# Drawing On Hindsight

/When sports journos wait until after a contest has been decided before declaring a group of winning punters to be "savvy", I find it hard not to be at least a little cynical about the aptness of the label.

So when, on Sunday, I read in the online version of the SMH that a posse of said savvy punters had foxed the bookies and cleaned up on the draw, collectively winning as I recall about $1m at prices ranging from $34 to $51, I did wonder how many column-inches would have been devoted to those same punters had the margin been anything different when the final siren sounded on Saturday. I'm fairly certain it would have been the number that has '1' as its next-door, up the road neighbour on Integer Street.

If there is something about a game that makes it more likely to end in a draw, clearly bookies don't know about it. Whenever I've checked the pre-game odds for a draw on TAB Sportsbet they've invariably been $51, regardless of the head-to-head prices.

A moment's modelling - a binary logit with game outcome, draw or not, on the left, and the favourite's implied probability on the right - confirms that there's been no statistically significant information content in the head-to-head prices for the period 1999 to 2010.

The fitted model is this:

logit(Probability of a Draw) = -4.228 - 2.904 x (Implied Probability of the Favourite - 50%)

Ignoring for a moment the fact that the -2.904 coefficient on the Implied Probability can't be ruled out on statistical grounds from in reality being zero, this model suggests that the probability of a draw when two equally-matched teams meet is about 1.44% or roughly 1.5 times more likely than the average probability of a draw during the period that the data covers.

(There have been 21 draws in the 2,220 games that have been played since the start of 1999, which yields a probability of about 0.95%. This is a little lower than the frequency with which draws have occurred across the competition's entire 13,762 game history, during which there have been 148 draws giving a 1.08% rate.)

Even at a 1.44% probability, a price of $51 for a draw is well short of fair value.

Using this model, a draw is more likely for a particular game than the all-time average of 1.08% when the implied probability of the favourite is less than 60%, which equates to a price of about $1.60.

Fitting a similar model using teams' MARS Rating difference at the time of their meeting produces yet another model without a statistically significant coefficient (other than the intercept).

In this case the fitted model is:

**logit(Probability of a Draw) = -4.561 - 0.004483 x (Absolute Difference in MARS Ratings)**

Using this model, when teams with equal MARS Ratings clash, the probability of a draw is just 1.03%. For any other sized Ratings difference, the probability is lower - though not by much.

To summarise: there appears to be no obvious pre-game basis on which to declare that a $51 price for the draw represents value.

Naturally, things change once the game is in progress.

For the following chart I created three more binary logits, one using the margin in the teams' scores at quarter-time, another using the margin at half-time, and the third using the margin at three-quarter time as bases on which to assess the probability that the game will end in a draw.

Also, to cater for the possibility that the probability of a draw responds in a non-linear way to these margins, I again used Multivariate Adaptive Regression Splines, which I introduced in this earlier blog.

For these models I used data from all 13,762 games spanning 1897 to 2010.

Looking first at the red line, we can see that the probability of a draw varies relatively little with the quarter-time margin. It's highest, 1.43%, when the margin at quarter time is 5 points or fewer. The probability progressively declines for larger margins and at 50 points is down to 0.27% or about 1 in 370.

Turning to the empirical data, the largest margin at quarter-time for any game that's ended in a draw is 38 points. This occurred in Round 16 of 1964 when Carlton faltered after leading Collingwood 41-3 at the first change, surrendering the lead by 3 points at half-time, regaining the lead by 27 points at three-quarter time, then failing to score a point in the final term to finish tied at 79. Either this game was played in a gale or the ground was as sloped as a church roof.

We could also - more crudely but without the need to construct a model - demonstrate the information content of the quarter-time margin in terms of predicting draws by noting that in about 55% of the games that have finished as draws, the margin at quarter-time has been 9 points or fewer. Only 43% of games that have not finished as draws have had such a margin.

Next consider the green line. This line suggests that the probability of a draw is highest when the scores are tied at half-time margin in which case the probability is 2.59%. The probability then progressively declines for larger margins until the margin reaches 28 points after which the probability is steady at 0.37% or about 1 in 270.

The largest margin at half-time for any game that's ended in a draw is 42 points. This was a far more recent game, taking place during Round 16 of 2009 when Richmond led the Roos 66-24 at half-time only to stumble, losing the final term 26-10 to wind up tied at 85.

Empirically, we can also note that in about 53% of the games that have finished as draws, the margin at half-time has been 8 points or less. By comparison, only 30% of games that have not finished as draws have had such a margin at half-time. The fact that 53/30 (or 1.8) is higher than 55/43 (or 1.3, the equivalent ratio using the quarter-time figures) suggests that the half-time margin is more predictive of the likelihood of an eventual draw than the quarter-time margin.

Finally, consider the purple line. Not unexpectedly, the greatest information about the likelihood of a draw resides in knowledge of the margin at the final change. If we know that this margin is zero then our best estimate of the probability of a draw is 3.70%. For the third-quarter margin too, the probability of a draw progressively declines for larger margins, quite rapidly initially, and ultimately bottoming out at a margin of 35 points at which it becomes and stays 0.22% or about 1 in 450.

The largest margin at three-quarter time for any game that's ended in a draw is 34 points, which was the case in the Hawthorn v South Melbourne clash from Round 2 of 1968. In that game, Hawthorn kicked 47 points to South Melbourne's 20 in the third term to head into the final term leading 110-76. South then proceeded to kick 50 points to the Hawks' 16 in that last term to tie things up at 126.

Again a snapshot of the empirical data reinforces what the modelling tells us. In about 51% of the games that have finished as draws, the margin at three-quarter time has been 8 points or fewer. By comparison, only 21% of games that have not finished as draws have had such a margin at three-quarter time. The ratio 51:21 (2.4) far exceeds the equivalent ratios for half-time (53:30) and for quarter-time (55:43) margins.

Now the margins at each change in Saturday's Grand Final were 6 points, 24 points and 8 points respectively, for which the associated probabilities of a draw using the models above were 1.38%, 0.49% and 1.96%.

Based on those probabilities and the knowledge that draws have historically occurred in about 1.08% of games, a price of $51 for the draw was:

- below fair value at the start of the game (you'd have wanted $95 or more)
- below fair value at quarter-time (you'd have wanted $75 or more)
- ridiculously below fair value at half-time (you'd have wanted $205, a lobotomy, and a bookie that could keep a straight face)
- just about break-even if it could have been secured at the three-quarter time break.

If savviness is making wagers with sub-zero expectations that occasionally win money, I think I'll pass.