New approaches to Mean Field Game Theory

The present thesis studies new possible approaches to Mean Field Game inspired by Mechanism Design and Mean Field Theory with moderate interactions. The structure of the thesis is the following. The first part of the thesis studies mean field games in which the particles interact like in mean field theory and in mean field theory with moderate interactions. Chapter 2 studies competitions in which the agents interact in a mean field way and convergence of the system to a proper mean field equation is proved. Chapter 3, starting from the ideas of chapter 2, consider the situation in which the agents interact locally. The second part of the thesis deals with the idea, inspired by Mechanism Design, of introducing, in the context of mean field games, an external player, the Principal, who can choose the rules of the game in order to achieve a specic outcome. Chapter 4 introduces the Principal in the context of classical mean field games; optimality for the payoff of the Principal is studied. Chapter 5 consider the presence of the Principal in the context of mean field games with mean field interactions: existence of minimum for Principal's payoff is proved.