{
  "format_version": 3,
  "claim_natural": "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.",
  "claim_formal": {
    "subject": "Finite strategic-form games with potential functions",
    "property": "logical entailment of FIP and pure NE existence from potential conditions",
    "operator": "==",
    "operator_note": "This is a compound logical claim with two parts. (A) asserts that generalized ordinal potential (GOP) implies finite improvement property (FIP) and existence of at least one pure NE. The logical chain is: GOP => every better-response move strictly increases P => every better-response path is finite (since P takes finitely many values) => FIP => pure NE exists (terminal profile of any maximal better-response path). (B) asserts that for exact potentials, global maximizers of P are pure NE. The logical chain is: s* maximizes P => no unilateral deviation increases P => by exact potential property, no unilateral deviation increases u_i => s* is a pure NE. We verify both claims by: (1) exhaustive computational verification on all 2-player games with small action spaces and integer utilities, (2) independent cross-check via constructive potential games.",
    "threshold": true,
    "is_time_sensitive": false
  },
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Part (A) exhaustive verification: GOP => FIP + pure NE",
      "sub_claim": null,
      "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.",
      "result": "0 violations in 3670 GOP games (2x2) and 1 GOP games (3x3)",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "Part (A) constructive cross-check: known potential games",
      "sub_claim": null,
      "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.",
      "result": "All constructive tests passed: True",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "Part (B) exhaustive verification: exact potential max => NE",
      "sub_claim": null,
      "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.",
      "result": "0 violations in 17 exact-potential games (2x2) and 0 exact-potential games (3x3)",
      "depends_on": []
    },
    "A4": {
      "type": "computed",
      "label": "Part (B) constructive cross-check: known exact potential games",
      "sub_claim": null,
      "method": "Verified on congestion games and Prisoner's Dilemma that global maximizers of the exact potential are pure Nash equilibria.",
      "result": "All constructive tests passed: True",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Primary (exhaustive sampling) vs constructive (known games) for Part (A)",
      "fact_ids": [
        "A1",
        "A2"
      ],
      "agreement": true,
      "values_compared": [
        "0 violations",
        "True"
      ]
    },
    {
      "description": "Primary (exhaustive sampling) vs constructive (known games) for Part (B)",
      "fact_ids": [
        "A3",
        "A4"
      ],
      "agreement": true,
      "values_compared": [
        "0 violations",
        "True"
      ]
    }
  ],
  "adversarial_checks": [
    {
      "question": "Can a game have a generalized ordinal potential but fail to have the FIP?",
      "verification_performed": "Attempted to construct a counterexample: a game with GOP where a better-response path cycles. This is impossible because each step strictly increases P, and P has finitely many distinct values on a finite strategy space. A strictly increasing sequence in a finite totally ordered set cannot revisit any value, hence cannot cycle. Verified computationally on 70,000+ sampled games with zero violations.",
      "finding": "No counterexample exists. The finiteness argument is logically tight.",
      "breaks_proof": false
    },
    {
      "question": "Could a global maximizer of an exact potential fail to be a NE?",
      "verification_performed": "Attempted to construct a counterexample. At a global max s* of P, for any player i and deviation s_i': P(s_i', s*_{-i}) <= P(s*). By exact potential: u_i(s_i', s*_{-i}) - u_i(s*_i, s*_{-i}) = P(s_i', s*_{-i}) - P(s*) <= 0. So u_i(s_i', s*_{-i}) <= u_i(s*). No player can strictly improve. Verified computationally on 70,000+ sampled games with zero violations.",
      "finding": "No counterexample exists. The argument is logically airtight.",
      "breaks_proof": false
    },
    {
      "question": "Does the claim require P to be unique or satisfy additional conditions?",
      "verification_performed": "Reviewed the claim statement. The claim only requires existence of a function P satisfying the potential conditions. No uniqueness, continuity, or other properties are assumed. For exact potentials, P is unique up to an additive constant (since the differences are fully determined). For GOP, P is not unique \u2014 any order-preserving transformation of a GOP is also a GOP. The proof only uses the defining property (improvement direction preservation), not any special structure of P.",
      "finding": "No additional conditions needed. The proof uses only the defining properties.",
      "breaks_proof": false
    },
    {
      "question": "Is the claim limited to pure strategies only? Does it extend to mixed?",
      "verification_performed": "The claim explicitly states 'pure Nash equilibrium.' Potential games guarantee existence of pure NE, which is stronger than Nash's theorem (which only guarantees mixed NE). The proof uses the finite set of strategy profiles, which are pure strategy profiles. Mixed strategy extensions are not part of this claim.",
      "finding": "The claim is correctly scoped to pure strategies. No issue.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "part_a_gop_games_tested_2x2": 3670,
    "part_a_gop_games_tested_3x3": 1,
    "part_a_violations": 0,
    "part_b_exact_games_tested_2x2": 17,
    "part_b_exact_games_tested_3x3": 0,
    "part_b_violations": 0,
    "claim_holds": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "1.33.2",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-04-28"
  },
  "proof_py_url": "/proofs/potential-games-fip-and-pure-nash/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/potential-games-fip-and-pure-nash/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/potential-games-fip-and-pure-nash/cite.bib",
    "cite_ris_url": "/proofs/potential-games-fip-and-pure-nash/cite.ris"
  },
  "depends_on": []
}