I've raised an eyebrow or two more than once when I've seen the TAB bookmaker post two markets with the same head-to-head prices but different line market handicaps priced at even-money.

One explanation would be that these two markets are completely or mostly separate, which would allow the prices in each to be set and fluctuate independently, but I find that possibility unlikely. It might well afford a bettor an arbitrage opportunity across the two markets, which is a rare enough situation when multiple bookmakers are considered, but surely even rarer when the prices are being set within the same betting establishment.

If, instead, the prices is these markets are being deliberately and rationally set in this way, that must lead an inquisitive bettor to wonder about the conclusion that every game's final margin is drawn from a Normal distribution with the same, universal standard deviation, such as I've suggested in this blog from 2009 or this one from 2013. If the head-to-head prices are the same for two games then the implicit home team probability's are the same (assuming that overround is levied equally on the two favourites and, hence, the two underdogs, in the two markets) and, with a fixed standard deviation, that means the appropriate even-money handicap is the same for both games too.

In more-recent years, such as in this more complex analysis from 2014, I've begun to explore the idea that the standard deviation of the final game margin might not be a fixed constant across games of all types, and started to describe games where, at least theoretically, a different standard deviation might prevail.

That latter blog post includes a chart, a prettier and slightly different version of which I include here, both based on the theoretical scoring model I described in that 2014 post.

What we see here a scatter plot, produced using the theoretical scoring models in that same blog, of the standard deviation of the game margin against the expected margin for different, realistic underlying home team and away team Scoring Shot assumptions. The data points shown in this chart are only for games where the expected total score lies in the 120 to 220 point range, which experience suggests spans the reasonable range contemplated by a sane bookmaker. (The chart in the earlier blog, by the way, used Absolute Error, not SD, on the y-axis. It was also decidedly uglier.)

By taking note of the colours of each point, which are determined by the expected total score for the game being simulated, one thing that's immediately apparent from this chart is that larger standard deviations are associated with larger expected total scores, and smaller deviations with smaller expected total scores. For example, for games with expected scores under 145 points (coloured red), standard deviations never exceed 32 points, whereas for games with expected scores of 195 points or more (coloured purple), standard deviations are never below 36.

With that in mind, it's instructive to consider what handicap a bookmaker might apply to a home team to which he attaches some given probability victory, depending on the associated standard deviation, which we'll constrain to the range 30 to 40 (still assuming that margins are drawn from a Normal distribution).

The table at right shows, for example, that a home team whose estimated victory probability is 20% might reasonably be associated with line market handicaps of anywhere between 25.5 and 33.5 points, depending on the assumed standard deviation (which, in turn, is based on an assumed total score).

On the far right of the table, each home-team probability is associated with a specific price that depends on the assumed embedded overround. Note that I define the overround embedded in a single team's price as the value of O that can be derived from the equation:

What's clear (and consistent with empirical observation) is that long- and short-priced favourites are associated with wider ranges of possible handicaps, while near-equal favourites have much narrower handicap ranges. A home team assessed as 85% chances, for example, might endure a handicap of anything between 31.5 and 41.5 points, whereas one assessed as 55% chances is likely to endure a handicap in the much narrower range of 3.5 to 5.5 points.

It's also instructive to take these same equations and solve them for a different variable, in particular to take a given expected margin and convert it to a probability for different standard deviations.

That's what the table at right does and shows, for example, that a bookmaker believing a home team to be 36-point underdogs might reasonably be assessed at anywhere between 12% and 18% chances of victory, depending on the assumed underlying total game score and hence standard deviation of the game margin.

On the far right of this table I've converted the probabilities shown at left to actual market prices given different embedded overrounds and assuming a standard deviation of 36 points (ie the fourth column from the left-hand side of the table).

One aspect of this table that's surprising to me is how small an expected margin is required with an assumed standard deviation of 36 to drive the home team price to levels under the minimum \$1.01 that the TAB sets as a floor. For example, \$1.01 isn't sufficiently small to embed a 5% overround in the home team's price for anything more than a 9-goal favourite. This is, of course, a partial explanation for the favourite-longshot bias.

### CONCLUSION AND CONFIRMATORY ANALYSIS

The key conclusion from this analysis for me is that the observed phenomenon of different line market handicaps for games with identical head-to-head prices is consistent with the belief that game margins are heteroskedastic.

Whilst we've struggled to find empirical evidence for heteroskedasticity in the earlier analyses (linked above), those analyses have not included real-world total score expectations as a predictor of a game's standard deviation. This is because I don't have the necessary data.

We have, of course, demonstrated the possibility of such a link between total scoring and margin standard deviations theoretically, but that result depends, in turn, on the appropriateness of the underlying model of home team and away team scores. The fit and general appropriateness of those models is also empirically untestable without a dataset of bookmaker pre-game markets for individual team scores. This is another set of data that I don't have.

If anyone has such time series - of pre-game under/over markets and/or pre-game team score markets - and would be willing to share them, I'd be very grateful. In the meantime, I've some Google-Fu to be practising.