The Bias in Line Betting Revisited

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.

What Makes a Team's Start Vary from Week-to-Week?

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:

• Change in home status = +1 if a team is playing at home this week and played away or at a neutral venue in the previous week
• Change in home status = -1 if a team is playing away or at a neutral venue this week and played at home in the previous week
• Change in home status = 0 otherwise

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.

From Theory to Practicality: Fitting a Model

(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.

All of which Means Exactly What Now?

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.

Proving the Bias

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.

Do Margins Behave Like Starts?

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.

Another Approach to the Source of the Bias

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.

To Conclude

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:

1. Away teams suffer larger reductions in their handicaps for a given previous weeks' HAM
2. For teams with near-zero previous week HAMs, starts only adjust by about 11 points when a team's home status changes but margins change by about 13 points. This favours home teams because the increase in their expected margin exceeds the expected decrease in their start, and works against away teams for the opposite reason.

If you've made it this far, my sincere thanks. I reckon your brain's earned a spell; mine certainly has.

Another Day, Another Model

In the previous blog I developed models for predicting victory margins and found that the selection of a 'best' model depended on the criterion used to measure performance.

This blog I'll review the models that we developed and then describe how I created another model, this one designed to predict line betting winners.

The Low Average Margin Predictor

The model that produced the lowest mean absolute prediction error MAPE was constructed by combining the predictions of two other models. One of the constituent models - which I collectively called floating window models - looked only at the victory margins and bookie's home team prices for the last 22 rounds, and the other constituent model looked at the same data but only for the most recent 35 rounds.

On their own neither of these two models produce especially small MAPEs, but optimally combined they produce an overall model with a 28.999 MAPE across seasons 2008 and 2009 (I know that the three decimal places is far more precision than is warranted, but any rounding's going to nudge it up to 29 which just doesn't have the same ability to impress. I consider it my nod to the retailing industry, which persists in believing that price proximity is not perceived linearly and so, for example, that a computer priced at $999 will be thought meaningfully cheaper than one priced at $1,000).

Those optimal weightings were created in the overall model by calculating the linear combination of the underlying models that would have performed best over the most recent 26 weeks of the competition, and then using those weights for the current week's predictions. These weights will change from week to week as one model or the other tends to perform better at predicting victory margins; that is what gives this model its predictive chops.

This low MAPE model henceforth I shall call the Low Average Margin Predictor (or LAMP, for brevity).

The Half Amazing Margin Predictor

Another model we considered produced margin predictions with a very low median absolute prediction error. It was similar to the LAMP but used four rather than two underlying models: the 19-, 36-, 39- and 52-round floating window models.

It boasted a 22.54 point median absolute prediction error over seasons 2008 and 2009, and its predictions have been within 4 goals of the actual victory margin in a tick over 52% of games. What destroys its mean absolute prediction error is its tendency to produce victory margin predictions that are about as close to the actual result as calcium carbonate is to coagulated milk curd. About once every two-and-a-half rounds one of its predictions will prove to be 12 goals or more distant from the actual game result.

Still, its median absolute prediction error is truly remarkable, which in essence means that its predictions are amazing about half the time, so I shall name it the Half Amazing Margin Predictor (or HAMP, for brevity).

In their own highly specialised ways, LAMP and HAMP are impressive but, like left-handed chess players, their particular specialities don't appear to provide them with any exploitable advantage. To be fair, TAB Sportsbet does field markets on victory margins and it might eventually prove that LAMP or HAMP can be used to make money on these markets, but I don't have the historical data to test this now. I do, however, have line market data that enables me to assess LAMP's and HAMP's ability to make money on this market, and they exhibit no such ability. Being good at predicting margins is different from being good at predicting handicap-adjusted margins.

Nonetheless, I'll be publishing LAMP's and HAMP's margin predictions this season.

HELP, I Need Another Model

Well if we want a model that predicts line market winners we really should build a dedicated model for this, and that's what I'll describe next.

The type of model that we'll build is called a binary logit. These can be used to fit a model to any phenomenon that is binary - that is, two-valued - in nature. You could, for example, fit one to model the characteristics of people who do or don't respond to a marketing campaign. In that case, the binary variable is campaign response. You could also, as I'll do here, fit a binary logit to model the relationship between home team price and whether or not the home team wins on line betting.

Fitting and interpreting such models is a bit more complicated than fitting and interpreting models fitted using the ordinary least squares method, which we covered in the previous blog. For this reason I'll not go into the details of the modelling here. Conceptually though all we're doing is fitting an equation that relates the Home team's head-to-head price with its probability of winning on line betting.

For this modelling exercise I have again created 47 floating window models of the sort I've just described, one model that uses price and line betting result data only the last 6 rounds, another that use the same data for the last 7 rounds, and so on up to one that uses data from the last 52 rounds.

Then, as I did in creating HAMP and LAMP, I looked for the combination of floating window models that best predicts winning line bet teams.

The overall model I found to perform best combines 24 of the 47 floating window models - I'll spare you the Lotto-like list of those models' numbers here. In 2008 this model predicted the line betting winner 57% of the time and in 2009 it predicted 64% of such winners. Combined, that gives it a 61% average across the two seasons. I'll call this model the Highly Evolved Line Predictor (or HELP), the 'highly evolved' part of the name in recognition of the fact that it was selected because of its fitness in predicting line betting winners in the environment that prevailed across the 2008 and 2009 seasons.

Whether HELP will thrive in the new environment of the 2010 season will be interesting to watch, as indeed will be the performance of LAMP and HAMP.

In my previous post I drew the distinction between fitting a model and using it to predict the future and explained that a model can be a good fit to existing data but turn out to be a poor predictor. In that context I mentioned the common statistical practice of fitting a model to one set of data and then measuring its predictive ability on a different set.

HAMP, LAMP and HELP are somewhat odd models in this respect. Certainly, when I've used them to predict they're predicting for games that weren't used in the creation of any of their underlying floating window models. So that's a tick.

They are, however, fitted models in that I generated a large number of potential LAMPs, HAMPs and HELPs, each using a different set of the available floating window models, and then selected those models which best predicted the results of the 2008 and 2009 seasons. Accordingly, it could well be that the superior performance of each of these models can be put down to chance, in which case we'll find that their performances in 2010 will drop to far less impressive levels.

We won't know whether or not we're witnessing such a decline until some way into the coming season but in the meantime we can ponder the basis on which we might justify asserting that the models are not mere chimera.

Recall that each of the floating window models use as predictive variables nothing more than the price of the Home team. The convoluted process of combining different floating window models with time-varying weights for each means that, in essence, the predictions of HAMP, LAMP and HELP are all just sophisticated transformations of one number: the Home team price for the relevant game.

So, for HAMP, LAMP and HELP to be considered anything other than statistical flukes it needs to be the case that:

1. the TAB Sportsbet bookie's Home team prices are reliable indicators of Home teams' victory margins and line betting success
2. the association between Home team prices and victory margins, and between Home team prices and line betting outcomes varies in a consistent manner over time
3. HAMP, LAMP and HELP are constructed in such a way as to effectively model these time-varying relationships

On balance I'd have to say that these conditions are unlikely to be met. Absent the experience gained from running these models live during a fresh season then, there's no way I'd be risking money on any of these models.

Many of the algorithms that support MAFL Funds have been developed in much the same way as I've described in this and the previous blog, though each of them is based on more than a single predictive variable and most of them have been shown to be profitable in testing using previous seasons' data and in real-world wagering.

Regardless, a few seasons of profitability doesn't rule out the possibility that any or all of the MAFL Fund algorithms haven't just been extremely lucky.

That's why I'm not retired ...

Does The Favourite Have It Covered?

You've wagered on Geelong - a line bet in which you've given 46.5 points start - and they lead by 42 points at three-quarter time. What price should you accept from someone wanting to purchase your wager? They also led by 44 points at quarter time and 43 points at half time. What prices should you have accepted then?

In this blog I've analysed line betting results since 2006 and derived three models to answer questions similar the one above. These models take as inputs the handicap offered by the favourite and the favourite's margin relative to that handicap at a particular quarter break. The output they provide is the probability that the favourite will go on to cover the spread given the situation they find themselves in at the end of some quarter.

The chart below plots these probabilities against margins relative to the spread at quarter time for 8 different handicap levels.

Negative margins mean that the favourite has already covered the spread, positive margins that there's still some spread to be covered.

The top line tracks the probability that a 47.5 point favourite covers the spread given different margins relative to the spread at quarter time. So, for example, if the favourite has the spread covered by 5.5 points (ie leads by 53 points) at quarter time, there's a 90% chance that the favourite will go on to cover the spread at full time.

In comparison, the bottom line tracks the probability that a 6.5 point favourite covers the spread given different margins relative to the spread at quarter time. If a favourite such as this has the spread covered by 5.5 points (ie leads by 12 points) at quarter time, there's only a 60% chance that this team will go on to cover the spread at full time. The logic of this is that a 6.5 point favourite is, relatively, less strong than a 47.5 point favourite and so more liable to fail to cover the spread for any given margin relative to the spread at quarter time.

Another way to look at this same data is to create a table showing what margin relative to the spread is required for an X-point favourite to have a given probability of covering the spread.

So, for example, for the chances of covering the spread to be even, a 6.5 point favourite can afford to lead by only 4 or 5 (ie be 2 points short of covering) at quarter time and a 47.5 point favourite can afford to lead by only 8 or 9 (ie be 39 points short of covering).

The following diagrams provide the same chart and table for the favourite's position at half time.

Finally, these next diagrams provide the same chart and table for the favourite's position at three-quarter time.

I find this last table especially interesting as it shows how fine the difference is at three-quarter time between likely success and possible failure in terms of covering the spread. The difference between a 50% and a 75% probability of covering is only about 9 points and between a 75% and a 90% probability is only 9 points more.

To finish then, let's go back to the question with which I started this blog. A 46.5 point favourite leading by 42 points at three-quarter time is about a 69.4% chance to go on and cover. So, assuming you backed the favourite at $1.90 your expected payout for a 1 unit wager is 0.694 x 0.9 - 0.306 = +0.32 units. So, you'd want to be paid 1.32 units for your wager, given that you also want your original stake back too.

A 46.5 point favourite leading by 44 points at quarter time is about an 85.5% chance to go on and cover, and a similar favourite leading by 43 points at half time is about an 84.7% chance to go on to cover. The expected payouts for these are +0.62 and +0.61 units respectively, so you'd have wanted about 1.62 units to surrender these bets (a little more if you're a risk-taker and a little less if you're risk-averse, but that's a topic for another day ...)