Estimating Fair Head-to-Head Prices : Part I

You'll recall that the total overround embedded in the head-to-head market, ignoring the possibility of a draw, is calculated by summing the reciprocal of the head-to-head prices for each team. So, for example, if the head-to-head prices for a game were $1.20 / $4.60, the overround would be 1/1.2 + 1/4.6, which is 105.1%. Some subtract 1 from this figure and would report this overround as +5.1%.

The overround will, for any sane bookmaker, be greater than 100% and the amount by which it exceeds 100% is a measure of the guaranteed profit that the bookmaker can secure if he induces wagers in the optimal proportion on each team. It's pretty easy to show that, ignoring draws, he can make a guaranteed profit of (1-1/Overround), also known as the vig, if he induces wagers on each team in proportion to the implicit probability of their opponents. So, in the example above, he'd want 79.3% of wagers (ie 4.6/(4.6+1.2)) on the favourites, and 20.7% on the underdogs to guarantee himself a vig of of 1-1/1.051, or about 4.8% of the amount wagered, whichever team won. This same calculation provides an estimate of the rate at which you should expect to lose money in the long-run when wagering in markets with such an overround if your wagering is effectively at random. 

Knowing the total vig then is clearly useful; knowing how it's distributed across the two participating teams could be, at the risk of offending the MasterCard brand police, priceless.

For some time now I've been perplexed about how to separately estimate the vig in each team's price and have therefore been forced to resort to the mindless, default assumption that it's been levied equally on both teams. Experience, and the probable existence of a favourite-longshot bias, suggests that such even-handedness on the part of the TAB bookmaker is unlikely to reflect reality, especially when he's likely trying to induce different proportions of wagering on the two teams.

A chance reading of the proceedings from a conference on maths in sport this week sparked an idea: perhaps the head-to-head prices and line market handicap might be combined to come up with estimates of team-specific vig.

HAMs, from a Favourite's Viewpoint, are Distributed Normal(0,sigma)

We know from previous analyses that Handicap-Adjusted Margins (HAMs) are, to a very good approximation at least, Normally distributed with a mean very close to zero if we calculate them from the Favourite's viewpoint. (If we calculate them from the Home team's viewpoint the mean's a little higher because the TAB bookmaker has historically given Home teams about 2.5 points too much start.)

Additional analysis I've done over the past few days shows that the spread of HAMs from the Favourite's viewpoint is broadly unrelated to the handicap set for that game. It's not the case then, for example and as you might hypothesise, that games expected to be one-sided (ie with a large handicap) are more likely to produce larger HAMs than are games that are expected to be more closely-fought. 

To put some numbers to my claim of broad unrelatedness, here are the relevant correlations: 

  • Correlation(Handicap for Favourite, HAM for Favourite) = -0.069
  • Correlation(Handicap for Favourite, abs(HAM for Favourite)) = +0.034

Taking the larger of these correlations in absolute terms, which is that between the handicap for the Favourite and the HAM from the Favourite's viewpoint, we find that these two variables share only about one-half of 1 percent of their variability (ie the square of -0.069). That's what I'd call broadly unrelated - 5th cousins, 3 times removed at best.

Knowing this fact allows us to assume that the variance, and hence standard deviation, of HAMs from the Favourite's viewpoint is a constant across games - in technical terms, that the deviations are homoscedastic. Using data from Seasons 2006 to the present, after excluding a number of games for reasons that I'll describe in the next blog, I estimate that this constant standard deviation is about 37.2 points per game, which roughly accords with the estimate I obtained in earlier blogs on the normality of HAMs.

The assumption of homoscedasticity, combined with the assumption of zero mean, allows us to fully define the Normal distribution for HAMs for every game.

The Logic

So, how does knowing that HAMs from the Favourite's viewpoint are distributed N(0,37.2) help when coupled with information about the line market?

Well if we know the complete distribution of the handicap-adjusted margin, we can make statements about the unadjusted margin, and hence about the victory probabilities of both teams.

We can say, for example, that the Favourite's probability of victory is equal to Phi(H+0.5, 37.2), where Phi is the CDF of the Normal Distribution and H is the handicap being imposed on the Favourite, which will be negative. The 0.5 adjustment is needed to cater for the probability associated with a drawn outcome: we assume that the probability of a draw is the probability of a HAM between H-0.5 and H+0.5.

Diagramatically, we have this:

Armed with this insight we can now calculate the "true" probability of the Favourite winning. The reciprocal of this probability is the fair market price for the Favourite and the relationship between this fair price and the actual price can be used to calculate the vig on the Favourite.

We can also calculate the "true" probability of a draw, and of the underdog being victorious. Thus we can calculate the vig for each of these two wagers too.

A Worked Example

Consider last week's markets for the Sydney v Western Bulldogs clash where had: 

  • Head-to-head: Sydney $1.23 / Dogs $4.20 (the draw was at $51).
  • Line: Sydney -27.5

So we have H = -27.5 and we need to calculate:

  • For Sydney's head-to-head chances : Phi(-27.5+0.5,37.2), which is 77.45%
  • For the chance of a draw : Phi(-27.5+0.5,37.2) - Phi(-27.5-0.5,37.2), which is 0.82%
  • For the Dog's head-to-head chances : 1-Sydney's chances-Chance of a draw, which is 21.73%

The fair prices are therefore: 

  • Sydney: 1/0.7745 = $1.29, which makes the actual price about 4.7% lower than the fair price
  • Draw: 1/0.0082 = $122.47, which makes the actual price about 58% lower than the fair price
  • Dogs: 1/(1-0.7745-0.0082) = $4.60, which makes the actual price about 8.7% lower than the fair price

In this particular game then the Dogs represented better value - at least as measured by the size of the estimated vig, if not the final result - than did the Swans.

What's Next

In the next blog I'll use this approach to calculate the vig on the Favourites and Underdogs in games from across seasons 2006 to the present, looking for examples where the vig has been disproportionately levied, and for examples where it appears to have been negative.