MAT406H5F Mathematical Introduction to Game Theory

Fall 2018

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Class Location & Time: Tue, 10 AM - 12 PM; Thu, 2 PM - 3 PM; DH2080

Tutorials: Wed 5 PM- 6 PM, IB385

Instructor: Ilia Binder (, DH3026.
Office Hours: Tue 2 PM - 3 PM and Thu 10 AM-11 AM

Teaching Assistant: James Belanger, (
Office Hours:  Tue 1-2 PM, DH2027.

Required Text: Anna R. Karlin and Yuval Peres. Game Theory, Alive.  American Mathematical Society, 2017).

Online book:

Thomas S. Ferguson. Game Theory.

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. We will also carefully discuss the Sprague-Grundy value. After a brief discussion of partisan combinatorial games, we will talk about 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 6: An introduction: preview of "coming attractions".
September 11: Impartial and partisan games. N- and P- positions. The games of Chomp! and Nim. Nim-sum. Karlin-Peres, section 1.1.1; Ferguson, sections I.1, I.2.1, I.2.2.
September 13: Bouton's theorem. Sprague-Grundy function. Karlin-Peres, section 1.1.2; Ferguson, sections I.2, I.3.
September 18: Sum of combinatorial games. Sprague-Grundy theorem. Lasker's game. Ferguson, section I.4.
September 20: Partizan games. The game of Hex. Karlin-Peres, section 1.2.1.
September 25: The game of Hex. Zero-sum games: examples and definition, strategic form, geometric properties of the set of mixed strategies. Karlin-Peres, sections 1.2.1, 1.2.2, 2.1, 2.2; Ferguson, sections II.1.1, II.1.2.
September 27: Von Neumann Theorem, response to a fixed strategy. Saddle points. Karlin-Peres, sections 2.3, 2.4.1; Ferguson, sections II.1.3, II.1.4, II.2.1.
October 2: Von Neumann Theorem: proof of the 2x2 case and in general. The Separation Theorem. 2xm and nx2 games. Karlin-Peres, section 2.6; Ferguson, sections II.2.3, II.3.5, II.2.2, II.2.4.
October 4: Domination. Symmetric games. Karlin-Peres, section 2.4.3; Ferguson, sections II.2.3, II.3.5.
October 16: The Principle of Indifference. Use of symmetry. Poker-like games. Kuhn tree. Converting games given in Extensive form to matrix form. Karlin-Peres, sections 2.4.3, 2.4.4; Ferguson, sections II.3, II.5.
October 18: Midterm review.
October 23: Midterm.
October 25: General sum games: definition, strategic and extensive form, safety levels, Nash equilibrium. Ferguson, section III.1; Karlin-Peres, section 4.1.
October 30: Mixed Nash equilibria. Finding Nash equilibria for 2x2 games. Brouwer fixed point Theorem. Ferguson, sections III.2.1, III.2.2; Karlin-Peres, sections 4.2, 5.1.
November 1: Proof of Brouwer Fixed point theorem using Game of Hex. Proof of Nash Theorem for two players. Karlin-Peres, sections 5.1, 5.4.
November 6: Methods for finding Nash Equilibria. Cournot's, Bertrand and Stackeleberg models of Duopoly. Cooperative games. Ferguson, sections III.2.3, III.2.4, III.3.1, III.3.2, III.3.3, III.4.1.
November 8: Cooperative games: feasible payoffs. Pareto-optimal payoffs. Solving TU games. Ferguson, sections III.4.1, III.4.2.
November 13: Solving TU games, Nash solution of NTU games. Ferguson, sections III.4.2, III.4.3.
November 15: Shapley λ-transfers solution of NTU games. Ferguson, section III.4.3.
November 20: Games in Coalitional form. Relation to the strategic form. Constant sum games.Imputations and core. Essential games. Karlin-Peres, section 12.1, 12.2; Ferguson, sections IV.1, IV.2.
November 22: Shapley Value. Karlin-Peres, section 12.3; Ferguson, sections IV.3.
November 27: Shapley Value. Shapley-Shubik power index. Voting mechanisms. Karlin -Peres, sections 12.3, 13.1; Ferguson, section IV.3.
November 29: Arrow's fairness criteria. Arrow Theorem: examples. Arrow Theorem: the proof. Karlin-Peres, sections 13.1, 13.2, 13.3, 13.6, 13.7.
December 4: Proof of Arrow Thoerem. Final review Karlin-Peres, section 13.7


Assignment #1, due September 19.
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 26.
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 3.
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 17.
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 7.
Recommended problems (do not turn in!): Ferguson, Part III, problems 1.6.1, 1.6.2; 2.5.4, 2.5.5, 2.5.6.

Assignment #6, due November 14.
Recommended problems (do not turn in!): Ferguson, Part III, problems 3.5.1, 3.5.4, 3.5.6.

Assignment #7, due November 21.
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 28.
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 a two-hour in-class midterm test on Tuesday, October 23. 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, including the recommended 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 13, 5-7pm, at IB210. 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: Tuesday, December 11, 1 - 2. Location: DH3000 .

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.

E-mail policy.
E-mails must originate from a address and contain the course code MAT406 in the subject line. Please include your full name and student number in your e-mail.

Academic Integrity.
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 ( 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.