{
  "abstract": "For some integer $\\ell\\geq 2$, consider $m=2\\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\\ell$ and $n\\to \\infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $Θ_{\\ell,s} (n^{-s(\\ell-1)/\\ell})$. Next, if $\\min\\{s,n-s\\}\\to\\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $Θ(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $Θ_\\ell(n^{-(\\ell-1)/\\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03814",
  "author_total": 2,
  "authors": [
    {
      "name": "Zachary Berchenko"
    },
    {
      "name": "Kaizhao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4b0frprsypke0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03814",
  "fulltext_url": null,
  "id": "4b0frprsypke0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03814",
  "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",
      "econ.TH"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.03814",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4b0frprsypke0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03814",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03814"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On the Cardinality of the Smith Set Under Impartial Culture with Many Alternatives",
  "title_text": "On the Cardinality of the Smith Set Under Impartial Culture with Many Alternatives",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4b0frprsypke0",
  "venue": "arXiv",
  "versions": []
}