MAFL's been very active in the SuperMargin market this season and somewhat successful there, which got me to wondering about the overround Investors might be facing each time they wade into that market.
Previous analyses have provided compelling empirical evidence that the final victory margin in any game can be modelled as a random variable drawn from a Normal distribution with some fixed mean, which we can estimate empirically using the Bookmaker's pre-game head-to-head market prices, his pre-game handicap in the line market, or by using MAFL's team MARS Ratings and the Interstate Status of the clash, and a fixed standard deviation, which I've variously estimated as being somewhere in the 30 to 38 points range (see, for example, this older post or this more-recent post on the topic).
Firstly, let's consider a game where the true expected margin of victory is, very conveniently, exactly in the middle of any of the 10-point buckets in the SuperMargin market - say, for example, it's for a home team victory by 24.5 points. The likelihood of the actual score landing in the 20 to 29 point bucket for the home team can then be estimated, employing the assumption about final margins being Normally distributed discussed above, as the probability density of Normal distribution with mean 24.5 between the values of +19.5 and +29.5.
The value of that probability depends on the value of the standard deviation that we choose, and I've provided the answer for a range of such values in the top section of the table below.
Smaller standard deviations make it more likely that the final score will end up closer to its expected value and so maximises the probability of it landing in the bucket containing that expected margin.
If we choose a standard deviation of 30 points we find that the probability of the final score landing in the 10-point bucket of interest - the 20 to 29 point bucket in the example I gave earlier - is about 13%.
As we increase the standard deviation this probability declines reaching a low of 10.5% for a standard deviation of 38 points.
When the expected margin for a game falls into any of the 10-point buckets, the TAB Bookmaker usually offers a price of $7 for that bucket. The column at the far right of the table shows the overround embedded in that price for different assumptions about the standard deviation.
Note that I've defined overround, as I did in this earlier blog, as the value that satisfies the following equation:
- Price Offered = Fair Price / (1 + Overround) or, equivalently
- Price Offered = 1 / (True Probability x (1 + Overround))
So, even if the true value of the standard deviation is as low as 30 points, a $7 price has 8% embedded overround. If, on the other hand, it's as high as 38 points then the overround is a whopping 36%.
The table also includes the results of calculations made using the same Normality assumption estimating how likely it is that the final result will land in either of the neighbouring buckets, assuming that each of these buckets is also 10-points wide. Roughly speaking, and much as I think you'd expect, it's about twice as likely that the final margin will land in the most-likely bucket than it will land in either of its neighbours.
In the lower half of the table above I've provided the same analysis but assuming that the true expected margin is in the 1 to 9 point bucket - specifically that this true margin is for a 4.5 point win or a 4.5 point loss by the home team - recognising that, in such cases, the TAB Bookmaker generally offers a $7.50 price for this bucket.
That increase in price actually does not fully compensate for the 1-point reduction in the bucket width from 10 to 9 points, as evidenced by the higher levels of embedded overround recorded in the table.
Overrounds of 10-20% appear to be the norm in SuperMargin wagers, making this a particularly difficult market in which to generate returns, especially in the longer term, if the Bookmaker is well-calibrated.
Bookmaker calibration aside, under the assumptions I've made above, which represent the idealised situation where the true expected margin is exactly mid-bucket and so maximise the probability attaching to that bucket, no bucket can logically have a true probability in excess of about 11-13%. Give that, positive expectation cannot exist in any bucket, however mispriced, if it's priced at less than about $8.
This result might explain why wagering using Combo_NN2's predictions has been quite successful this year and why wagering using Bookie_9's predictions has not. Combo_NN2 tends to select buckets other than the bookmaker's most-expected bucket, which means that it often secures prices higher than $7 and, assuming that the TAB Bookmaker has mispriced the market in the range that Combo_NN2 has selected, means that wagering with positive expectation is at least plausible. Bookie_9, in contrast, almost always wagers on buckets priced at $7 which, logically, can never have a positive expectation.
All of which is food for thought for wagering in the SuperMargin market next year ...