{
  "abstract": "Many classes of two-player zero-sum stochastic games have the orderfield property: if all payoffs and transition probabilities lie in a subfield of $\\mathbb{R}$, so does the undiscounted value. Absorbing games fail this property, and Oliu-Barton and Vigeral [Absorbing games with irrational values, Oper. Res. Lett. 51 (2023) 555--559] conjectured the precise extent of the failure: every real algebraic number $α$ of degree $m\\geq 1$ over $\\mathbb{Q}$ is the undiscounted value of a rational $m\\times m$ absorbing game. We prove this conjecture, and in fact within a special subclass of absorbing games: for every such $α$, the game realizing it is strictly absorbing and recursive, i.e., every action pair is absorbing with positive probability and all non-absorbing stage payoffs are zero; when $α>0$ it can moreover be taken positive recursive, with positive absorbing payoffs. As a corollary, the set of undiscounted values of rational $m\\times m$ absorbing games is exactly the set of real algebraic numbers of degree at most $m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11583",
  "author_total": 2,
  "authors": [
    {
      "name": "Ali Asadi"
    },
    {
      "name": "Krishnendu Chatterjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdtd31nyfxcn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11583",
  "fulltext_url": null,
  "id": "fdtd31nyfxcn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11583",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1803|Game Theory and Applications"
    ]
  ],
  "openalex": "W7212229134",
  "pdf_url": "https://arxiv.org/pdf/2609.11583",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdtd31nyfxcn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.11583",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11583"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Values of Absorbing Recursive Games Express All Real Algebraic Numbers",
  "title_text": "Values of Absorbing Recursive Games Express All Real Algebraic Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/fdtd31nyfxcn0",
  "venue": "arXiv",
  "versions": []
}