Using Risk-Equalising Probabilities for the Margin Predictors

With the exception of Combo_NN_2, all of the Margin Predictors rely on an algorithm that takes Bookmaker Implicit Probabilities as an input in some form: 

• Bookie_3 and Bookie_9 use Bookmaker Implicit Probabilities directly
• ProPred_3 and ProPred_7 use the outputs of the ProPred algorithm, which uses a log transform of Bookmaker Implicit Probabilities as one input
• WinPred_3 and WinPred_7 use the outputs of the WinPred algorithm, which also uses a log transform of Bookmaker Implicit Probabilities as one input
• H2H_U3, H2H_U10, H2H_A3 and H2H_A7 use the outputs of the Head-to-Head algorithm, which uses Bookmaker Implicit Probabilities as one input
• Combo_7 uses Bookmaker Implicit Probabilities directly as well as via its use of the outputs of the Head-to-Head Algorithm
• Combo_NN_2 uses Bookmaker Implicit Probabilities directly as well as via its use of the outputs of the ProPred, WinPred and H2H algorithms

For this short blog I've switched, in all of the underlying algorithms, the Implicit Probabilities calculated using the Risk-Equalising Approach as replacements for those calculated using the Overround-Equalising Approach and then compared the resulting MAPEs for seasons 2007 to 2012 for all the Margin Predictors.

Overall, all Margin Predictors except Bookie_3 benefit from the switch, however modestly. Bookie_9, which now will serve as a co-predictor in the MAFL Margin Fund, benefits most, knocking over one quarter of a point per game off its MAPE.

The uniformity of these improvements is made slightly more remarkable by the realisation that the Margin Predictors, built using Eureqa, were optimised for the probability outputs of the underlying algorithms when those algorithms were using Overround-Equalising Implicit Probabilities. So, for example, the equation for Bookie_9, which is:

Predicted Home Team Margin = 2.2205129 + 17.729506 * ln(Home Team Bookmaker Probability/(1-Home Team Bookmaker Probability)) + 2*Home Team Bookmaker Probability

was created by Eureqa to minimise the historical MAPE of this equation when the Home Team Bookmaker Probabilities being used were those calculated assuming Overround-Equalisation. The 0.26 points per game reduction in the MAPE is being achieved without re-optimising this equation but, instead, simply by replacing the Home Team Probabilities with those calculated using a Risk-Equalising Approach.

Bookie_3 is the one Margin Predictor that responds poorly to the switch of probabilities without an accompanying re-optimisation in Eureqa. When I performed such a re-optimisation, Eureqa came up with this remarkably simple equation:

Predicted Home Team Margin = 21 * ln(Home Team Bookmaker Probability/(1-Home Team Bookmaker Probability))

This predictor has an MAPE of 29.22 points per game, which is extraordinarily low for such an easy-to-use predictor.

CONCLUSION

Virtually every algorithm used in MAFL has now been shown to benefit, however slightly, from using Implicit Probabilities calculated using the Risk-Equalising instead of the Overround-Equalising Approach. Naturallly, this makes me wonder if there's an even better way ...

Maybe next year I'll look for it.

Bookmaker Implicit Probabilities: Empirical Value of the Risk-Equalising Approach

A few blogs back I developed the idea that bookmakers might embed overround in each team's price not equally but instead such that the resulting head-to-head market prices provide insurance for a fixed (in percentage point terms) calibration error of equivalent size for both teams. Since then I've made only passing comment about the empirical superiority of this approach (which I've called the Risk-Equalising Approach) relative to the previous approach (which I've called the Overround-Equalising Approach).

Click to read more ...

Determining Bookmaker Implicit Probabilities: The Risk-Equalising Approach

In the previous blog I developed a new way of divining a bookmaker's probability assessments of the two teams by assuming that he believes his maximum calibration error - the (negative) difference between his probability assessment for a team and its true probability of victory - is the same for each team in percentage point terms, and that he levies overround on each team's price so as to ensure that it will still deliver an expected profit even if his probability assessment is maximally in error.

Click to read more ...

Measuring Bookmaker Calibration Errors

We've found ample evidence in the past to assert that the TAB Bookmaker is well-calibrated, by which I mean that teams he rates as 40% chances tend to win about 40% of the time, teams he rates as 90% chances tend to win about 90% of the time and, more generally, that teams he rates as X% chances tend to win about X% of the time.

Click to read more ...

Does an Extra Day's Rest Matter in the Home and Away Season?

Whenever the draw for a new season is revealed there's much discussion about the teams that face one another only once, about which teams need to travel interstate more than others, and about which teams are asked to play successive games with fewer days rest. There is in the discussion an implicit assumption that more days rest is better than fewer days rest but, to my knowledge, this is never supported by empirical analysis. It is, like much of the discussion about football, considered axiomatic. In this blog we'll assess how reasonable that assumption is.

Click to read more ...