# SuperMargin Implications? Yes, They Are Atrocious.

/In a recent blog I developed an empirical model of AFL scoring in which I assumed that the Scoring Shots generated by Home and Away teams could be modelled by a bivariate Negative Binomial and that the conversion of these shots into Goals could be modelled by Beta Binomials.

That model allows me to draw implications about the distribution of aspects of AFL games such as the margin between the Home team and the Away team score (the game margin) and the aggregate of the Home team and the Away team score, based on assumptions about the pre-game expected scores or scoring shot production of each team.

For example, if I were to assume that the pre-game expectation were that Sydney would beat Richmond by 95.5 to 74 then I could use the model to make inferences about the actual final game margin and aggregate score.

In the SuperMargin market the Bookmaker offers prices for different ranges of final game margins, and in this blog I'll be investigating what the model suggests fair prices for those ranges might be.

Consider this week's Sydney v Richmond game where the TAB Bookmaker has the Swans as 21.5 point favourites and is offering even money in the Overs/Unders market at 169.5 points. We can infer from this that he expects the Swans to win by 95.5 to 74, from which we can calculate his Scoring Shot expectations (assuming a 53% conversion rate for both teams) and then use the scoring model I've developed to make inferences.

The left-hand block of columns in the table below present the results.

We see, for example, that - based on 1,000,000 simulations of the scoring model - the Swans' probability of winning by 20 to 29 points, the most likely SuperMargin outcome, is estimated to be 11.7%. A fair price for that bucket would therefore be $8.56, but the TAB is currently offering only $7, which represents a 22% overround.

That overround figure is the lowest of any bucket on offer, some of them having levels of overround that would make even the sharpest-toothed loan shark blush. SuperMargin wagering seems unattractively challenging if the Bookmaker's game margin assessments are well-calibrated.

In fact, if you cast your eyes across the range of fair prices you can see that $7 is never fair for any bucket, and even a price of $10 carries a positive expectation only if the Bookmaker's assessment of the true margin is in error by a couple of buckets, so this form of wagering looks more generally unattractive unless you've compelling reasons to believe the Bookmaker is grossly in error in his assessment of the true expected game margin.

The figures at the foot of the table provide information of a similar kind about the size of the errors a Bookmaker could make in other betting forms and still enjoy an expectation of profit. For our Sydney v Richmond game it shows that the Bookmaker could have set a Line market handicap anywhere in the 19.5 to 23.5 point range and still be comfortable that a bettor, knowing the true expected margin was 21.5 points, would not enjoy a positive expectation betting at $1.90. This is because, according to the modelled results, Sydney's probability of winning by any margin in that range lies between 1/1.9 and 1-1/1.9, so there's no positive expectation in taking either side of the wager.

Similarly, the figures below this reveal the safe Over/Under ranges for a Bookmaker offering $1.90 or, as the TAB, $1.87 in this market. You can see the motivation for offering only $1.87 when you recognise that it expands the safe range by two points in comparison to that which applies to a $1.90 price.

The remainder of the table provides the same information for a range of hypothetical pre-game expected scores, from which I'd draw your attention to a few things:

- Higher expected aggregate scores tend to broaden the range of feasible buckets, most notably for buckets more distant from the most likely bucket. Compare, for example, the columns for the 110-90 scenario to the 90-70 scenario. These both, clearly, have the same expected game margin, but the 60 to 69 bucket moves from being a 5.3% proposition to a 4.8% proposition and the -50 to -59 bucket moves from being a 1.5% to a 0.9% proposition.
- Never does anything less than $8 represent value.
- The Draw is never value at less than about $90.
- Buckets at the extreme, especially in games where the expected margin is relatively small in absolute terms, will always carry a huge overround impost given that prices above $501 for buckets are very rare.
- The 1 to 9 and -1 to -9 buckets suffer from being 1 point smaller than (almost) every other bucket.
- At $1.90, the Bookmaker generally has a 3- or 4-point buffer in the Line market.
- At $1.90, he has usually a 3-point buffer in the Over/Under market, while at $1.97 that usually expands to a 5-point buffer.

(I swear, by the way, that I didn't concoct this analysis solely for the purpose of inflicting the appalling pun that is this blog's title on an unsuspecting audience - but I will admit to being childishly proud of it.)