In recent decades, ideas and methods from statistical physics have become valuable tools for understanding self-organizing collective phenomena and the macroscopic statistical patterns emerging in large-scale socio-economic and biological systems composed of interacting agents. This course is designed to introduce students to the modeling of agent-based systems and their applications across interdisciplinary fields. A central focus is on examining how simple interactions between individual agents—using frameworks like game theory and basic behavioral rules—can give rise to non-trivial collective behaviors, including market trends, opinion polarization, social segregation, and animal flocking. The course will explore how micro-level interactions link to macro-level outcomes, providing students with a robust understanding of these processes.

With this course, students will acquire both theoretical knowledge and computational skills to develop agent-based models and analyze collective phenomena emerging in real-world systems involving rational and bounded-rational agents, with applications ranging from biological systems to socio-economic ones.

First-level degree knowledge of linear algebra and calculus is required. Basic familiarity with Python or Julia programming language is not mandatory — students will learn Julia language in other courses during the same semester — but it can be beneficial, especially in the final part of the course.

1. Foundations of Individual Decision-Making (15h) - Preferences and choice; utility function representation - Decisions under risk: lotteries and Expected Utility Theory - Sequential decision problems, separability and information processing - Optimal Planning and its relation to Optimal Control, - Bellman equation and Dynamic Programming - Markov Decision Processes - Introduction to Reinforcement Learning 2. Multi-Agent Interactions (25h) - Normal-form games: pure strategies, utility function and payoff tables strategic dominance, best-response and Nash equilibria, mixed strategies, Nash theorem; special classes of games - Extensive form and repeated games: game tree, information sets, backward induction, equilibrium refinements, and Folks theorems - Bounded rationality, social learning and herding behavior - Population dynamics and evolutionary game theory 3. Agent-Based Modeling and Computational Applications (20h) Illustrative case studies carried out using computer simulations include: - Evolution of competition, cooperation and mutualism - Opinion dynamics, the emergence of consensus and social norms - Collective motion in swarms and flocks - Spatial segregation and pedestrian dynamics - Modeling financial markets: price formation, bubbles and crashes

The course is characterized by classroom-taught lectures mostly delivered using the blackboard, computer presentations and computational notebooks. Exercises (analytical and computational ones) will be proposed and solved during the classes.

- Lecture notes and reading material provided by the teacher. - D.P. Bertsekas, “Dynamic Programming and Optimal Control, Vol. I”, Athena Scientific, 1995. - K. Leyton-Brown and Y. Shoham, “Essentials of Game Theory: A concise, multidisciplinary Introduction”, Morgan&Calypool Pubs, 2008. - A. Mas-Colell, M.D. Whinston, and J.R. Green, “Microeconomic theory, volume 1”. Oxford University Press New York, 1995. - M. Maschler, E. Solan, S. Zamir, “Game Theory”, Cambridge Univ. Press, 2013. - M. Broom, J. Rychtar, “Game-theoretical models in biology”, Chapman & Hall – CRC, 2013. - D. Easley and J. Kleinberg, “Networks, Crowds, and Markets”, Cambridge Univ. Press, 2010. - F. Slanina, “Essentials of Econophysics Modelling”, Oxford Univ. Press, 2014. - J.-P. Bouchaud, “Crises and collective socio-economic phenomena: simple models and challenges”, J. Stat. Physics, 2013. - C. Castellano, S. Fortunato, V. Loreto, “Statistical Physics of Social Dynamics”, Review of Modern Physics, 2009. - T. Vicsek, A. Zafeiris, “Collective Motion”, Physics Reports, 2012. - T. Lux, “Applications of Statistical Physics to Finance and Economics”, Handbook of research and complexity, 2009. - M. Jusup et al., “Social Physics”, Physics Reports, 2022. - R. Axtell, J.D. Farmer, “Agent-Based Modeling in Economics and Finance: Past, Present and Future”, J. Economic Literature, 2025.

Slides; Dispense; Esercizi;

E' possibile sostenere l’esame in anticipo rispetto all’acquisizione della frequenza

Modalità di esame: Prova orale obbligatoria; Elaborato progettuale in gruppo;

Exam: Compulsory oral exam; Group project;

... The exam is designed in order to assess the students understanding of the topics covered in the course, their ability to identify and use the appropriate modeling techniques and apply these concepts to exemplary problems and case studies taylored on interdisciplinary applications. The evaluation process is structured in two parts. The first part is based on a group project that students have to submit by a prearranged date and consists of a Jupyter Notebook or a short written report (approx. 5 pages) developing a simple computational application of the ideas and models seen in the course. The topic of this project is chosen by the students among a set of alternatives provided by the teacher before the end of the classes. The submitted report should provide a motivation for the problem under study, a description of modeling assumptions and computational approaches adopted, a brief discussion of the numerical results. The evaluation of the project (30% of the final grade) focuses on the understanding of the modeling assumptions and the interpretation of the results. Coding skills are not directly evaluated. Although the project is carried out in small groups, it will be then discussed individually during the first part of the oral exam. In this part, the knowledge of the problem under study, its phenomenology and modeling properties will be futher ascertained (20% of the final grade). The second part of the oral exam consists of theoretical questions covering the core topics addressed in the lectures (concepts, derivations, calculations) and will assess the ability of the candidate to navigate through them and apply these concepts (50% of the final grade). The total duration of the oral exam typically ranges between 60-90 min. The final grade is the weighted average of the results obtained in different parts of the exam.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.