The students will become familiar with: - basic concepts of strategic reasoning and non-cooperative decision making in two-person games and multi-player setups (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. - 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. Approximately 20% of the lecture time will be devoted to discuss examples and applications and to solve exercises. 
Homework problems will be also proposed after completing each topic and 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.

- 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. - Lecture notes and reading material provided by the teacher. - 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.

Exam: Paper-based written test with video surveillance of the teaching staff;

Assessment is performed through one written exam composed of a series of problems to be solved applying the ideas and techniques learned in the course. The problems are similar to those proposed as homework during the course and will cover topics from both game theory and optimal control. More specific information will be provided by the teachers at the beginning of the course. Duration of the exam: 2h-2.5h. Students can consult lecture notes/personal notes during the exam. 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.

Exam: Written test; Paper-based written test with video surveillance of the teaching staff;

Assessment is performed through one written exam composed of a series of problems to be solved applying the ideas and techniques learned in the course. The problems are similar to those proposed as homework during the course and will cover topics from both game theory and optimal control. More specific information will be provided by the teachers at the beginning of the course. Duration of the exam: 2h-2.5h. Students can consult lecture notes/personal notes during the exam. 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.