Mean field games with heterogeneous players: From portfolio optimization to network effects

Mean Field Games (MFG) are the infinite-population analogue of symmetric stochastic differential games with interacting players. By considering a limiting model with a continuum of players, the theory of MFG provides a more tractable representation and can effectively approximate a broad class of perfectly symmetric stochastic dynamic games. This thesis studies games with heterogeneous players, the heterogeneity being expressed either through a type parameter or through asymmetric interactions among players, and aims at understanding under which condition the MFG approximation remains valid for such games and, if it fails, to find a substitute model. In many real-life settings, players do not view themselves as exchangeable and accurate models should incorporate this heterogeneity. We first adapt the MFG paradigm to model more heterogeneous agents by introducing a type parameter in a financial problem that has gained huge interest in the recent years: the competitive Merton problem under relative performance criteria. By deriving a closed-form solution for the finitely many player investment-consumption problem, we show how the risk tolerance and competitivity of the investors influence their optimal strategy in equilibrium. Moreover, this thesis contributes to a very recent line of work bridging MFG theory and network games by studying n-player stochastic dynamic games in which interactions are governed by a graph. For games with perfectly symmetric players, the MFG approximation can be rigorously justified under suitable assumptions for two main reasons: On the one hand, the equilibria of n-player games can be shown to converge to the MFG limit. On the other hand, a solution of the continuum model may be used to construct approximate equilibria for the corresponding n-player model. This thesis extends these results in two cases: first, for games on general graph sequences in the setting of a specific yet rich linear-quadratic model and second, for general games on dense graph sequences. For linear-quadratic games, we show that the MFG is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. We show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence. Finally, since graphons have been shown to be the correct limit object for converging dense graph sequences, we develop the theory of graphon-based analogues of MFG. We propose a new formulation of graphon games based on a single typical player's label-state distribution. We show how our notion of graphon equilibrium can be used to construct approximate equilibria for large finite games set on any (weighted, directed) graph which converges in cut norm. The lack of players' exchangeability necessitates a careful definition of approximate equilibrium, allowing heterogeneity among the players' approximation errors, and we show how various regularity properties of the model inputs and underlying graphon lead naturally to different strengths of approximation