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INTRODUCTION TO GAME THEORY 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 classes) 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 gerrymandering) (apportionment of the House of Representatives, 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 ) References 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. Assessment 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%). 2