{
  "abstract": "Fair allocation problems on a continuous simplex are often approached through Sperner-type discretizations: agents label mesh vertices according to their preferences and a fully labelled elementary triangle gives a discrete localization of a fair outcome. We develop a continuous alternative for three agents and three alternatives by representing preferences with multiplicatively consistent reciprocal pairwise-comparison matrices. Two independent ratios per agent induce exact linear indifference boundaries and turn envy-free allocation into a finite-dimensional feasibility problem. For the six possible assignments, we introduce an assignment score $Φ$ built from the agents' consistent priority vectors and prove that an incentive solution component is nonempty if and only if its assignment is the unique minimizer of $Φ$, whereas a disincentive solution component is nonempty if and only if its assignment is the unique maximizer. Consequently, the strict solution region is either empty or a unique convex preference region whose closure lies in the interior of the simplex. The weak solution set is never empty and degenerates exactly when $Φ$ has ties. We also compare the exact continuous formulation with Sperner-based discretization, including the boundary convention required in the disincentive model. Under exact consistency, the model uses six independent pairwise thresholds, whereas a regular triangular mesh with resolution $m$ contains $(m+1)(m+2)/2$ vertices. The method therefore exchanges a stronger parametric preference assumption for an exact continuous solution set and a resolution-independent elicitation stage.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38880",
  "author_total": 4,
  "authors": [
    {
      "name": "Luis Ángel Calvo Pascual"
    },
    {
      "name": "Susana Merchán Rubira"
    },
    {
      "name": "Diego de Pereda Sebastián"
    },
    {
      "name": "Javier Rodrigo Hitos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8c2bwnzejeptt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38880",
  "fulltext_url": null,
  "id": "8c2bwnzejeptt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38880",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38880",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8c2bwnzejeptt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38880",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38880"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization",
  "title_text": "Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8c2bwnzejeptt",
  "venue": "arXiv",
  "versions": []
}