{
  "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 $\\ell(x)=ax+b$ and an added chord has latency $\\ell_{*}(x) = cx+d$, where $a > 0$ and $b,c,d \\ge 0$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28938",
  "author_total": 2,
  "authors": [
    {
      "name": "Andy Lu"
    },
    {
      "name": "Steven J. Miller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dftxq28wy5jg0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28938",
  "fulltext_url": null,
  "id": "dftxq28wy5jg0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28938",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "3313|Transportation Planning and Optimization"
    ]
  ],
  "openalex": "W7204950340",
  "pdf_url": "https://arxiv.org/pdf/2608.28938",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dftxq28wy5jg0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28938",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28938"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Braess' Paradox in Uniform Affine Grid Networks",
  "title_text": "Braess' Paradox in Uniform Affine Grid Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dftxq28wy5jg0",
  "venue": "arXiv",
  "versions": []
}