Braess' Paradox in Uniform Affine Grid Networks
Abstract
Braess' Paradox is the phenomenon in which adding an edge to a congestion network increases total travel time. We study the paradox in directed rectangular grids where every edge shares the latency function ℓ(x)=ax+b and an added chord has latency ℓ*(x) = cx+d, where a > 0 and b,c,d ≥ 0. Using an analogy with electrical networks, we bound the change in total travel time. We then give a necessary and sufficient condition for a chord to induce the paradox for some choice of nonnegative coefficients and compute the exact proportion of such chords in all grids with dimensions at most $100$. Such chords are scarce, and the fraction is maximized near an aspect ratio of 2:1. We then improve the established 4/3 upper bound on the Braess Ratio to one depending only on the grid dimensions, approaching $1.207$ on squares and 4/3 on thin grids. Finally, we prove any ratio-maximizing chord must have zero latency.
Related
- Network Topology That Excludes Braess's Paradox and Maintains Monotonicity in Flows Over Time
- The Kemeny-Braess Paradox in Random Graphs
- Affine Geometry of Gaussian ReLU Networks via Conditional Kac-Rice Formulas
- Sharp Approximation Rates for Neural Networks with Affine Latent Parameterizations
- Recurrent Neural Networks and (Time-Uniform) Universal Approximation of Bayesian Filters
BibTeX
@misc{lu2026braess,
title = {{Braess' Paradox in Uniform Affine Grid Networks}},
author = {Andy Lu and Steven J. Miller},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28938},
eprint = {2608.28938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28938},
}
RIS
TY - UNPB
TI - Braess' Paradox in Uniform Affine Grid Networks
AU - Andy Lu
AU - Steven J. Miller
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28938
UR - https://arxiv.org/abs/2608.28938
AB - Braess' Paradox is the phenomenon in which adding an edge to a congestion network increases total travel time. We study the paradox in directed rectangular grids where every edge shares the latency function ℓ(x)=ax+b and an added chord has latency ℓ_*(x) = cx+d, where a > 0 and b,c,d ≥ 0. Using an analogy with electrical networks, we bound the change in total travel time. We then give a necessary and sufficient condition for a chord to induce the paradox for some choice of nonnegative coefficients and compute the exact proportion of such chords in all grids with dimensions at most $100$. Such chords are scarce, and the fraction is maximized near an aspect ratio of 2:1. We then improve the established 4/3 upper bound on the Braess Ratio to one depending only on the grid dimensions, approaching $1.207$ on squares and 4/3 on thin grids. Finally, we prove any ratio-maximizing chord must have zero latency.
ER -