Infinite Horizon Mean-Field Terminal Value Problem
Abstract
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.
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BibTeX
@misc{aydın2026infinite,
title = {{Infinite Horizon Mean-Field Terminal Value Problem}},
author = {Uğur Aydın and Tamer Başar and Naci Saldi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12768},
eprint = {2609.12768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12768},
}
RIS
TY - UNPB
TI - Infinite Horizon Mean-Field Terminal Value Problem
AU - Uğur Aydın
AU - Tamer Başar
AU - Naci Saldi
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12768
UR - https://arxiv.org/abs/2609.12768
AB - 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.
ER -