Chapter 8 · Worked solutions
Chapter 8 workings: Stripping the Vig
Chapter eight's three exercises worked in full: a three-way market stripped two ways, an over/under market where Shin needs no solver, and a specimen correct-score audit run through all four methods.
Two of the four methods are phone-calculator jobs. Multiplicative divides each implied probability by the booksum. Additive subtracts an equal slice of the overround from each outcome, and the margin-proportional-to-odds shortcut gives the same answer in decimal-odds form. Power and Shin need a solver, which is what the margin remover is for. Every figure below can be reproduced there.
The exercises
A fat three-way book
A soft firm has the favourite at 8/11, the draw 12/5 and the outsider 4/1. Size the margin, remove it by division and by the shortcut, and find where the two answers differ most.
Convert
8/11 claims 11/19, 12/5 claims 5/17, 4/1 claims 1/5: 57.89 per cent, 29.41 per cent, 20.00 per cent.
Size the book
Booksum 107.31 per cent, overround 7.31 points, margin 6.81 per cent. That is a generous helping of rent for a match-odds market.
Multiplicative
Divide each by 1.0731: 53.95 / 27.41 / 18.64.
Shortcut
Take 7.31 ÷ 3 = 2.44 points off each: 55.46 / 26.98 / 17.56. In decimal form, nO ÷ (n − MO) gives fair prices of 1.80, 3.71, 5.69.
Where they part company
At the ends. The shortcut hands the favourite 1.51 points more than multiplicative and takes 1.07 off the outsider. The draw barely moves. The fatter the margin, the wider the split, which is why the choice of method matters most where the rent is highest.
| Outcome | Price | Implied | Multiplicative | Shortcut | Fair decimal | About |
|---|---|---|---|---|---|---|
| Home | 8/11 | 57.89% | 53.95% | 55.46% | 1.80 | 4/5 |
| Draw | 12/5 | 29.41% | 27.41% | 26.98% | 3.71 | 11/4 |
| Away | 4/1 | 20.00% | 18.64% | 17.56% | 5.69 | 9/2 |
The answer
Booksum 107.31, margin 6.81 per cent. Fair favourite about 4/5 by the shortcut.
Over/under, and Shin for free
Over 2.5 goals 4/5, under evens. Strip it both ways, then say what Shin would give without running it.
Convert
4/5 claims 55.56 per cent, evens claims 50.00 per cent; together they sum to 105.56 per cent.
Multiplicative, dividing by the booksum
52.63 / 47.37.
Additive, which is the shortcut
5.56 ÷ 2 = 2.78 points off each: 52.78 / 47.22.
Shin, by theory
With only two outcomes, Shin's solution lands exactly on the additive answer, so you can write it down without iterating. Running the solver anyway gives z = 0.0556 and 52.78 / 47.22, identical to the additive figures.
| Outcome | Price | Implied | Multiplicative | Additive | Shin |
|---|---|---|---|---|---|
| Over 2.5 | 4/5 | 55.56% | 52.63% | 52.78% | 52.78% |
| Under 2.5 | evens | 50.00% | 47.37% | 47.22% | 47.22% |
The answer
Multiplicative 52.63 / 47.37. Additive, shortcut and Shin all 52.78 / 47.22.
Specimen correct-score audit
Run all four methods on a correct-score market, after testing the longest price for the additive danger condition (β − 1) ÷ πᵢ > n and predicting where multiplicative flatters.
Where this market came from
It is a specimen I built for this page, not a firm's quote. Fair probabilities come from the chapter 16 goal expectancies (2.2425 and 1.188), Shin's model shades them towards the long prices with an insider share of 0.01, and each price is then rounded in the bookmaker's favour to the nearest coupon price. Because I chose the shading, this audit cannot tell you which method is closest to the truth. It shows how they behave. Use a live coupon for your own.
Size it
19 outcomes, booksum 125.59 per cent, margin 20.38 per cent. A three-way market at that level would be a scandal; on a correct-score board it is a normal Saturday.
Test the longest price
The longest line is Any other draw at 125/1, claiming 0.79 per cent. The test value is 0.2559 ÷ 0.0079 = 32.2, against n = 19. It fails, so additive would push it below zero. 0-3 comes within reach of the limit.
Predict, then check
Multiplicative takes the same proportion off every line, so it leaves the most probability on the longest prices, which is where the shading put the extra rent. Measured against Shin, it is most generous to Any other draw, 0-3, 3-3 (by factors of 2.24, 1.52, 1.24).
Power against Shin
On the chapter's own 1X2 market (4/6, 14/5, 9/2) the two solvers never differ by more than 0.9 per cent of each other. Here they part company on the long lines: for Any other draw, power says 0.51 per cent and Shin says 0.28 per cent, a ratio of 1.80. With 19 outcomes and 25.59 points of overround, the choice of method stops being a rounding detail. Power solves to k = 1.0922; Shin to z = 0.0150.
| Scoreline | Price | Implied | Multiplicative | Additive | Power | Shin |
|---|---|---|---|---|---|---|
| 0-0 | 22/1 | 4.35% | 3.46% | 3.00% | 3.26% | 3.22% |
| 0-1 | 20/1 | 4.76% | 3.79% | 3.42% | 3.60% | 3.59% |
| 0-2 | 28/1 | 3.45% | 2.75% | 2.10% | 2.53% | 2.43% |
| 0-3 | 66/1 | 1.49% | 1.19% | 0.15% | 1.01% | 0.78% |
| 1-0 | 10/1 | 9.09% | 7.24% | 7.74% | 7.29% | 7.45% |
| 1-1 | 9/1 | 10.00% | 7.96% | 8.65% | 8.09% | 8.26% |
| 1-2 | 14/1 | 6.67% | 5.31% | 5.32% | 5.19% | 5.28% |
| 1-3 | 33/1 | 2.94% | 2.34% | 1.59% | 2.13% | 1.99% |
| 2-0 | 9/1 | 10.00% | 7.96% | 8.65% | 8.09% | 8.26% |
| 2-1 | 8/1 | 11.11% | 8.85% | 9.76% | 9.07% | 9.26% |
| 2-2 | 12/1 | 7.69% | 6.12% | 6.35% | 6.07% | 6.20% |
| 2-3 | 28/1 | 3.45% | 2.75% | 2.10% | 2.53% | 2.43% |
| 3-0 | 12/1 | 7.69% | 6.12% | 6.35% | 6.07% | 6.20% |
| 3-1 | 11/1 | 8.33% | 6.64% | 6.99% | 6.63% | 6.77% |
| 3-2 | 18/1 | 5.26% | 4.19% | 3.92% | 4.01% | 4.03% |
| 3-3 | 40/1 | 2.44% | 1.94% | 1.09% | 1.73% | 1.56% |
| Any other home win | 7/2 | 22.22% | 17.69% | 20.88% | 19.35% | 19.23% |
| Any other away win | 25/1 | 3.85% | 3.06% | 2.50% | 2.85% | 2.78% |
| Any other draw | 125/1 | 0.79% | 0.63% | below zero | 0.51% | 0.28% |
| Total | 125.59% | 100.00% | 100.00% | 100.00% | 100.00% |
The answer
Booksum 125.6 per cent; Any other draw fails the additive test; multiplicative is kindest to the longest lines; power and Shin split.
These are the workings. The argument for them, and the exercises themselves, are in chapter 8 of the book, Stripping the Vig.
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