无限时域均值场终端值问题
Infinite Horizon Mean-Field Terminal Value Problem
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
- Bilkent University（比尔肯特大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出均值场终端值问题（MFTVP），用于求解无限时域非平稳均值场均衡，并利用其唯一性在弱条件下近似平稳均衡，适用于多均衡场景。
AI中文摘要：
我们引入一类有限和无限时域离散时间控制问题，用于研究离散时间下的宏观系统，称之为均值场终端值问题（MFTVP）。MFTVP以终端Q函数作为输入，并从此终端Q函数出发，寻求最优策略和状态测度的不定向后演化，这是离散时间均值场博弈（MFG）的无限时域非平稳均值场均衡（MFE）所必需的。我们研究了该问题的存在性和唯一性性质。作为应用，利用MFTVP的唯一性性质，我们证明在平均成本设定下（以及在折扣因子足够小的折扣设定下），MFTVP可作为中间步骤，用于开发有限时域MFE与平均成本MFE（分别为平稳MFE）之间的近似方案。与使用无限时域MFG作为中间问题的方法相比，当折扣因子足够小（或在平均成本设定下，当漂移因子足够小）时，MFTVP可结合正则化器用于近似平稳MFE。假设MFTVP解的唯一性，我们对系统Lipschitz参数施加的限制弱于正则化平稳MFG可用的压缩条件，并且我们的近似方案即使在存在多个平稳MFE的情况下仍然适用。
英文摘要：
We introduce a class of finite- and infinite-horizon discrete-time control problems for studying macroscale systems in discrete time, which we refer to as mean-field terminal value problems (MFTVPs). An MFTVP takes a terminal \(Q\)-function as input and, starting from this terminal \(Q\)-function, seeks an indefinite backward evolution of optimal policies and state measures, as required for an infinite-horizon non-stationary mean-field equilibrium (MFE) of a discrete-time mean-field game (MFG). We study the existence and uniqueness properties of this problem. As an application, using the uniqueness properties of MFTVPs, we show that, in the average-cost setting (and in the discounted setting for sufficiently small discount factors), MFTVPs can be used as an intermediate step to develop approximation schemes between finite-horizon MFEs and average-cost MFEs (respectively, stationary MFEs). In contrast to approaches that use infinite-horizon MFGs as intermediate problems, MFTVPs can be employed when the discount factor is sufficiently small (or, in the average-cost setting, when the drift factor is sufficiently small), together with a regularizer, to approximate a stationary MFE. Assuming uniqueness of the solution of a MFTVP, the restrictions we impose on the Lipschitz parameters of the system are weaker than the contraction conditions available for regularized stationary MFGs, and our approximation scheme remains applicable even in the presence of multiple stationary MFEs.
