ChatGPT analysis

Yep — 4 / 58 / 36 / 1 was too aggressive, mainly because it implicitly assumed “the champ almost always plays at least one West team” and it underweighted the big structural blocker:

• If the NFC champ is a West team (~60% by your NFC-title market among the main contenders), then X=3 is impossible, and X=0 becomes pretty common (because the other West teams can get knocked out before the champ sees them).

Below is my most accurate model using only the info you gave, and I cross-validated it with a few different win-prob methods.

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What I modeled (using your inputs)

Target: X= “How many NFC West teams the NFC champion beats.”

Playoff mechanics: 7-team NFC bracket + reseeding.

Seeds:

• West teams (SEA/LAR/SF) sampled to match your seed marginals (unique-seed joint built by “product of marginals + rejection,” i.e., maximum-entropy consistent with your inputs).

• Remaining seeds filled by the main non-West contenders you showed (CHI/PHI/GB/TB).

Win probability model(s):

I ran 4 versions, then blended them.

All of them include home-field = 61.8% when teams are equal (your request), and some incorporate your “to make divisional / to make conference champ / to win NFC / #1 seed” markets as team-strength signals.

Final mixing: I compute P(X=k\mid \text{champ}=t) from sims, then mix using your NFC title market weights (so champion identity matches your market).

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Cross-validation results (distribution of X)

Model outputs

(Percent)

Model X=0 X=1 X=2 X=3

HCA-only (p(home)=0.618) 23.7 49.4 24.4 2.5

Composite strength (Div+CC+NFC+#1) 19.3 51.0 27.0 2.7

Title-odds strength (NFC market only) 18.4 53.2 26.2 2.2

GPF table strength (your rankings) 21.6 49.0 26.6 2.8

Best single-number estimate (blended)

Taking a simple average across the 4:

✅ Final (ensemble):

• X=0: 20.7%

• X=1: 50.6%

• X=2: 26.0%

• X=3: 2.6%

• EV: ~1.10 West teams beaten

So yes: the “mass on 1–2” is real, but “0” is not rare.

Your prior 9% on “3” is the piece that’s hardest to support structurally.

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The key structural explanation (why “3” is small)

Using your NFC champ market (among these teams), West wins the NFC ~59.8%.

When the champ is West, X=3 is impossible, and “0” gets a lot of weight.

From the composite model (representative):

If champ is NFC West (SEA/LAR/SF)

• X=0: 29%

• X=1: 61%

• X=2: 10%

• X=3: 0%

If champ is NOT NFC West (CHI/PHI/GB/TB)

• X=0: 4%

• X=1: 36%

• X=2: 53%

• X=3: 7%

So “3” only comes from non-West champs, and even then it requires a very specific path: the champ must face West teams in WC + DIV + CC, which is hard with reseeding and because West teams often eliminate each other before the champ can.

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What would move your distribution toward your prior (especially X=3)?

One of these has to be true:

1. Non-West champs are much more likely than your market suggests, or

2. West teams are frequently seeded in a way that lets a non-West champ face them in all three rounds (West producing a WC opponent and a DIV opponent and a CC opponent without West-vs-West collisions), or

3. West has 4 playoff teams (not your current setup).

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If you want the next refinement that actually matters: I’d add a better joint seed model (correlations: e.g., if SEA is #1, what does that imply about LAR/SF being 5/6/7?) and then rerun the same framework. That’s the lever most likely to change X=0 vs X=1 (and slightly X=3).