibonacci Nim, Moore's Nim, Dawson's Chess, Dots and Boxes, and Dominotion.
Martin Chlond's games. A simple Nim game.
Matrix game solver.
Bimatrix game solver.

Week 1 (Jan. 12--16). We gave an overview of the subject and covered Section 1 of Part I (Take-Away Games). Assignment for next week: page I-6, Exercises 2, 3, 4a.
Week 2 (Jan. 21--23). We covered Section 2 of Part I (The game of Nim). Assignment for next week: page I-11, Exercises 2, 3, 5.
Week 3 (Jan. 26--30). We covered through page I-22, omitting Section 3.4. Assignment for next week: page I-19, Exercise 5, and page I-26, Exercises 1, 2, 3.
Week 4 (Feb. 2--6). We finished Chapter 4 of Part I and began Part II, nearly finishing Section 1.1. Assignment for next week: page I-26, Exercises 5, 10; page II-7, Exercise 4.
Week 5 (Feb. 9--13). Information on the game of le her. We nearly finished Section 2.4 of Part II. Assignment for next week: page II-14, Exercises 4, 5, 7.
Week 6 (Feb. 18--20). We solved the game of baccarat chemin de fer and finished Section II.2.4. Assignment for next week: page II-15, Exercises 9, 10, 11.
Week 7 (Feb. 23--27). We covered Part II Section 3 through symmetric games. Assignment for next week: page II-29, Exercises 1, 3, 6. And prepare for midterm exam on the material we have covered through Part II, Section 3.2.
Week 8 (Mar. 2--6). We finished Chapter 3 of Part II (Indifference Principle). Assignment for next week: page II-29, Exercises 8, 9, 11.
Week 9 (Mar. 9--13). We proved the minimax theorem using linear algebra instead of linear programming (as in the book). We got to about page II-51. Assignment for next week: Enjoy the Spring break. Remember, no class Monday March 23.
Week 10 (Mar. 25--27). We finished Part II, Chapter 5, and started Part III, Chapter 1. Assignment for next week: p. II-55, Exercises 5, 6; p. III-6, Exercise 3.
Week 11 (Mar. 30--Apr. 3). We got to the prisoner's dilemma on page III-10. Assignment for next week: p. III-13, Exercises 2, 3, 4.
Week 12 (Apr. 6--10). We covered only Cournot's model in Section 3.3 of Part III. We will begin cooperative games next. Assignment for next week: p. III-14, Exercises 6 and 7; p. III-23, Exercise 1.
Week 13 (Apr. 13--17). We nearly finished Nash's theory of cooperative NTU games. Last assignment, due Apr. 27: p. III-39, Exercises 2*, 3, 5. *There are two problems labeled 2, and it is the first 2 that is intended. Don't forget the project, due Wednesday, Apr. 22.
Week 14 (Apr. 20--24). We finished Part III and started Part IV. No additional assignments.
Week 15 (Apr. 27--29). We finished Chapter 2 of Part IV on imputations and the core. So we've covered Chaps. 1--4 of Part I, Chapters 1--5 of Part II (omitting the linear programming material in Chapter 4), Chapters 1--4 of Part III (covering only the Cournot model in Chapter 3), and Chapters 1 and 2 in Part IV. This is what you are responsible for on the final.

Final Exam scheduled for Wednesday, May 6, 1:00--3:00. You may bring one crib sheet to the exam. You may not use (and will not need) a calculator. Projects will be returned at that time.

Recommended assignment for Part IV. Ex. 1, 3, page IV-5; Ex. 2, page IV-10.