{
  "schemaVersion": 1,
  "generatedOn": "2026-09-04",
  "generatedBy": "scripts/generate-pko-bounty-value-data.mjs",
  "title": "PKO bounty branch-value threshold evidence bundle",
  "scope": "A transparent, synthetic two-outcome threshold model for off-table study. It does not prescribe a range or supply tournament-state values.",
  "formula": {
    "callExpectedValue": "EV(call) = q * (W + B + H) + (1 - q) * L",
    "breakEvenProbability": "q* = (F - L) / (W + B + H - L)",
    "decisionWhenDenominatorPositive": "Call is not worse than fold when q >= q*.",
    "variables": {
      "q": "Probability of the exact modeled win-and-eliminate branch, from 0 through 1.",
      "F": "Complete continuation value after folding, including all modeled future prize payouts, existing-bounty value, and later-knockout paths.",
      "W": "Complete continuation value after calling, winning, and eliminating, including all modeled future prize payouts, existing-bounty value, and later-knockout paths; already net of the call cost and excluding only current immediate B and any explicitly isolated incremental transfer H.",
      "B": "Immediate bounty value actually paid in the modeled operator format; never the opponent’s displayed total bounty by default.",
      "H": "Separately modeled incremental value assigned to future effects of the progressive amount added to the caller’s own head; it may be positive, zero, or negative, and is neither face value nor automatic cash. If W already includes this incremental transfer, H must be zero.",
      "L": "Complete continuation value after the modeled call-and-lose branch, including all modeled future prize payouts, existing-bounty value, and later-knockout paths; already net of the call cost."
    }
  },
  "unitRule": "F, W, B, H, and L must use one declared value unit. Do not add cash, chips, prize-pool equity, or ICM outputs until they have been converted to the same value basis.",
  "workedCase": {
    "id": "worked_case",
    "label": "Worked synthetic case",
    "unit": "synthetic value units",
    "probabilityWinAndEliminate": 0.6,
    "foldValue": 120,
    "winContinuationValue": 165,
    "immediateBountyValue": 50,
    "addedHeadValue": 0,
    "loseContinuationValue": 0,
    "note": "All branch values use one synthetic unit. H uses a zero incremental-transfer baseline; W still includes every other modeled future pathway.",
    "winBranchValue": 215,
    "callExpectedValue": 129,
    "callMinusFold": 9,
    "decision": "call_higher_value",
    "threshold": {
      "numerator": 120,
      "denominator": 215,
      "threshold": 0.5581395348837209,
      "thresholdRounded12": 0.558139534884,
      "exactFraction": "24/43",
      "inequality": "q >= threshold",
      "feasibility": "bounded",
      "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
    }
  },
  "branchLedger": [
    {
      "row": 1,
      "rowType": "benchmark",
      "isSubtotal": false,
      "choice": "fold",
      "outcome": "Fold benchmark",
      "probability": 1,
      "branchValue": 120,
      "probabilityWeightedValue": 120,
      "contributesToCallEv": false
    },
    {
      "row": 2,
      "rowType": "outcome",
      "isSubtotal": false,
      "choice": "call",
      "outcome": "Win and eliminate",
      "probability": 0.6,
      "branchValue": 215,
      "probabilityWeightedValue": 129,
      "contributesToCallEv": true
    },
    {
      "row": 3,
      "rowType": "outcome",
      "isSubtotal": false,
      "choice": "call",
      "outcome": "Lose",
      "probability": 0.4,
      "branchValue": 0,
      "probabilityWeightedValue": 0,
      "contributesToCallEv": true
    },
    {
      "row": 4,
      "rowType": "subtotal",
      "isSubtotal": true,
      "choice": "call",
      "outcome": "Blended call EV",
      "probability": null,
      "branchValue": null,
      "probabilityWeightedValue": 129,
      "contributesToCallEv": false
    }
  ],
  "branchLedgerRule": "Only rows with rowType \"outcome\" and contributesToCallEv true are additive. The blended call-EV row is a non-additive subtotal; the fold row is a separate benchmark.",
  "sensitivity": {
    "description": "Only the synthetic immediate bounty value B changes; F=120, W=165, H=0, and L=0 stay fixed.",
    "rows": [
      {
        "immediateBountyValue": 0,
        "addedHeadValue": 0,
        "winBranchValue": 165,
        "breakEvenProbability": 0.7272727272727273,
        "breakEvenFraction": "8/11",
        "breakEvenPercent": 72.727272727,
        "unit": "synthetic value units",
        "synthetic": true
      },
      {
        "immediateBountyValue": 25,
        "addedHeadValue": 0,
        "winBranchValue": 190,
        "breakEvenProbability": 0.631578947368421,
        "breakEvenFraction": "12/19",
        "breakEvenPercent": 63.157894737,
        "unit": "synthetic value units",
        "synthetic": true
      },
      {
        "immediateBountyValue": 50,
        "addedHeadValue": 0,
        "winBranchValue": 215,
        "breakEvenProbability": 0.5581395348837209,
        "breakEvenFraction": "24/43",
        "breakEvenPercent": 55.813953488,
        "unit": "synthetic value units",
        "synthetic": true
      },
      {
        "immediateBountyValue": 75,
        "addedHeadValue": 0,
        "winBranchValue": 240,
        "breakEvenProbability": 0.5,
        "breakEvenFraction": "1/2",
        "breakEvenPercent": 50,
        "unit": "synthetic value units",
        "synthetic": true
      }
    ]
  },
  "fixtures": [
    {
      "id": "worked_case",
      "label": "Worked synthetic case",
      "unit": "synthetic value units",
      "probabilityWinAndEliminate": 0.6,
      "foldValue": 120,
      "winContinuationValue": 165,
      "immediateBountyValue": 50,
      "addedHeadValue": 0,
      "loseContinuationValue": 0,
      "note": "All branch values use one synthetic unit. H uses a zero incremental-transfer baseline; W still includes every other modeled future pathway.",
      "winBranchValue": 215,
      "callExpectedValue": 129,
      "callMinusFold": 9,
      "decision": "call_higher_value",
      "threshold": {
        "numerator": 120,
        "denominator": 215,
        "threshold": 0.5581395348837209,
        "thresholdRounded12": 0.558139534884,
        "exactFraction": "24/43",
        "inequality": "q >= threshold",
        "feasibility": "bounded",
        "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
      }
    },
    {
      "id": "exact_break_even",
      "label": "Exact threshold check",
      "unit": "synthetic value units",
      "probabilityWinAndEliminate": 0.5581395348837209,
      "foldValue": 120,
      "winContinuationValue": 165,
      "immediateBountyValue": 50,
      "addedHeadValue": 0,
      "loseContinuationValue": 0,
      "note": "q is set to the exact H=0 threshold.",
      "winBranchValue": 215,
      "callExpectedValue": 120,
      "callMinusFold": 0,
      "decision": "tie",
      "threshold": {
        "numerator": 120,
        "denominator": 215,
        "threshold": 0.5581395348837209,
        "thresholdRounded12": 0.558139534884,
        "exactFraction": "24/43",
        "inequality": "q >= threshold",
        "feasibility": "bounded",
        "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
      }
    },
    {
      "id": "no_immediate_bounty",
      "label": "Remove immediate bounty",
      "unit": "synthetic value units",
      "probabilityWinAndEliminate": 0.6,
      "foldValue": 120,
      "winContinuationValue": 165,
      "immediateBountyValue": 0,
      "addedHeadValue": 0,
      "loseContinuationValue": 0,
      "note": "Same branch values and q; only B changes from 50 to zero.",
      "winBranchValue": 165,
      "callExpectedValue": 99,
      "callMinusFold": -21,
      "decision": "fold_higher_value",
      "threshold": {
        "numerator": 120,
        "denominator": 165,
        "threshold": 0.7272727272727273,
        "thresholdRounded12": 0.727272727273,
        "exactFraction": "8/11",
        "inequality": "q >= threshold",
        "feasibility": "bounded",
        "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
      }
    },
    {
      "id": "nonzero_lose_branch",
      "label": "Nonzero lose continuation",
      "unit": "synthetic value units",
      "probabilityWinAndEliminate": 0.45,
      "foldValue": 100,
      "winContinuationValue": 170,
      "immediateBountyValue": 30,
      "addedHeadValue": 0,
      "loseContinuationValue": 25,
      "note": "Synthetic side-value illustration; L need not be zero if the modeled losing state retains value.",
      "winBranchValue": 200,
      "callExpectedValue": 103.75,
      "callMinusFold": 3.75,
      "decision": "call_higher_value",
      "threshold": {
        "numerator": 75,
        "denominator": 175,
        "threshold": 0.42857142857142855,
        "thresholdRounded12": 0.428571428571,
        "exactFraction": "3/7",
        "inequality": "q >= threshold",
        "feasibility": "bounded",
        "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
      }
    },
    {
      "id": "explicit_head_sensitivity",
      "label": "Separately modeled H sensitivity",
      "unit": "synthetic value units",
      "probabilityWinAndEliminate": 0.55,
      "foldValue": 120,
      "winContinuationValue": 165,
      "immediateBountyValue": 50,
      "addedHeadValue": 8,
      "loseContinuationValue": 0,
      "note": "H=8 is an explicit synthetic sensitivity assumption, not the displayed added bounty face value.",
      "winBranchValue": 223,
      "callExpectedValue": 122.65,
      "callMinusFold": 2.6500000000000057,
      "decision": "call_higher_value",
      "threshold": {
        "numerator": 120,
        "denominator": 223,
        "threshold": 0.5381165919282511,
        "thresholdRounded12": 0.538116591928,
        "exactFraction": "120/223",
        "inequality": "q >= threshold",
        "feasibility": "bounded",
        "explanation": "The win-and-eliminate branch is worth more than the lose branch, so higher q raises call EV."
      }
    }
  ],
  "operatorBoundary": "PokerStars documents common PKO splits and explicit exceptions. Enter the cash paid now under the actual tournament rules; do not assume every operator or event uses a 50/50 split.",
  "sources": [
    {
      "name": "PokerStars: Poker Tournament Types — Progressive Knockouts",
      "url": "https://www.pokerstars.com/poker/tournaments/types/",
      "accessed": "2026-09-04",
      "use": "Official operator-specific description of common PKO prize/bounty allocation, the usual progressive split, and named exceptions."
    },
    {
      "name": "PokerStars Help: What are Knockout tournaments?",
      "url": "https://www.pokerstars.com/help/articles/trn-knockout/213397/",
      "accessed": "2026-09-04",
      "use": "Official operator-specific description of immediate versus added-on-head bounty treatment and payment of the winner’s own bounty."
    },
    {
      "name": "George T. Gilbert: The Independent Chip Model and Risk Aversion",
      "url": "https://arxiv.org/abs/0911.3100",
      "accessed": "2026-09-04",
      "use": "Primary mathematical source showing that tournament expected value under ICM is state-dependent and a fair chip bet need not preserve payout equity."
    }
  ],
  "boundaries": [
    "Every published number and row is synthetic; the bundle contains no hand history, player observation, operator lobby scrape, or solver output.",
    "q is not automatically raw showdown equity. Ties, multiway pots, side pots, failure to cover the opponent, or winning without an elimination require more branches.",
    "F, W, and L are complete continuation-state values that include all modeled future prize payouts, existing-bounty value, and later-knockout paths. W and L are already net of the call cost; adding the wager again double-counts it.",
    "B is only the immediate value actually received on this knockout under the event’s rules. The displayed bounty may include a progressive portion that is not paid now.",
    "H uses a zero incremental-transfer baseline. A declared future-state model may instead produce a positive or negative finite H; W still includes every other modeled future prize payout, existing-bounty value, and later-knockout path. H must not be set equal to the amount added to one’s head merely because that number is visible, and if W already includes this incremental transfer, H must be zero to prevent double-counting.",
    "The threshold is descriptive algebra, not a range recommendation. The model does not estimate q, F, W, L, future bounty access, opponent strategy, or ICM values.",
    "When the threshold denominator W+B+H-L is zero there is no unique q threshold. If it is negative, the threshold inequality reverses and the branch labels or valuations deserve scrutiny."
  ]
}