Course coordinator: Balazs Sziklai
No. of Credits: 3, and no. of ECTS credits: 6
Prerequisites: Basic Linear Algebra
Course Level: introductory PhD
Brief introduction to the course: Game theory studies strategic decision making when
players have interest of conflict and/or can create value by cooperating with each other. As
this vague definition suggests game theory is a vast field. During the semester we will study
various topics including: matching theory and the mathematical aspects of fair division and
social choice.
The goals of the course:
The main goal is to introduce the fundamental tools in game theory.
The learning outcomes of the course:
By the end of the course, students are enabled to do independent study and research in fields
touching on the topics of the course. In addition, they develop some special expertise in the
topics covered, which they can use efficiently in other mathematical fields, and in applications,
as well. They also learn how the topic of the course is interconnected to various other fields in
mathematics, and in science, in general.
More detailed display of contents:
1. Introduction (brief history and branches of game theory, graph theory, complexity
2. Combinatorial games (chess, Hex, Chomp, Nim-type games, pursuit games)
3. Matching theory I. (college admission and its variants, Gale-Shapley algorithm)
4. Matching theory II. (kidney exchange program, house allocation problems)
5. Social choice I. (axiomatic judgment aggregation, May’s theorem on majority voting,
Condorcet-paradox, Arrow’s impossibility theorem)
6. Social choice II. (Gibbard- Satterthwaite theorem, group identification)
7. Apportionment problem
8. Cooperative game theory I. (transferable utility games, characterization of the Shapleyvalue, convex games, nucleolus)
9. Cooperative game theory II. (bankruptcy problem, a riddle from the Talmud, a hydraulic
proof of the Aumann-Maschler theorem)
10. Cooperative game theory III. (cost sharing games, an algorithm for the nucleolus of the
airport game)
11. Cake-cutting (different methods of dividing a cake: Cut and Choose, Banach-Knasteralgorithm, etc.)
12. Traffic routing and scheduling games (Wardrop-model, Braess-paradox, standard
scheduling games )
Arrow, K.: Social Choice and Individual Values, Wiley, New York, (1951)
Peleg, B. and Sudhölter, P.: Introduction to the Theory of Cooperative Games, SpringerVerlag, Heidelberg (2007)
Aumann, R. and Maschler, M.: Game theoretic analysis of a bankruptcy problem from the
Talmud, Journal of Economic Theory, 36, (1985), pp. 195-213.
Gale, D. and Shapley, L. S.: College Admissions and the Stability of Marriage, The American
Mathematical Monthly, Vol. 69, No. 1. (1962), pp. 9-15.
Kasher, A. and Rubinstein, A.: On the question “Who is a J?”, a social choice approach.
Logique et Analyse 160, (1997), pp. 385-395.
Some homework will be assigned during the semester.
Final exam: Written exam, in which students are expected to solve excercises, describe game theoretic
algorithms, and present minor proofs.
Grading: Your final grade will be based on attendance and homework (20%) and on the score on the
final +exam (80%).