Margin removal explained for betting

In one sentence

Margin removal turns a bookmaker's inflated implied probabilities into fair ones that add up to 100%, using a rule for how the margin was spread across outcomes.

How it works

If a book adds up to 104.8%, you need to shrink the prices back to 100% before treating them as fair. The question is how much to take off each outcome. Different methods make different assumptions about where the bookmaker put the margin.

The proportional method takes the same share off everyone. The power method raises every implied probability to a power slightly above 1, which shrinks small probabilities more. Shin's method (Shin 1993) assumes a small share of money comes from insiders, and the bookmaker protects himself by loading the longshots. The odds-ratio method scales everyone's odds-against by the same factor.

The last three all take more margin off longshots than the proportional method does, which matches the favourite-longshot bias seen in real markets.

The maths

\text{Proportional: } p_i = \frac{q_i}{B}

\text{Power: } p_i = q_i^{\,k}, \quad \text{with } k \text{ chosen so } \sum p_i = 1

\text{Shin: } p_i = \frac{\sqrt{z^2 + 4(1-z)\,q_i^2 / B} - z}{2(1-z)}

\text{Odds-ratio: } \frac{p_i}{1-p_i} = \frac{1}{c}\cdot\frac{q_i}{1-q_i}

q i: raw implied probability, 1 ÷ odds.
B: the book total, the sum of all q.
p i: the fair probability after removing the margin.
k: the power, found by trial so the fair chances add to 1.
z: Shin's estimate of the share of insider money, found the same way.
c: the odds-ratio constant, again solved so the fair chances add to 1.

In plain English: every method rescales the prices so they add to 100%, but they disagree on how much to shave off favourites versus outsiders.

Worked betting example

Football match at a bookmaker: Home 1.50, Draw 4.20, Away 7.00.

Raw implied: 66.67%, 23.81%, 14.29%. Book total 104.76%, an overround of 4.76%.
Proportional: divide each by 1.0476. Fair chances 63.64%, 22.73%, 13.64%. Fair odds 1.571, 4.40, 7.33.
Power: solving gives k = 1.0554. Fair chances 65.19%, 21.99%, 12.83%. Fair odds 1.534, 4.548, 7.797.
Shin: solving gives z = 0.0241. Fair chances 64.71%, 22.34%, 12.95%. Fair odds 1.545, 4.475, 7.724.
Odds-ratio: solving gives c = 1.0957. Fair chances 64.61%, 22.19%, 13.20%. Fair odds 1.548, 4.506, 7.574.

The methods disagree by about 1.5 percentage points on the favourite. If your model says Home is 65%, it looks like value under proportional (63.6%) and like no value under power (65.2%). The method you choose can decide the bet.

Where it's good

Getting a fair benchmark from bookmaker prices to test your model against.
Pricing Correct Score and other many-outcome markets, where longshots carry heavy margin and proportional removal overstates them.
Building closing line value measures from bookmaker closing prices.
Combining several bookmakers into a single fair consensus price.
Comparing bookmaker fair prices with exchange mid-prices.

Limitations and pitfalls

No method is correct for every market. Test which one best predicts results in your own data, using log loss on historical outcomes.
Two-way markets give almost identical answers across methods; the choice matters most in many-outcome markets such as Correct Score or horse races.
Shin's insider story is a model, not a fact. It fits racing reasonably, but that is no proof of why bookmakers price as they do.
Bookmakers shade prices for commercial reasons (popular teams, promotions), which none of these methods capture.
Exchange prices near 100% need little margin removal; use the back-lay mid-price instead.
Fair probabilities are the market's view without margin, not the truth.

How to build it

scipy.optimize.brentq solves for k, z and c in a few lines; the Python package shin and R package implied also do it.
Data: complete market snapshots with every selection priced at the same moment.
Practical tip: compare methods by log loss on a season of results before settling on one; in many-outcome markets power and Shin often beat proportional.

Overround: the margin these methods remove.
Implied probability: the raw inputs.
Favourite-longshot bias: why uneven removal fits the data better.
Model vs market: testing your model against fair prices.
Closing line value: uses margin-free closing prices as the benchmark.