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  "slug": "pmms-three-agents-eight-goods",
  "title": "Pairwise maximin share allocations for three agents exist up to eight goods",
  "shortTitle": "PMMS for three agents up to eight goods",
  "url": "https://evidencepress.org/releases/pmms-three-agents-eight-goods/",
  "oneLine": "Three people can always share up to eight indivisible items under pairwise maximin share fairness; nine is the smallest number of items where it can fail.",
  "abstract": "A computer-assisted candidate proof shows that every instance with three agents, nonnegative additive valuations and at most eight indivisible goods has a complete pairwise maximin share (PMMS) allocation. With the published nine-good counterexamples, nine is the least number of goods in a three-agent counterexample. A written reduction turns a hypothetical eight-good counterexample into ten propositional formulas through a normal form; six- and seven-good existence is re-proved in the same framework and injected. Every formula is re-derived clause by clause by a separate verifier, and every refutation is a trimmed LRAT certificate accepted by two checkers. The paper also proves a sufficient condition for PMMS with 3q-1 goods involving one balanced agent, and reports hard eight-good instances that defeat natural constructions.",
  "datePublished": "2026-10-10",
  "dateModified": "2026-10-10",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.23284137",
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  "authors": [
    "Anonymous"
  ],
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      "note": "Published GitHub/Zenodo assets verified against the local SHA-256 ledger; availability is not proof or peer review."
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      "Anthropic Claude models (Claude Code): encoding, generator, verifier, certificates, paper and publication",
      "OpenAI gpt-5.6-sol: three adversarial reviews",
      "AI research agents in the supplied seven-good package; exact originating model not established"
    ],
    "humanRole": "Publication authority; no human research or verification contribution asserted.",
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  "attributionBoundary": null,
  "problem": {
    "name": "Least number of goods in a three-agent PMMS counterexample",
    "url": "https://arxiv.org/abs/2609.08954"
  },
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  "metadataProvenance": "Produced, revised and internally checked between 7 and 10 October 2026 from a supplied research package; recorded at publication on 10 October 2026. Discovery effort is not reconstructed.",
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  "keywords": [
    "fair division",
    "pairwise maximin share",
    "PMMS",
    "indivisible goods",
    "SAT",
    "LRAT certificates"
  ],
  "keyResults": [
    "Theorem 1: every three-agent instance with nonnegative additive valuations and at most eight goods has a complete PMMS allocation.",
    "Corollary: with the nine-good counterexamples of Gölz and of Bei, Lin, Liu, Luan and Tao, the least counterexample size is nine.",
    "Ten eight-good normal-form formulas, each with 252,930 variables, are refuted by trimmed LRAT certificates accepted by lrat-check and a standard-library Python checker; every formula is re-derived clause by clause.",
    "Proposition 7: one q-balanced agent and a second agent c with 3v_c(P) ≤ v_c(M) for the third agent's top q−1 goods P guarantee PMMS with 3q−1 goods."
  ],
  "reviews": [],
  "evidencePackage": "Written reduction with complete clause schemas; fourteen formulas with variable dictionaries; independent clause verifier; trimmed LRAT certificates with two checkers' receipts; nine-good enumeration; generator controls; one-command replay; manifest, claim ledger and AI index.",
  "openProblems": [
    "A human-readable proof of the eight-good theorem; the hardest instances found have no balanced agent.",
    "Explain why the pair template fails on the instance of Observation 3(a) and find a construction that covers such instances.",
    "Unaffiliated replay of the certificates and formalisation of the written reduction.",
    "The four-agent analogue, where the smallest known counterexample has 52 goods."
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  "relatedWorks": [
    {
      "citation": "Gölz (2026), PMMS allocations need not exist for 3 agents with additive valuations. Nine-good counterexample and the remark that nine is believed minimal or almost minimal.",
      "url": "https://arxiv.org/abs/2609.08954",
      "doi": "10.48550/arXiv.2609.08954"
    },
    {
      "citation": "Bei, Lin, Liu, Luan and Tao (2026), Non-existence of PMMS allocations and a 4/3-PMMS guarantee for additive chores. A second nine-good goods instance (Table 2).",
      "url": "https://arxiv.org/abs/2609.10493",
      "doi": "10.48550/arXiv.2609.10493"
    },
    {
      "citation": "Aziz (2026), Pairwise maximin share allocations need not exist. Four agents, 52 goods.",
      "url": "https://arxiv.org/abs/2609.06282",
      "doi": "10.48550/arXiv.2609.06282"
    },
    {
      "citation": "Amanatidis, Markakis and Ntokos (2020), Multiple birds with one stone: beating 1/2 for EFX and GMMS via envy cycle elimination. Theorem 18 and Corollary 19 give existence for at most n+2 goods.",
      "url": "https://doi.org/10.1609/aaai.v34i02.5545",
      "doi": "10.1609/aaai.v34i02.5545"
    },
    {
      "citation": "Akrami, Mayorov, Mehlhorn, Srinivas and Weidenbach (2026), A counterexample to EFX via SAT-solving, v3. Closest methodological precedent: subset-order encoding, Lean-formalised reduction, DRAT checking.",
      "url": "https://arxiv.org/abs/2604.18216v3",
      "doi": "10.48550/arXiv.2604.18216"
    },
    {
      "citation": "Caragiannis, Kurokawa, Moulin, Procaccia, Shah and Wang (2019), The unreasonable fairness of maximum Nash welfare. Definition of PMMS.",
      "url": "https://doi.org/10.1145/3355902",
      "doi": "10.1145/3355902"
    }
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