Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Matematică Computaţională - în limba maghiară
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MO264	Game Theory	

Semester	Hours: C+S+L	Category	Status	
3	2+2+0	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
KASSAY Gabor	Ph.D.	Prof.	Mathematical Analysis and Optimization	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
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. 

IV. Course contents
Formulation and description of a noncooperative game, matrix games, n-person’s games, mixed strategies, saddle points, optimal strategies, examples. Modelling practical problems within the framework of game theory. Classical and new minimax theorems. Equivalence between different minimax theorems. Methods for solving matrix games. n-person’s noncooperative games, examples, existence results (Nash’s theorems). 

V. 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—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.

VI. Thematic of didactic activities per weeks
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’s minimax theorem.
  
Course 6:  Classical minimax results: Wald and Ville’s theorems.
 
Course 7: Kakutani and Kneser’s results.
 
Course 8:  Ky Fan and König’s results.
 
Course 9:  Sion’s minimax theorems and its different elementary proofs.

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

Course 11:  n-person’s games. Nash equilibrium points.  John Nash’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.  

Schedule of the seminars: discussing the homeworks, proving some elementary properties needed for the course, presentations of some related topics by the students

VII. Didactic methods used
Bibliographical materials specific to the studied topics are used (books, papers, Internet resources). As well, computers (departmental network) loaded with the relevant system software and the development environments, are used for the students work on the programming projects assignments. 

VIII. 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%.

IX. Additional references
Sion M (1958): On general minmax theorems, Pacific J.Math 8:171—176/

Frenk JBG, Kassay G (2006 : The level set method of Joo and its use in minmax theory, Mathematical Programming Ser.A 105:145-155

Wald A (1945): Generalization of a theorem by von Neumann concerning zero-sum two-person games, Annals of Mathematics 46 (2):281—286.

Kneser H (1952): Sur un theoreme fondamental de la theorie des jeux, Comptes Rendus Acad.Sci.Paris 234: 2418—2420.

	Date, 30.09.2008				                           Course responsible,
								               Prof. univ.dr. KASSAY Gabor
            
            Dean,                                                                                         Department chair,         
            Prof. univ.dr. TAMBULEA Leon                                          Prof. univ. dr. COBZAS Stefan