Advanced devig methods

The multiplicative and additive methods spread the margin "mechanically". But bookmakers don't spread their margin evenly: they put proportionally more on outsiders. This is the favourite-longshot bias, well documented in most betting markets.

The consequence: the multiplicative method overestimates the probability of outsiders, and can make you see value where there isn't any.

The power method

Find the exponent $k$ such that:

\sum_i \left(\frac{1}{\text{odds}_i}\right)^k = 1, \qquad p_i = \left(\frac{1}{\text{odds}_i}\right)^k

Since $k > 1$ , small probabilities are reduced proportionally more than large ones. Simple, and often very effective.

Shin's method

An economic model (Shin, 1991-1993): the bookmaker protects itself against a proportion $z$ of insider bettors. The fair probability is:

p_i = \frac{\sqrt{z^2 + 4(1 - z)\,\dfrac{\pi_i^2}{S}} - z}{2(1 - z)}

where $\pi_i = 1/\text{odds}_i$ and $S = \sum \pi_i$ ; $z$ is found numerically so that $\sum p_i = 1$ . Well suited to "horse racing" type markets or highly asymmetric ones.

The odds ratio method

Assume the bookmaker applies the same odds ratio $c$ to every outcome:

\frac{\pi_i}{1 - \pi_i} = c \cdot \frac{p_i}{1 - p_i} \quad \Longrightarrow \quad p_i = \frac{\pi_i}{c + \pi_i - c\,\pi_i}

with $c$ found numerically.

Comparison

Example

Running example — Team A vs Team B (2.50 / 3.40 / 3.00): a balanced match.

Method	Team A	Draw	Team B
Multiplicative	38.9%	28.6%	32.4%
Power	39.1%	28.5%	32.4%
Shin	39.0%	28.5%	32.4%
Odds ratio	39.0%	28.6%	32.4%

Negligible differences.

Unbalanced match (1.25 / 6.00 / 11.00, 5.8% margin):

Method	Favourite	Draw	Outsider	Outsider fair odds
Multiplicative	75.6%	15.8%	8.6%	11.63
Power	78.4%	14.2%	7.3%	13.61
Shin	77.5%	15.0%	7.6%	13.22
Odds ratio	77.4%	14.7%	7.9%	12.65

Odds of 12.50 on the outsider show +7.5% EV with the multiplicative method… but −8% with the power method and −5.5% with Shin.

Which one to choose?

There's no universally best method. The right approach:

Gather a history of closing odds and results.
Compare the methods using a calibration metric (log loss, Brier score: see the next module).
Choose the method that predicts best, by sport and by market.

Interactive tool

Method

Outcome 1

Outcome 2

Educational tool. Results are theoretical estimates, not betting advice.

The power method

Find the exponent

k

such that:

\sum_i \left(\frac{1}{\text{odds}_i}\right)^k = 1, \qquad p_i = \left(\frac{1}{\text{odds}_i}\right)^k

Since

k > 1

, small probabilities are reduced proportionally more than large ones. Simple, and often very effective.

Shin's method

An economic model (Shin, 1991-1993): the bookmaker protects itself against a proportion

z

of insider bettors. The fair probability is:

p_i = \frac{\sqrt{z^2 + 4(1 - z)\,\dfrac{\pi_i^2}{S}} - z}{2(1 - z)}

where

\pi_i = 1/\text{odds}_i

and

S = \sum \pi_i

;

z

is found numerically so that

\sum p_i = 1

. Well suited to "horse racing" type markets or highly asymmetric ones.

The odds ratio method

Assume the bookmaker applies the same odds ratio

c

to every outcome:

\frac{\pi_i}{1 - \pi_i} = c \cdot \frac{p_i}{1 - p_i} \quad \Longrightarrow \quad p_i = \frac{\pi_i}{c + \pi_i - c\,\pi_i}

with

c

found numerically.