Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1014	Game Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Interdisciplinary Mathematics - in Hungarian	3	2+1+0	speciality	compulsory	
Optimization of Computational Models	3	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. KASSAY Gabor, Ph.D., kassay@math.ubbcluj.ro	
Aims	1.To show how problems coming from practice (for instance in economics) can be modeled within the framework of game theory. 
2.To present the basic theoretical results and development of the (noncooperative) game theory. 
3.To show algorithms for solving matrix games.
	
Content	Course 1: The concept of game. Situation of conflict, pure and mixed strategies, equilibrium point, optimal strategies. 
 
Course 2: Two-persons zero-sum games. Examples. Modelling games. Necessary and sufficient conditions for the existence of a saddle point.

Course 3:  Some elementary methods useful for proving game theoretical (minimax) results.
   
Course 4:  The duality theorem of linear programming. Intersection (KKM) theorems, minimax inequalities.  

Course 5: The first important result in game theory: John von Neumann&#8217;s minimax theorem.
  
Course 6:  Classical minimax results: Wald and Ville&#8217;s theorems.
 
Course 7: Kakutani and Kneser&#8217;s results.
 
Course 8:  Ky Fan and König&#8217;s results.
 
Course 9:  Sion&#8217;s minimax theorems and its different elementary proofs.

Course 10: Some new results within game theory: equivalent minimax results. 

Course 11:  n-person&#8217;s games. Nash equilibrium points.  John Nash&#8217;s theorems. 

Course 12:  Numerical methods for solving games: the graphics method.

Course 13: The simplex method of linear programming (background).

Course 14:  Solving games using the simplex method.
	
Bibliography	1.  J.P. Aubin:  Mathematical methods of game and economic theory, North Holland, Amsterdam, 1979.   

2. J.B.G Frenk, G. Kassay: Introduction to Convex and Quasiconvex Analysis, in: Handbook of Generalized Convexity and Monotonicity, Series:
Nonconvex Optimization and its Applications, Vol. 76, Hadjisavvas,
Nicolas; Komósi, Sándor; Schaible, Siegfried (Eds.), pp. 3-87
Springer, Berlin-Heidelberg-New York 2005.

3. J.B.G. Frenk, G. Kassay: On noncooperative games, minimax theorems and equilibrium problems, in: Pareto Optimality, Game Theory and
Equilibria, Athanasios Migdalas (Crete), Panos Pardalos
(Florida), Leonidas Pitsoulis (London) and Altannar Chinchuluun
(Florida) (Eds.), Springer Verlag, to appear in 2007.     

4. A.J. Jones: Game theory: mathematical models of conflict, Horwood Publishing, Chicester, 2000.

5. G. Kassay: The Equilibrium Problem and Related Topics, Risoprint, Cluj, 2000. 

6. J. Nash: Non-cooperative games, Ann. of Math. 54:286&#8212;295, 1951. 

7. J. von Neumann, O. Morgenstern: Theory of games and economic behavior, Princeton University Press, Princeton, 1944.

8. R.T. Rockafellar: Convex analysis, Princeton University Press, Princeton, 1972.

9. J. Szép,  F. Forgó: Introduction to the theory of games, Akadémiai Kiadó, Budapest, 1985.
	
Assessment	The activity ends with a written final. The exam subjects have theoretical questions from all the studied topics, and one problem, among the problems studied at the course and the seminar. The final grade: homeworks 30%, presentation/project within the seminars 30%, written exam 40%.