Trembling-Hand Perfection in Mean Field Games
Abstract
We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.
Related
- Mean-field quadratic BSDEs and related mean-field portfolio games of controls
- Mean field games with option to buy information
- Some Remarks on the Asymptotic Behavior of Mean Field Games of Controls
- Learning controls in mean field games: the fictitious play and related gradient descent
- Extended Graphon Mean-Field Games in Discrete Time
BibTeX
@misc{campi2026tremblinghand,
title = {{Trembling-Hand Perfection in Mean Field Games}},
author = {Luciano Campi and Luca Di Persio and Lucrezia Zorzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05083},
eprint = {2609.05083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05083},
}
RIS
TY - UNPB
TI - Trembling-Hand Perfection in Mean Field Games
AU - Luciano Campi
AU - Luca Di Persio
AU - Lucrezia Zorzi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05083
UR - https://arxiv.org/abs/2609.05083
AB - We introduce an admissible trembling-hand refinement for stochastic mean field games formulated through relaxed controlled martingale problems. A relaxed MFG equilibrium is perfect if it is the common Wasserstein limit of two sequences of joint control-state laws. The first consists of full-support population perturbations admissible for the state laws they generate; the second consists of exact optimal responses to those perturbed state laws. This admissibility condition prevents the control and state coordinates of a perturbation from being varied independently. Under the standing continuity, growth, and coercivity assumptions, we prove existence without requiring compact controls or bounded coefficients. The proof combines a perturbed fixed-point argument in the compact-bounded case with truncation, uniform moment estimates, compactness, and stability of controlled martingale problems. A one-dimensional model with exactly two relaxed mean field game equilibria shows that the refinement is genuinely selective: only one equilibrium is trembling-hand perfect.
ER -