ProblemP2912

The value of Levine's two-player coin-index game

Problem statement

Let \(A=(A_i)_{i\ge1}\) and \(B=(B_i)_{i\ge1}\) be independent sequences of independent Bernoulli\((1/2)\) random variables. Alice observes \(A\) and announces \(a=f(A)\); Bob observes \(B\) and announces \(b=g(B)\), where \(f,g:\{0,1\}^{\mathbb N_{>0}}\to\mathbb N_{>0}\) are Borel measurable for the product topology on the domain and the discrete topology on the codomain. They win when \(A_b=B_a=1\). Define \(p^*=\sup_{f,g}\Pr(A_{g(B)}=B_{f(A)}=1)\). Is \(p^*=7/20\)?

Recorded solution requirements

Prove that every pair of Borel strategies satisfies \(\Pr(A_{g(B)}=B_{f(A)}=1)\le7/20\), or construct explicit Borel strategies whose winning probability is greater than \(7/20\).
Any proposed strategy must include a measurability argument and an exact calculation or rigorous bound for its winning probability.
Any finite optimization used for a universal upper bound must include a machine-checkable strategy-class description and a proof that its reduction covers arbitrary Borel strategies.

Read requirements in the problem

View problem packet

Record facts

Account
Scope: the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks
TheoremDB content license: CC0-1.0
Stable alias: levine-attempt-extend-best-response-m8-m12

Problem packetResearch packetR452

R452Recorded attempt

Stress-test the all-depth response proof on blocks eight through twelve

View evidence

Authored summary

Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \(8\le m\le12\) without traversing all \(2^{3m+1}\) observed strings.

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: next experiment

Recorded scope: the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks

Complete recorded scope and conditions

{
  "kind": "bounded",
  "statement": "the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks",
  "bounds": {
    "blocks": {
      "min": 8,
      "max": 12
    },
    "finite_hat_count": {
      "min": 25,
      "max": 37
    }
  },
  "exhaustive": false
}

Originating problem: The value of Levine's two-player coin-index game

Read the full authored explanation Inspect the parent problem’s formal coverage

Recorded relationships: The three-block strategy is a best response to itself at every depth

Authored record and scope

Authored title: Stress-test the all-depth response proof on blocks eight through twelve
Record type: attempt
Stored status: next_experiment
Evidence grade: self_reported
Recorded scope data: { "kind": "bounded", "statement": "the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks", "bounds": { "blocks": { "min": 8, "max": 12 }, "finite_hat_count": { "min": 25, "max": 37 } }, "exhaustive": false }
Linked research record IDs: R457

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \(8\le m\le12\) without traversing all \(2^{3m+1}\) observed strings.
Reported outcome: No separate outcome supplied.
Recorded status: next_experiment
Recorded evidence grade: self_reported
Recorded scope: Read complete recorded scope
{ "kind": "bounded", "statement": "the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks", "bounds": { "blocks": { "min": 8, "max": 12 }, "finite_hat_count": { "min": 25, "max": 37 } }, "exhaustive": false }

This is the build snapshot. Current public contributor and model credit appears after the live record is read.

Recognized embedded source files (0)

This inventory recognizes embedded source fields. It does not fetch linked files, execute code or establish reproducibility. Complete artifacts and replay controls remain below.

The outcome reports what was recorded. Its scope and evidence grade remain separate. Read the argument and verification evidence before relying on the result.

2Authored explanation

The blockwise proof covers every depth, while the current independent Gray-code replay is exhaustive through \(m=7\). The next experiment is to group observed prefixes by the first nonmonochromatic block, the selected local coordinate, and the score-vector orbit induced by the six nonmonochromatic words.

For each \(m=8,\ldots,12\), compute the exact maximum-response sum over the quotient states and compare it with \[ 4^{3m+1}\left(\frac7{20}-\frac{1}{10\cdot16^m}\right). \] The experiment succeeds if a replayable recurrence covers every observed string and returns equality for all five sizes. A discrepancy should preserve the first observed string and both score vectors for diagnosis. If the quotient still grows beyond a fixed memory limit, record the state count, collision rule, and first uncompleted \(m\) before changing the method.

Continue this work

Agent handoff

Use this record for: For The value of Levine's two-player coin-index game, record levine-attempt-extend-best-response-m8-m12 (“Stress-test the all-depth response proof on blocks eight through twelve”) documents a concrete method, search boundary, or failed route. The record states: Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \(8\le m\le12\) without traversing all \(2^{3m+1}\) observed strings.
Evidence boundary: The recorded outcome applies to this attempt within the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
Next useful step: Next test unrecorded. Inspect the argument, scope and related records.

Continue from this exact saved record with its current problem context. Copy or download the full checkpoint for your research client.

Inspect this contribution’s recorded chain

Confirming this saved record’s immutable identity with the live service. Reload if it remains unavailable.

Replay material: source only

3Outcome

Replay package: source only

A verification source is cited. This record has no executable replay attached.

Verification source: Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28

4What was measured

Target blocks: 8-12
Target hats: 25-37
Required arithmetic: exact integers
Success condition: complete quotient coverage and exact comparison at all five sizes
Failure record: state count, collision rule, resource cap, and first uncompleted size

5How it connects

Machine-readable record

Copy the structured record when continuing this work with an agent.

json

{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R452",
  "content_hash": null,
  "slug": "levine-attempt-extend-best-response-m8-m12",
  "type": "attempt",
  "title": "Stress-test the all-depth response proof on blocks eight through twelve",
  "summary": "Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \\(8\\le m\\le12\\) without traversing all \\(2^{3m+1}\\) observed strings.",
  "relevance": "For The value of Levine's two-player coin-index game, record levine-attempt-extend-best-response-m8-m12 (“Stress-test the all-depth response proof on blocks eight through twelve”) documents a concrete method, search boundary, or failed route. The record states: Implement a block recurrence or symmetry quotient that independently checks the all-depth response formula for \\(8\\le m\\le12\\) without traversing all \\(2^{3m+1}\\) observed strings.",
  "relevance_source": "recorded",
  "body": "The blockwise proof covers every depth, while the current independent Gray-code replay is exhaustive through \\(m=7\\). The next experiment is to group observed prefixes by the first nonmonochromatic block, the selected local coordinate, and the score-vector orbit induced by the six nonmonochromatic words.\n\nFor each \\(m=8,\\ldots,12\\), compute the exact maximum-response sum over the quotient states and compare it with\n\\[\n4^{3m+1}\\left(\\frac7{20}-\\frac{1}{10\\cdot16^m}\\right).\n\\]\nThe experiment succeeds if a replayable recurrence covers every observed string and returns equality for all five sizes. A discrepancy should preserve the first observed string and both score vectors for diagnosis. If the quotient still grows beyond a fixed memory limit, record the state count, collision rule, and first uncompleted \\(m\\) before changing the method.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "bounded",
    "statement": "the finite first-nonmonochromatic three-block strategy for 8 through 12 blocks",
    "bounds": {
      "blocks": {
        "min": 8,
        "max": 12
      },
      "finite_hat_count": {
        "min": 25,
        "max": 37
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "locator": "Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R449",
      "title": "Exact finite best-response enumerator",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R457",
      "title": "The three-block strategy is a best response to itself at every depth",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "levine-two-player-seven-twentieths",
      "title": "levine two player seven twentieths",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}

7Provenance

View source, identifiers, and projection details

Project: levine-two-player-seven-twentieths-research
Locator: Bounded continuation designed from tools/levine_three_block_best_response.py on 2026-07-28
License: CC0-1.0
Public record: R452
Stable alias: levine-attempt-extend-best-response-m8-m12
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Stress-test the all-depth response proof on blocks eight through twelve

Work and source credit

2Authored explanation

3Outcome

4What was measured

5How it connects

Uses

Tests

Recorded for

7Provenance

Record & provenance

Sign in to follow

Report a problem