The basic method

The default method used by the function implied_probabilities() is called the basic method. This is the simplest and most common method for converting bookmaker odds into probabilities, and is obtained by dividing the naive probabilities (the inverted odds) by the sum of the inverted odds. If p_i is the true underlying probability for outcome i, and r_i is the corresponding inverted odds, then the probabilities are computed as

p_i = r_i / sum(r)

This method tend to be the least accurate of the methods in this package. I have also seen this normalization method been referred to as the multiplicative method.

The implied_probabilities() function return a list with the proper probabilities (as a matrix) and the bookmaker margins.

In the examples below are three sets of bookmaker odds from three football matches.

library(implied)

# One column for each outcome, one row for each race or match.
my_odds <- rbind(c(4.20, 3.70, 1.95),
                 c(2.45, 3.70, 2.90),
                 c(2.05, 3.20, 3.80))
colnames(my_odds) <- c('Home', 'Draw', 'Away')

res1 <- implied_probabilities(my_odds)

res1$probabilities
#>           Home      Draw      Away
#> [1,] 0.2331556 0.2646631 0.5021813
#> [2,] 0.3988848 0.2641264 0.3369888
#> [3,] 0.4586948 0.2938514 0.2474538

res1$margin
#> [1] 0.02118602 0.02326112 0.06346277

Margin Weights Proportional to the Odds

This method is from Joseph Buchdahl’s Wisom of the Crowds document, and assumes that the margin applied by the bookmaker for each of the outcome is proprtional to the probabilitiy of the outcome. In other words, the excessive probabilties are unevenly applied in a way that is reflects the favorite-longshot bias.

The probabilities are calculated from the bookmaker odds O using the following formula

p_i = (n - M * O_i) / n * O_i

where n is the number of outcomes, and M is the bookmaker margin.

res2 <- implied_probabilities(my_odds, method = 'wpo')

res2$probabilities
#>           Home      Draw      Away
#> [1,] 0.2310332 0.2632083 0.5057585
#> [2,] 0.4004096 0.2625166 0.3370739
#> [3,] 0.4666506 0.2913457 0.2420036

# The margins applied to each outcome.
res2$specific_margins
#>            Home       Draw       Away
#> [1,] 0.03056706 0.02683049 0.01396320
#> [2,] 0.01936444 0.02953607 0.02300299
#> [3,] 0.04533211 0.07260878 0.08741297

The odds ratio method

The odds ratio method is also from the Wisdom of the Crowds document, but is originally from an article by Keith Cheung. This method models the relationship between the proper probabilities and the improper bookmaker probabilties using the odds ratio (OR) function:

OR = p_i (1 - r_i) / r_i (1 - p_i)

This gives the probabilities

p_i = r_i / OR + r_i - (OR * r_i)

where the odds ratio OR is selected so that sum(p_i) = 1.

res3 <- implied_probabilities(my_odds, method = 'or')

res3$probabilities
#>           Home      Draw      Away
#> [1,] 0.2320045 0.2636413 0.5043542
#> [2,] 0.3996913 0.2633868 0.3369219
#> [3,] 0.4634417 0.2919028 0.2446556

# The odds ratios converting the proper probablities to bookmaker probabilities.
res3$odds_ratios
#> [1] 1.034456 1.035814 1.102631

The power method

The power method models the bookmaker probabilities as a power function of the proper probabilities. This method is also described in the Wisdom of the Crowds document, where it is referred to as the logarithmic method.

p_i = r_i^(1/k)

where k is selected so that sum(p_i) = 1.

res4 <- implied_probabilities(my_odds, method = 'power')

res4$probabilities
#>           Home      Draw      Away
#> [1,] 0.2311414 0.2630644 0.5057941
#> [2,] 0.4003156 0.2627189 0.3369655
#> [3,] 0.4667137 0.2908986 0.2423877

# The inverse exponents (n) used to convert the proper probablities to bookmaker probabilities.
res4$exponents
#> [1] 0.9797666 0.9788117 0.9419759

The additive method

The additive method removes the margin from the naive probabilities by subtracting an equal amount of of the margin from each outcome. The formula used is

p_i = r_i - ((sum(r) - 1) / n)

If there are only two outcomes, the additive method and Shin’s method are equivalent.

res5 <- implied_probabilities(my_odds, method = 'additive')

res5$probabilities
#>           Home      Draw      Away
#> [1,] 0.2310332 0.2632083 0.5057585
#> [2,] 0.4004096 0.2625166 0.3370739
#> [3,] 0.4666506 0.2913457 0.2420036

One problem with the additive method is that it can produce negative probabilities, escpecially for outcomes with low probabilties. This can often be the case when there are many outcomes, for example in racing sports. If this happens, you will be given a warning. Here is an example taken from Clarke et al (2017):

my_odds2 <- t(matrix(1/c(0.870, 0.2, 0.1, 0.05, 0.02, 0.01)))
colnames(my_odds2) <- paste('X', 1:6, sep='')

res6 <- implied_probabilities(my_odds2, method = 'additive')
#> Warning in implied_probabilities(my_odds2, method = "additive"): Probabilities outside the 0-1 range produced at 1 instances.

res6$probabilities
#>             X1        X2         X3          X4          X5          X6
#> [1,] 0.8283333 0.1583333 0.05833333 0.008333333 -0.02166667 -0.03166667

Balanced books and Shin’s method

The two methods referred to as “balanced book” and Shin’s method are based on the assumption that there is a small proportion of bettors that actually knows the outcome (called inside traders), and the rest of the bettors reflect the otherwise “true” uncertainty about the outcome. The proportion of inside traders is denoted Z.

The two methods differ in what assumptions they make about how the bookmakers react to the pressence of inside traders. Shin’s method is derived from the assumption that the bookmakers tries to maximize their profits when there are inside traders. The balanced books method assumes the bookmakers tries to minimize their losses in the worst case scenario if the least likely outcome were to acctually occur.

We can not know what the insiders know, but both methods gives an estimate of the proportion of insiders.

res7 <- implied_probabilities(my_odds, method = 'shin')

res7$probabilities
#>           Home      Draw      Away
#> [1,] 0.2315811 0.2635808 0.5048382
#> [2,] 0.4000160 0.2629336 0.3370505
#> [3,] 0.4645977 0.2919757 0.2434266

# The estimated proportion of inside traders.
res7$zvalues
#> [1] 0.01054734 0.01157314 0.03187455

# Balanced books
res8 <- implied_probabilities(my_odds, method = 'bb')

res8$probabilities
#>           Home      Draw      Away
#> [1,] 0.2299380 0.2624575 0.5076046
#> [2,] 0.4011989 0.2616832 0.3371179
#> [3,] 0.4710196 0.2899698 0.2390106

# The estimated proportion of inside traders.
res8$zvalues
#> [1] 0.01059301 0.01163056 0.03173139

The Jensen–Shannon distance method

This method sees the improper bookmaker probabilities as a noisy version of the true underlying probabilities, and uses the Jensen–Shannon (JS) distance as a measure of how noisy the bookmaker probabilities are.

For the sake of finding the denoised probabilities p_i, each outcome i is modeled as a binomial variable, with outcomes i and NOT i. These have probabilities p_i and 1-p_i, with corresponding improper bookmaker probabilities r_i and 1-r_i. For a given noise-level D, as measued by the symmetric JS distance, the underlying probabilities can be found by solving the JS distance equation for p_i:

D = 0.5 * BKL(p_i, m_i) + 0.5 * BKL(r_i, m_i)

where m_i = (p_i + r_i) / 2

and

BKL(x, y) = x * log(x/y) + (1-x) * log((1-x)/(1-y))) + y * log(y/x) + (1-y) * log((1-y)/(1-y))

is the “binomial” Kullback–Leibler divergence.

The solution is found numerically by finding the value of of D so that sum(p_i) = 1.

The method was developed by Christopher D. Long (twitter: @octonion), and described in a series of Twitter postings.

# Balanced books
res9 <- implied_probabilities(my_odds, method = 'jsd')

res9$probabilities
#>           Home      Draw      Away
#> [1,] 0.2315189 0.2634117 0.5050694
#> [2,] 0.4000505 0.2629589 0.3369906
#> [3,] 0.4650456 0.2915675 0.2433869

# The estimated noise (JS distance)
res9$distance
#> [1] 0.005485371 0.005849245 0.016099204