COARSE CORRELATED EQUILIBRIA IN CONTINUOUS-TIME MEAN FIELD GAMES

Mean field games (MFGs), an active theme of research since the mid 2000's, arise as the limit formulation of symmetric N-player games with mean field interactions, and they act as a treatable approximation of such games, when the numbers of players is large enough. The usual notion of MFG solution can be regarded as the infinitely many players counterpart of the concept of Nash equilibrium in the underlying N-player game, and the connection between Nash equilibria and solutions to the MFG can be made rigorous via convergence arguments. The goal of this thesis is to introduce more general equilibria with better properties in continuous-time MFGs. In game theory literature, coarse correlated equilibria (CCEs) have been proposed as an alternative to Nash equilibria. CCEs can be seen as generalizations of the Nash equilibrium concept, incorporating a correlation device that allows agents to adopt correlated strategies without requiring cooperation. CCEs generalize Aumann's concept of correlated equilibria as well as Nash equilibria in both pure and mixed strategies. Among the nice features of CCEs, they are shown to be able to lead to higher payoffs than Nash equilibria in standard game theory, to be computationally ``easier'' and more natural to be learned by the players. In this thesis, we introduce CCEs in continuous-time MFGs and study their properties. In each chapter a different framework is considered and different problems are addressed, each requiring specific methodologies. In particular, the first chapter presents abstract results on the existence of CCEs in MFGs and on the approximation of CCEs in the underlying N-player games by means of the former ones. Subsequent chapters focus on detailed studies of specific MFG models, in which it is possible to analytically compute CCEs (or at least some of them) and compare them with more classical solution concepts. In Chapter 1 we consider MFGs driven by additive Wiener noise, with general drift and cost functions. The interaction term is given by a flow of probability measures, which appears both in the drift and in the cost functions. We introduce CCEs in both continuous time stochastic differential games and MFGs. The notion of coarse correlated solution to the MFG is justified by proving an approximation result. An existence result is also presented, whose proof relies on a minimax theorem. In Chapter 2 we consider linear-quadratic MFGs. The interaction term is given by a flow of first order moments, which appears in the payoff functional only. A methodology to compute CCEs in such class of MFGs is provided and, through the study of a simple yet important example with applications in environmental economics, we show that there exist infinitely many CCEs for the MFG which both yield higher payoffs than the classical MFG solutions and are more efficient with respect to the environmental goals. Moreover, we provide instances of MFGs that do not admit any MFG solution, whereas infinitely many CCEs exist. In Chapter 3 we consider a simple class of stationary MFGs of singular control. The reward, of ergodic type, is given by the long-time average of an expected utility functional. The interaction term is given by the stationary mean of the distribution of the representative player, which appears only in the reward. We provide constructive existence results, as well as approximation results and comparison with both MFG solutions and mean field control solutions. Finally, we show that CCEs may exist even when MFG solutions do not.