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.

翻译：我们研究离散时间设定下无限时域时间不一致平均场博弈（MFG）。首先给出该MFG的经典均衡及其存在性结果。当时域一致时，该经典均衡与MFG文献中研究的传统均衡概念一致。接着我们证明，尽管该均衡在应用于相关N-参与人博弈时能产生近似最优策略，但这仅局限于预先承诺意义下。因此，从N-参与人博弈中理性参与人的视角看，它无法充当真正意义上的近似均衡策略。为解决这一局限性，我们提出MFG和N-参与人博弈中新的相容均衡概念。我们证明MFG中的相容均衡确实能作为N-参与人博弈的近似相容均衡。此外，我们分析N-参与人博弈的相容均衡当N趋于无穷时收敛至相容MFG均衡的过程。