World vs Model · Methodology
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⚠️ 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.
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.
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:
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.
Three independent structural sources, none of which require football knowledge:
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.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.
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:
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).wc-live) — re-evaluated daily, not just on matchdays (the board refreshes ~every 3 hours). It trades only on an explicit, cost-gated rule set (src/wc_active.py), so the only difference from Buy & Hold is disciplined action — never churn. With aligned = (fair − price)·sign the position's still-favourable edge and buffer the half-spread:aligned > buffer — the gap that justified the trade is still open.aligned < −buffer — the model now thinks the leg loses, so exit early rather than realise a bigger loss at settlement. The ±buffer band is hysteresis: a leg that merely wobbles around fair is not cut, so we don't whipsaw the spread.|aligned| ≤ buffer) has ~no edge left; closing it just pays a spread to sit in cash, so we ride it to its free settlement — and only close it early to rotate into a clearly better edge.edge_cand − aligned > 2·cost + buffer (the swap must beat its round trip). Freed capital is redeployed same-side (a long replaces a long, a short a short), so the book stays dollar-neutral by construction.A riskless inconsistency is the cleanest any-day trigger. The test: "does acting on the model's updates beat just holding, net of churn?" Its changelog is the rebalance 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.
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):
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:
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 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.
Scoring the bracket. That projection is also kept honest. We register, pre-tournament, every side's forecast at each knockout rung (advance / reach-QF / SF / final / win) and score it two ways as results land. (1) A round-weighted points race — each side fills the bracket with its top-k teams per rung and a correctly placed team scores that rung's weight (×1 / ×4 / ×8 / ×16 / ×32). This race is Informed (Elo) vs Market: the zero-knowledge model only re-shapes the market's own prices with a monotonic favorite-longshot power, so its ranking — and therefore its whole bracket — is identical to the market's by construction; it cannot disagree about who goes how far, so showing it as a third bracket would just mirror the market. (2) The rigorous companion: Brier at every rung (the same proper score as the books) — and this is where the zero-knowledge model does score separately, because Brier grades probability magnitudes, which its power correction changes even when the ranking doesn't move. One bracket is a single high-variance draw while the Brier averages over all of them. Both come from the one pre-registered, timestamped ledger (src/worldcup_register.py::bracket_scorecard), so a skeptic can recompute either.
As it unfolds (the living companion). The frozen bracket is the commitment; alongside it (src/wc_evolution.py) we surface how the picture moves once games are played — which stays dormant pre-tournament. Two views: the biggest forecast moves (how far the informed model's live re-forecast has drifted from its frozen kickoff title odds — the model visibly changing its mind as the Elo updates on results), and the biggest surprises, ranked by surprisal = −log₂(p) in bits (a coin-flip that lands is 1 bit, a 1-in-8 is 3): underdogs who got through against the odds and favourites who flopped, each tagged with whether the model or the market was less surprised — i.e. who actually saw it coming. World Cups reliably spring a few; this is where you watch the two forecasters react in real time, on top of the frozen call.
The ensemble — Bayesian model averaging (src/wc_bma.py). Rather than commit to one of our two models upfront we run both and present a third forecast voice: the weighted average, where the weights are derived from each model's per-rung Brier track record. The mechanics, deliberately simple:
0.5 (no data → no evidence to favour anyone). As forecasts at each rung resolve we update the per-(model, rung) Brier and recompute weights as w_m(L) ∝ exp(−η · Brier_m(L)), with η=10 and a small "prior count" of imaginary 0.25-Brier forecasts blended in to shrink toward 50/50 when data is sparse.−∞ catastrophe at p=0 outcome=1), and it's robust to a single catastrophic miss. Strict BMA is recovered with log-loss; we trade a small theoretical purity for practical robustness.We are deliberately loud about the limitations — the credibility is in the caveats.
What's missing / weak today
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.src/wc_bracket.py: the 16 fixed R32 fixtures + the full 495-row best-third contingency table), so the matchups you see are the real fixtures, not a guess. The Monte-Carlo probabilities (win %, reach %) are still computed over a balanced seeded bracket that avoids same-group R32 rematches — so a team's precise path difficulty in the numbers remains an approximation, even though the rendered path is now exact.How we'd improve it
ledger/wc_results.json; pre-tournament, with no results, it is the day-0 forecast). The Dixon–Coles goals correction is also in. The displayed projection now uses the exact official slot table (incl. best-third contingencies, src/wc_bracket.py); the remaining gap is to drive the Monte-Carlo probabilities through those same official slots (today they run over a balanced seeded bracket).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).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 (\(S \in \{1,\tfrac12,0\}\), \(K = 24\)):
$$E = \frac{1}{1 + 10^{-(R_A - R_B + h)/400}}, \qquad R \leftarrow R + K\,\ln\!\big(|\Delta\text{goals}| + 1\big)\,(S - E).$$
Reading the logistic. Only the rating gap matters, not absolute ratings; the 400 is a log-odds scale chosen so that every 400 points multiplies the odds by 10. Useful reference points (stronger side's win probability):
| Gap | 0 | 100 | 200 | 300 | 400 | 600 | 800 |
|---|---|---|---|---|---|---|---|
| P(win) | 50% | 64% | 76% | 85% | 91% | 97% | 99% |
| Odds | 1:1 | 1.8:1 | 3.2:1 | 5.6:1 | 10:1 | 32:1 | 100:1 |
Going the other way — implied rating gap from a probability — \(\Delta R = 400\,\log_{10}\!\frac{p}{1-p}\). So a market price of 65% implies the favourite has about 108 Elo points of edge; 75% implies about 191. Handy for sanity-checking what a market is "really saying" in Elo terms.
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. The Monte-Carlo 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 for the probabilities. The Most-likely-outcome projection shown on the board is separate and exact: it pours the model's group standings into FIFA's official Round-of-32 slot table and the full 495-row best-third contingency (src/wc_bracket.py), so the rendered fixtures are the real ones.
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:
$$\text{Brier} = \frac{1}{n}\sum_i (p_i - o_i)^2, \qquad \text{skill} = \text{Brier}_{\text{market}} - \text{Brier}_{\text{model}} .$$
| Piece | Module |
|---|---|
| De-vig / favorite–longshot / no-arb detectors | src/consistency.py |
| Full nested ladder + per-level overround + arb scan | src/worldcup_markets.py |
| Informed Elo model (Elo + Poisson + Dixon–Coles Monte Carlo) | src/worldcup_fundamental.py, src/worldcup_sim.py |
| Official FIFA 2026 knockout slot table + projected bracket | src/wc_bracket.py |
| Paper positions, mark-to-live, resolve PnL | src/worldcup_positions.py |
| Active book rotation rules (sell / take-profit / cut / switch, cost-gated) | src/wc_active.py |
| Conviction-weighted sizing + the four books + the HTML board | src/worldcup_board.py |
| Pre-registered predictions, frozen books, scorecard + the bracket score | src/worldcup_register.py, ledger/ |
| "As it unfolds" — forecast moves + surprise (surprisal) ranking | src/wc_evolution.py |
| The ensemble — Bayesian model averaging, per-rung weights | src/wc_bma.py |
| Record a played result → live re-forecast | src/feed_result.py |