Some blogs almost write themselves. This hasn't been one of them.
It all started when I read a journal article - to which I'd now link if I could find the darned thing again - that suggested a bias in NFL (or maybe it was College Football) spread betting markets arising from bookmakers' tendency to over-correct when a team won on line betting. The authors found that after a team won on line betting one week it was less likely to win on line betting next week because it was forced to overcome too large a handicap.
Naturally, I wondered if this was also true of AFL spread betting.
In the process of investigating that question, I wound up wondering about the process of setting handicaps in the first place and what causes a team's handicap to change from one week to the next.
Logically, I reckoned, the start that a team receives could be described by this equation:
Start received by Team A (playing Team B) = (Quality of Team B - Quality of Team A) - Home Status for Team A
In words, the start that a team gets is a function of its quality relative to its opponent's (measured in points) and whether or not it's playing at home. The stronger a team's opponent the larger will be the start, and there'll be a deduction if the team's playing at home. This formulation of start assumes that game venue only ever has a positive effect on one team and not an additional, negative effect on the other. It excludes the possibility that a side might be a P point worse side whenever it plays away from home.
With that as the equation for the start that a team receives, the change in that start from one week to the next can be written as:
Change in Start received by Team A = Change in Quality of Team A + Difference in Quality of Teams played in successive weeks + Change in Home Status for Team A
To use this equation for we need to come up with proxies for as many of the terms that we can. Firstly then, what might a bookie use to reassess the quality of a particular team? An obvious choice is the performance of that team in the previous week relative to the bookie's expectations - which is exactly what the handicap adjusted margin for the previous week measures.
Next, we could define the change in home status as follows:
This formulation implies that there's no difference between playing away and playing at a neutral venue. Varying this assumption is something that I might explore in a future blog.
(well, actually there's a bit more theory too ...)
Having identified a way to quantify the change in a team's quality and the change in its home status we can now run a linear regression in which, for simplicity, I've further assumed that home ground advantage is the same for every team.
We get the following result using all home-and-away data for seasons 2006 to 2010:
For a team (designated to be) playing at home in the current week:
Change in start = -2.453 - 0.072 x HAM in Previous Week - 8.241 x Change in Home Status
For a team (designated to be) playing away in the current week:
Change in start = 3.035 - 0.155 x HAM in Previous Week - 8.241 x Change in Home Status
These equations explain about 15.7% of the variability in the change in start and all of the coefficients (except the intercept) are statistically significant at the 1% level or higher.
(You might notice that I've not included any variable to capture the change in opponent quality. Doubtless that variable would explain a significant proportion of the otherwise unexplained variability in change in start but it suffers from the incurable defect of being unmeasurable for previous and for future games. That renders it not especially useful for model fitting or for forecasting purposes.
Whilst that's a shame from the point of view of better modelling the change in teams' start from week-to-week, the good news is that leaving this variable out almost certainly doesn't distort the coefficients for the variables that we have included. Technically, the potential problem we have in leaving out a measure of the change in opponent quality is what's called an omitted variable bias, but such bias disappears if the the variables we have included are uncorrelated with the one we've omitted. I think we can make a reasonable case that the difference in the quality of successive opponents is unlikely to be correlated with a team's HAM in the previous week, and is also unlikely to be correlated with the change in a team's home status.)
Using these equations and historical home status and HAM data, we can calculate that the average (designated) home team receives 8 fewer points start than it did in the previous week, and the average (designated) away team receives 8 points more.
Okay, so what do these equations tell us?
Firstly let's consider teams playing at home in the current week. The nature of the AFL draw is such that it's more likely than not that a team playing at home in one week played away in the previous week in which case the Change in Home Status for that team will be +1 and their equation can be rewritten as
Change in Start = -10.694 - 0.072 x HAM in Previous Week
So, the start for a home team will tend to drop by about 10.7 points relative to the start it received in the previous week (because they're at home this week) plus about another 1 point for every 14.5 points lower their HAM was in the previous week. Remember: the more positive the HAM, the larger the margin by which the spread was covered.
Next, let's look at teams playing away in the current week. For them it's more likely than not that they played at home in the previous week in which case the Change in Home Status will be -1 for them and their equation can be rewritten as
Change in Start = 11.276 - 0.155 x HAM in Previous Week
Their start, therefore, will tend to increase by about 11.3 points relative to the previous week (because they're away this week) less 1 point for every 6.5 points lower their HAM was in the previous week.
Away teams, therefore, are penalised more heavily for larger HAMs than are home teams.
This I offer as one source of potential bias, similar to the bias that was found in the original article I read.
As a simple way of quantifying any bias I've fitted what's called a binary logit to estimate the following model:
Probability of Winning on Line Betting = f(Result on Line Betting in Previous Week, Start Received, Home Team Status)
This model will detect any bias in line betting results that's due to an over-reaction to the previous week's line betting results, a tendency for teams receiving particular sized starts to win or lose too often, or to a team's home team status.
The result is as follows:
logit(Probability of Winning on Line Betting) = -0.0269 + 0.054 x Previous Line Result + 0.001 x Start Received + 0.124 x Home Team Status
The only coefficient that's statistically significant in that equation is the one on Home Team Status and it's significant at the 5% level. This coefficient is positive, which implies that home teams win on line betting more often than they should.
Using this equation we can quantify how much more often. An away team, we find, has about a 46% probability of winning on line betting, a team playing at a neutral venue has about a 49% probability, and a team playing at home has about a 52% probability.
That is undoubtedly a bias, but I have two pieces of bad news about it. Firstly, it's not large enough to overcome the vig on line betting at $1.90 and secondly, it disappeared in 2010.
We now know something about how the points start given by the TAB Sportsbet bookie responds to a team's change in estimated quality and to a change in a team's home status. Do the actual game margins respond similarly?
One way to find this out is to use exactly the same equation as we used above, replacing Change in Start with Change in Margin and defining the change in a team's margin as its margin of victory this week less its margin of victory last week (where victory margins are negative for losses).
If we do that and run the new regression model, we get the following:
For a team (designated to be) playing at home in the current week:
Change in Margin = 4.058 - 0.865 x HAM in Previous Week + 8.801 x Change in Home Status
For a team (designated to be) playing away in the current week:
Change in Margin = -4.571 - 0.865 x HAM in Previous Week + 8.801 x Change in Home Status
These equations explain an impressive 38.7% of the variability in the change in margin. We can simplify them, as we did for the regression equations for Change in Start, by using the fact that the draw tends to alternate team's home and away status from one week to the next.
So, for home teams:
Change in Margin = 12.859 - 0.865 x HAM in Previous Week
While, for away teams:
Change in Margin = -13.372 - 0.865 x HAM in Previous Week
At first blush it seems a little surprising that a team's HAM in the previous week is negatively correlated with its change in margin. Why should that be the case?
It's a phenomenon that we've discussed before: regression to the mean. What these equations are saying are that teams that perform better than expected in one week - where expectation is measured relative to the line betting handicap - are likely to win by slightly less than they did in the previous week or lose by slightly more.
What's particularly interesting is that home teams and away teams show mean regression to the same degree. The TAB Sportsbet bookie, however, hasn't always behaved as if this was the case.
Bringing the Change in Start and Change in Margin equations together provides another window into the home team bias.
The simplified equations for Change in Start were:
Home Teams: Change in Start = -10.694 - 0.072 x HAM in Previous Week
Away Teams: Change in Start = 11.276 - 0.155 x HAM in Previous Week
So, for teams whose previous HAM was around zero (which is what the average HAM should be), the typical change in start will be around 11 points - a reduction for home teams, and an increase for away teams.
The simplified equations for Change in Margin were:
Home Teams: Change in Margin = 12.859 - 0.865 x HAM in Previous Week
Away Teams: Change in Margin = -13.372 - 0.865 x HAM in Previous Week
So, for teams whose previous HAM was around zero, the typical change in margin will be around 13 points - an increase for home teams, and a decrease for away teams.
Overall the 11-point v 13-point disparity favours home teams since they enjoy the larger margin increase relative to the smaller decrease in start, and it disfavours away teams since they suffer a larger margin decrease relative to the smaller increase in start.
Historically, home teams win on line betting more often than away teams. That means home teams tend to receive too much start and away teams too little.
I've offered two possible reasons for this:
If you've made it this far, my sincere thanks. I reckon your brain's earned a spell; mine certainly has.
Across the 111 Grand Finals in VFL/AFL history - excluding the two replays - only 18 of them, or about 1-in-6, has seen the team finishing 1st on the home-and-away ladder play the team finishing 3rd.
This year, of course, will be the nineteenth.
Far more common, as you'd expect, has been a matchup between the teams from 1st and 2nd on the ladder. This pairing accounts for 56 Grand Finals, which is a smidgeon over half, and has been so frequent partly because of the benefits accorded to teams finishing in these positions by the various finals systems that have been in use, and partly no doubt because these two teams have tended to be the best two teams.
In the 18 Grand Finals to date that have involved the teams from 1st and 3rd, the minor premier has an 11-7 record, which represents a 61% success rate. This is only slightly better than the minor premiers' record against teams coming 2nd, which is 33-23 or about 59%.
Overall, the minor premiers have missed only 13 of the Grand Finals and have won 62% of those they've been in.
By comparison, teams finishing 2nd have appeared in 68 Grand Finals (61%) and won 44% of them. In only 12 of those 68 appearances have they faced a team from lower on the ladder; their record for these games is 7-5, or 58%.
Teams from 3rd and 4th positions have each made about the same number of appearances, winning a spot about 1 year in 4. Whilst their rates of appearance are very similar, their success rates are vastly different, with teams from 3rd winning 46% of the Grand Finals they've made, and those from 4th winning only 27% of them.
That means that teams from 3rd have a better record than teams from 2nd, largely because teams from 3rd have faced teams other than the minor premier in 25% of their Grand Final appearances whereas teams from 2nd have found themselves in this situation for only 18% of their Grand Final appearances.
Ladder positions 5 and 6 have provided only 6 Grand Finalists between them, and only 2 Flags. Surprisingly, both wins have been against minor premiers - in 1998, when 5th-placed Adelaide beat North Melbourne, and in 1900 when 6th-placed Melbourne defeated Fitzroy. (Note that the finals systems have, especially in the early days of footy, been fairly complex, so not all 6ths are created equal.)
One conclusion I'd draw from the table above is that ladder position is important, but only mildly so, in predicting the winner of the Grand Final. For example, only 69 of the 111 Grand Finals, or about 62%, have been won by the team finishing higher on the ladder.
It turns out that ladder position - or, more correctly, the difference in ladder position between the two grand finalists - is also a very poor predictor of the margin in the Grand Final.
This chart shows that there is a slight increase in the difference between the expected number of points that the higher-placed team will score relative to the lower-placed team as the gap in their respective ladder positions increases, but it's only half a goal per ladder position.
What's more, this difference explains only about half of one percentage of the variability in that margin.
Perhaps, I thought, more recent history would show a stronger link between ladder position difference and margin.
Quite the contrary, it transpires. Looking just at the last 20 years, an increase in the difference of 1 ladder position has been worth only 1.7 points in increased expected margin.
Come the Grand Final, it seems, some of your pedigree follows you onto the park, but much of it wanders off for a good bark and a long lie down.
Creating the recent blog on predicting the Grand Final margin based on the difference in the teams' MARS Ratings set me off once again down the path of building simple models to predict game margin.
It usually doesn't take much.
Firstly, here's a simple linear model using MARS Ratings differences that repeats what I did for that recent blog post but uses every game since 1999, not just Grand Finals.
It suggests that you can predict game margins - from the viewpoint of the home team - by completing the following steps:
One interesting feature of this model is that it suggests that home ground advantage is worth about 10 points.
The R-squared number that appears on the chart tells you that this model explains 21.1% of the variability is game margins.
You might recall we've found previously that we can do better than this by using the home team's victory probability implied by its head-to-head price.
This model says that you can predict the home team margin by multiplying its implicit probability by 105.4 and then subtracting 48.27. It explains 22.3% of the observed variability in game margins, or a little over 1% more than we can explain with the simple model based on MARS Ratings.
With this model we can obtain another estimate of the home team advantage by forecasting the margin with a home team probability of 50%. That gives an estimate of 4.4 points, which is much smaller than we obtained with the MARS-based model earlier.
(EDIT: On reflection, I should have been clearer about the relative interpretation of this estimate of home ground advantage in comparison to that from the MARS Rating based model above. They're not measuring the same thing.
The earlier estimate of about 10 points is a more natural estimate of home ground advantage. It's an estimate of how many more points a home team can be expected to score than an away team of equal quality based on MARS Rating, since the MARS Rating of a team for a particular game does not include any allowance for whether or not it's playing at home or away.
In comparison, this latest estimate of 4.4 points is a measure of the "unexpected" home ground advantage that has historically accrued to home teams, over-and-above the advantage that's already built into the bookie's probabilities. It's a measure of how many more points home teams have scored than away teams when the bookie has rated both teams as even money chances, taking into account the fact that one of the teams is (possibly) at home.
It's entirely possible that the true home ground advantage is about 10 points and that, historically, the bookie has priced only about 5 or 6 points into the head-to-head prices, leaving the excess of 4.4 that we're seeing. In fact, this is, if memory serves me, consistent with earlier analyses that suggested home teams have been receiving an unwarranted benefit of about 2 points per game on line betting.
Which, again, is why MAFL wagers on home teams.)
Perhaps we can transform the probability variable and explain even more of the variability in game margins.
In another earlier blog we found that the handicap a team received could be explained by using what's called the logit transformation of the bookie's probability, which is ln(Prob/(1-Prob)).
Let's try that.
We do see some improvement in the fit, but it's only another 0.2% to 22.5%. Once again we can estimate home ground advantage by evaluating this model with a probability of 50%. That gives us 4.4 points, the same as we obtained with the previous bookie-probability based model.
A quick model-fitting analysis of the data in Eureqa gives us one more transformation to try: exp(Prob). Here's how that works out:
We explain another 0.1% of the variability with this model as we inch our way to 22.6%. With this model the estimated home-ground advantage is 2.6 points, which is the lowest we've seen so far.
If you look closely at the first model we built using bookie probabilities you'll notice that there seems to be more points above the fitted line than below it for probabilities from somewhere around 60% onwards.
Statistically, there are various ways that we could deal with this, one of which is by using Multivariate Adaptive Regression Splines.
(The algorithm in R - the statistical package that I use for most of my analysis - with which I created my MARS models is called earth since, for legal reasons, it can't be called MARS. There is, however, another R package that also creates MARS models, albeit in a different format. The maintainer of the earth package couldn't resist the temptation not to call the function that converts from one model format to the other mars.to.earth. Nice.)
The benefit that MARS models bring us is the ability to incorporate 'kinks' in the model and to let the data determine how many such kinks to incorporate and where to place them.
Running earth on the bookie probability and margin data gives the following model:
Predicted Margin = 20.7799 + if(Prob > 0.6898155, 162.37738 x (Prob - 0.6898155),0) + if(Prob < 0.6898155, -91.86478 x (0.6898155 - Prob),0)
This is a model with one kink at a probability of around 69%, and it does a slightly better job at explaining the variability in game margins: it gives us an R-squared of 22.7%.
When you overlay it on the actual data, it looks like this.
You can see the model's distinctive kink in the diagram, by virtue of which it seems to do a better job of dissecting the data for games with higher probabilities.
It's hard to keep all of these models based on bookie probability in our head, so let's bring them together by charting their predictions for a range of bookie probabilities.
For probabilities between about 30% and 70%, which approximately equates to prices in the $1.35 to $3.15 range, all four models give roughly the same margin prediction for a given bookie probability. They differ, however, outside that range of probabilities, by up to 10-15 points. Since only about 37% of games have bookie probabilities in this range, none of the models is penalised too heavily for producing errant margin forecasts for these probability values.
So far then, the best model we've produced has used only bookie probability and a MARS modelling approach.
Let's finish by adding the other MARS back into the equation - my MARS Ratings, which bear no resemblance to the MARS algorithm, and just happen to share a name. A bit like John Howard and John Howard.
This gives us the following model:
Predicted Margin = 14.487934 + if(Prob > 0.6898155, 78.090701 x (Prob - 0.6898155),0) + if(Prob < 0.6898155, -75.579198 x (0.6898155 - Prob),0) + if(MARS_Diff < -7.29, 0, 0.399591 x (MARS_Diff + 7.29)
The model described by this equation is kinked with respect to bookie probability in much the same way as the previous model. There's a single kink located at the same probability, though the slope to the left and right of the kink is smaller in this latest model.
There's also a kink for the MARS Rating variable (which I've called MARS_Diff here), but it's a kink of a different kind. For MARS Ratings differences below -7.29 Ratings points - that is, where the home team is rated 7.29 Ratings points or more below the away team - the contribution of the Ratings difference to the predicted margin is 0. Then, for every 1 Rating point increase in the difference above -7.29, the predicted margin goes up by about 0.4 points.
This final model, which I think can still legitimately be called a simple one, has an R-squared of 23.5%. That's a further increase of 0.8%, which can loosely be thought of as the contribution of MARS Ratings to the explanation of game margins over and above that which can be explained by the bookie's probability assessment of the home team's chances.
During the week I'm sure I'll have a number of attempts at predicting the result of the Grand Final - after all, the more predictions you make about the same event, the better your chances of generating at least one that's remembered for its accuracy, long after the remainder have faded from memory.
In this brief blog the entrails I'll be contemplating come from a review of the relationship between Grand Finalists' MARS Ratings and the eventual result for each of the 10 most recent Grand Finals.
Firstly, here's the data:
In seven of the last 10 Grand Finals the team with the higher MARS Rating has prevailed. You can glean this from the fact that the rightmost column contains only three negative values indicating that the team with the higher MARS Rating scored fewer points in the Grand Final than the team with the lower MARS Rating.
What this table also reveals is that:
From the raw data alone it's difficult to determine if there's much of a relationship between the Grand Finalists' MARS Ratings and their eventual result. Much better to use a chart:
The dots each represent a single Grand Final and the line is the best fitting linear relationship between the difference in MARS Ratings and the eventual Grand Final score difference. As well as showing the line, I've also included the equation that describes it, which tells us that the best linear predictor of the Grand Final margin is that the team with the higher MARS Rating will win by a margin equal to about 1.06 times the difference in the teams' MARS Ratings less a bit under 1 point.
For this year's Grand Final that suggests that Collingwood will win by 1.062 x 26.1 - 0.952, which is just under 27 points. (I've included this in gray in the table above.)
One measure of the predictive power of the equation I've used here is the proportion of variability in Grand Final margins that it's explained historically. The R-squared of 0.172 tells us that this proportion is about 17%, which is comforting without being compelling.
We can also use a model fitted to the last 10 Grand Finals to create what are called confidence intervals for the final result. For example, we can say that there's a 50% chance that the result of the Grand Final will be in the range spanning a 5-point loss for the Pies to a 59-point win, which demonstrates just how difficult it is to create precise predictions when you've only 10 data points to play with.
Previously, on at least a few occasions, I've looked at the topic of the Favourite-Longshot Bias and whether or not it exists in the TAB Sportsbet wagering markets for AFL.
A Favourite-Longshot Bias (FLB) is said to exist when favourites win at a rate in excess of their price-implied probability and longshots win at a rate less than their price-implied probability. So if, for example, teams priced at $10 - ignoring the vig for now - win at a rate of just 1 time in 15, this would be evidence for a bias against longshots. In addition, if teams priced at $1.10 won, say, 99% of the time, this would be evidence for a bias towards favourites.
When I've considered this topic in the past I've generally produced tables such as the following, which are highly suggestive of the existence of such an FLB.
Each row of this table, which is based on all games from 2006 to the present, corresponds to the results for teams with price-implied probabilities in a given range. The first row, for example, is for all those teams whose price-implied probability was less than 10%. This equates, roughly, to teams priced at $9.50 or more. The average implied probability for these teams has been 9%, yet they've won at a rate of only 4%, less than one-half of their 'expected' rate of victory.
As you move down the table you need to arrive at the second-last row before you come to one where the win rate exceed the expected rate (ie the average implied probability). That's fairly compelling evidence for an FLB.
This empirical analysis is interesting as far as it goes, but we need a more rigorous statistical approach if we're to take it much further. And heck, one of the things I do for a living is build statistical models, so you'd think that by now I might have thrown such a model at the topic ...
A bit of poking around on the net uncovered this paper which proposes an eminently suitable modelling approach, using what are called conditional logit models.
In this formulation we seek to explain a team's winning rate purely as a function of (the natural log of) its price-implied probability. There's only one parameter to fit in such a model and its value tells us whether or not there's evidence for an FLB: if it's greater than 1 then there is evidence for an FLB, and the larger it is the more pronounced is the bias.
When we fit this model to the data for the period 2006 to 2010 the fitted value of the parameter is 1.06, which provides evidence for a moderate level of FLB. The following table gives you some idea of the size and nature of the bias.
The first row applies to those teams whose price-implied probability of victory is 10%. A fair-value price for such teams would be $10 but, with a 6% vig applied, these teams would carry a market price of around $9.40. The modelled win rate for these teams is just 9%, which is slightly less than their implied probability. So, even if you were able to bet on these teams at their fair-value price of $10, you'd lose money in the long run. Because, instead, you can only bet on them at $9.40 or thereabouts, in reality you lose even more - about 16c in the dollar, as the last column shows.
We need to move all the way down to the row for teams with 60% implied probabilities before we reach a row where the modelled win rate exceeds the implied probability. The excess is not, regrettably, enough to overcome the vig, which is why the rightmost entry for this row is also negative - as, indeed, it is for every other row underneath the 60% row.
Conclusion: there has been an FLB on the TAB Sportsbet market for AFL across the period 2006-2010, but it hasn't been generally exploitable (at least to level-stake wagering).
The modelling approach I've adopted also allows us to consider subsets of the data to see if there's any evidence for an FLB in those subsets.
I've looked firstly at the evidence for FLB considering just one season at a time, then considering only particular rounds across the five seasons.
So, there is evidence for an FLB for every season except 2007. For that season there's evidence of a reverse FLB, which means that longshots won more often than they were expected to and favourites won less often. In fact, in that season, the modelled success rate of teams with implied probabilities of 20% or less was sufficiently high to overcome the vig and make wagering on them a profitable strategy.
That year aside, 2010 has been the year with the smallest FLB. One way to interpret this is as evidence for an increasing level of sophistication in the TAB Sportsbet wagering market, from punters or the bookie, or both. Let's hope not.
Turning next to a consideration of portions of the season, we can see that there's tended to be a very mild reverse FLB through rounds 1 to 6, a mild to strong FLB across rounds 7 to 16, a mild reverse FLB for the last 6 rounds of the season and a huge FLB in the finals. There's a reminder in that for all punters: longshots rarely win finals.
Lastly, I considered a few more subsets, and found:
FLB: done.
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