{
  "abstract": "In 2013 Cooper and Dutle invented a dueling scenario where Alice and Bob shoot at each other until one is hit. Each shot is successful with some fixed probability $p$, $0 < p < 1$. The shooting order is given by a greedy algorithm, where at each step a shot is assigned to the player whose current probability of success is smaller. Cooper and Dutle observed that as $p \\rightarrow 0$, the resulting sequence of shots (by Alice or Bob) converges to the infinite Thue-Morse sequence $\\mathbf{t}$, but left the speed of convergence as an open problem. In this note we determine the speed of this convergence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.00194",
  "author_total": 1,
  "authors": [
    {
      "name": "Jeffrey Shallit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fby9anh6rv1sz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.00194",
  "fulltext_url": null,
  "id": "fby9anh6rv1sz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.00194",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7160067900",
  "pdf_url": "https://arxiv.org/pdf/2605.00194",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fby9anh6rv1sz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.00194",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.00194"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The speed of convergence in the Cooper-Dutle dueling game",
  "title_text": "The speed of convergence in the Cooper-Dutle dueling game",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fby9anh6rv1sz",
  "venue": "arXiv",
  "versions": []
}