On Time‐Inconsistency in Mean‐Field Games

We investigate an infinite‐horizon time‐inconsistent mean‐field game (MFG) in a discrete time setting. We first present a classic equilibrium for the MFG and its associated existence result. This classic equilibrium aligns with the conventional equilibrium concept studied in MFG literature when the context is time‐consistent. Then we demonstrate that while this equilibrium produces an approximate optimal strategy when applied to the related N$N$‐agent games, it does so solely in a precommitment sense. Therefore, it cannot function as a genuinely approximate equilibrium strategy from the perspective of a sophisticated agent within the N$N$‐agent game. To address this limitation, we propose a new consistent equilibrium concept in both the MFG and the N$N$‐agent game. We show that a consistent equilibrium in the MFG can indeed function as an approximate consistent equilibrium in the N$N$‐agent game. Additionally, we analyze the convergence of consistent equilibria for N$N$‐agent games toward a consistent MFG equilibrium as N$N$ tends to infinity.