# Noncooperative Games and Applications

August 16, 2016A basic problem in optimization theory is to find the maximum value of a payoff function, over a given domain.

Game theory, on the other hand, is concerned with the more complex situation where two or more agents are present. Each player can choose among his set of available options seeks to maximize his own payoff, which depends also on the choices of all other agents.

When the game is not instantaneous, but takes place over an interval of time, this usually leads to a differential game. Optimal strategies by the various players (in feedback form) can be found by solving a suitable system of PDEs.

Our main interest lies in the mathematical properties of the system of Hamilton-Jacobi PDEs that provides these optimal strategies. In particular, the existence and uniqueness of solutions, and stability w.r.t. perturbations.

Among various applications, we have studied:

- A game theoretical model of the limit order book, in a stock market.
- Optimal debt management, in a model where the current interest rate is related to the future bankruptcy risk.