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

Abstract We determine the order of thek-core in a large class of dense graph sequences. Let$$G_n$$be a sequence of undirected,n-vertex graphs with edge weights$$\{a^n_{i,j}\}_{i,j \in [n]}$$that converges to a graphon$$W\colon[0,1]^2 \to [0,+\infty)$$in the cut metric. Keeping an edge (i,j) of$$G_n$$with probability$${a^n_{i,j}}/{n}$$independently, we obtain a sequence of random graphs$$G_n({1}/{n})$$. Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of thek-core of random graphs$$G_n({1}/{n})$$. Our result can also be used to obtain the threshold of appearance of ak-core of ordern.