# Medium field games

C. Bertucci

*This course takes place in the second semester, 3 hours per week*.

The aim of this course is to present the new medium-field game theory, notably through some of its applications in finance. The aim of this course is to present the new theory of mean-field games, in particular through some of its applications in finance. Mean-field games are games involving infinite number of “small” players, i.e. who have only a marginal influence on the game. influence on the game. We will see in particular through two examples that we will follow the course (one from a crypto-currency market and one from an optimal liquidation problem), why such games are We will see in particular through two examples that we will follow during the course (one from a crypto-currency market and one from an optimal liquidation problem) why such games are natural models in finance. From a mathematical point of view, this theory is essentially based on optimal control (stochastic) control and game theory. Moreover, we will interpret some equations equations characterising equilibria in a mean-field game as a form of dynamic programming, where each dynamic programming, where each player takes into account the behaviour of the other players.

Although most of the mathematical concepts will be reintroduced, it is strongly recommended that you familiar with Bellman’s dynamic programming (having taken the optimization course in the first semester for of the first semester for example). No knowledge of game theory is required. This course will be largely devoted to modelling issues, including The course will be devoted largely to modelling issues, including the structural assumptions behind stability in these games.

- Preliminaries: game theory and optimization
- A first example from crypto-currencies
- Population evolution and equilibrium equations in a mean field game
- Example of optimal liquidation
- Master equation
- Additional modelling (learning procedure, presence of a majority player, …)