World Cup 2026 — methodology ("World vs Model")

⚠️ A research & education experiment — NOT gambling and NOT an encouragement to gamble. We hold no positions and invest no real capital. Every figure is a paper simulation of market structure, not investment returns, advice, or a solicitation. Prediction-market platforms like Polymarket are restricted or banned in some jurisdictions (e.g. Singapore) — know your local laws.

1. What we are trying to do

Beat the market at the World Cup using zero football knowledge.

We never forecast a match, a player, or a team. We take the market's own prices and ask a narrower, more honest question: are those prices internally consistent and unbiased — and if not, can we harvest the inconsistency? The entire edge, if any, comes from the structure of the prices, not from knowing anything about football.

Then we keep score in public. Because these markets resolve, every call is timestamped and graded out-of-sample against what actually happened. The point isn't to look clever — it's to find out, on the record, whether a purely structural strategy can beat a liquid crowd.

2. Why bounded (prediction) markets are interesting for this

A "bounded" market is one that settles to a known value — here, 0 or 1 (a team advances, or it doesn't). That boundedness is exactly what makes zero-knowledge strategies viable, in four ways an ordinary stock doesn't offer:

Ground truth arrives. A stock never tells you its "true value." A prediction market resolves — so a forecast is falsifiable and skill is measurable, not just assertable. This is what lets us run an honest scorecard instead of a vibe.
Hard mathematical constraints exist for free. Prices are probabilities. Sub-events nest (winning ⊆ reaching the final ⊆ reaching the semi …). Mutually-exclusive outcomes must sum to the number of slots. These create no-arbitrage relationships that need no domain model — pure logic flags a mispricing.
There are well-documented behavioral biases. The favorite–longshot bias (longshots systematically overpriced, favorites underpriced) is one of the oldest and most-replicated anomalies in betting markets — horse racing, sports, elections, for decades. It's a shape in the price distribution we can correct without any view on a specific team.
Breadth. One tournament is hundreds of correlated markets (advance / QF / SF / final / win, per team). That breadth lets us build a market-neutral long/short book instead of a single winner pick — and breadth is the cleanest lever there is (Grinold's law: information ratio ≈ skill × √breadth).

3. The market surface: a nested ladder

Our single market data source is Polymarket. We read its live, real-money World Cup prices through the public Gamma API — no private feeds, no broker. Those prices are the "world" / market side of every comparison on the board; the crowd's money is our benchmark, and we link straight out to the underlying Polymarket market on every team. (We are not affiliated with Polymarket, and — see the banner — it is restricted in some jurisdictions; the links are reference-only.) Because everything keys off one transparent, reproducible price source, there is no separate "data source" tab to maintain: it is Polymarket, end to end.

Polymarket lists the World Cup as several separately-traded events. Per team they form a nested ladder of "how far do you go," each level with a known number of slots:

Round	Slots (prices must sum to)
Advance to knockouts	32
Reach quarter-final	8
Reach semi-final	4
Reach final	2
Win the Cup	1

≈240 markets vs 48 winner-only — 5× the breadth, and where the cleanest structural tells live.

4. Why we think there's an edge — and why it could work

Three independent structural sources, none of which require football knowledge:

Favorite–longshot shape correction (the main edge). We de-vig each level to its slot count, then apply a power-law shape correction p → pᵖᵒʷᵉʳ (renormalized). On a planted-bias test this beats the de-vigged market by ~0.7% Brier. It's small — but it's robust and directional: it buys favorites and fades longshots, the historically profitable side of the bias. Why it could work: the bias is behavioral (longshot lottery-love + margin structure) and has persisted for decades; we exploit the distribution shape, which is far more stable than any single price.
No-arbitrage logic (riskless when present). If a team is priced higher to reach the final than to reach the semi-final, that's impossible — buy the cheap leg, sell the dear one. The nested ladder and the slot-sum constraint give us these checks for free.
Breadth + market-neutrality. We go long the most under-priced and short the most over-priced across all 240 markets, dollar-neutral. Why it could work even if we whiff on the champion: the book doesn't need the winner right — it needs the cross-section to be mispriced on average. A long/short book monetizes relative mispricing, so gross-of-cost alpha can survive being wrong on any single market.

We measure all of this against the market's own price as the baseline — skill-vs-market, the only comparison that matters. Beating a coin flip is meaningless; beating the de-vigged crowd is the whole game.

5. Sizing & the four books (2 models × 2 styles)

A fixed $1,000 paper bankroll per book, conviction-weighted (stake ∝ |edge|, capped per trade), dollar-neutral, net of the half-spread. Each ticket holds signed YES-shares: unrealized = shares × (price − entry), realized = shares × (outcome − entry). Every book is frozen at day 0 into the ledger (ledger/wc_*.jsonl) and marked to live, so each shows real running Entry → Now → PnL — not a re-sized proposal.

Two trading styles, run identically for both models so the comparison is symmetric:

Buy & Hold (wc-core) — entered once at day 0, every ticket held to its market's resolution. The clean test of "was the pre-tournament view right?" PnL settles in waves (advance → QF → SF → final → win).
Active Trading (wc-live) — rebalanced each matchday: settle what resolved, close the rest at the current price, redeploy the compounding capital into the freshest edges. The test of "does acting on the model's updates beat just holding?" Its matchday changelog is the timeline.

Crossed with the two models that produce the edges — the zero-knowledge structural model and the informed Elo model (§5b) — that is four books, each its own tab with its own PnL: Buy & Hold, Active, Elo · Buy & Hold, Elo · Active. Comparing across the grid is itself the finding: if Active doesn't beat Buy & Hold net of churn the daily updates are noise; if the Elo books don't beat the zero-knowledge ones, football knowledge didn't help — and we'd report either honestly.

5b. The fundamental model (Elo) — an independent second opinion

The structural book above is derived from the market, so by construction it can only disagree with the crowd a little. To get a genuinely independent view we run the engine's own simulation — Elo + Poisson goals over the verified 2026 bracket — on real per-team World Football Elo ratings (eloratings.net, dated, reproducible), not seeded from Polymarket.

Four disclosed choices make it honest (all priors from football, not tuned to the market):

Real ratings, swappable and dated — so anyone can reproduce or update them.
A host bonus (+60 Elo) for the three 2026 co-hosts (USA, Canada, Mexico) — home advantage is worth roughly this much in recent international football, and without it the model absurdly faded the USA to advance at home.
A knockout-only shrink (×0.6): raw match-Elo over-concentrates the favorite when compounded through the knockout (the top team's title prob balloons to ~30%). A single-elimination bracket is far higher-variance than the match-Elo implies, so we flatten the spread for the knockout only; the group round-robin keeps (near-)raw Elo.
Rating uncertainty (±70 Elo per simulation): an Elo number is a point estimate of an uncertain true strength, so every run jitters each rating. This integrates over that uncertainty, so the model stops printing false-precision 0%/100% for minnows and giants. The favorite lands at a plausible ~16%. (We checked: a group-stage shrink does not explain the model's advance-level disagreements — those are real.)

What it finds (live): a clean results-vs-reputation split — the Elo model rates Spain, Argentina and the in-form South-Americans (Colombia, Ecuador, Uruguay) higher than the market, and fades the reputation teams (France, England, Portugal, Germany) whose results have lagged their squad value. That's a real, defensible disagreement.

How the Elo model sizes and places its bets. The Elo book is sized exactly like the structural book — conviction-weighted and dollar-neutral: long the markets the model thinks are under-priced, short the over-priced, each side getting half a fixed $1,000 paper bankroll, stake ∝ |edge| and capped per trade so no single bet dominates. The crucial difference is where the edge comes from:

Edge = Elo-model probability − de-vigged market probability — both are proper probabilities (each round sums to its slots), so the bookmaker's margin doesn't contaminate the edge.
Price you'd pay = the raw market price (with vig), used as the entry/cost and for max ↑/↓.
It is a separate paper book (its own colour — violet — on the board, so you can tell at a glance which model the column/tab is showing), scored by the same out-of-sample scorecard.

Which of its bets to trust: the win-level disagreements (results-vs-reputation) are the defensible part; the advance-level bets are larger but less reliable — the market usually knows a team's specific group context better than a single backward-looking Elo number.

The honest caveat, stated on the board itself: a model built on public ratings is usually less sharp than a liquid, heavily-traded market — so a big gap is more likely the model being cruder than the crowd being wrong. Big disagreement ≠ edge. Both books are scored by the same live scorecard; that's what adjudicates. (src/worldcup_fundamental.py.)

The most-likely outcome map. The same 20k simulations also drive a projection on the board: the projected group stage (each team's most-likely finishing rank + advance %, from the group finishing-position distribution) and a knockout pyramid (the most-likely Quarter-finalists → Semi-finalists → Finalists → Champion, read off the reach-round probabilities). These are the model's probabilities, shown in its violet colour — a picture of what the informed model expects, not a market consensus.

6. What's missing, and how we'd improve it (read this part)

We are deliberately loud about the limitations — the credibility is in the caveats.

What's missing / weak today

Costs & capacity. Polymarket spreads are wide and depth is thin; our capacity.py analysis puts the break-even capacity near \$0 — the spread alone can exceed the gross daily edge. We now net a flat ~1¢ half-spread off every book edge (sub-spread gaps are sized to zero), so the paper books are no longer purely gross. This is still a flat approximation — real per-market spreads vary and a full slippage/impact model isn't wired in — but it removes the worst of the gross-edge illusion.
The model's loud "edges" can be its blind spots. A big model-vs-market gap is frequently the model missing something the crowd prices, not alpha. We've closed two of the worst: the 2026 co-hosts now get a +60 Elo home bump (the model no longer fades the USA to advance at home), and rating uncertainty (±70 Elo per sim) stops the false-precision 0%/100% extremes for minnows and giants. Advance-level bets are still excluded from the books for the same reason.
One tournament = tiny sample. Single-event variance is enormous; one World Cup cannot validate a strategy. Confidence is LOW; the value is the scorecard accruing over many events.
The favorite–longshot knob is illustrative. The power exponent is a single global constant, not fit from data. It should be learned from resolved outcomes (recalibration) and may differ by round and by time-to-resolution.
No independent information. By design we use only the market's own prices. A durable edge vs a sharp market may require some orthogonal signal; we keep our fundamental Elo/Poisson model strictly separate to preserve the zero-knowledge purity, which caps the ceiling.
Risk model is crude. "Dollar-neutral" is not risk-neutral — the nested markets are highly correlated, so the book's true risk and the "max ↑/↓" envelope are loose bounds.
Breadth is largely an illusion. The Elo book's edges almost all express one view — Elo's results-based rating vs the market's reputation pricing (fade France/England/Portugal/Germany, back Spain/Colombia/Ecuador/Uruguay). They are highly correlated, so the effective number of independent bets is close to one, and Grinold's IR ≈ IC·√K (which assumes K independent signals) would badly overstate the information ratio. We present the rows as one correlated position, not as breadth.
Data completeness & timing. Partial fetches distort the de-vig; "reach-QF"-type markets resolve progressively while we treat them statically; some levels (reach-final) carry huge vig (live overrounds run advance ≈ 0%, reach-SF ≈ +17%, reach-final ≈ +30%), so any "edge" on the thin middle rungs is the least real. (The Elo forecast itself now re-forecasts live — see below.)
The bracket is a balanced approximation, not the official draw. We seed winners apart and avoid same-group R32 rematches, but we do not reproduce the exact 2026 slot table (the best-third placement contingencies), so a team's precise path difficulty is approximate.
The goals model has no team-specific attack/defence. It is a Dixon–Coles-corrected Poisson (a realistic ~27% draw rate), but scoring rates come only from the Elo gap — not from a team's actual goals-for/against, so a heavy-scoring or defensive side isn't individually captured.
The Elo model's advance-level edges are real but unreliable. They are not a shrink artifact (we verified a group shrink barely moves them) — raw Elo genuinely rates some teams high within their group (e.g. Panama's Elo is 2nd in Group L), while the market strongly disagrees. Those are legitimate model disagreements, but the market is usually the sharper one there, so the win-level disagreements (results-vs-reputation) are the more defensible story.

How we'd improve it

Fit the bias from settled outcomes (bounded.py::fit_recalibration) per round, and weight by time-to-resolution (IC rises as τ→0).
Per-market, dynamic costs — we now net a flat half-spread; next is real per-market spread + depth/impact and capacity gating, so sizing respects each market's true liquidity.
Live re-forecasting is wired — played results Elo-update the ratings and completed group games are held fixed, so the Elo forecast moves with the tournament (it reads ledger/wc_results.json; pre-tournament, with no results, it is the day-0 forecast). The Dixon–Coles goals correction is also in. The remaining bracket gap is the exact official slot table (incl. best-third contingencies); we currently use a balanced seeded approximation.
Stack more decorrelated zero-knowledge signals (calibration, no-arb, round-clustering) to lift breadth (IR ≈ IC·√K).
Covariance-aware risk (RMT-denoised or hierarchical risk parity) instead of naive dollar- neutral, so sizing respects the nested correlations.
The independent fundamental model is now built (src/worldcup_fundamental.py, §5b) — shown as a separate, clearly-labelled section so readers see structure-only vs Elo side by side. The shrink is now knockout-only (the group stage trusts raw Elo). Next for it: real form/injury adjustments to the ratings (so its advance-level disagreements are better-founded).

7. Honesty guardrails

Confidence is LOW for any single tournament. Small, unproven edges; huge single-event variance; ~zero realistic capacity. The deliverable is a falsifiable record, not a return.
Skill is scored vs the market, with the market price captured as the baseline.
No domain knowledge is injected anywhere. A team-specific judgement would be a bug.

The math — formulas & techniques

A compact, reproducible reference. Prices are probabilities $p \in (0,1)$; each round has $S$ slots (advance 32, QF 8, SF 4, final 2, win 1). Everything is plain arithmetic over a handful of disclosed priors — none fit to the market: the favorite–longshot $\alpha$, the Elo shrinks $\kappa$, the host bonus, the rating-uncertainty $\sigma$, the Dixon–Coles $\rho$, and the cost half-spread $c$.

De-vig & overround. Raw prices sum to more than the slots (the bookmaker's margin); we rescale each level proportionally to its slot count:

$$q_i = \frac{p_i}{\sum_j p_j}\,S, \qquad \text{overround} = \frac{\sum_j p_j}{S} - 1 .$$

Zero-knowledge model — favorite–longshot correction. A power transform of the shape, renormalised back to the slots, with $\alpha = 1.15$:

$$\text{model}_i = \frac{p_i^{\alpha}}{\sum_j p_j^{\alpha}}\,S, \qquad \text{edge}_i = \text{model}_i - q_i .$$

$\alpha > 1$ lifts favorites and trims longshots — the historically profitable direction of the bias.

No-arbitrage (nested). For each team the deeper rounds can't exceed the shallower ones:

$$P(\text{win}) \le P(\text{final}) \le P(\text{SF}) \le P(\text{QF}) \le P(\text{advance}) .$$

Any $P(\text{deeper}) > P(\text{shallower}) + \text{tol}$ is a riskless mispricing.

Cost (half-spread) netting. Every edge shown in a book is net of a flat half-spread $c = 0.01$ (≈ 1 cent, a typical Polymarket tick):

$$\text{edge}_{\text{net}} = \operatorname{sign}(\text{edge})\cdot\max\!\big(|\text{edge}| - c,\ 0\big).$$

A gap that doesn't clear the cost to trade it is sized to zero — we only act on edges that survive the spread. (The model-vs-market comparison table still shows the gross disagreement; netting applies where money is at stake.)

Sizing — the paper book. Conviction-weighted and dollar-neutral, on the net edge. Each side (long/short) gets half a bankroll $B$, allocated by $|\text{edge}|$ and capped per trade:

$$\text{stake}_i = \min\!\left(\frac{B}{2}\cdot\frac{|\text{edge}_i|}{\sum_j|\text{edge}_j|},\ \ \text{cap}\right), \quad \text{cap} = \tfrac{B}{2}\cdot 0.15 ,$$

$$\text{shares}_i = \pm\,\frac{\text{stake}_i}{\max(\kappa_i,\ \varepsilon)}, \quad \kappa_i = p_i\ (\text{long})\ \text{ or }\ 1-p_i\ (\text{short}).$$

Mark-to-market $\text{PnL} = \text{shares}\cdot(\text{price}_{\text{now}} - \text{entry})$; realised uses the settled outcome $o \in \{0,1\}$. A binary bet can't lose more than its stake: $\max\!\downarrow = -\,\text{stake}$, $\max\!\uparrow = |\text{shares}| - \text{stake}$.

The informed (Elo) model. Match win probability (Elo expected score), with a home bonus $h$, and the rating update from a result ($S \in \{1,\tfrac12,0\}$, $K = 24$):

$$E = \frac{1}{1 + 10^{-(R_A - R_B + h)/400}}, \qquad R \leftarrow R + K\,\ln\!\big(|\Delta\text{goals}| + 1\big)\,(S - E).$$

Group-stage scoreline — a Dixon–Coles-corrected double Poisson (the $\rho$ term lifts the low-score draws that independent Poisson under-produces, giving a realistic $\sim 27\%$ draw rate):

$$g_A \sim \text{Poisson}(\lambda_A),\quad \lambda_A = \mu\,e^{+b(R_A - R_B + h)},\quad \lambda_B = \mu\,e^{-b(R_A - R_B + h)},$$

with $\mu = 1.35$, $b = 0.0028$, $\rho = -0.05$; the DC factor $\tau_\rho$ reweights the $(0\text{-}0,\ 1\text{-}0,\ 0\text{-}1,\ 1\text{-}1)$ cells.

Disclosed priors on the ratings: a host bonus of $+60$ Elo for the three 2026 co-hosts (added before the shrink), and a knockout-only shrink toward the field mean $\bar{R}$:

$$R' = \bar{R} + \kappa\,(R - \bar{R}), \qquad \kappa_{\text{group}} = 1.0,\quad \kappa_{\text{KO}} = 0.6.$$

Rating uncertainty. Each of the $N$ simulations perturbs every team's rating by $\varepsilon \sim \mathcal{N}(0,\sigma^2)$, $\sigma = 70$ Elo (the same $\varepsilon$ for a team's group and knockout ratings), integrating over our uncertainty about true strength — so a favorite reads $\sim 99\%$, not a false 100%, to advance.

Bracket. Qualifiers are seeded into a balanced bracket (winners take the protected top seeds, then runners-up, then best-thirds, by rating within each tier), so winners are kept apart and same-group teams don't meet in the Round of 32 — a structural approximation, not the exact official slot table.

Monte-Carlo reach probabilities. Over $N = 20{,}000$ sims, $P(\text{level}) = \text{count}/N$. The Monte-Carlo error — how precisely the sim estimates its own answer — is

$$\mathrm{SE} \approx \sqrt{\frac{p(1-p)}{N}} \approx 0.2\text{–}0.3\%.$$

That is not the forecast's error bar: the real uncertainty is dominated by the inputs (the Elo ratings, the host bonus, the bracket path, and everything the model can't see — form, injuries, motivation), which is far larger. So a big model-vs-market gap is not "signal because it clears the MC noise"; treat it as a disagreement the scorecard adjudicates, not a proven edge. The rating-uncertainty band is a first, partial attempt to reflect this.

Scoring (the public scorecard). Out-of-sample, always vs the market — lower Brier is better, positive skill means the model beat the crowd:

$$\text{Brier} = \frac{1}{n}\sum_i (p_i - o_i)^2, \qquad \text{skill} = \text{Brier}_{\text{market}} - \text{Brier}_{\text{model}} .$$

Where this lives in code

Piece	Module
De-vig / favorite–longshot / no-arb detectors	`src/consistency.py`
Full nested ladder + per-level overround + arb scan	`src/worldcup_markets.py`
Paper positions, mark-to-live, resolve PnL	`src/worldcup_positions.py`
Conviction-weighted sizing + the HTML board	`src/worldcup_board.py`
Two books — Buy & Hold vs Active Trading	`src/worldcup_books.py`
Bounded-market calibration / time-to-resolution signals	`src/bounded.py`
Cost / capacity analysis	`src/capacity.py`
Pre-registered, timestamped hypothesis	`docs/preregistration-worldcup.md`