Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


A.Y. Credits
2021/2022 8
Lecturer Email Office hours for students
Gian Italo Bischi
Teaching in foreign languages
Course with optional materials in a foreign language English
This course is entirely taught in Italian. Study materials can be provided in the foreign language and the final exam can be taken in the foreign language.

Assigned to the Degree Course

Economics and Management (LM-77)
Date Time Classroom / Location
Date Time Classroom / Location

Learning Objectives

The course aims to define and apply some mathematical methods and tools, in particular the theory of dynamical systems and the theory of games, to model the time evolution of social and economic systems. The systems considered are often adaptive systems, influenced by the presence of boundedly rational, heterogeneous and interacting agents. Some lessons will be devoted to a deeper analysis of some mathematical concepts already introduced in the framework of the course in General Mathematics in order to use them in more advanced applications. The formal methods studied in this course will give a general understanding of the setup of a mathematical model in economics and how the results obtained should be critically analyzed. This is obtained both through the analysis of some models given in the literature and by examples and exercises proposed in classrooom notes.At the end of the course the students should be able to built and analyze mathematical models expressed by the formalism of dynamical systems and/or the theory of games, and a sufficient capability of using mathematical and logical tools to describe in a schematic way the behaviour of complex situations. These mathematical tools should enhance the approach towards the description and the management of time evolving complex economic systems, and favour the ability to interface with experts in mathematics and computer science in order to study the behaviour of economic systems by an interdisciplinary approach.


Part 1. Dynamical systems in discrete and continuous time.
One-dimensional and two-dimensional linear dynamical systems with constant coefficients in continuous time and discrete time. Classification of equilibrium points. Eigenvalues and eigenvectors, phase diagrams. Basins of attraction.Nonlinear dynamical systems: equilibrium points and their stability through linear approximations. Local bifurcations. Periodic solutions, Limit cycles. Logistic map, period doubling route to chaos. Features of deterministic chaos. Elements of dynamical systems in more than two dimensions.Part 2. Introduction to decision theory and game theory.
Representations of games in normal form and extensive form. Dominated strategies, best reply, Nash equilibrium in pure and mixed strategies. Examples and problems of inefficiency, multiplicity of Nash equilibria. Case of zero-sum games. Evolutionary games with one population and two populations of players. Replicator dynamics.Part 3. Examples and applications.
Cobweb model, endogenous business cycle models, Cournot, Bertrand and Stackelberg games, models of financial markets with heterogeneous agents, dynamic oligopoly games, models of population dynamics, hawk-dove games with replicator dynamics, adaptive models in dynamic games with boundedly rational agents

Learning Achievements (Dublin Descriptors)

Dublin descriptorsKnowledge and understanding: At the end of the course the students should grasp the main parts of the programApplying knowledge and understanding: At the end of the course the students should be able to apply the methods of game theory and the theory of dynamical systems to the solution of problems and exercises as well as to the translation of real life situations into mathematical terms and symbols, especially for social, economic and environmental systems.Making judgements: At the end of the course the students should be able to see links and analogies between different systems and situations, as well as to manage complex systems and situations by using tools from mathematics, logic, graphical methods.Communication: At the end of the course the students should be able to communicate what they learned by using a proper language and terminology, according to different kinds of audiences.Lifelong learning skills: The students must be able to get new concepts related to the ones exposed in the classroom from books, lectures and internet-based devices autonomously, following personal paths for understanding

Teaching Material

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Supporting Activities

During the course practical lessons for exercises and study of mathematical models by numerical methods will be given by Dr. Lorenzo Cerboni Baiardi.

Teaching, Attendance, Course Books and Assessment


Lessons at the blackboard and computer aided exercises

Course books

Textbooks (see also blended learning):Bischi G.I., Lamantia F. and Radi D. Qualitative Methods in Continuous and Discrete Dynamical Systems, chapter 1 in "Qualitative Theory of Dynamical Systems, Tools and Applications for Economic Modelling", Bischi, Panchuk and Radi (Eds), Springer-Verlag 2016, ISBN 978-3-319-33276-5. Also available at http://urbino2015.gecomplexity-cost.eu/index.php?id=315Martin J. Osborne An Introduction To Game TheoryKen Binmore, Fun and Games, chapter 9.


Oral exam

Disabilità e DSA

Le studentesse e gli studenti che hanno registrato la certificazione di disabilità o la certificazione di DSA presso l'Ufficio Inclusione e diritto allo studio, possono chiedere di utilizzare le mappe concettuali (per parole chiave) durante la prova di esame.

A tal fine, è necessario inviare le mappe, due settimane prima dell’appello di esame, alla o al docente del corso, che ne verificherà la coerenza con le indicazioni delle linee guida di ateneo e potrà chiederne la modifica.

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