Vito Fragnelli
University of Eastern Piedmont - Department of Advanced Sciences and Technologies

Course Type

Type B

Calendar

11/6 h. 14.00-18.00
12-15/6 h. 9.00-13.00
Room 103

Room

Program

Cooperative games

Game theory allows accounting the interactions of the agents involved in a real world situation, that often may
strongly influence the outcome of the situation and the payoff of the agents.
In this course, after introducing the preliminaries and the notations of game theory, the interactions among non-
cooperating agents will be considered, with particular attention to the strategical behavior of the agents and to the
notion of Nash equilibrium.
Then, the possibility of cooperation among the agents will be accounted, leading to cooperative games, both
without and with transferable utility. In the first case, the Nash bargaining problem will be analyzed, jointly with
the most classical developments. In the second case, classical set solutions and point solutions will be considered.
Some applications of game theory to real world situations will be presented, devoting particular attention to
bankruptcy and project management problems. Operations research games, i.e. classical operations research
problems with more than one decision-maker, conclude the course.

Short program
Introduction - Representation of a Game - Extensive Form - Strategic Form - Characteristic Form - Solution
Concept
Non-cooperative Games: Nash Equilibrium - Mixed Strategies - Refinements of Nash Equilibrium
Cooperative Games, NTU Games: The Bargaining Problem - Nash Axiomatic Solution - Other solutions
Cooperative Games, TU Games: Set Solutions Games - Imputations - Core - Balancedness - Point Solutions -
Shapley Value - Axiomatization of the Shapley Value - Nucleolus - Power Indices
Bankruptcy problems and games
Project management problems and games
Operations Research Games - Linear Programming Games - Sequencing Games - Assignment Games (Bilateral
Market) - Network Games

Main References
Borm P, Hamers H, Hendrickx R (2001) Operations Research Games: A Survey, TOP 9 : 139-216.
Branzei R., Ferrari G., Fragnelli V., Tijs S. (2011) A Bonus-Malus Approach to Project Management. Central
European Journal of Operations Research 19 : 495-512
Fragnelli V, Tadei R (2005) Operations Research Games, Bollettino dellâ€™Unione Matematica Italiana, sez. A 8 :
107-122.
Gillies DB (1953) Some Theorems on n-person Games, PhD Thesis, Princeton, Princeton University Press.
Nash JF (1950a) Equilibrium Points in n-person Games, Proceedings of the National Academy of Sciences of the
United States of America 36 : 48-49.
Nash JF (1950b) The bargaining problem, Econometrica 18 : 155-162.
O'Neill B. (1982) A Problem of Rights Arbitration from the Talmud. Mathematical Social Sciences 2, 345-371
Schmeidler D (1969) The Nucleolus of a Characteristic Function Game, SIAM Journal of Applied Mathematics
17 : 1163-1170.
Shapley LS (1953) A Value for n-Person Games, in Contributions to the Theory of Games, Vol II (Annals of
Mathematics Studies 28) (Kuhn HW, Tucker AW eds.), Princeton University Press, Princeton : 307-317
Thomson W. (2003) Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: a Survey.
Mathematical Social Sciences 45, 249-297