Monotonicity, Uniqueness and Frank--Wolfe Dynamics in Atomic Splittable Congestion Games
Abstract
We study atomic splittable congestion games with n≥2 players and nondecreasing C2 resource costs f with convex x↦ xf(x). We characterize the largest resource cost class for which the Nash equilibrium is unique. This characterization combines a curvature inequality involving the first two derivatives and the number of players with strict increase of the scalar marginal costs x↦ f(x)+xf'(x). The proof rests on first characterizing the largest resource cost classes for which the associated variational inequality operator is monotone, strictly monotone, or strongly monotone. We also draw a perhaps surprising connection to learning dynamics: the same curvature inequality characterizes universal local and global stability of Euclidean-regularized Frank--Wolfe dynamics on arbitrary compact convex strategy spaces. Both the uniqueness and stability characterizations assume closure under positive affine transformations. Both characterizations remain exact for network games; the necessity constructions use acyclic directed graphs whose arc costs can be chosen strictly positive. Finally, we show that on parallel-link networks, every interior equilibrium of the regularized Frank--Wolfe dynamics is locally exponentially stable even without the curvature condition.
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BibTeX
@misc{harks2026monotonicity,
title = {{Monotonicity, Uniqueness and Frank--Wolfe Dynamics in Atomic Splittable Congestion Games}},
author = {Tobias Harks},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.17684},
eprint = {2607.17684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17684},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monotonicity, Uniqueness and Frank--Wolfe Dynamics in Atomic Splittable Congestion Games
AU - Tobias Harks
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2607.17684
UR - https://arxiv.org/abs/2607.17684
AB - We study atomic splittable congestion games with n≥2 players and nondecreasing C^2 resource costs f with convex x↦ xf(x). We characterize the largest resource cost class for which the Nash equilibrium is unique. This characterization combines a curvature inequality involving the first two derivatives and the number of players with strict increase of the scalar marginal costs x↦ f(x)+xf'(x). The proof rests on first characterizing the largest resource cost classes for which the associated variational inequality operator is monotone, strictly monotone, or strongly monotone. We also draw a perhaps surprising connection to learning dynamics: the same curvature inequality characterizes universal local and global stability of Euclidean-regularized Frank--Wolfe dynamics on arbitrary compact convex strategy spaces. Both the uniqueness and stability characterizations assume closure under positive affine transformations. Both characterizations remain exact for network games; the necessity constructions use acyclic directed graphs whose arc costs can be chosen strictly positive. Finally, we show that on parallel-link networks, every interior equilibrium of the regularized Frank--Wolfe dynamics is locally exponentially stable even without the curvature condition.
ER -