Bookmaker Overround: Relating Team Overround to Victory Probability

In the previous blog I described a general framework for thinking about Bookmaker overround.

There I discussed, in the context of the two-outcome case, the choice of a functional form to describe a team's overround in terms of its true probability as assessed by the Bookmaker. As one very simple example I suggested oi = (1-pi), which we could use to model a Bookmaker who embeds overround in the price of any team by setting it to 1 minus the team's assessed probability of victory. 

Whilst we could choose just about any function, including the one I've just described, for the purpose of modelling Bookmaker overround, choices that fit empirical reality are actually, I now realise, quite circumscribed. This is because of the observable fact that the total overround in any head-to-head market, T, appears to be constant, or close to it, in every game regardless of the market prices, and hence the underlying true probability assessments, of the teams involved. In other words, the total overround in the head-to-head market when 1st plays last is about the same as when 1st plays 2nd.

So, how does this constrain our choice of functional form? Well we know that T is defined as 1/m1 + 1/m2 - 1, where mi is that market price for team i, and that mi = 1/(pi(1+oi)), from which we can determine that: 

• T = p1(o1 - o2) + o2

If T is to remain constant across the full range of values of p1 then, we need the derivative with respect to p1 of the RHS of this equation to be zero for all values of p1. This implies that the functions chosen for o1 and o2 must satisfy the following equality: 

• p1(o1' - o2') + o2' = o2 - o1 (where the dash signifies a derivative with respect to p1).

I doubt that many functional forms o1 and o2 (both of which we're assuming are functions of p1, by the way) exist that will satisfy this equation for all values of p1, especially if we also impose the seemingly reasonable constraint that o1 and o2 be of equivalent form, albeit it that o1 might be expressed in terms of p1 and o2 in terms of (1-p1), which we can think of as p2.

Two forms that do satisfy the equation, the proof of which I'll leave as an exercise for any interested reader to check, are: 

• The Overround-Equalising approach : o1 = o2 = k, a constant, and
• The Risk-Equalising approach : o1 = e/p1; o2 = e/(1-p1), with e a constant 

There may be another functional form that satisfies the equality above, but I can't find it. (There's a rigorous non-existence proof for you.) Certainly oi = 1 - pi, which was put forward earlier, doesn't satisfy it, and I can postulate a bunch of other plausible functional forms that similarly fail. What you find when you use these forms is that total overround changes with the value of p1.

So, if we want to choose functions for o1 and o2 that produce results consistent with the observed reality that total overround remains constant across all values of the assessed true probability of the two teams it seems that we've only three options (maybe four): 

1. Assume that the Bookmaker follows the Overround-Equalising approach
2. Assume that the Bookmaker follows the Risk-Equalising approach
3. Assume that the Bookmaker chooses one team, say the favourite or the home team, and establishes its overround using a pre-determined function relating its overround to its assessed victory probability. He then sets a price for the other team that delivers the total overround he is targetting. This is effectively the path I followed in this earlier blog where I described what's come to be called the Log Probability Score Optimising (LPSO) approach.

A fourth, largely unmodellable option would be that he simultaneously sets the market prices of both teams so that they together produce a market with the desired total overround while accounting for his assessment of the two team's victory probabilities so that a wager on either team has negative expectation. He does this, we'd assume, without employing a pre-determined functional form for the relationship between overround and probability for either team. 

If these truly are the only logical options available to the Bookmaker then MAFL, it turns out, is already covering the complete range since we track the performance of a Predictor that models its probability assessments by following an Overround-Equalising approach, of another Predictor that does the same using a Risk-Equalising approach, and of a third (Bookie_LPSO) that pursues a strategy consistent with the third option above. That's serendipitously neat and tidy.

The only area for future investigation would be then to seek a solution superior to LPSO for the third approach described above. Here we could use any of the functional forms I listed in the previously blog, but could only apply them to the determination of the overround for one of the teams - say the home team or the favourite - with the price and hence overround for the remaining team determined by the need to produce a market with some pre-specified total overround.

That's enough for today though ...

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.