In the previous blog I investigated a number of additional approaches to determining the Bookmaker's Implicit Probability - and, by mathematical implication, his embedded overround - for each team based on observed head-to-head market prices. Identifying a "good" approach to converting head-to-head prices into implicit probabilities has practical benefits for MAFL because it will:
- enhance MAFL modelling, which uses implicit probabilities extensively as inputs
- enhance MAFL wagering, because it will improve estimates of the implicit profit margin built into the head-to-head prices of each team (rather than into the market as a whole)
Using the Log Probability Score (LPS), my preferred probability score performance metric, there were four candidates for "Best Approach". Two other Approaches were within a honey-producing insect's procreational appendage of being just as good, but sport is, we know, a game of fifth decimal places.
We need a way to choose amongst these four strongest candidates - those that returned a Log Probability Score of 0.1944 bits per game - and it turns out that one of them provides implicit probability predictions that can be used to create a simple but very accurate Margin Predictor. Specifically, we can construct:
Predicted Home Team Margin = 19.83089 x ln(Prob(Home Team win)/(1-Prob(Home Team win)))
where Prob(Home Team win) = 1/Home Team Price - 1.0281%
This Margin Predictor has an all-season MAPE (across seasons 2007 to 2012) of 29.07 points per game, and the following season-by-season and round-by-round results:
That's an impressive Margin Predictor, and is one which would rank amongst the best I've ever produced for this timeframe. The other three Approaches with 0.1944 Log Probability Scores produce Margin Predictors with the same general form that aren't quite as good as this one.
So, let me anoint this approach to deriving implicit probabilities as the preferred one from the previous blog and label it the LPS-Optimising Approach, which means that we now have three approaches, as follows:
For which we have:
- p1 = 1/m1 - k [or, equivalently, m1 = 1/(p1+k) ]
- Ov1 = k/p1
- Ov2 = (T-k)/(1-p1)
where m1 is the Home team head-to-head price; m2 is the Away team head-to-head price; T is the Total Overround in the head-to-head prices, given by 1/m1 + 1/m2 - 1; Ov1 is the overround embedded in the Home team's price; Ov2 the overround embedded in the Away team's price; and we've empirically determined the optimal k to be 1.0281%.
In this approach we imagine that the Bookmaker makes an assessment of the Home team's chances and then formulates the Home team's price as the inverse of the sum of its probability and about 1%. He then sets the overround on the Away team's price to ensure that the market as a whole has overround equal to T.
For which we have:
- p1 = 1/m1 - T/2 [or, equivalently, m1 = 2/(2p1 + T)]
- Ov1 = T/p1
- Ov2 = T/(2(1-p1))
(The Risk-Equalising Approach is the LPS-Optimising Approach with k = T/2.)
In this approach we imagine that the Bookmaker makes an assessment of the Home team's chances and then formulates the Home team price and that of the Away team to ensure that these prices provide a buffer against the same-sized calibration errors and that the total overround in the market is T. Those assumptions lead to the equations shown above (see this blog for the derivation.)
For which we have:
- p1 = m2 / (m1 + m2)
- Ov1 = Ov2 = T
In this approach we imagine that the Bookmaker makes an assessment of the Home team's chances and then formulates the Home team's price and that of the Away team to ensure that the ratio of these prices is the inverse of the ratio of these probabilities and that the total overround in the market is T.
COMPARING THE APPROACHES
Armed with the equations above you can take the observed head-to-head market prices in any contest and calculate for all three approaches the implicit probabilities and embedded overround figures for both teams.
I've done just that for a range of prices for the Home team in contests where the total overround in the market is exactly 5%, which is about the average overround for all contests in 2012.
The first set of columns provides the Implicit Probability for the Home team for the three Approaches. So, for example, if the Home team were priced at $1.80, the LPS-Optimised Approach would take this to mean that the Bookmaker assessed the Home team's chances as 54.5%, the Risk-Equalising Approach would assess them a little lower at 53.1%, while the Overround-Equalising Approach would assess them lowest of all at 52.9%.
As you can see, the differences amongst the three approaches are all quite small across the entire range of Home team prices. The LPS-Optimised Approach always provides a higher implicit probability estimate for the Home team than does the Risk-Equalising Approach at the same price - because it assumes there is less overround embedded in the Home team's price - while the Overround-Equalising Approach provides a higher implicit probability estimate for the Home team for prices above about $5.
The next set of columns provides the embedded overround in the Home team's price for each of the approaches. The LPS-Optimised Approach returns quite low estimates of the overround for prices below about $3 and only reaches double-digit overround figures for Home team prices in the $10 range and higher.
The Risk-Equalising Approach is already at a 2.6% overround estimate for a Home team price as low as $1.01 but increases this estimate only slowly, reaching 10% at a Home team price of about $3.65. Heavy underdog Home teams are estimated as carrying significant overround. Meantime, the Overround-Equalising Approach, true to its name, estimates a 5% overround in the Home team's price regardless of what that price is.
Finally, the third set of columns provides the embedded overround in the Away team's price for each of the approaches. The LPS-Optimised Approach estimates very high levels of overround in the prices of Away teams that are rank underdogs, reducing this estimate below 10% by about the time the Home team price reaches $1.65, and then slowly reducing it further as the Home team's price increases, though never providing an estimated overround below 4%.
For the Risk-Equalising Approach, the overround story for the Away team is the exact mirror-image of that of the Home team while, for the Overround-Equalising Approach, a flat 5% estimate regardless of price is its considered opinion.
We now have:
- a new input to consider in MAFL modelling and analysis: the LPS-Optimising Implicit Probabilities
- another Probability Predictor to follow, the LPS-Optimising Predictor
- another Margin Predictor to track, the LPS-Optimising Margin Predictor
I've not yet decided whether or not to add these measures to the MAFL Tipster Leaderboard but, at the very least, we'll be checking in on them occasionally during the course of the season.