Web page: http://www.math.toronto.edu/ilia/MAT406.2015Fall/.
Class Location & Time: Tue, 11:00 AM - 1:00 PM and Thu, 1:00 PM - 2:00 PM; IB200
Tutorials: Wed 5:00 PM- 6:00 PM, IB 390
Instructor: Ilia Binder (firstname.lastname@example.org), Deerfield Hall 3026.
Office Hours: Tue 2:00 PM-3:00 PM, Thu 11:00 AM-12:00 PM
Teaching Assistant: Charlene Chu, (email@example.com).
Office Hours: Fri, 10:30-11:30am, Deerfield Hall 3015.
Prerequisites: MAT102H5, MAT223H5, STA256H5.
Prerequisites will be checked, and students not meeting them will be removed from the course by the end of the second week of classes. If a student believes that s/he does have the necessary background material, and is able to prove it (e.g., has a transfer credit from a different university), then s/he should submit a 'Prerequisite/Corequisite Waiver Request Form'.
The course will discuss the mathematical aspects of the game theory, an important area of Mathematics/Probability with multiple applications to Economics, Political Science, and Evolutionary Biology, to name a few.
The course will start with the discussion of impartial combinatorial games: subtraction game, Nim, and Chomp, will discuss the Sprague-Grundy value. After a brief discussion of partisan combinatorial games, we will discuss the zero-sum games and von Neuman's minimax theorem. We will discuss various methods for solving such games. The next big topic will be the general sum games and Nash equilibrium. Other topics will include the coalition games and Shapley value, applications of Game theory to voting (such as Arrow theorem), auctions, and stochastic games.
Topics covered in class.
September 8: The introduction. Definition of a combinatorial game. Impartial and partisan games. N- and P- positions. Ferguson, section I.1.
September 10: N- and P- positions. The games of Chomp! and Nim. Peres, section 2.1; Ferguson, section I.2.
September 15: Bouton's theorem. The sum of combinatorial games. Sprague-Grundy function and theorem. Ferguson, sections I.2, I.3, I.4.
September 17: Sprague-Grundy theorem. Examples of using Sprague-Grundy function. Ferguson, section I.4.
September 22: Partisan Games. Zero-sum games: examples and definition, strategic form, geometric properties of the set of mixed strategies. Peres, sections 2.2, 3.1; Ferguson, sections II.1.1, II.1.2, II.1.3.
September 24: Zero-sum games: von Neumann Theorem, response to a fixed strategy, Saddle points. Ferguson, sections II.1.4, II.2.1
September 29: Proof of von Neumann Theorem for 2x2 games and in general. The Separation Theorem. Ferguson, section II.2.2; Peres, section 3.2
October 1: 2xm and nx2 games. Domination. Ferguson, sections II.2.3, II.2.4; Peres, section 3.3
October 6: Symmetric games. The Principle of Indifference. Use of symmetry. Ferguson, section II.3; Peres, section 3.4.
October 8: Poker-like games. Kuhn tree. Ferguson, sections II.5.1, II.5.2, II.5.3, II.5.4.
October 13: Converting Poker-like games to matrix form. General sum games: definition, strategic and extensive form, safety levels, Nash equilibrium. Ferguson, sections II.5, III.1, III.2.1; Peres, sections 4.1, 4.2.
October 15: Midterm review.
October 20: Midterm.
October 22: Nash Theorem. Brower fixed point Theorem. Peres, section 4.6.
October 27: Mixed Nash equilibria. Finding Nash equilibria for 2x2 games. Nash Theorem. Proofs of Sperner's Lemma and No-retraction Thorem. Ferguson, sections III.2.2; Peres, sections 4.2, 4.6.
October 29: Proof of Nash Theorem. Using domination. Ferguson, sections III.2.4; Peres, section 4.6.
November 3: A method for finding some Nash Equilibria. Models of Duopoly. Cooperative games. Ferguson, section III.3.
November 5: Cooperative games: feasible payoffs. Pareto-optimal payoffs. Ferguson, section III.4.1.
November 10: Cooperative games : solution for TU games, Nash solution of NTU games. Shapley λ-transfers solution of NTU games. Ferguson, sections III.4.2, III.4.3.
November 12: Games in Coalitional form. Relation to the strategic form. Ferguson, section IV.1.
November 17: Constant sum games. Imputations and core. Essential games. Shapley Value. Ferguson, sections IV.1.3, IV.2, IV.3.3.
November 19: Shapley Value: uniqueness. Shapley-Shubik power index. Ferguson, sections IV.3.2, IV.3.4.
November 24: Arrow Theorem: examples and the proof. Peres, section 8.3.
November 26: Final review.
Assignment #1, due September 23.
Recommended problems (do not turn in!): Ferguson, Part I, problems 1.5.1, 1.5.4, 1.5.8a, 2.6.2a,b, 2.6.3.
Assignment #2, due September 30.
Recommended problems (do not turn in!): Ferguson, Part I, problems 3.5.2, 3.5.3, 3.5.8; 4.5.1, 4.5.3.
Assignment #3, due October 7.
Recommended problems (do not turn in!): Ferguson, Part II, problems 1.5.2, 1.5.3; 2.6.1, 2.6.2, 2.6.8.
Assignment #4, due October 14.
Recommended problems (do not turn in!): Ferguson, Part II, problems 2.6.4, 2.6.9; 3.7.2, 3.7.4, 3.7.8.
Assignment #5, due November 4.
Recommended problems (do not turn in!): Ferguson, Part III, problems 1.6.1, 1.6.2; 2.6.4, 2.6.5, 2.6.6.
Assignment #6, due November 11.
Recommended problems (do not turn in!): Ferguson, Part III, problems 3.5.1, 3.5.4, 3.5.6.
Assignment #7, due November 18.
Recommended problems (do not turn in!): Ferguson, Part III, problems 4.5.3, 4.5.4, 4.5.5, 4.5.6.
Assignment #8, due November 25.
Recommended problems (do not turn in!): Ferguson, Part IV, problems 1.5.1, 1.5.4, 2.5.1, 2.5.8, 3.5.3.
Midterm Test. There will be an in-class midterm test on Tuesday, October 20. No aides are allowed for this test.
The test will cover combinatorial and zero-sum games, roughly the first two chapters of Ferguson.
Recommended preparation (do not turn in): all homework problems and the following problems from Ferguson: I.2.6.4, I.3.5.1, I.3.5.6, I.4.5.8, II.2.6.10, II.3.7.1, II.3.7.3, II.3.7.14, II.5.9.1, II.5.9.3.
Final exam. The final exam will be held on Thursday, December 10, 5-7pm, at RAWC Gym C. You will be allowed to use one one-sided letter-sized page of notes. Textbooks or calculators are not allowed for this exam.
Recommended preparation (do not turn in): all homework problems, practice problems for midterm, and the following problems from Ferguson: III.2.5.2, III.2.5.7, III.3.5.3, III.3.5.7, III.4.5.1, IV.1.5.2, IV.2.5.2, IV.3.5.2, IV.3.5.7.
Last year Final Exam.
Additional office hours: Thursday, December 3, 12-2, at DH3050.
Grading. Grades will be based on eight homework assignments (3% each), Midterm test (31%), and Final exam (45%). I will also occasionally assign bonus problems.
Late work. Extensions for homework deadlines will be considered only for medical reasons. Late assignments will lose 20% per day. Submission on the day the homework is due but after the tutorial is considered to be one day late. Special consideration for late assignments or missed exams must be submitted via e-mail within a week of the original due date. There will be no make-up midterm tests or final. Justifiable absences must be declared on ROSI, undocumented absences will result in zero credit.
Honesty and fairness are fundamental to the University of Toronto’s mission. Plagiarism is a form of academic fraud and is treated
very seriously. The work that you submit must be your own and cannot contain anyone elses work or ideas without proper
attribution. You are expected to read the handout How not to plagiarize (http://www.writing.utoronto.ca/advice/using-sources/how-not-to-plagiarize) and to be familiar with the Code of behaviour on academic matters, which is linked from the UTM calendar under the link Codes and policies.