Monte Carlo Bracket Skill

WC 2026 format (do NOT use the old 32-team bracket)

• 48 teams, 12 groups (A–L) of 4.
• Group stage: 4-team round-robin, 3/1/0 points. Equal-points cohorts use available FIFA fields:

head-to-head points → head-to-head GD → head-to-head goals → overall GD → overall GF. Missing conduct/latest-ranking data falls back visibly to model rating, then RNG only for equal ratings.

• Advancement: top 2 of each group (24) + 8 best third-placed teams = 32 → Round of 32.
• Knockout: R32 → R16 → QF → SF → Final (single elimination).
• Encoded in football/data/wc2026.json, elo_seed.json, and wc2026_bracket.json (official

R32 template, fixed tree, and all 495 best-third group combinations from Annex C).

Match engine (poisson_xg.py)

• lambdas_from_elo(elo_home, elo_away, home_advantage): supremacy = (elo_home + adv - elo_away) * 0.004; split around avg_total_goals=2.6; clamp to >= 0.05.
• scoreline_matrix / outcome_probabilities: independent Poisson grid (truncated at 10 goals); tril = home win, diagonal = draw, triu = away win.
• Home advantage at the World Cup is 0 (neutral venues) except for the hosts if you choose to model it.

Elo (elo.py)

• expected_score(Ra, Rb, H) = 1 / (1 + 10^(-((Ra+H)-Rb)/400)). Used for the knockout shootout coin flip on a drawn tie.

Sims

• group_sim.simulate_group_once(rng, teams, ratings) -> ranked standings (composed by the bracket sim).
• group_sim.simulate_group_stage(...) -> all 12 groups, contextual best thirds, per-team

p_auto_advance, p_best_third_advance, combined p_advance, and fallback counts.

• bracket_sim.simulate_tournament(groups, ratings, n_iter, seed) -> per-team {reach_r32..win} + champion.

Fixture conditioning preserves provider match_id, stage, and explicit winner. Deduplicate by ID, otherwise stage class + pairing; never let a same-pair knockout overwrite its group result. Level knockout scores lock only with an explicit penalty winner.

Invariants to preserve (tested)

• Automatic-advance mass over each group's 4 teams == 2.
• Best-third advance mass across all 12 groups == 8.
• reach_r32 mass across all teams == 32 (exactly 32 qualify per iteration).
• win mass == 1 (one champion per iteration).
• Round probabilities monotone (reaching a later round implies reaching earlier ones).
• Champion title-probability stable within ±2% across seeds at ~10k iterations.
• Always pass a seed for reproducible output; np.random.default_rng(seed).

Official allocation and fallback caveat

Do not strength-seed or rebuild a generic 1-vs-N bracket. _build_r32 must use the committed official template and 495-row allocation. Conduct/latest-ranking inputs remain unavailable; keep the model-rating fallback counted and exposed rather than silently calling it an official field. Head-to-head criteria are currently applied once to the equal-points cohort rather than reapplied iteratively to a shrinking tied subset; retain this explicit modeling approximation.