连续时间带正则化的平均场博弈的与时域无关的收缩性质
Horizon-Independent Contraction for Continuous-Time Discounted Regularized Mean-Field Games
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中文总结 AI 辅助
该研究针对带折扣和熵正则化的非平稳连续时间平均场博弈，证明大折扣率下有限时域MFG满足与时域无关的收缩条件，推导了有限与无限时域均衡的收敛率及相关误差界。
中文摘要 AI 辅助
我们研究带折扣和熵正则化的非平稳连续时间平均场博弈（MFGs）的收缩性质。代表性智能体的状态根据受控连续时间马尔可夫链演化，状态空间和动作空间均为有限。与无折扣情形不同，我们证明在足够大的折扣率下，有限时域MFGs满足与时域无关的收缩条件，该条件也对应无限时域非平稳收缩条件。作为副产品，我们得到有限时域与无限时域平均场均衡间的显式收敛率。对每个有限时域，我们进一步从主导策略误差传播的正算子谱半径推导得到改进的收缩准则，并证明其时域极限与与时域无关的收缩因子一致。最后，我们给出带折扣与无折扣有限时域正则化均衡间的显式误差界。
英文摘要
We study contraction properties of non-stationary continuous-time mean-field games (MFGs) under discounting and entropy regularization. The state of the representative agent evolves according to a controlled continuous-time Markov chain, and both the state and action spaces are finite. In contrast to the undiscounted case, we show that, under a sufficiently large discount rate, finite-horizon MFGs admit a horizon-independent contraction condition, which also coincides with the corresponding infinite-horizon non-stationary contraction condition. As a byproduct, we obtain an explicit convergence rate between finite- and infinite-horizon mean-field equilibria. For each finite horizon, we further derive a refined contraction criterion from the spectral radius of a positive operator that majorizes the propagation of policy errors, and show that its large-horizon limit agrees with the horizon-independent contraction factor. Finally, we provide an explicit error bound between discounted and undiscounted finite-horizon regularized equilibria.
