Removing the margin: four de-vig methods
The overround tells you how much margin to remove but not where it sits. Compare proportional, additive, power and Shin adjustments, see where each one breaks, and learn what published comparisons found.
Learning objectives
- Apply proportional, additive, power and Shin adjustments to a three-way market and report fair probabilities and fair odds.
- Explain how each method splits the margin between favourites and long shots.
- Identify when the additive method produces an impossible probability.
- Summarise what Štrumbelj (2014) and Clarke, Kovalchik & Ingram (2017) found, and what their results do not show.
One excess, many ways to remove it
A book of 104.81% tells you 4.81 points of probability must go. It does not tell you how much to take from each outcome. Every de-vig method is a small model of how the bookmaker spread the margin, and different models give different fair probabilities from the same quotes. Write πᵢ = 1/oᵢ, β = Σπᵢ and n for the number of outcomes.
Proportional: pᵢ = πᵢ / βAdditive: pᵢ = πᵢ − (β − 1) / nPower: pᵢ = πᵢᵏ, with k chosen so that Σ πᵢᵏ = 1Shin: pᵢ = (√(z² + 4(1 − z) · πᵢ² / β) − z) / (2(1 − z)), with z chosen so that Σ pᵢ = 1Where each method puts the margin
Proportional cuts favourite and long shot by the same relative amount. If bookmakers in fact load more margin onto long shots, as the favourite–longshot pattern suggests, proportional leaves long-shot probabilities too high. Additive removes an equal number of points from every outcome, which is a large relative cut for a long shot and can push it below zero. Power shrinks small probabilities proportionally more and always returns valid probabilities.
Shin comes from a pricing model in which a bookmaker protects itself against bettors with inside information Shin (1992), with an estimation approach developed in a follow-up paper Shin (1993). It also takes relatively more from long shots. Reading z as a literal share of insider money is disputed; you can use the method as a fitting device without accepting that story.
| Method | Home | Draw | Away | Fair odds (H / D / A) |
|---|---|---|---|---|
| Proportional | 45.43% | 28.06% | 26.50% | 2.201 / 3.563 / 3.773 |
| Additive | 46.02% | 27.81% | 26.17% | 2.173 / 3.596 / 3.820 |
| Power (k = 1.0461) | 46.02% | 27.80% | 26.18% | 2.173 / 3.597 / 3.819 |
| Shin (z = 0.0241) | 45.87% | 27.87% | 26.26% | 2.180 / 3.588 / 3.808 |
In a fairly balanced market the methods disagree by about half a percentage point on the favourite. The home fair odds run from 2.173 to 2.201, a 1.28% gap. That looks small, but it is the same size as many claimed edges, so the choice of method is not a detail.
When the additive method breaks
Now take a lopsided hypothetical market: Glenrock 1.10, draw 5.00, Millford 60.0. Raw implied probabilities are 90.91%, 20.00% and 1.67%; the book is 112.58%. Additive subtracts 4.19 points from each outcome, which leaves Millford at −2.53%, an impossible probability. The other methods stay valid but disagree sharply about the long shot.
| Method | Glenrock | Draw | Millford | Millford fair odds |
|---|---|---|---|---|
| Proportional | 80.75% | 17.77% | 1.48% | 67.5 |
| Additive | 86.72% | 15.81% | −2.53% | invalid |
| Power (k = 1.3399) | 88.01% | 11.57% | 0.42% | 241.3 |
| Shin (z = 0.1056) | 84.89% | 14.88% | 0.23% | 436.3 |
What the evidence says
In practice, run all valid methods and treat their spread as part of your uncertainty. If a price looks attractive under every method, the method is not driving your conclusion. If it only looks attractive under proportional, be suspicious: of the valid methods, proportional is the most generous to long shots, because it assumes they carry no extra share of the margin. The word “fair” in “fair odds” always means margin-free under a method, never true.
Worked example
Worked example · De-vig Kestrel Park v Dunmore four ways
Hypothetical match-result quotes: Kestrel Park 2.10, draw 3.40, Dunmore 3.60 (from the previous lesson).
- Raw probabilities and book
π = 0.47619, 0.29412, 0.27778; β = Σπβ = 1.04809 - Proportional
Divide each π by 1.0480945.43%, 28.06%, 26.50% - Additive
Subtract (1.04809 − 1) / 3 = 0.01603 from each π46.02%, 27.81%, 26.17% - Power
Find k with 0.47619ᵏ + 0.29412ᵏ + 0.27778ᵏ = 1 by bisectionk = 1.0461 → 46.02%, 27.80%, 26.18% - Shin
Find z so the Shin probabilities sum to 1z = 0.0241 → 45.87%, 27.87%, 26.26% - Spread of fair home odds
Lowest 2.173 (power), highest 2.201 (proportional): 2.201 / 2.173 − 1≈ 1.28%
Every method agrees the home side is roughly a 45–46% chance, and the non-proportional methods move about half a point of probability from the outsiders to the favourite. If your own estimate is within that band, the de-vig method alone can decide whether a price looks attractive, so treat a sub-1.5% “edge” in this market as no clear edge.
Practice · Break the additive method
Open the Remove the Margin lab with the preset 1.10 / 5.00 / 60.0. Record the book sum and each method's probability for the long shot. Then switch back to 2.10 / 3.40 / 3.60 and compare how far apart the methods are.
Lab 2 · Remove the margin
Synthetic · deterministic| Method | Home | Draw | Away |
|---|---|---|---|
| Raw implied (with margin) | 90.91% | 20.00% | 1.67% |
| Proportional | 80.75% (1.238) | 17.77% (5.629) | 1.48% (67.545) |
| Additive | Not applicable | Not applicable | Not applicable |
| Power k = 1.3399 | 88.01% (1.136) | 11.57% (8.640) | 0.41% (241.250) |
| Shin z = 0.1056 | 84.89% (1.178) | 14.88% (6.719) | 0.23% (436.282) |
- Proportional: Divides every implied probability by the book sum. Simple, but assumes the margin is spread in proportion to probability.
- Additive: Removes an equal amount from every outcome. Easy to explain; it can push long shots below zero. For these prices it produces an impossible probability.
- Power: Raises every implied probability to a common power. Takes relatively more margin from long shots.
- Shin: Derived from Shin's insider-trading model. Takes more margin from long shots; performed well in Štrumbelj (2014).
Reading the result. The methods agree on the favourite more than on long shots. None of them recovers the “true” probability: they are different assumptions about how the bookmaker spread its margin. Differences of a point or two can decide whether a price looks like value, so state which method you used.
Educational calculator. Uses only the numbers you enter or seeded synthetic data — no live prices, no staking, no real money.
Show what you should find
The book sum is 112.58%. Additive gives the long shot −2.53% and is flagged invalid. Proportional gives 1.48% (fair odds 67.5), power 0.42% (241.3) and Shin 0.23% (436.3). With 2.10 / 3.40 / 3.60 all four methods are valid and within about 0.6 points of each other on the favourite. Lopsided markets magnify method disagreement; balanced ones hide it.
Common mistakes
- ✗ Treating one method's output as the true probability.Every method encodes an assumption about how margin is distributed. Report the method and check whether your conclusion survives the others.
- ✗ Using the additive method near long shots.An equal cut can exceed a long shot's whole implied probability. Use power or Shin in lopsided markets.
- ✗ Reading Shin's z as a measured share of insider money.That interpretation is disputed. Treat z as a fitted parameter that makes the probabilities sum to one.
- ✗ De-vigging a market whose outcomes are not mutually exclusive.Anytime-goalscorer outcomes can all happen or none can. These methods assume one and only one outcome occurs.
When it fails
Supporting research
- Štrumbelj (2014): On determining probability forecasts from betting oddsInternational Journal of Forecasting · Peer-reviewed empiricalShin-based probabilities were more accurate than basic normalisation or regression in the samples studied, and bookmakers differed in quality as probability sources.
- Clarke, Kovalchik & Ingram (2017): Adjusting Bookmaker's Odds to Allow for OverroundAmerican Journal of Sports Science · Peer-reviewed empiricalThe power method outperformed normalisation on three non-football datasets, was comparable to or better than Shin and always gave valid probabilities.
- Shin (1992): Prices of State Contingent Claims with Insider Traders, and the Favourite-Longshot BiasThe Economic Journal · Peer-reviewed theoryThe theoretical origin of the Shin method: a bookmaker guarding against insiders sets prices with a favourite–longshot pattern.
- Shin (1993): Measuring the Incidence of Insider Trading in a Market for State-Contingent ClaimsThe Economic Journal · Peer-reviewed empiricalThe estimation approach later used to convert odds into probabilities under Shin's model; the insider interpretation of its parameter is disputed.
Check your understanding
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