Does extra rest between matches help? (Not measurably)

Status: Not shipped. See decision gate at the bottom. Backtest date: 2026-05-21 Reproducer: scripts/backtest_rest_day_ablation.py Persisted output: data/wc2026/rest_day_ablation.json Reliability diagrams: web/public/research/img/rest-day-ablation-{baseline,treatment}.png

Hypothesis

Sports-science literature reports a measurable effect of recovery time on football performance: better-rested teams score slightly more goals than fatigued ones. The expected magnitude is small but consistent across studies (Mohr et al. 2012; Dellal et al. 2013; FIFA technical reports on tournament congestion).

The Dixon-Coles fit on ~49k international matches absorbs this implicitly by averaging across rest states. The question this note answers is: does a small, transparent predict-time multiplicative adjustment on the DC expected goal rate, conditioned on each team's days-since-last-match, improve the ensemble's Brier / log-loss / ECE on held-out matches?

Model form:

λ_adjusted = λ_baseline × exp(α × (rest_days − REF_REST))

with one scalar α learned from training data, the same α applied to both sides per match, and a defensive clamp [0.7, 1.3] on the multiplicative factor.

This mirrors the multiplicative-adjustment pattern already used by scripts/weather_adjustment.py for venue altitude and heat.

Backtest setup

Fit:

• Baseline λ_h, λ_a from a Dixon-Coles fit on the training window (scripts/fit_dixon_coles.py defaults, no rest term).
• α estimated by maximising the Poisson log-likelihood of the observed goals (home + away pooled — 85,646 per-side observations) under g ~ Poisson(λ_baseline × exp(α × (rest − REF))).
• Each arm of the test (baseline; treatment) re-fits its own per-class isotonic calibrator on training-window ensemble outputs, so calibration drift caused by the multiplier is fairly counted in.

Result

Estimated α

The sign is positive and the effect is statistically very precise. A team that has had 2 extra days of rest beyond the population median (i.e. 7 days vs 5) gets a multiplicative bump of exp(0.00381 × 2) ≈ 1.0076 — about three-quarters of a percent more goals expected. At the tail (cap of 30 days, e.g. a team's first match in months), the bump is ≈ 1.103.

Test-set metrics on the common 2,904-match subset

Calibrated-vs-calibrated deltas (the production comparison):

• Δ Brier: −0.00016 (treatment 0.00016 better)
• Δ log-loss: −0.00035 (treatment fractionally better)
• Δ ECE: +0.27pp (treatment 0.27pp worse)

Reliability diagrams

Both arms track the y=x line at roughly the same distance; the treatment arm's slight ECE regression is visible at the higher-probability bins.

Sensitivity of α to choice of REF

The MLE fits a single scalar α with λ_baseline (from DC) held fixed — there's no jointly-fitted intercept. That makes α weakly dependent on the reference rest, since shifting REF changes the constant-offset portion of the multiplier that α has to absorb. Sensitivity check on the training data (DC baseline λ):

In every case the implied multipliers are small and the test-set verdict (Brier and ECE deltas below the ship threshold) is the same. The headline α reported in this note uses REF=5 because that matches the typical within-FIFA-window rest the adjustment is meant to model; a jointly-fitted intercept would resolve the REF dependence but adds a parameter that the DC baseline already covers.

Why a statistically-significant α doesn't translate into Brier lift

Two reasons, both substantive:

1. The DC fit has already absorbed it. The DC parameters are fitted on the same pool of matches whose rest distribution they're being asked to explain post-hoc. Any systematic rest effect that's confounded with team identity (some federations schedule less aggressively, some always rest more before tournaments) is already in the per-team α_t / β_t ratings. The marginal rest signal that survives is the within-team variation, which is small.
2. Most matches sit near the reference rest. A 5-day reference and a real-world distribution heavily concentrated between 3 and 8 days means the multiplicative factor for the median match is exp(0.00381 × (rest − 5)) ≈ 1.00 ± a few tenths of a percent. Even at the tails the factor is bounded enough that the per-fixture P(H/D/A) shifts by a fraction of a percentage point.

Both of these are intelligible failure modes — they imply the underlying physical effect is real (the α-fit confirms it) but that the predictive lift is below the noise floor on the held-out window we have.

Decision

Gate (defined in advance):

Verdict: do not ship. The effect is real and detectable in the goal-likelihood, but the per-fixture P(H/D/A) shifts it produces don't materially move Brier or log-loss on the test set, and they cost a small amount of calibration. Adding a predict-time multiplier comes with maintenance and failure-mode cost (clamps, missing-rest handling, the isotonic re-fit dependency); the value side has to clearly clear that bar, and it doesn't.

What this backtest can't tell you

• The α magnitude could be larger for fitness-sensitive subgroups. A pooled fit on all 86k per-side observations averages across friendlies (low intensity, less rest sensitivity) and major-tournament knockouts (high intensity, more rest sensitivity). A subgroup model fit only on tournament matches might find a larger α and could move the needle in the production WC-only forecast. Out of scope for this v0 ablation.
• The 3,040-match test window includes a lot of friendlies. The metrics published here weight every match equally. A tournament-only test slice has ~80 matches per WC cycle and very different Brier dynamics; the small Brier improvement we did see (−0.00016) could be larger or smaller on that slice. The current test set is too narrow to evaluate that.
• Calibration matters more for the published-probabilities use case than Brier. The +0.27pp ECE regression is what really blocks the decision. A calibrator re-fit on a larger window (e.g. last 3 years instead of last year) might absorb the rest-adjusted distribution shift; we haven't explored that.
• The model form is the simplest possible. A single-scalar α applied uniformly is a strong constraint. A spline on rest_days, a separate α for home / away, an interaction with the team's strength rating, a discontinuity around 2-day "no rest" minimums — none of these have been tried. The negative result here is for the v0 form only.

Files touched

• scripts/backtest_rest_day_ablation.py — new
• tests/test_backtest_rest_day_ablation.py — new (12 tests, all passing)
• data/wc2026/rest_day_ablation.json — persisted output (gitignored)
• web/public/research/img/rest-day-ablation-baseline.png — reliability diagram (baseline arm)
• web/public/research/img/rest-day-ablation-treatment.png — reliability diagram (treatment arm)
• documentation/research-notes/rest-day-ablation.md — this file

scripts/rest_adjustment.py was not added. The ensemble's predict path is unchanged.

Reproducing

.venv/bin/python scripts/backtest_rest_day_ablation.py

(requires data/raw/intl/results.csv from python scripts/pull_intl_results.py and the existing DC + HP + Elo fit infrastructure.)