  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 $N$-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 $N$-agent game. To address this limitation, we
propose a new consistent equilibrium concept in both the MFG and the $N$-agent
game. We show that a consistent equilibrium in the MFG can indeed function as
an approximate consistent equilibrium in the $N$-agent game. Additionally, we
analyze the convergence of consistent equilibria for $N$-agent games toward a
consistent MFG equilibrium as $N$ tends to infinity.
