Robust mean-field games under entropy-based uncertainty

In this article, we introduce a new class of entropy-penalized robust mean field game problems in which the representative agent is opposed to Nature. The agent's objective is formulated as a min-max stochastic control problem, in which Nature distorts the reference probability measure at an entropic cost. As a consequence, the distribution of the continuum of agents represented by the player is given by the effective measure induced by Nature. Existence of a mean-field game equilibrium is established via a Schauder fixed point argument. To ensure uniqueness, we introduce a joint flat anti-monotonicity and displacement monotonicity condition, extending the classical Lasry-Lions monotonicity framework. Finally, we present two classes of N -player games for which the mean-field game limit yields $\epsilon$-Nash equilibria.