AI 中文总结
本文针对有限时间连续时间时间不一致平均场博弈，通过构造集值最佳响应/一致性映射并应用Kakutani-Fan-Glicksberg不动点定理，确立了其松弛均衡的存在性。
AI 中文摘要
本文研究有限时间连续时间时间不一致平均场博弈的松弛均衡存在性。我们直接在松弛反馈策略与总体流的乘积空间上展开研究：策略分量配备Young测度的稳定拓扑，总体分量被限制在具有一致矩和时间正则性界的Wasserstein连续流的紧凸集内。对任意策略-流对，我们为相关辅助值函数建立一致Sobolev和Hölder估计，并证明其在策略的Young测度收敛及总体流的一致Wasserstein收敛下的稳定性；同时建立诱导总体流映射的连续性，该过程需对Fokker-Planck方程应用对偶论证，因Young测度收敛仅产生受控漂移的弱-*收敛。随后我们构造具有非空紧凸值的集值最佳响应/一致性映射，并应用Kakutani-Fan-Glicksberg不动点定理，所得不动点既满足个人内博弈的均衡响应条件，又满足平均场一致性条件，从而确立松弛均衡的存在性。
英文摘要
This paper studies the existence of relaxed equilibria for finite-horizon continuous-time time-inconsistent mean field games. We work directly on the product space of relaxed feedback policies and population flows. The policy component is endowed with the stable topology of Young measures, while the population component is restricted to a compact convex set of Wasserstein-continuous flows with uniform moment and time-regularity bounds. For every policy-flow pair, we establish uniform Sobolev and Hölder estimates for the associated auxiliary value function and prove its stability under Young-measure convergence of policies and uniform Wasserstein convergence of population flows. We also establish continuity of the induced population-flow map. The latter requires a duality argument for the Fokker-Planck equations because Young-measure convergence yields only weak-$*$ convergence of the controlled drifts. We then construct a set-valued best-response/consistency map with nonempty compact convex values and apply the Kakutani-Fan-Glicksberg fixed-point theorem. The resulting fixed point satisfies both the equilibrium response condition of the intra-personal game and the mean field consistency condition, , thereby establishing the existence of a relaxed equilibrium
