Math 432, Fall 2017
Dr. Jim Haglund, email@example.com
Course webpage: http://www.math.upenn.edu/~jhaglund/432/
Office hours: (week of Dec. 18 only): M 10:30-noon in DRL 4E2B.
Lecture: MWF 10-10:50am in DRL A4
Grader: Adam Clearwater, firstname.lastname@example.org
Office hours: M 3:30-4:30pm, R 11am-noon in DRL 3C13
Office Phone: 215-898-2949
Course: Game Theory is a relatively new and exciting field of mathematics which has applications to economics and many other areas. This course covers standard topics in game theory such as combinatorial games, two person (zero-sum and general-sum) games, noncooperative games, and Nash's equilibrium theorem. We will focus on the mathematics involved.
Text: The text for this course is "
Game Theory" by Thomas S. Ferguson, available free online at his
UCLA webpage. Some auxillary sources which may be helpful to consult,
on reserve in the Math/Physics Library in DRL 3N1, are
"Winning Ways for your Mathematical Plays" by Berlekamp, Conway, and Guy, and
"A Course in Game Theory" by M. J. Osborne and A. Rubenstein.
Notes on Group Actions and Invariance
Exams and Grades:
There will be three midterm exams and a final exam. Midterm exam 1 counts for 15% of your grade,
while Midterm 2 and Midterm 3 each count for 20% of your grade. The final exam counts for
35% of your grade, while hw and quizzes together count for 10%.
Midterm 1 will be on Wednesday, September 27,
Midterm 2 on Wednesday, November 1, and Midterm 3 on Wednesday,
Nov. 29. All three midterms are during our normal lecture time, from 10-10:50am in our
normal lecture room - DRL A4. The final exam will
be on Tuesday, Dec. 19 from 9:00-11:00am in ??.
Midterm 1: Covers all of Part I: Impartial Combinatorial Games. Games you may be asked to find winning moves for
include Nim, Lasker's Nim, Moore's Nim_k,
Misere Nim, Subtraction Games,
Directed Graph Games, Kayles, Turtles, Twins, Mock Turtles, Ruler, Turning Corners, Tartan Games,
Green Hackenbush, sums of games. SG functions for Kayles and Turning Corners will be provided, if they appear in a question. You
should know the SG functions for Lasker's Nim and Mock Turtles.
You may be asked to
prove a given pattern for an SG function holds using induction. You may
also be asked to explain parts of any of the proofs of theorems discussed in class, such as why in the sum of games G1 + G2,
the SG-value of a position (x,y) equals the Nim sum of the SG-value of x in G1 and the SG-value of y in G2,
the proof of the Colon Principle from Green Hackenbush, or proving something by induction.
Midterm 1 is closed book; no notes, calculators, cell phones, etc. are allowed.
Midterm 2: Covers pages 1-57 (Sections 1 through 5) of Part II: Two-Person Zero-Sum Games.
Topics include Strategic form of Games, Matrix Games: Saddle Points,
Domination, Solving 2 by n and m by 2 games, Latin square games,
Principle of Indifference, Nonsingular matrix games,
Diagonal games and other special matrix games, Invariance,
Simplex method (also known as the pivot method). Ficticious play. Extensive form of a game.
Things you may need to prove: anything on page 2 of the handout above titled
"Notes on Group Actions and Invariance".
Midterm 2 is closed book; no notes, calculators, cell phones, etc. are allowed.
Midterm 2 Solutions
Midterm 3: Covers all of Part III: Two-person general sum games.
Topics include General-Sum Strategic and Extensive form games, Safety levels, PSE's and SE's, Models of Monopoly and Dupoly
(Cournot, Bertrand, Stackelberg), TU cooperative game solutions, Nash axioms and Nash solution of NTU games, All-in Player poker.
You will not be asked to prove things on Midterm 3, although you may be asked to derive solutions using the methods in class
and in the book, and you will have to show your work.
Midterm 3 is closed book; no notes, calculators, cell phones, etc. are allowed.
Midterm 3 Solutions
Final Exam Tuesday, Dec. 19 from 9:00-11:00am in DRL A6. Alternate seating.
The final exam is cummulative. It Covers all of Part I, all of Part II up to and including section 5.9,
all of Part III, and all of Part IV except for sections 3.2 and 4.3.
The notes on Group Actions and Invariance will not be part of the final exam. The only proofs you need to know involve mathematical induction.
You will be provided with a copy of the table in the book
on Nim multiplication used
in solving Tartan games. You will also be provided with the Sprague-Grundy (SG) values for any individual games
from Part I you are asked about, although you may be asked to prove the formulas for the SG values (using induction).
The final exam does not include anything on fictitious play. For the final exam you are allowed one side of a standard 8.5" by 11" sheet of