Advanced Topics in Probability Theory
A.Y. 2026/2027
Learning objectives
The course presents fundamental methods and results in game theory, with particular emphasis on static games, dynamic games, and stochastic games in discrete and continuous time.
The aim is to introduce students to the language and mathematical tools of game theory, highlighting its connections with advanced probability, stochastic control, and stochastic calculus. Special attention will also be devoted to the use of game theory as a tool for the analysis of concrete models arising in economics and finance.
The aim is to introduce students to the language and mathematical tools of game theory, highlighting its connections with advanced probability, stochastic control, and stochastic calculus. Special attention will also be devoted to the use of game theory as a tool for the analysis of concrete models arising in economics and finance.
Expected learning outcomes
Students are expected to become familiar with the main classes of static, dynamic, and stochastic games, as well as with the main notions of equilibrium.
They should be able to formulate strategic interaction problems in random environments, apply dynamic programming methods and verification theorems to analyse stochastic games in discrete and continuous time, and connect the abstract theory with selected applications in economics and finance.
They should be able to formulate strategic interaction problems in random environments, apply dynamic programming methods and verification theorems to analyse stochastic games in discrete and continuous time, and connect the abstract theory with selected applications in economics and finance.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
1) Static games and equilibrium theory
Normal-form games. Strict and weak dominance, iterated elimination of dominated strategies. Best replies and Nash equilibrium. Zero-sum games, value of the game, optimal strategies and minimax theorems. Examples: prisoner's dilemma, matching pennies, coordination games, Cournot and Bertrand oligopoly.
2) Deterministic dynamic games and repeated games
Extensive-form games. Perfect and imperfect information. Backward induction. Subgame-perfect equilibria. Mixed and behavioral strategies. Repeated games over finite and infinite horizons, stationary strategies, punishments, cooperation and unilateral deviations. Connection with dynamic programming.
3) Discrete-time stochastic games
Dynamic games with random state. Markov chains controlled by several players. History-dependent and Markovian strategies. Finite-horizon, discounted and long-run criteria. Zero-sum games: lower and upper value, Shapley operator, Bellman-Isaacs equations, existence of the value and Markovian optimal strategies. Nonzero-sum games: Markovian Nash equilibria, coupled Bellman equations, examples of non-uniqueness.
4) Continuous-time stochastic differential games
Stochastic differential equations controlled by several players. Open-loop and feedback strategies. Cost functionals over finite and infinite horizons. Zero-sum games, Isaacs condition, Hamilton-Jacobi-Bellman-Isaacs equations. Verification theorems for regular solutions and an introduction to viscosity solutions. Nonzero-sum games and coupled HJB systems. Stochastic linear-quadratic games.
5) Applications to economics and finance
Models of dynamic competition between firms. Strategic investment, optimal consumption and portfolio choice with strategic interaction. Timing games, strategic optimal stopping and real options. Depending on time, an introduction to mean field games and models with many agents.
Normal-form games. Strict and weak dominance, iterated elimination of dominated strategies. Best replies and Nash equilibrium. Zero-sum games, value of the game, optimal strategies and minimax theorems. Examples: prisoner's dilemma, matching pennies, coordination games, Cournot and Bertrand oligopoly.
2) Deterministic dynamic games and repeated games
Extensive-form games. Perfect and imperfect information. Backward induction. Subgame-perfect equilibria. Mixed and behavioral strategies. Repeated games over finite and infinite horizons, stationary strategies, punishments, cooperation and unilateral deviations. Connection with dynamic programming.
3) Discrete-time stochastic games
Dynamic games with random state. Markov chains controlled by several players. History-dependent and Markovian strategies. Finite-horizon, discounted and long-run criteria. Zero-sum games: lower and upper value, Shapley operator, Bellman-Isaacs equations, existence of the value and Markovian optimal strategies. Nonzero-sum games: Markovian Nash equilibria, coupled Bellman equations, examples of non-uniqueness.
4) Continuous-time stochastic differential games
Stochastic differential equations controlled by several players. Open-loop and feedback strategies. Cost functionals over finite and infinite horizons. Zero-sum games, Isaacs condition, Hamilton-Jacobi-Bellman-Isaacs equations. Verification theorems for regular solutions and an introduction to viscosity solutions. Nonzero-sum games and coupled HJB systems. Stochastic linear-quadratic games.
5) Applications to economics and finance
Models of dynamic competition between firms. Strategic investment, optimal consumption and portfolio choice with strategic interaction. Timing games, strategic optimal stopping and real options. Depending on time, an introduction to mean field games and models with many agents.
Prerequisites for admission
Students attending the course are expected to know the contents of an undergraduate probability course in mathematics and of an advanced probability course. Familiarity is required with discrete-time martingales, Markov chains, stochastic integration with respect to Brownian motion and stochastic differential equations. No previous knowledge of game theory is required.
Teaching methods
Classroom lectures and exercise sessions. Attendance is not compulsory, but strongly recommended. Exercise sessions will be devoted to the explicit computation of equilibria, guided proofs of verification results, examples of zero-sum games, linear-quadratic models and applications to economics and finance.
Teaching Resources
- R. Laraki, J. Renault, S. Sorin, Mathematical Foundations of Game Theory, Springer, 2019.
- T. Başar, G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1999.
- J. Filar, K. Vrieze, Competitive Markov Decision Processes, Springer, 1997.
- J. Renault, A tutorial on Zero-sum Stochastic Games, 2019.
- W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer.
- R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications, Vol. I-II, Springer, 2018.
- A. Bressan, Noncooperative Differential Games, lecture notes.
- E. Dockner, S. Jørgensen, N. Van Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000.
- T. Başar, G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1999.
- J. Filar, K. Vrieze, Competitive Markov Decision Processes, Springer, 1997.
- J. Renault, A tutorial on Zero-sum Stochastic Games, 2019.
- W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer.
- R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications, Vol. I-II, Springer, 2018.
- A. Bressan, Noncooperative Differential Games, lecture notes.
- E. Dockner, S. Jørgensen, N. Van Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, 2000.
Assessment methods and Criteria
The final examination consists of an oral exam. During the exam, students will be asked to present selected topics from the course syllabus, in order to assess their knowledge and understanding of the material and their ability to apply it.
MATH-03/B - Probability and Mathematical Statistics - University credits: 6
Exercises: 12 hours
Lessons: 35 hours
Lessons: 35 hours
Professor:
Campi Luciano
Shifts:
Turno
Professor:
Campi LucianoProfessor(s)