MAT406H5F Mathematical Introduction to Game Theory

Fall 2016

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Class Location & Time: Tue, 11:00 AM - 1:00 PM; DH2080 and Thu, 12:00 PM - 1:00 PM; IB240

Tutorials: Wed 5:00 PM- 6:00 PM, DV2094A

Instructor: Ilia Binder (, DH3026.
Office Hours: Tue and Thu 10:00 AM-11:00 AM

Teaching Assistant: James Belanger, (
Office Hours:  Wed, 1-2pm; DH2027 (Math Help room).

Online books:
  1. Thomas S. Ferguson. Game Theory.
  2. Anna R. Karlin and Yuval Peres. Game Theory, Alive.

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: The introduction. Definition of an impartial combinatorial game. Impartial and partisan games. N- and P- positions. Ferguson, section I.1.
September 8: N- and P- positions. The games of Chomp! and Nim. Karlin-Peres, section 1.1.1; Ferguson, sections I.2.1, I.2.2.
September 13: Bouton's theorem. The sum of combinatorial games. Sprague-Grundy function and theorem. Karlin-Peres, section 1.1.2; Ferguson, sections I.2, I.3, I.4.
September 15: Sprague-Grundy theorem. Examples of using Sprague-Grundy function. Partisan games. The game of Hex. Karlin-Peres, section 1.2.2; Ferguson, section I.4.
September 20: Partisan Games. Zero-sum games: examples and definition, strategic form, geometric properties of the set of mixed strategies. Karlin-Peres, sections 1.2.1, 2.1, 2.2; Ferguson, sections II.1.1, II.1.2, II.1.3.
September 22: Zero-sum games: von Neumann Theorem, response to a fixed strategy, Saddle points. Karlin-Peres, sections 2.3, 2.4.1; Ferguson, sections II.1.4, II.2.1
September 27: Proof of von Neumann Theorem for 2x2 games and in general. The Separation Theorem. Karlin-Peres, section 2.6; Ferguson, section II.2.2.
September 29: 2xm and nx2 games. Domination. Symmetric games. Karlin-Peres, section 2.4.3; Ferguson, sections II.2.3, II.2.4, II.3.5
October 4: Symmetric games. The Principle of Indifference. Use of symmetry. Karlin-Peres, sections 2.4.3, 2.4.4; Ferguson, section II.3.
October 6: Poker-like games. Kuhn tree. Converting Poker-like games to the strategic form. Ferguson, section II.5.
October 18: 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; Karlin-Peres, sections 4.1, 4.2.
October 20: Midterm review.
October 25: Midterm.
October 27: Mixed Nash equilibria. Finding Nash equilibria for 2x2 games. Ferguson, sections III.2.2; Karlin-Peres, section 4.1.
November 1: Nash Theorem. 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 3: Methods for finding Nash Equilibria. Cournot model of Duopoly. Ferguson, sections III.2.3, III.2.4, III.3.1.
November 8: A method for finding some Nash Equilibria. Bertrand and Stackeleberg models of Duopoly. Cooperative games: feasible payoffs. Pareto-optimal payoffs. Solving TU games. Ferguson, sections III.3.2, III.3.3, III.4.1, III.4.2.
November 10: Cooperative games : examples of 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. Games in Coalitional form. Relation to the strategic form. Constant sum games. Karlin-Peres, section 12.1; Ferguson, sections III.4.3, IV.1.
November 17: Imputations and core. Essential games. Shapley Value. Karlin-Peres, sections 12.2, 12.3; Ferguson, sections IV.2, IV.3.3.
November 22: Shapley Value. Shapley-Shubik power index. Karlin -Peres, section 12.3; Ferguson, sections IV.3.
November 24: Arrow's fairness criteria. Arrow Theorem: examples. Karlin-Peres, sections 13.1, 13.2.
November 29: Arrow Theorem: examples and the proof. Final review Karlin-Peres, sections 13.3, 13.6, 13.7.
December 1: Final review.


Assignment #1, due September 21.
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 28.
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 5.
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 19.
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 9.
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 16.
Recommended problems (do not turn in!): Ferguson, Part III, problems 3.5.1, 3.5.4, 3.5.6.
Please disregard problem 4 -- you will get the full credit for it.

Assignment #7, due November 23.
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 30.
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 25. 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.
Additional office hours: Thursday, October 20, 3-4pm, at DH3026.

Final exam. The final exam will be held on Tuesday, December 13, 1-3pm, at TFC Cafeteria. 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: Monday, December 12, 10 - 12, DH4001.
Additional office hours by TA: Saturday, December 10, 12 - 1, Math Help Room.

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.

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.