§ VI · Long-form

8 min readlast revised 2026-04-22snapshot 2026-06-15T03:47Z

Market Layer

De-vigging, edge, and the power method. How raw bookmaker decimal odds become de-vigged implied probabilities; and why the proportional method is wrong.

By The 45% Problem project

Contents

From odds to a fair probability

Bookmaker decimal odds are not directly probabilities. The implied probability of an outcome quoted at decimal odds $o_i$ is the inverse, $r_i = 1/o_i$ , but for any commercial book the implied probabilities across mutually exclusive outcomes do not satisfy $\sum_i r_i = 1$ . The overround $\pi = \sum_i r_i - 1$ is the bookmaker's gross margin: positive for a sportsbook like Pinnacle (overround), negative for an exchange like Betfair or a prediction market like Polymarket (underround once commission is netted).

De-vigging is the math that strips out the margin and recovers an estimator $q_i$ of the true risk-neutral probability the book actually believes. Without de-vigging, any model-vs-market comparison is contaminated by margin and the result is not a probability statement at all.

Pinnacle is the canonical book in this project. Its de-vigged probability vector is what the model is compared against. Betfair Exchange and Polymarket are also de-vigged, but only as cross-references for the Volatility Gate and the price-discovery rules that live there; they are never used as $q$ in the edge calculation. This is a deliberate scope choice: book-specific bias corrections are calibrated only on Pinnacle, and applying $q$ from a non-calibrated book would introduce uncontrolled error into the edge metric.

The implementation lives in market/devig.py, market/edge_calculator.py, and market/market_pipeline.py. Every constant on this page is read from evaluation/pre_reg_constants.yaml and was sealed in the OSF pre-registration before any 2026 forecast was published.

The Power method

The de-vigging machinery follows Shin (1993) and Štrumbelj (2014).

Why not proportional de-vigging

The naive estimator $q_i = r_i / \sum_j r_j$ distributes the overround uniformly across outcomes. This is provably wrong on every sportsbook studied since Shin (1993): the longshot side of any market carries a disproportionate share of the overround, because books charge more vig where adverse selection from informed bettors is highest, and because retail bettors systematically over-pay for tail outcomes (the favorite-longshot bias). Proportional de-vigging therefore inflates $q$ on longshots and deflates it on favorites, exactly the wrong direction relative to the empirical bias the model is trying to surface.

Štrumbelj (2014) compares the Power method against Shin's original quadratic-surplus method and against proportional de-vigging on a corpus of more than 100,000 two-way and three-way markets. The Power method dominates Shin in out-of-sample log-loss and dominates the proportional estimator by a wide margin on three-way markets like 1X2.

The root-finding equation

The Power method posits that there exists a single market exponent $z$ such that the de-vigged probabilities are obtained by raising raw probabilities to that exponent and the result sums to one by construction:

\sum_{i=1}^{n} r_i^{\,z} = 1, \qquad q_i = r_i^{\,z}

Equivalently, $z$ is the unique root of

F(z) = \sum_{i=1}^{n} r_i^{\,z} - 1 = 0

For an overround book like Pinnacle, $\sum_i r_i > 1$ at $z = 1$ , so the root sits at $z > 1$ (raising $r_i < 1$ to a higher power decreases the sum). For an underround exchange, the root sits at $z < 1$ . The implementation brackets accordingly: $z \in (1, 20)$ for overround, $z \in (10^{-9}, 1)$ for underround. Brent's method converges in six to eight iterations with $\mathrm{xtol} = 10^{-8}$ and a hard cap of 100 iterations.

If Brent's method fails to bracket the root, the function raises DeviggingNotConverged rather than silently coercing to a fallback. This is intentional: failure to converge on well-formed odds indicates upstream data corruption, and a silent fallback would mask the upstream bug.

The de-vigged probabilities

Once $z$ is found, $q_i = r_i^{z}$ is the de-vigged probability for outcome $i$ , and the vector sums to $1$ by construction (that is what we solved for). The Power method implicitly corrects the favorite-longshot bias through $z$ alone; we do not stack a separate FLB adjustment on top, which would double-count.

Anomaly bands and out-of-band routing

The fitted $z$ is monitored against pre-registered acceptance bands. For Pinnacle, $z$ values outside $[1.00, 1.20]$ are anomalous: a typical Pinnacle overround between two and twelve percent produces $z \in [1.02, 1.12]$ , and values outside that range with margin usually indicate either a misparsed book or a market in the middle of price discovery. For exchanges, the symmetric band is $z \in [0.80, 1.00]$ .

A market with out-of-band $z$ is not de-vigged at all. The pipeline emits a Z_OUT_OF_BAND warning, and the orchestrator routes the market to the Volatility Gate with the same reason code, where it is suppressed before reaching the sizer. This is a deliberate escalation: a book whose Power-method exponent does not sit inside its calibrated band is one we cannot trust to compare the model against.

Pinnacle bias corrections

The 2010 to 2022 World Cup calibration corpus (the same 347 major-tournament matches that constrain every other calibrated component in the project) surfaced two systematic Pinnacle distortions on top of the Power-method de-vigging. Both are applied as fixed additive corrections to specific legs of specific market classes, with proportional renormalization of the remaining legs so the vector still sums to one. Both magnitudes are sealed in pre_reg_constants.yaml::market.pinnacle_bias and cannot be re-fit on 2026 data. Re-fitting would constitute look-ahead bias and would invalidate the entire evaluation chain. The small corpus also means the bias-correction magnitudes carry meaningful estimation error of their own; we treat them as point values after pre-registration rather than re-estimating.

The knockout draw under-pricing

Pinnacle systematically under-prices the draw by approximately 1.4 probability points on neutral-venue knockout matches that go to ninety minutes. The likely structural cause is that the draw on a knockout market is a near-pure ninety-minute construct that does not carry tournament-progression information once shootouts are factored in, and Pinnacle's quote concentrates on the progression price rather than the ninety-minute price.

The correction is

\Delta_{\text{draw}} = +0.014

added to $q_{\text{draw}}$ on any knockout-stage market, with the win and loss legs renormalized proportionally to keep $\sum_i q_i = 1$ . The correction applies to all knockout stages: round of 32, round of 16, quarterfinal, semifinal, third-place playoff, and final.

The group-stage host premium

In group-stage markets where one of the two teams is a tournament host, Pinnacle exhibits a small "host-nation premium" of roughly 0.6 probability points on the host's win line. This is most plausibly recreational money rather than information: home crowds attract retail bets at the closing line, and Pinnacle's quote drifts up just enough that the host's de-vigged probability is consistently above the structural fair value the model identifies.

The correction is

\Delta_{\text{host}} = -0.006

subtracted from $q_{\text{host\_win}}$ on group-stage markets where the host is identified as one of the listed teams. The remaining legs renormalize proportionally.

Why these corrections are pre-registered constants

A 1.4 pp adjustment on draws and a 0.6 pp adjustment on host wins are not large fixes, and we are not claiming to repair Pinnacle. We are correcting two specific, consistent biases that the calibration corpus identified, and we sealed the magnitudes before the tournament so the corrections cannot be tuned post hoc on 2026 data. The pre-registration is the credibility move; the magnitudes themselves are secondary.

For Betfair Exchange and Polymarket, no book-specific correction is applied. Both are de-vigged with the Power method only and used only as cross-references for the price-discovery rules in the Volatility Gate. They are never the $q$ in the edge metric, so applying a non-calibrated correction to them would have no downstream consumer in any case.

The edge metric

After de-vigging and bias correction, the edge is computed.

The additive edge

For each outcome $i$ in a given market, the edge is the additive difference between the model probability and the de-vigged bias-corrected Pinnacle probability:

E_i = p_{\text{model},i} - q_{\text{devigged},i}

Reported in probability points. $E_i > 0$ means the model thinks the outcome is more likely than Pinnacle does; $E_i < 0$ means the book is higher.

The choice of additive edge over multiplicative or log-odds edge is deliberate. The additive form is the quantity that is statistically tractable for the Diebold-Mariano and Nyberg tests on the Evaluation page, and it has a stable interpretation across markets of different favorite strength. The Kelly transformation $f_{\text{full}} = (p \cdot o - 1)/(o - 1)$ is applied later, downstream of the gate, on the same $E_i$ value.

The variance-adjusted standardized edge

Raw $E_i$ is a point estimate. The Phase 5 Monte Carlo engine produces ten thousand runs per match, so we have a sampling distribution over $p_{\text{model},i}$ with empirical standard error $\sigma_p = \sqrt{p(1 - p)/N_{\text{MC}}}$ . The de-vigging step has its own uncertainty $\sigma_q$ from the bootstrap below. The standardized edge combines them:

E^{*}_i = \frac{E_i}{\sqrt{\sigma_p^{\,2} + \sigma_q^{\,2}}}

$E^{*}$ is logged as a sidecar column in forecast_log.jsonl and is the quantity used for two downstream purposes: sorting flagged outcomes by statistical strength when displaying the divergence terminal, and Phase 7 Z-score aggregation against historical baselines. The flagging decision, however, uses raw $|E|$ against the pre-registered threshold; $E^{*}$ is informational, not gating.

The σ_q bootstrap

The de-vigging uncertainty $\sigma_q$ is estimated by parametric bootstrap. Fifty resamples are drawn from a tick-noise model on the decimal odds: each $o_i$ is perturbed by a uniform draw on $[-0.005, +0.005]$ (half a probability point at fifty-percent implied probability is the calibrated Pinnacle minimum tick). For each resample the Power method is re-solved on the noisy odds, and $\sigma_q$ is the empirical standard deviation of the resulting $q_i$ across the fifty resamples.

The bootstrap captures rounding and tick-level uncertainty in the closing line, not the deeper uncertainty in whether the bias-corrected probability is the right one. The bias-correction constants $\Delta_{\text{draw}}$ and $\Delta_{\text{host}}$ are themselves estimated from the 2010 to 2022 corpus and carry their own calibration error, but the project treats them as known after pre-registration. Adding the bias-correction uncertainty to $\sigma_q$ would couple $E^{*}$ to a quantity we have already committed to as fixed.

Edge thresholds and market classes

After computation, edges are flagged for downstream sizing only when they exceed a market-class threshold.

The two-threshold structure

The pre-registered thresholds are asymmetric across market classes:

Market class	Threshold $\varepsilon$
Mainline (1X2, match winner, group winner, tournament winner)	0.030 (3.0 pp)
Derivative (over/under 2.5, BTTS, correct score, stage of elimination, both teams to score)	0.050 (5.0 pp)

The asymmetry reflects two facts. Derivative markets carry higher de-vigging uncertainty because their lower liquidity produces a wider effective tick and a larger $\sigma_q$ . The model probability for derivative markets is also a downstream functional of the bivariate Poisson surface and inherits more model error than the 1X2 marginal. A larger $\varepsilon$ on derivatives is the protection against flagging on noise that a thinner market generates by construction.

The values 3.0 pp and 5.0 pp are sealed in pre_reg_constants.yaml::market.edge_threshold_* and cannot be tuned on 2026 data.

Market class assignment

The mainline classes are: 1x2, match_winner, group_winner, tournament_winner. The derivative classes are: over_under, btts, correct_score, stage_of_elimination, both_teams_score. New market classes are not added during the tournament. A market with an unrecognized class falls back to the conservative derivative threshold; an unrecognized class also produces a warning that the market schema needs updating, but the fallback prevents a parse error from blocking the pipeline.

The flagging rule

A leg is flagged when

|E_i| > \varepsilon

Flagged legs surface as EdgeFlag(flagged=True, reason_code="ABOVE_THRESHOLD") on the divergence terminal as candidates for hypothetical bets. The flag status alone does not size a stake. The Volatility Gate consumes the flag next, decides whether the surrounding microstructure is quiet enough to act on it, and only then does the Kelly sizer compute a recommended fraction. All three layers (this page, the gate, and the sizer) must agree before any non-zero stake fraction is recommended in forecast_log.jsonl.

Two-sided flags as a misspecification signal

A market in which two opposing legs are simultaneously flagged is not two opportunities. It is a warning. If the model thinks the home win is under-priced and also thinks the away win is under-priced, $p_{\text{model}}$ is inconsistent with the de-vigged Pinnacle vector in a way that no honest reading of the math can resolve. The pipeline detects this via sign analysis on the flagged legs of the same market and sets is_two_sided_flag=True on those flags. The Volatility Gate's Rule 3 (price-discovery cross-book) also tends to fire on these markets, which catches them as suppressions before the sizer sees them. We surface the boolean explicitly because the underlying pattern is rare and worth auditing in Phase 7 review.

What this layer does not do

The market layer ends at flagged edges. Things that are deliberately out of scope here:

No Kelly sizing, no per-market caps, no per-event or per-day caps. All sizing logic and bankroll management live in Volatility Gate (where the fifth suppression rule is sizing).
No suppression of flagged edges based on news, price-discovery, or liquidity. The five suppression rules also live in Volatility Gate.
No CLV computation. CLV is the Evaluation page's metric. The market layer logs the open and close de-vigged probabilities for every match; the inferential procedure runs in evaluation/clv_tracker.py.
No proportional fallback if the Power method fails to converge. The pipeline raises DeviggingNotConverged on bad odds rather than silently using a less accurate estimator.
No book-specific bias correction beyond Pinnacle. Betfair Exchange and Polymarket get Power-method de-vigging only and are used only as cross-references for the gate's price-discovery rules.
No live re-fitting of $\Delta_{\text{draw}}$ or $\Delta_{\text{host}}$ during the tournament. Both are pre-registered constants from the 2010 to 2022 calibration corpus and changing them requires an OSF amendment.
No book-specific edge thresholds. $\varepsilon$ is set by market class, not by book.
No nightly behavioural-bias testing. The seven pre-registered bias hypotheses (favorite-longshot, sentiment, recency, anchoring, late-money, stage-of-elimination, manager-sacking) live in market/bias_tests.py and run against the cumulative forecast_log.jsonl. They are an evaluation activity that consumes the artifact this layer produces; the Evaluation page is the right home for them.

Where to go next

Volatility Gate: the five suppression rules that decide which flagged edges are quiet enough to act on, plus the Kelly sizer and bankroll machinery.
Evaluation: Brier, log-loss, RPS, the Diebold-Mariano machinery, and the Closing Line Value tests that grade the edges this layer produces.
Models: where $p_{\text{model}}$ comes from, including the four shadow models (M0 to M3) and the cross-validation battery that decided M★.
Pre-registration: the OSF DOI, the signed Git tag, and the sealed pre_reg_constants.yaml that locks every threshold and bias correction on this page.
Notation: the symbol table for $r_i$ , $q_i$ , $z$ , $E_i$ , $E^{*}_i$ , $\varepsilon$ , $\sigma_p$ , $\sigma_q$ , and related quantities.

§ VI · Long-form

8 min readlast revised 2026-04-22snapshot 2026-06-15T03:47Z

Market Layer

De-vigging, edge, and the power method. How raw bookmaker decimal odds become de-vigged implied probabilities; and why the proportional method is wrong.

By The 45% Problem project

Contents

From odds to a fair probability

The Power method

The de-vigging machinery follows Shin (1993) and Štrumbelj (2014).

Why not proportional de-vigging

The root-finding equation

\sum_{i=1}^{n} r_i^{\,z} = 1, \qquad q_i = r_i^{\,z}

Equivalently, $z$ is the unique root of

F(z) = \sum_{i=1}^{n} r_i^{\,z} - 1 = 0

The de-vigged probabilities

Anomaly bands and out-of-band routing

Pinnacle bias corrections

The knockout draw under-pricing

The correction is

\Delta_{\text{draw}} = +0.014

The group-stage host premium

The correction is

\Delta_{\text{host}} = -0.006

subtracted from $q_{\text{host\_win}}$ on group-stage markets where the host is identified as one of the listed teams. The remaining legs renormalize proportionally.

Why these corrections are pre-registered constants

The edge metric

After de-vigging and bias correction, the edge is computed.

The additive edge

For each outcome $i$ in a given market, the edge is the additive difference between the model probability and the de-vigged bias-corrected Pinnacle probability:

E_i = p_{\text{model},i} - q_{\text{devigged},i}

Reported in probability points. $E_i > 0$ means the model thinks the outcome is more likely than Pinnacle does; $E_i < 0$ means the book is higher.

The variance-adjusted standardized edge

E^{*}_i = \frac{E_i}{\sqrt{\sigma_p^{\,2} + \sigma_q^{\,2}}}

Market class	Threshold $\varepsilon$
Mainline (1X2, match winner, group winner, tournament winner)	0.030 (3.0 pp)
Derivative (over/under 2.5, BTTS, correct score, stage of elimination, both teams to score)	0.050 (5.0 pp)

The values 3.0 pp and 5.0 pp are sealed in pre_reg_constants.yaml::market.edge_threshold_* and cannot be tuned on 2026 data.

Market class assignment

The flagging rule

A leg is flagged when

|E_i| > \varepsilon

Two-sided flags as a misspecification signal

What this layer does not do

The market layer ends at flagged edges. Things that are deliberately out of scope here:

No Kelly sizing, no per-market caps, no per-event or per-day caps. All sizing logic and bankroll management live in Volatility Gate (where the fifth suppression rule is sizing).
No suppression of flagged edges based on news, price-discovery, or liquidity. The five suppression rules also live in Volatility Gate.
No CLV computation. CLV is the Evaluation page's metric. The market layer logs the open and close de-vigged probabilities for every match; the inferential procedure runs in evaluation/clv_tracker.py.
No proportional fallback if the Power method fails to converge. The pipeline raises DeviggingNotConverged on bad odds rather than silently using a less accurate estimator.
No book-specific bias correction beyond Pinnacle. Betfair Exchange and Polymarket get Power-method de-vigging only and are used only as cross-references for the gate's price-discovery rules.
No live re-fitting of $\Delta_{\text{draw}}$ or $\Delta_{\text{host}}$ during the tournament. Both are pre-registered constants from the 2010 to 2022 calibration corpus and changing them requires an OSF amendment.
No book-specific edge thresholds. $\varepsilon$ is set by market class, not by book.
No nightly behavioural-bias testing. The seven pre-registered bias hypotheses (favorite-longshot, sentiment, recency, anchoring, late-money, stage-of-elimination, manager-sacking) live in market/bias_tests.py and run against the cumulative forecast_log.jsonl. They are an evaluation activity that consumes the artifact this layer produces; the Evaluation page is the right home for them.

Where to go next

Volatility Gate: the five suppression rules that decide which flagged edges are quiet enough to act on, plus the Kelly sizer and bankroll machinery.
Evaluation: Brier, log-loss, RPS, the Diebold-Mariano machinery, and the Closing Line Value tests that grade the edges this layer produces.
Models: where $p_{\text{model}}$ comes from, including the four shadow models (M0 to M3) and the cross-validation battery that decided M★.
Pre-registration: the OSF DOI, the signed Git tag, and the sealed pre_reg_constants.yaml that locks every threshold and bias correction on this page.
Notation: the symbol table for $r_i$ , $q_i$ , $z$ , $E_i$ , $E^{*}_i$ , $\varepsilon$ , $\sigma_p$ , $\sigma_q$ , and related quantities.