Fourth-Down Threshold Methodology
Methods and sources for “Not All Field Goals Weigh the Same.” The case is Houston’s two fourth-and-one decisions against Cincinnati on September 20, 2026. Every model uses only games played before that date.
- The two decisions
- Data and information boundary
- Models
- Break-even calculation
- Model refits
- What changed the threshold
- Where each number comes from
- Business examples
- Figures
- Limitations
The two decisions
| Earlier fourth-and-one | Later fourth-and-one | |
|---|---|---|
| Time | 2nd quarter, 6:58 | 3rd quarter, 7:33 |
| Score (Houston-Cincinnati) | 0-0 | 3-10 |
| Spot | Cincinnati 18 | Cincinnati 40 |
| Choice | Went for it; David Montgomery stopped for a one-yard loss | 58-yard field goal attempt; short |
| Estimated conversion probability | 70% | 71% |
| Central break-even conversion probability | 52% (95% of refits: about 29-69%) | 28% (95% of refits: about 18-48%) |
Both plays are verified in the official gamebook and in nflverse play-by-play (game 2026_02_CIN_HOU, plays 1547 and 2846). The roof was closed. The Week 1 comparison is Fairbairn’s made 58-yard field goal against Buffalo on fourth-and-four from the Buffalo 39 with 6:48 left, which gave Houston a 31-30 lead (nflverse 2026_01_BUF_HOU, play 3724). Ryans’s explanation that Fairbairn had made one from the same spot comes from his Monday, September 21 press remarks as reported by the Houston Chronicle.
Data and information boundary
Play-by-play comes from nflverse for the 2016 through 2026 seasons, with each file’s URL, retrieval time and SHA-256 recorded in the research manifest. Models and estimates use only games dated before September 20, 2026. The outcomes of the two plays, and every later play, are not used in any decision calculation.
Fairbairn’s 2025 regular-season record from 50 yards or longer (9 of 13, all between 50 and 57 yards) and Houston’s two converted fourth-and-one runs in Week 1 were counted from the same play-by-play by analysis/d6_draft6_evidence.py.
Models
Win probability. XGBoost classifier on whether the team in possession wins, trained on regulation plays. The primary specification includes the closing point spread, decayed over game time; a second specification uses game state only. An expected-points model was also fitted as a check. Models were validated by training on seasons through 2024 and testing on 2025.
Fourth-and-one conversion. Logistic model of conversion on field position, fitted to fourth-and-one attempts from 2016 through the cutoff. Alternatives (league rates, team-strength adjustments and play-type splits) ranged from 69 to 82 percent. The highest values come from quarterback sneaks, which are a play choice rather than a property of the situation.
Field goal. League distance model for a 58-yard attempt with the roof closed: 66.7 percent. Estimates that shrink toward Fairbairn’s own record gave 67.4 to 68.9 percent.
Break-even calculation
For each option the model values every outcome as Houston’s win probability in the resulting game state: conversion or failure for going for it, make or miss for the field goal, and the expected result of a punt. The break-even conversion probability is the value p at which going for it ties the best kicking option:
p × WP(convert) + (1 - p) × WP(fail) = WP(best kick)
This is the article’s general rule, C / (V + C), with V = WP(convert) - WP(best kick) and C = WP(best kick) - WP(fail). At the Cincinnati 40 the central values were 41.7 percent after a conversion, 24.5 percent after a failure, 29.3 percent for the field goal and 29.3 percent for the punt, giving a break-even of 27.9 percent. A made kick would have left Houston at 32.5 percent and a miss at 22.9 percent.
Model refits
The win-probability model was refitted 200 times on bootstrap samples of games. At the Cincinnati 40, going for it was the best option in every refit under both win-probability specifications and the expected-points model. Assuming a certain make, going for it at the estimated conversion rate still beat the field goal in 97 percent of refits. At the Cincinnati 18, the middle 95 percent of refits put the break-even between about 29 and 69 percent, and going for it was best in 98 percent of them, by a small margin. At the Cincinnati 40 the same interval was about 18 to 48 percent, about as wide, but entirely below the 71 percent conversion estimate.
What changed the threshold
The two situations differ in score, field position and clock. Changing these one at a time in sequence gives effects that depend on the order chosen, so the reported decomposition averages over all orderings (a Shapley decomposition) and repeats it across 200 refits. The score change, from tied to trailing by seven, lowered the threshold by about 25 percentage points (95 percent refit interval -40 to -4) and was negative in every ordering. Field position and clock effects were near zero and changed sign with the ordering.
Where each number comes from
All files are in research/nfl-fourth-down-thresholds/ in the pascal-site repository. The claim ledger, docs/CLAIM_EVIDENCE_LEDGER.md, maps each claim to its source and publication status.
| Article statement | Value | Source |
|---|---|---|
| 58-yard make probability | 67% (alternatives 67-69%) | results/fg_estimates.csv |
| Win probability after a made kick | 32.5% | results/decision_values.csv |
| Win probability of going for it | 36.8% | results/decision_values.csv |
| Going beats a certain make | 97% of refits | results/validation/uncertainty_reps.csv |
| Make-versus-miss value, Buffalo and Cincinnati | 25.1 and 9.6 points | results/validation/v3_week1_contrast.csv |
| Conversion probability, Cincinnati 40 | 71% | results/conv_model.csv |
| Break-even, Cincinnati 40 | 28% (95% of refits about 18-48%) | results/decision_values.csv, results/validation/v6_interval_stability.csv |
| Break-even, Cincinnati 18 | 52% (95% of refits about 29-69%) | results/decision_values.csv, results/validation/v6_interval_stability.csv |
| Fairbairn, 2025 regular season, 50 yards or longer | 9 of 13 | results/draft6/fairbairn_2025_long_fg.csv |
| Houston fourth downs before the decision | two converted fourth-and-one runs in Week 1 | results/draft6/hou_prior_fourth_downs.csv |
| Score effect on the threshold | lowered it by about 25 points | results/validation/v6_shapley.csv |
Business examples
The merger, product-expansion and investment scenarios are hypothetical illustrations, and their figures are arithmetic, not data. The expected payoffs are 0.2 × $20,000,000 - 0.8 × $500,000 = $3,600,000 and 0.9 × $100,000 - 0.1 × $2,000,000 = -$110,000. The Facebook example refers to Peter Thiel’s initial $500,000 investment in 2004 and the 2012 share sales (about $640 million in the May IPO and $395.8 million after the August lockup expired) that took his total cash-out to just over $1 billion, as reported by Julianne Pepitone and Stacy Cowley, “Facebook’s first big investor, Peter Thiel, cashes out,” CNNMoney, August 20, 2012. That total is gross: some proceeds went to investors in the funds that held the shares. The example illustrates asymmetric payoff in hindsight and is not a probability estimate. The ABS values (0.292, 0.142 and 0.194 expected runs) come from the Pricing an ABS Challenge methodology.
Figures
All four figures are generated by scripts/insights/field-goals-figures/build.py, which reads its values from the files named here; no figure value is typed by hand. Figure 1 plots the field-goal value at each make probability from results/validation/v3_caseB_fg_sweep.csv against the going-for-it and punt values in results/decision_values.csv. Figure 2 uses results/validation/v3_week1_contrast.csv. Figure 3 computes V and C for Houston from results/decision_values.csv against the better kicking option, uses the published CT-S1 values for the 1-2 count, and computes the business threshold from the article’s hypothetical investment ($500,000 / ($20,000,000 + $500,000) = 2.4%). Figure 4 is the arithmetic of the article’s two hypothetical investments.
Limitations
The conclusions hold under these models. Win-probability values for a single game state carry real model uncertainty, which is why the refit ranges are reported. The official distance on both plays was one yard; the earlier play was described afterward as fourth-and-inches, and no measured distance is public, so its conversion estimate may be understated. The models do not see personnel, injuries or play calls. Nothing here shows what Houston’s staff believed, and no public source connects the earlier failed attempt with the later decision. League behavior in similar situations is context, not evidence of the right decision.