Existence of Relaxed Equilibrium for Time-Inconsistent Mean Field Games: A Set-Valued Fixed Point Approach

Abstract

This paper studies the existence of relaxed equilibria for finite-horizon continuous-time time-inconsistent mean field games. We work directly on the product space of relaxed feedback policies and population flows. The policy component is endowed with the stable topology of Young measures, while the population component is restricted to a compact convex set of Wasserstein-continuous flows with uniform moment and time-regularity bounds. For every policy-flow pair, we establish uniform Sobolev and H\"older estimates for the associated auxiliary value function and prove its stability under Young-measure convergence of policies and uniform Wasserstein convergence of population flows. We also establish continuity of the induced population-flow map. The latter requires a duality argument for the Fokker-Planck equations because Young-measure convergence yields only weak-$*$ convergence of the controlled drifts. We then construct a set-valued best-response/consistency map with nonempty compact convex values and apply the Kakutani-Fan-Glicksberg fixed-point theorem. The resulting fixed point satisfies both the equilibrium response condition of the intra-personal game and the mean field consistency condition, , thereby establishing the existence of a relaxed equilibrium