{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R457",
  "content_hash": null,
  "slug": "levine-claim-three-block-self-best-response-all-depths",
  "type": "claim",
  "title": "The three-block strategy is a best response to itself at every depth",
  "summary": "For every finite truncation, and for the infinite Borel strategy, no unilateral strategy change improves the winning probability against the fixed symmetric three-block strategy.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-claim-three-block-self-best-response-all-depths (“The three-block strategy is a best response to itself at every depth”) records a bound, answer, status fact, or structural consequence. The record states: For every finite truncation, and for the infinite Borel strategy, no unilateral strategy change improves the winning probability against the fixed symmetric three-block strategy.",
  "relevance_source": "recorded",
  "body": "Bouquet and coauthors describe pointwise conditional-score maximization as the general way to compute a response to one fixed finite strategy. Specializing that principle to the recursive three-block strategy gives a closed formula at every depth.\n\nFix the three-block strategy for one player. For a block number \\(r\\ge0\\) and local output \\(q\\in\\{0,1,2\\}\\), let \\(E_{r,q}\\) be the event that the first \\(r\\) blocks are monochromatic and block \\(r\\) is the first nonmonochromatic block, with local output \\(q\\). Each of the three output classes contains two words, so\n\\[\n\\Pr(E_{r,q})=4^{-(r+1)}=:w_r.\n\\]\nFor a proposed response coordinate \\((t,p)\\), the count or probability of a win against choices outside block \\(t\\) has the same baseline for every \\(p\\). Inside block \\(t\\), the numbers of 1s in coordinate \\(p\\) among the two words mapped to \\(q\\) form the matrix\n\\[\nL=\\begin{pmatrix}2&1&0\\\\1&0&2\\\\0&2&1\\end{pmatrix}.\n\\]\nThus, for an observed block \\(y=(y_0,y_1,y_2)\\), the response score at \\((t,p)\\) is a common baseline plus \\(w_t/2\\) times the corresponding entry of\n\\[\n\\bigl(y_0-y_2,\\ y_2-y_1,\\ y_1-y_0\\bigr).\n\\]\n\nBoth monochromatic words give the zero vector. On each nonmonochromatic word, the strategy's displayed local choice has advantage \\(1\\), and the other two advantages are \\(0\\) and \\(-1\\). For an observed sequence whose first nonmonochromatic block is \\(t\\), all earlier blocks therefore give zero advantage. Every later block has advantage at most \\(w_s/2\\), where \\(w_s\\le w_t/4\\), while the chosen coordinate has advantage \\(w_t/2\\). It is the pointwise best response. If all inspected blocks are monochromatic in a finite truncation, every coordinate ties and the fallback coordinate is optimal.\n\nThe same calculation applies to the infinite strategy. Its first nonmonochromatic block is finite almost surely. Define its output to be coordinate 1 on the null set of sequences with no such block; every response coordinate ties on that set. Pointwise maximization proves optimality among deterministic Borel responses, and averaging shows that private randomized responses cannot do better. This mutual best-response result is a local equilibrium statement. It gives no global upper bound on pairs that change both strategies.",
  "status": "supported",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "family",
    "statement": "every finite truncation and the infinite symmetric first-nonmonochromatic three-block strategy",
    "family": "three-block strategy at every finite block depth and its almost-surely terminating infinite limit"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2508.01737",
      "locator": "Bouquet et al., Section 4.5, especially Theorem 36 and Proposition 37"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2508.01737",
    "locator": "Bouquet et al., Section 4.5, especially Theorem 36 and Proposition 37"
  },
  "models": [],
  "relations": [
    {
      "slug": "R452",
      "title": "Stress-test the all-depth response proof on blocks eight through twelve",
      "object_type": "attempt",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R449",
      "title": "Exact finite best-response enumerator",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R456",
      "title": "The truncated strategy is a best response to itself through seven blocks",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R455",
      "title": "The checked interval is 7/20 through 81/224, and equality at 7/20 remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}