Analytic and probabilistic methods in Mean Field Games

Differential games fall into the category of N-player games, in which different "agents" each try to minimise their own cost (or maximise payoff), which is dependent on the strategies of the whole agent population. An important concept in game theory is that of a Nash equilibrium, a scenario in which no agent is benefited from unilaterally deviating from their strategy.

The existence of a Nash equilibrium is naturally desirable in the perspective of differential games. Such differential games with large numbers of agents appear in many real life applications. A typical example is that of the stock market, where each investor is an individual agent looking to maximise their own profit, but the strategy of any individual agent to buy or sell a given stock depends on the strategies of the others.

Real life complex problems involve differential games with very large number of agents. In the ex amples above, stock markets could have hundreds of millions of interacting agents. In such scenarios the solvability of the optimisation problems, and finding equilibria becomes increasingly difficult, be- cause of the curse of dimensionality.

Mean field games (MFGs) theory overcomes this difficulty. Similarly to models in statistical mechanics aiming to describe large systems of interacting particles, as the number of agents gets very large, we can describe the behaviour of the collection of agents via their distributions, i.e. flows of probability measures. In the context of game theory, studying such models has been initiated less than 20-years by two groups, Lasry-Lions in the mathematical community, and by Huang-Caines--Malhamé in the engineering community.

Ever since, this theory has attracted many prominent scientists, initiated profound research programmes and has found a tremendous number of applications.

The purpose of this PhD project is to study various problems arising in MFG, using cutting edge tools from the theory calculus of variations and optimal control theory, partial differential equations, stochastic analysis and probability.