The students will become familiar with: - basic concepts of strategic reasoning and non-cooperative decision making in two-person games and multi-player setups (possibly also in networks); - optimal planning and dynamic programming techniques both in discrete and continuous systems; - relation between repeated games and optimal control problems; - Pontryagin’s maximum principle and its applications to control and dynamic games. Students will learn how to mathematically model control problems and analyse strategic decision-making in the presence of selfish individuals, with particular attention to the type of problems arising in the study of complex systems.

Basics of calculus, linear algebra and probability theory. Basic properties of ordinary differential equations. Elements of functional analysis (operators and functionals on Banach spaces) are desirable but not mandatory. Basic concepts of graph theory.

1. Optimal Control (20h) - Elements of constrained and unconstrained smooth optimization: Lagrangian multipliers, Karush-Kuhn-Tucker optimality conditions, existence of global minima. - Optimal control problem formulation for finite-dimensional continuous-time problems. - Pontryagin’s maximum principle (PMP) in the Hamiltonian and Lagrangian forms. - PMP and calculus of variations. - PMP for linear systems, time-optimality of the “bang-bang” control. - Sufficient conditions of optimality, existence of global optima. - Dynamic programming. - Bellman’s principle in deterministic discrete-time systems; Hamilton-Jacobi-Bellman equation in continuous-time systems. - Dynamic programming for stochastic discrete systems, Markov Decision Processes. 2. Game theory (40 h) - Elements of decision theory: preference relations, utility function, decision under risk and ignorance, sequential decision problems, relation to optimal control and reinforcement learning. - Games in normal form: dominance, best-response, rationalizability, Nash equilibria, correlated eq., solution concepts for zero-sum games. - Computational aspects of normal-form games: algorithms for finding equilibria in zero-sum and general sum two-player games, computational complexity, price of anarchy, relation to linear and integer programming. - Extensive-form games with perfect information: subgame-perfect equilibria, backward induction, repeated games. - Extensive-form games with imperfect and incomplete information: sequential equilibria, bayesian games. - Multi-agent games: games with global interaction, coordination and congestion games, cooperation, networks games. - Learning and dynamical equilibrium selection.

Frontal lectures, using mainly blackboard and computer presentations (in the case of online course, use of slides and tablet whiteboard). Approximately 25% of the lecture time will be devoted to discussing examples and applications and solving exercises. 
Homework problems will be also proposed after completing each topic and possibly solved after a few lectures, in order to provide the students a way to see practical applications of the theory and test their level of understanding of the exposed notions.

- Lecture notes and reading material provided by the teacher. - K.Leyton-Brown and Y. Shoham, Essentials of Game Theory: A concise, multidisciplinary Introduction, Morgan&Calypool Pubs, 2008. - M.J. Osborne and A. Rubinstein, A course in Game Theory, The MIT Press, 1994. - N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani, Algorithmic Game Theory, Cambridge Univ. Press, 2007. - D. Easley and J. Kleinberg, Networks, Crowds, and Markets, Cambridge Univ. Press, 2010. - M.O. Jackson, Social and Economic Networks, Princeton Univ. Press 2008. - D.P. Bertsekas, Dynamic Programming and Optimal Control, Vol. I, Athena Scientific, 1995. - D. Lieberzon, "Calculus of Variations and Optimal Control Theory", Princeton University Press, 2012. - E.B. Lee, L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, 1967. - T. Basar and G.J. Olsder, Dynamic Non-cooperative Game Theory, ser. “Classics in Applied Mathematics”, vol. 23, SIAM, 1999.

Lecture slides; Lecture notes; Exercises; Exercise with solutions ;

You can take this exam before attending the course

Exam: Written test; Compulsory oral exam;

The assessment is performed through a written exam plus a mandatory oral exam. The written exam is composed of a fixed number (e.g. 6) of multiple-choice questions and a few open problems (e.g. 3), to be solved by applying the ideas and techniques learned in the course. More specific information and examples of past exams will be provided by the teachers during the course. Duration of the exam: 2.30h. Students can consult lecture notes/personal notes during the exam. There will be also a short (20-25 min) oral exam for students achieving in the written exam a grade equal to or larger than 18/30. The oral exam will focus on parts of the material covered during the course and specified beforehand. The objective of the final exams is two-fold: to evaluate the theoretical knowledge of the subjects covered in the lectures and to assess the ability of the students to apply these concepts in order to solve practical problems. Both elements will have equivalent weight on the final grade: the exact weight of each part (question/problems) of the written exam will be clearly specified beforehand.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.