AIMS AND CONTENT

LEARNING OUTCOMES

Knowledge of the main models and solutions used in game theory. Capability to model real situations with these formal tools. Critical analysis of the assumptions of the theory and of its limits of applicability.

TEACHING METHODS

Teaching consists of lectures, for a total of 54 hours. Lecture notes are available on the Aulaweb page of the course.

SYLLABUS/CONTENT

PART 1: Theory of non-cooperative games

Introduction to game theory, its motivations, comparison with decision theory. Examples of application.

Definition of game. Non-cooperative games and cooperative games. Strictly dominant and strictly dominated actions. Examples of non-cooperative games: the "prisoner's dilemma", the "battle of the sexes", games of pure coordination.

Representation of a non-cooperative game: extended form and strategic form. Payments function. Information state. Social situation. Games with perfect and imperfect information. Pure and mixed strategies. Games with perfect and imperfect memory. Examples. 

Utility theory. Preference relations. Lotteries. Von Neumann-Morgenstern utility functions. Examples: ultimatum game, prisoner's dilemma with altruism.

Application of game theory to property law: state of nature and presence of a social contract.

Game form and games in strategic form. Solution of a non-cooperative game. Nash equilibrium. Zero-sum games in normal form: Nash equilibria and saddle points.

Nash equilibria in Cournot and Bertrand duopoly models (homogeneous and non-homogeneous goods).

Zero-sum games without Nash equilibria in pure strategies. Minimum guaranteed win for the first player.  Maximum guaranteed loss for the second player. Mixed strategies: minimax theorem, Nash theorem. Determination of the Nash equilibrium in mixed strategies.

Correlated strategies. Correlated equilibria. Types of players.

Some ways to achieve a given Nash equilibrium: cheap talk, conventions, focal points. Strict dominance with respect to payoffs and risk. Cases in which a Nash equilibrium is not an "obvious way" to play.

Strictly dominated strategies and weakly dominated strategies. Iterated elimination of strictly dominated strategies. Independence from the order of elimination. Rationalizable strategies.

Trembling hand equilibrium. Evolutionary equilibrium.

Possible inefficiency of the Nash equilibrium. Braess paradox and Wardrop equilibrium.

Future rationality. Backward induction principle. Subgames. Zermelo's theorem.

Perfect Nash equilibrium in subgames. Existence theorem. Calculation methods. Relation to the backward induction principle for perfect information games.

Rationalization of past choices. Forward induction. Games with incomplete information. Harsanyi transformation. Bayesian games. Nash-Bayesian equilibria. Example of computation of a Nash-Bayesian equilibrium in pure strategies.

Example of computation of a Nash-Bayesian equilibrium in mixed strategies. Solving zero-sum two-player games through linear programming.

PART 2: Theory of cooperative games

Cooperative games with transferable and non-transferable utility. Utility and risk functions: risk aversion, risk propensity, risk neutrality. Relationship with cooperative games with transferable utility. Characteristic form of a cooperative game with transferable utility. Coalitions. Characteristic function. Superadditivity, subadditivity, additivity, cohesiveness.

Characteristic function for a transferable utility game. Examples.

Non-transferable utility cooperative games. Examples of construction of the characteristic function. Two-player bargaining problem. Nash solution and its axiomatic characterization. Derivation of the Nash solution. Other solutions (Kalai-Smorodinsky, egalitarian solution) and their axiomatic characterization. Further solutions (lambda-egalitarian, with equal areas, dictatorial, utilitarian).

Cooperative games with transferable utility. Examples of construction of the characteristic function. Properties (monotonic, convex, simple games). Set solutions and point solutions.

Imputations. Essential and inessential games. Dominance of imputations.

Stable sets.

Core. Its relationship with linear programming. Balanced collections. Minimal balanced collections. Bondareva-Shapley theorem. Proof of the Bondareva-Shapley theorem.

Examples of transferable utility games and their cores. Bankruptcy game. Fixed tree game. Weighted majority game. Production game. Assignment game.

Point solutions of a game with transferable utility. Shapley value and its axiomatic characterizations. Calculation of the Shapley value. Airport game. 

Indices of: Banzhaf-Coleman, normalized Banzhaf-Coleman, Deegan-Packel, Holler, Johnston.

Vector of the excesses. Lexicographic order. Nucleolus. Kopelowitz algorithm for the computation of the nucleolus.

EXERCISES

Solved recapitulation exercises on non-cooperative and cooperative game theory.

RECOMMENDED READING/BIBLIOGRAPHY

Bibliographical references

Books (in Italian) 

E. Barbuto, Teoria dei giochi, modelli e strategie per massimizzare le probabilità di vincita e minimizzare i rischi, 2007.
B. Chiarini, Un mondo in conflitto, teoria dei giochi applicata, 2017.  
F. Colombo, Introduzione alla teoria dei giochi, 2003. 
M. Li Calzi, Teoria dei giochi, 140 esercizi commentati e risolti,1995.
F. Patrone, Decisori (razionali) interagenti, una introduzione alla teoria dei giochi, 2006 (available online on the author’s website).

Books (in English)        

D. Bauso, Game theory with engineering applications, 2016.       
N. van Long, A survey of dynamic games in economics, 2010.
MJ Osborne, A. Rubinstein, A course in game theory, 1994 (available online on the website of one of the authors).    
B. Salanié, The economics of contracts, 2005.
S. Tijs, Introduction to game theory, 2003.      

Other lecture notes (online):     

A. Agnetis, Introduction to game theory.  
G. Barbarino, Notes on game theory.      
P. Garibaldi, M. Bruschi, Introduction to game theory and oligopoly models.     
V. Fragnelli, Game theory.       

Solved exercises:

H. H. Nax, B. S. R. Pradelski, Solved exercises for "Introduction to game theory".
R. Lucchetti, A primer in game theory with solved exercises.  
Collections of solved exercises from the site of Prof. F. Patrone.

TEACHERS AND EXAM BOARD

Exam Board

GIORGIO STEFANO GNECCO (President)

SILVANO CINCOTTI (President Substitute)

MARCELLO SANGUINETI (President Substitute)

LESSONS

LESSONS START

https://corsi.unige.it/8734/p/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam will take place in one of the following ways (chosen by the student).

1) Written examination with two exercises respectively on non-cooperative and cooperative games and with two theoretical questions (in both cases, the topics denoted by *, which are reported on the Aulaweb page of the course, are excluded).

2) Written examination with two exercises respectively on non-cooperative and cooperative games (excluding the topics denoted by *) plus preparation and discussion of a presentation (for example in the form of slides, to be presented shortly after the written examination, or on another date to be agreed) on a topic of game theory among those indicated on the Aulaweb page of the course.

The pre-appeal will take place on December 22, 2021, from 2pm to 5 pm, in room B6. After that, presentations will follow (until 6.30 pm), for those who will want to do them that day.

Exam dates:

Monday 10, January 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Monday 17, January 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Friday 11, February 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Friday 10, June 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Friday 8, July 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Monday 5, September 2022, 9am-12pm (room to be defined). Eventual presentations will follow (until 1.30pm).

Group presentations (maximum 2 people) are allowed as long as the contribution of each student is well identified, and each student presents his or her own part of the presentation.

Please book the exam on Aulaweb, or by sending an e-mail to the teacher. The location of the exam will be reported on Aulaweb.

ASSESSMENT METHODS

The exam will verify the actual acquisition and understanding of the knowledge acquired during the course.