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：本文研究离散时间设定下的无限时域时间不一致平均场博弈。我们首先提出平均场博弈的经典均衡及其存在性结果。当背景为时间一致时，该经典均衡与平均场博弈文献中研究的传统均衡概念相一致。随后我们证明，虽然该均衡在应用于相关$N$人博弈时能产生近似最优策略，但这仅是在预先承诺的意义上成立。因此，从$N$人博弈中具有前瞻性智能体的视角来看，它无法作为真正的近似均衡策略发挥作用。为克服这一局限，我们在平均场博弈和$N$人博弈中提出了一种新的一致性均衡概念。我们证明平均场博弈中的一致性均衡确实可以作为$N$人博弈中的近似一致性均衡。此外，我们分析了当$N$趋于无穷时，$N$人博弈的一致性均衡向一致性平均场博弈均衡的收敛性。