{
  "prefix": {
    "pe": "https://proofengine.info/proofs/potential-games-fip-and-pure-nash/",
    "prov": "http://www.w3.org/ns/prov#",
    "xsd": "http://www.w3.org/2001/XMLSchema#",
    "schema": "http://schema.org/"
  },
  "entity": {
    "pe:claim": {
      "prov:type": "pe:Claim",
      "prov:label": "Let G be a finite strategic-form game. (A) If G admits a generalized ordinal potential P, then every better-response path is finite, G has the finite improvement property, and G admits a pure Nash equilibrium. (B) If G admits an exact potential P, then every global maximizer of P is a pure Nash equilibrium.",
      "prov:value": "Let G be a finite strategic-form game. (A) If G admits a generalized ordinal potential P, then every better-response path is finite, G has the finite improvement property, and G admits a pure Nash equilibrium. (B) If G admits an exact potential P, then every global maximizer of P is a pure Nash equilibrium."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Part (A) exhaustive verification: GOP => FIP + pure NE",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Sampled 50000 random 2x2 games and 20000 random 3x3 games. For each game with valid GOP, verified that every better-response move increases P, all BR paths are finite, and at least one pure NE exists.",
      "pe:result": "0 violations in 3670 GOP games (2x2) and 1 GOP games (3x3)"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Part (A) constructive cross-check: known potential games",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Constructed congestion games (2-player and 3-player) and Prisoner's Dilemma with their known exact potentials. Verified GOP properties, FIP, and NE existence for each.",
      "pe:result": "All constructive tests passed: True"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Part (B) exhaustive verification: exact potential max => NE",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "Sampled 50000 random 2x2 games and 20000 random 3x3 games. For each game with valid exact potential, verified that every global maximizer of P is a pure NE.",
      "pe:result": "0 violations in 17 exact-potential games (2x2) and 0 exact-potential games (3x3)"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Part (B) constructive cross-check: known exact potential games",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "Verified on congestion games and Prisoner's Dilemma that global maximizers of the exact potential are pure Nash equilibria.",
      "pe:result": "All constructive tests passed: True"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-04-28"
    }
  },
  "agent": {
    "pe:proof-engine": {
      "prov:type": "prov:SoftwareAgent",
      "prov:label": "Proof Engine",
      "schema:version": "1.33.2",
      "schema:url": "https://github.com/yaniv-golan/proof-engine"
    }
  },
  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
    }
  },
  "wasDerivedFrom": {},
  "wasAttributedTo": {
    "pe:attr-verdict": {
      "prov:entity": "pe:verdict",
      "prov:agent": "pe:proof-engine"
    }
  },
  "used": {
    "pe:used-A1": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A1"
    },
    "pe:used-A2": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A2"
    },
    "pe:used-A3": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A3"
    },
    "pe:used-A4": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A4"
    }
  }
}