How would you characterise the Grand Finals that you've witnessed? As low-scoring, closely fought games; as high-scoring games with regular blow-out finishes; or as something else?
First let's look at the total points scored in Grand Finals relative to the average points scored per game in the season that immediately preceded them.
Apart from a period spanning about the first 25 years of the competition, during which Grand Finals tended to be lower-scoring affairs than the matches that took place leading up to them, Grand Finals have been about as likely to produce more points than the season average as to produce fewer points.
One way to demonstrate this is to group and summarise the Grand Finals and non-Grand Finals by the decade in which they occurred.
There's no real justification then, it seems, in characterising them as dour affairs.
That said, there have been a number of Grand Finals that failed to produce more than 150 points between the two sides - 49 overall, but only 3 of the last 30. The most recent of these was the 2005 Grand Final in which Sydney's 8.10 (58) was just good enough to trump the Eagles' 7.12 (54). Low-scoring, sure, but the sort of game for which the cliche "modern-day classic" was coined.
To find the lowest-scoring Grand Final of all time you'd need to wander back to 1927 when Collingwood 2.13 (25) out-yawned Richmond 1.7 (13). Collingwood, with efficiency in mind, got all of its goal-scoring out of the way by the main break, kicking 2.6 (20) in the first half. Richmond, instead, left something in the tank, going into the main break at 0.4 (4) before unleashing a devastating but ultimately unsuccessful 1.3 (9) scoring flurry in the second half.
That's 23 scoring shots combined, only 3 of them goals, comprising 12 scoring shots in the first half and 11 in the second. You could see that many in an under 10s soccer game most weekends.
Forty-five years later, in 1972, Carlton and Richmond produced the highest-scoring Grand Final so far. In that game, Carlton 28.9 (177) held off a fast-finishing Richmond 22.18 (150), with Richmond kicking 7.3 (45) to Carlton's 3.0 (18) in the final term.
Just a few weeks earlier these same teams had played out an 8.13 (63) to 8.13 (63) draw in their Semi Final. In the replay Richmond prevailed 15.20 (110) to Carlton's 9.15 (69) meaning that, combined, the two Semi Finals they played generated 22 points fewer than did the Grand Final.
From total points we turn to victory margins.
Here too, again save for a period spanning about the first 35 years of the competition during which GFs tended to be closer fought than the average games that had gone before them, Grand Finals have been about as likely to be won by a margin smaller than the season average as to be won by a greater margin.
Of the 10 most recent Grand Finals, 5 have produced margins smaller than the season average and 5 have produced greater margins.
Perhaps a better view of the history of Grand Final margins is produced by looking at the actual margins rather than the margins relative to the season average. This next table looks at the actual margins of victory in Grand Finals summarised by decade.
One feature of this table is the scarcity of close finishes in Grand Finals of the 1980s, 1990s and 2000s. Only 4 of these Grand Finals have produced a victory margin of less than 3 goals. In fact, 19 of the 29 Grand Finals have been won by 5 goals or more.
An interesting way to put this period of generally one-sided Grand Finals into historical perspective is provided by this, the final graphic for today.
They just don't make close Grand Finals like they used to.
In Preliminary Finals since 2000 teams finishing in ladder position 1 are now 3-0 over teams finishing 3rd, and teams finishing in ladder position 2 are 5-0 over teams finishing 4th.
Overall in Preliminary Finals, teams finishing in 1st now have a 70% record, teams finishing 2nd an 80% record, teams finishing 3rd a 38% record, and teams finishing 4th a measly 20% record. This generally poor showing by teams from 3rd and 4th has meant that we've had at least 1 of the top 2 teams in every Grand Final since 2000.
Reviewing the middle table in the diagram above we see that there have been 4 Grand Finals since 2000 involving the teams from 1st and 2nd on the ladder and these contests have been split 2 apiece. No other pairing has occurred with a greater frequency.
Two of these top-of-the-table clashes have come in the last 2 seasons, with 1st-placed Geelong defeating 2nd-placed Port Adelaide in 2007, and 2nd-placed Hawthorn toppling 1st-placed Geelong last season. Prior to that we need to go back firstly to 2004, when 1st-placed Port Adelaide defeated 2nd-placed Brisbane Lions, and then to 2001 when 1st-placed Essendon surrendered to 2nd-placed Brisbane Lions.
Ignoring the replays of 1948 and 1977 there have been 110 Grand Finals in the 113-year history of the VFL/AFL history, with Grand Finals not being used in the 1897 or 1924 seasons. The pairings and win-loss records for each are shown in the table below.
As you can see, this is the first season that St Kilda have met Geelong in the Grand Final. Neither team has been what you'd call a regular fixture at the G come Grand Final Day, though the Cats can lay claim to having been there more often (15 times to the Saints' 5) and to having a better win-loss percentage (47% to the Saints' 20%).
After next weekend the Cats will move ahead of Hawthorn into outright 7th in terms of number of GF appearances. Even if they win, however, they'll still trail the Hawks by 2 in terms of number of Flags.
If the Grand Final were to be played this weekend, what prices would be on offer?
We can answer this question for the TAB Sportsbet bookie using his prices for this week's games, his prices for the Flag market and a little knowledge of probability.
Consider, for example, what must happen for the Saints to win the flag. They must beat the Dogs this weekend and then beat whichever of the Cats or the Pies wins the other Preliminary Final. So, there are two mutually exclusive ways for them to win the Flag.
In terms of probabilities, we can write this as:
Prob(St Kilda Wins Flag) =
Prob(St Kilda Beats Bulldogs) x Prob (Geelong Beats Collingwood) x Prob(St Kilda Beats Geelong) +
Prob(St Kilda Beats Bulldogs) x Prob (Collingwood Beats Geelong) x Prob(St Kilda Beats Collingwood)
We can write three more equations like this, one for each of the other three Preliminary Finalists.
Now if we assume that the bookie's overround has been applied to each team equally then we can, firstly, calculate the bookie's probability of each team winning the Flag based on the current Flag market prices which are St Kilda $2.40; Geelong $2.50; Collingwood $5.50; and Bulldogs $7.50.
If we do this, we obtain:
Next, from the current head-to-head prices for this week's games, again assuming equally applied overround, we can calculate the following probabilities:
Armed with those probabilities and the four equations of the form of the one above in bold we come up with a set of four equations in four unknowns, the unknowns being the implicit bookie probabilities for all the possible Grand Final matchups.
To lapse into the technical side of things for a second, we have a system of equations Ax = b that we want to solve for x. But, it turns out, the A matrix is rank-deficient. Mathematically this means that there are an infinite number of solutions for x; practically it means that we need to define one of the probabilities in x and we can then solve for the remainder.
Which probability should we choose?
I feel most confident about setting a probability - or a range of probabilities - for a St Kilda v Geelong Grand Final. St Kilda surely would be slight favourites, so let's solve the equations for Prob(St Kilda Beats Geelong) equal to 51% to 57%.
Each column of the table above provides a different solution and is obtained by setting the probability in the top row and then solving the equations to obtain the remaining probabilities.
The solutions in the first 5 columns all have the same characteristic, namely that the Saints are considered more likely to beat the Cats than they are to beat the Pies. To steal a line from Get Smart, I find that hard to believe, Max.
Inevitably then we're drawn to the last two columns of the table, which I've shaded in gray. Either of these solutions, I'd contend, are valid possibilities for the TAB Sportsbet bookie's true current Grand Final matchup probabilities.
If we turn these probabilities into prices, add a 6.5% overround to each, and then round up or down as appropriate, this gives us the following Grand Final matchup prices.
St Kilda v Geelong
St Kilda v Collingwood
Geelong v Bulldogs
Collingwood v Bulldogs
We've had a cracking finals series so far and there's the prospect of even better to come. Two matches that stand out from what we've already witnessed are the Lions v Carlton and Collingwood v Adelaide games. A quick look at the Round 22 MARS ratings of these teams tells us just how evenly matched they were.
Glancing down to the bottom of the 2009 column tells us a bit more about the quality of this year's finalists.
As a group, their average rating is 1,020.8, which is the 3rd highest average rating since season 2000, behind only the averages for 2001 and 2003, and weighed down by the sub-1000 rating of the eighth-placed Dons.
At the top of the 8, the quality really stands out. The top 4 teams have the highest average rating for any season since 2000, and the top 5 teams are all rated 1,025 or higher, a characteristic also unique to 2009.
Someone from among that upper eschelon had to go out in the first 2 weeks and, as we now know, it was Adelaide, making them the highest MARS rated team to finish fifth at the end of the season.
(Adelaide aren't as unlucky as the Carlton side of 2001, however, who finished 6th with a MARS Rating of 1,037.9)
There's been a pleasing neatness to the results of the finals so far, depicted below.
Close inspection will reveal that Week 1 saw the teams from ladder positions 7 and 8 knocked out and Week 2 left those fans of the teams from positions 5 and 6 with other things to do for the rest of September. Which, of course, leaves the teams from ladder spots 1 through 4 to contest for a ticket to the Granny. All very much as it should be.
The phenomenon of the top 4 teams playing off in the Prelims has been witnessed in every season since we've had the current finals system, bar 2001 and 2007 when, in both cases, the team from 6th took the place of the team from 3rd.
The Week 2 Semi-Finals have been footballing graveyards for the teams in the bottom half of the eight, as shown in the table below.
Teams from 5th have won only 4 of 10 Elimination Finals, while teams from 8th have won the other 6. None of the winning teams have proceeded beyond the Semis.
Teams from 7th have only survived into the second week once, and this victor proceeded no further than the second week, while teams from 6th have reached the Semis on 9 occasions, losing 7 and winning just 2.
The summary of performance by ladder position across all weeks of the finals now looks like this:
Overall, teams finishing 7th have the worst finals record, winning only 1 from 11 (9%). Teams from 3rd, 4th, 5th and 8th all also have sub-50% records.
Teams finishing 6th have an 11 and 10 record, though 9 of those 11 wins have come in the Elimination Final.
And, as you might expect, teams from 1st and 2nd have the best records. Teams from 1st position have an overall 20 and 9 (69%) record, and a 10 and 5 (67%) record for the last 2 weeks of the finals. Teams from 2nd position have a slightly better overall record of 20 and 7 (74%) but a slightly worse record of 10 and 6 (63%) across the last two weeks of the finals.
Looking ahead to Week 3 of the finals I can tell you that teams from 1st have a perfect 2-0 record against teams from 3rd, and teams from 2nd have an equally perfect 4-0 record against teams from 4th.
Though I doubt many of you are mourning the loss of the Funds that operated last year, I know I was curious to see how they would have performed had I allowed them to run around for another season.
Let me start with a comparison of last year's Recommended Portfolio with this year's:
Starting with the obvious, Investors would have been happier at this point of the season - up by 22% rather than 10% - had last year's Funds and weightings been in force to create the Recommended Portfolio this year.
They'd not, however, have been in this state for all of the season, just for a period from the first game of Round 7, when Essendon beat Hawthorn and paid $5, to the first game of Round 11, and then again from the last game of Round 20, when Essendon toppled St Kilda and paid $6.50, to the present.
Interspersed with the periods of euphoria would have been long periods of despair as the Fund dropped to 80% of its value on two occasions before rebounding to its current levels. The fluctuations in the value of the 2008 Recommended Portfolio encompass a trough of almost -20% at the end of Round 8 to a peak of almost 30% at the end of Round 20. This range of about 50% contrasts starkly with that produced by this year's Recommended Portfolio, which is a range a touch less than half that amount and which includes a low point of only about -5%.
It'll be no surprise to anyone that the major cause of the dramatic fluctuations in the performance of the 2008 Recommended Portfolio would have been the Heritage Fund. In the chart below I've mapped its performance against that of the New Heritage Fund.
Pay close attention to the y-axis on this chart. The Heritage Fund peaks in value early in the season, climbing to +63% on the back of a lucrative wager on Freo at $3.75 who knocked off the Blues at Gold Coast Stadium, before plummeting to -69% at the end of Round 12 after a series of large and unsuccessful wagers, thereafter recovering, plummeting again and then climbing to a peak of +28% after the first game of Round 21, one game after landing the Dons at $6.50.
Also contributing to the superior performance of the 2008 Recommended Portfolio is the extraordinary success of last year's Line Fund compared to Line Redux.
The impressive result for the Line Fund has been achieved on the basis of just 29 wagers, 19 of them successful. Line Redux, by comparison, has eked out its profit on the back of 70 wagers with just 37 of them successful, a scant 2 wins better than chance.
Last year's Chi Fund has also performed better than this year's Chi-squared - well, more accurately, it's performed less worse. It has even, albeit briefly, drifted into profit.
Amongst the remaining Funds, Hope has outshone them all, though Alpha and Beta have also produced solid gains. Prudence has generated consistent if unspectacular profits.
This year represents the 10th under the current system of finals, a system I think has much to recommend it. It certainly seems to - justifiably, I'd argue - favour those teams that have proven their credentials across the entire season.
The table below shows how the finals have played out over the 10 years:
This next table summarises, on a one-week-of-the-finals-at-a-time basis, how teams from each ladder position have fared:
Of particular note in relation to Week 1 of the finals is the performance of teams finishing 3rd and of those finishing 7th. Only two such teams - one from 3rd and one from 7th - have been successful in their respective Qualifying and Elimination Finals.
In the matchups of 1st v 4th and 5th v 8th the outcomes have been far more balanced. In the 1st v 4th clashes, it's been the higher ranked team that has prevailed on 6 of 10 occasions, whereas in the 5th v 8th clashes, it's been the lower ranked team that's won 60% of the time.
Turning our attention next to Week 2 of the finals, we find that the news isn't great for Adelaide or Lions fans. On both those occasions when 4th has met 5th in Week 2, the team from 4th on the ladder has emerged victorious, and on the 7 occasions that 3rd has faced 6th in Week 2, the team from 3rd on the ladder has won 5 and lost only 2.
Looking more generally at the finals, it's interesting to note that no team from ladder positions 5, 7 or 8 has made it through to the Preliminary Finals and, on the only two occasions that the team from position 6 has made it that far, none has progressed into the Grand Final.
So, teams only from positions 1 to 4 have so far contested Grand Finals, teams from 1st on 6 occasions, teams from 2nd on 7 occasions, teams from 3rd on 3 occasions, and teams from 4th only twice.
No team finishing lower than 3rd has yet won a Flag.
If the teams in the finals are the best teams in the competition (and the MARS ratings say otherwise, but nonetheless) then it seems to make sense to focus on the games in which they've faced off in assessing each team's relative chances. The tables below have been constructed including just those games.
The first table provides the aggregate record of each team:
I'll come back and analyse this some more in a moment, but let me note a couple of interesting things here in passing:
The second table provides summary information on the team matchups:
Now, looking at both tables together, let's consider the records of each team.
St Kilda
Geelong
Western Bulldogs
Collingwood
Carlton
Essendon
Adelaide
Brisbane Lions
Here's the detail for Week 1 of the Finals and the road thereafter.
So far we've learned that handicap-adjusted margins appear to be normally distributed with a mean of zero and a standard deviation of 37.7 points. That means that the unadjusted margin - from the favourite's viewpoint - will be normally distributed with a mean equal to minus the handicap and a standard deviation of 37.7 points. So, if we want to simulate the result of a single game we can generate a random Normal deviate (surely a statistical contradiction in terms) with this mean and standard deviation.
Alternatively, we can, if we want, work from the head-to-head prices if we're willing to assume that the overround attached to each team's price is the same. If we assume that, then the home team's probability of victory is the head-to-head price of the underdog divided by the sum of the favourite's head-to-head price and the underdog's head-to-head price.
So, for example, if the market was Carlton $3.00 / Geelong $1.36, then Carlton's probability of victory is 1.36 / (3.00 + 1.36) or about 31%. More generally let's call the probability we're considering P%.
Working backwards then we can ask: what value of x for a Normal distribution with mean 0 and standard deviation 37.7 puts P% of the distribution on the left? This value will be the appropriate handicap for this game.
Again an example might help, so let's return to the Carlton v Geelong game from earlier and ask what value of x for a Normal distribution with mean 0 and standard deviation 37.7 puts 31% of the distribution on the left? The answer is -18.5. This is the negative of the handicap that Carlton should receive, so Carlton should receive 18.5 points start. Put another way, the head-to-head prices imply that Geelong is expected to win by about 18.5 points.
With this result alone we can draw some fairly startling conclusions.
In a game with prices as per the Carlton v Geelong example above, we know that 69% of the time this match should result in a Geelong victory. But, given our empirically-based assumption about the inherent variability of a football contest, we also know that Carlton, as well as winning 31% of the time, will win by 6 goals or more about 1 time in 14, and will win by 10 goals or more a litle less than 1 time in 50. All of which is ordained to be exactly what we should expect when the underlying stochastic framework is that Geelong's victory margin should follow a Normal distribution with a mean of 18.8 points and a standard deviation of 37.7 points.
So, given only the head-to-head prices for each team, we could readily simulate the outcome of the same game as many times as we like and marvel at the frequency with which apparently extreme results occur. All this is largely because 37.7 points is a sizeable standard deviation.
Well if simulating one game is fun, imagine the joy there is to be had in simulating a whole season. And, following this logic, if simulating a season brings such bounteous enjoyment, simulating say 10,000 seasons must surely produce something close to ecstasy.
I'll let you be the judge of that.
Anyway, using the Wednesday noon (or nearest available) head-to-head TAB Sportsbet prices for each of Rounds 1 to 20, I've calculated the relevant team probabilities for each game using the method described above and then, in turn, used these probabilities to simulate the outcome of each game after first converting these probabilities into expected margins of victory.
(I could, of course, have just used the line betting handicaps but these are posted for some games on days other than Wednesday and I thought it'd be neater to use data that was all from the one day of the week. I'd also need to make an adjustment for those games where the start was 6.5 points as these are handled differently by TAB Sportsbet. In practice it probably wouldn't have made much difference.)
Next, armed with a simulation of the outcome of every game for the season, I've formed the competition ladder that these simulated results would have produced. Since my simulations are of the margins of victory and not of the actual game scores, I've needed to use points differential - that is, total points scored in all games less total points conceded - to separate teams with the same number of wins. As I've shown previously, this is almost always a distinction without a difference.
Lastly, I've repeated all this 10,000 times to generate a distribution of the ladder positions that might have eventuated for each team across an imaginary 10,000 seasons, each played under the same set of game probabilities, a summary of which I've depicted below. As you're reviewing these results keep in mind that every ladder has been produced using the same implicit probabilities derived from actual TAB Sportsbet prices for each game and so, in a sense, every ladder is completely consistent with what TAB Sportsbet 'expected'.
The variability you're seeing in teams' final ladder positions is not due to my assuming, say, that Melbourne were a strong team in one season's simulation, an average team in another simulation, and a very weak team in another. Instead, it's because even weak teams occasionally get repeatedly lucky and finish much higher up the ladder than they might reasonably expect to. You know, the glorious uncertainty of sport and all that.
Consider the row for Geelong. It tells us that, based on the average ladder position across the 10,000 simulations, Geelong ranks 1st, based on its average ladder position of 1.5. The barchart in the 3rd column shows the aggregated results for all 10,000 simulations, the leftmost bar showing how often Geelong finished 1st, the next bar how often they finished 2nd, and so on.
The column headed 1st tells us in what proportion of the simulations the relevant team finished 1st, which, for Geelong, was 68%. In the next three columns we find how often the team finished in the Top 4, the Top 8, or Last. Finally we have the team's current ladder position and then, in the column headed Diff, a comparison of the each teams' current ladder position with its ranking based on the average ladder position from the 10,000 simulations. This column provides a crude measure of how well or how poorly teams have fared relative to TAB Sportsbet's expectations, as reflected in their head-to-head prices.
Here are a few things that I find interesting about these results:
(In another blog I've used the same simulation methodology to simulate the last two rounds of the season and project where each team is likely to finish.)
If you were to consider each quarter of football as a separate contest, what pattern of wins and losses do you think has been most common? Would it be where one team wins all 4 quarters and the other therefore losses all 4? Instead, might it be where teams alternated, winning one and losing the next, or vice versa? Or would it be something else entirely?
The answer, it turns out, depends on the period of history over which you ask the question. Here's the data:
So, if you consider the entire expanse of VFL/AFL history, the egalitarian "WLWL / LWLW" cadence has been most common, occurring in over 18% of all games. The next most common cadence, coming in at just under 15% is "WWWW / LLLL" - the Clean Sweep, if you will. The next four most common cadences all have one team winning 3 quarters and the other winning the remaining quarter, each of which such cadences have occurred about 10-12% of the time. The other patterns have occurred with frequencies as shown under the 1897 to 2009 columns, and taper off to the rarest of all combinations in which 3 quarters were drawn and the other - the third quarter as it happens - was won by one team and so lost by the other. This game took place in Round 13 of 1901 and involved Fitzroy and Collingwood.
If, instead, you were only to consider more recent seasons excluding the current one, say from 1980 to 2008, you'd find that the most common cadence has been the Clean Sweep on about 18%, with the "WLLL / "LWWW" cadence in second on a little over 12%. Four other cadences then follow in the 10-11.5% range, three of them involving one team winning 3 of the 4 quarters and the other the "WLWL / LWLW" cadence.
In short it seems that teams have tended to dominate contests more in the 1980 to 2008 period than had been the case historically.
(It's interesting to note that, amongst those games where the quarters are split 2 each, "WLWL / LWLW" is more common than either of the two other possible cadences, especially across the entire history of footy.)
Turning next to the current season, we find that the Clean Sweep has been the most common cadence, but is only a little ahead of 5 other cadences, 3 of these involving a 3-1 split of quarters and 2 of them involving a 2-2 split.
So, 2009 looks more like the period 1980 to 2008 than it does the period 1897 to 2009.
What about the evidence for within-game momentum in the quarter-to-quarter cadence? In other words, are teams who've won the previous quarter more or less likely to win the next?
Once again, the answer depends on your timeframe.
Across the period 1897 to 2009 (and ignoring games where one of the two relevant quarters was drawn):
So, across the entire history of football, there's been, if anything, an anti-momentum effect, since teams that win one quarter have been a little less likely to win the next.
Inspecting the record for more recent times, however, consistent with our earlier conclusion about the greater tendency for teams to dominate matches, we find that, for the periods 1980 to 2008 (and, in brackets, for 2009):
In more recent history then, there is evidence of within-game momentum.
All of which would lead you to believe that winning the 1st quarter should be particularly important, since it gets the momentum moving in the right direction right from the start. And, indeed, this season that has been the case, as teams that have won matches have also won the 1st quarter in 71% of those games, the greatest proportion of any quarter.
Most major Australian sports have their iconic annual event. Cricket has its Boxing Day test, tennis and golf have their respective Australian Opens, rugby league has the State of Origin series, rugby union the Bledisloe, and AFL, it now seems, has the Tanking Debate, usually commencing near Round 15 or 16 and running to the end of the season proper.
The T-word has been all over the Melbourne newspapers and various footy websites this week, perhaps most startlingly in the form of Terry Wallace's admission that in Round 22 of 2007 in the Tigers clash against St Kilda, a game in which the Tigers led by 3 points at the final change but went on to lose by 10 points:
"while he had not "tanked" during the Trent Cotchin game in Round 22, 2007, he had let the contest run its natural course without intervention"
That stain on the competition's reputation (coupled, I'll admit, with he realisation that the loss cost MAFL Investors an additional return of about 13% for that year) makes it all the more apparent to me that the draft system, especially the priority draft component, must change.
Here's what I wrote on the topic - presciently as it turns out - in the newsletter for Round 19 of 2007.
If you’re a diehard AFL fan and completely conversant with the nuances of the Draft, please feel free to skip this next section of the newsletter.
I thought that a number of you might be interested to know why, in some quarters, there’s such a fuss around this time of year about “The Draft” and its potential impact on the commitment levels of teams towards the bottom of the ladder.
The Draft is, as Wikipedia puts it, the “annual draft of young talent” into the AFL that takes place prior to the start of each season. In the words of the AFL’s own website:
"In simple terms, the NAB AFL Draft is designed to give clubs which finished lower on the ladder the first opportunity to pick the best new talent in Australia. At season's end, all clubs are allocated draft selections. The club that finished last receives the first selection, the second last team gets the second selection and so on until the premier receives the 16th selection."
So, here’s the first issue: towards season’s end, those teams for whom all hopes of a Finals berth have long since left the stadium find that there’s more to be gained by losing games than there is by winning them.
Why? Well say, for example, that Richmond suddenly remembered what the big sticks are there for and jagged two wins in the last four games, leaping a startled Melbourne in the process, relegating them to position Spoon. The Tigers’ reward for such a stunning effort would be to (possibly – see below) hand Melbourne the sweetest of draft plums, the Number 1 draft pick, while relegating themselves to the Number 2 pick. Now, in truth, over the years, Number 1 picks have not always worked out better than Number 2 picks, but think about it this way: isn’t it always nicer to have first hand into the Quality Assortment?
Now entereth the notion of Priority Picks, which accrue to those teams who have demonstrated season-long ineptitude to the extent that they’ve accumulated fewer than 17 points over its duration. They get a second draft pick prior to everyone else’s second draft pick and then a third pick not that long after, once all the other Priority Picks have taken place. So, for example, if a team comes last and wins, say, four games, it gets Pick #1, Pick #17 (their Priority pick, immediately after all the remaining teams have had their first pick) and then Pick #18 (their true second round Pick). If more than one team is in entitled to Priority Picks then the Picks are taken in reverse ladder order.
Still with me?
Now, the final twist. If a team has proven its footballing inadequacy knows not the bounds of a single year, having done so by securing fewer than 17 points in each of two successive seasons, then it gets its Priority Pick before anyone else gets even their first round pick. Once again, if more than one team is in this situation, then the tips are taken in reverse ladder order.
So, what’s the relevance to this year? Well, last year Carlton managed only 14 points and this year they’re perched precipitously on 16 points. If they lose their next four games, their first three draft picks will be #1 (their Priority Pick), #4 (their first round pick), and #20 (their second round pick); if they win or even draw one or more of their remaining games and do this without leaping a ladder spot, their first three draft picks will be #3, #19 and #35. Which would you prefer?
I find it hard to believe that a professional footballer or coach could ever succumb to the temptation to “tank” games (as it’s called), but the game’s administrators should never have set up the draft process in such a way that it incites such speculation every year around this time.
I can think of a couple of ways of preserving the intent of the current draft process without so blatantly rewarding failure and inviting suspicion and rumour. We’ll talk about this some more next week.
*****
In the following week's newsletter, I wrote this:
With the Tigers and the Dees winning last week, I guess many would feel that the “tanking” issue has been cast aside for yet another season. Up to a point, that’s probably a fair assessment, although only a win by the Blues could truly muffle all the critics.
Regardless, as I said last week, it’s unfair to leave any of the teams in a position where they could even be suspected of putting in anything other than a 100% effort.
I have two suggestions for changes to the draft that would broadly preserve its intent but that would also go a long way to removing much of the contention that currently exists.
(1) Randomise the draft to some extent.
Sure, give teams further down the ladder a strong chance of getting early draft picks, but don’t make ladder position completely determine the pick order. One way to achieve this would be to place balls in an urn with the number of balls increasing as ladder position increased. So, for example, the team that finished 9th might get 5 balls in the urn; 10th might get 6 balls, and so on. Then, draw from this urn to determine the order of draft picks.
Actually, although it’s not strictly in keeping with the current spirit of the draft, I’d like to see this system used in such a way that marginally more balls are placed in the urn for teams higher up the ladder to ensure that all teams are still striving for victory all the way to Round 22.
(2) Base draft picks on ladder position at the end of Round 11, not Round 22.
Sides that are poor performers generally don’t need 22 rounds to prove it; 11 rounds should be more than enough. What’s more, I reckon that it’s far less likely that a team would even consider tanking say rounds 9, 10 and 11 when there’s still so much of the season to go that a spot in the Finals is not totally out of the question. With this approach I’d be happy to stick with the current notion that 1st draft pick accrues to the team at the foot at the ladder.
Under either of these new draft regimes, the notion of Priority Picks has to go. Let’s compensate for underperformance but not lavish it with silken opportunity.
****
My opinion hasn't changed. The changes to the draft for the next few years that have been made to smooth the entry of the Gold Coast into the competition probably mean that we're stuck with a version of the draft we have now for the next few years. After that though, we do have to fix it because it is broken.
Though there are numerous differences between the various football codes in Australia, two that have always struck me as arbitrary are AFL's awarding of 4 points for a victory and 2 from a draw (why not, say, pi and pi/2 if you just want to be different?) and AFL's use of percentage rather than points differential to separate teams that are level on competition points.
I'd long suspected that this latter choice would only rarely be significant - that is, that a team with a superior percentage would not also enjoy a superior points differential - and thought it time to let the data speak for itself.
Sure enough, a review of the final competition ladders for all 112 seasons, 1897 to 2008, shows that the AFL's choice of tiebreaker has mattered only 8 times and that on only 3 of those occasions (shown in grey below) has it had any bearing on the conduct of the finals.
Historically, Richmond has been the greatest beneficiary of the AFL's choice of tiebreaker, being awarded the higher ladder position on the basis of percentage on 3 occasions when the use of points differential would have meant otherwise. Essendon and St Kilda have suffered most from the use of percentage, being consigned to a lower ladder position on 2 occasions each.
There you go: trivia that even a trivia buff would dismiss as trivial.
Last year, you might recall, a spate of deliberately rushed behinds prompted the AFL to review and ultimately change the laws relating to this form of scoring.
Has the change led to a reduction in the number of behinds recorded in each game? The evidence is fairly strong:
So far this season we've seen 22.3 behinds per game, which is 2.6 per game fewer than we saw in 2008 and puts us on track to record the lowest number of average behinds per game since 1915. Back then though goals came as much more of a surprise, so a spectator at an average game in 1915 could expect to witness only 16 goals to go along with the 22 behinds. Happy days.
This year's behind decline continues a trend during which the number of behinds per game has dropped from a high of 27.3 per game in 1991 to its current level, a full 5 behinds fewer, interrupted only by occasional upticks such as the 25.1 behinds per game recorded in 2007 and the 24.9 recorded in 2008.
While behind numbers have been falling recently, goals per game have also trended down - from 29.6 in 1991, to this season's current average of 26.8. Still, AFL followers can expect to witness more goals than behinds in most games they watch. This wasn't always the case. Not until the season of 1969 had there been a single season with more goals than behinds, and not until 1976 did such an outcome became a regular occurrence. In only one season since then, 1981, have fans endured more behinds than goals across the entire season.
On a game-by-game basis, 90 of 128 games this season, or a smidge over 70%, have produced more goals than behinds. Four more games have produced an equal number of each.
As a logical consequence of all these trends, behinds have had a significantly smaller impact on the result of games, as evidenced by the chart below which shows the percentage of scoring attributable to behinds falling from above 20% in the very early seasons to around 15% across the period 1930 to 1980, to this season's 12.2%, the second-lowest percentage of all time, surpassed only by the 11.9% of season 2000.
(There are more statistical analyses of the AFL on MAFL Online's sister site at MAFL Stats.)
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 ...)
Okay, this is probably going to be a long blog so you might want to make yourself comfortable.
For some time now I've been wondering about the statistical properties of the Handicap-Adjusted Margin (HAM). Does it, for example, follow a normal distribution with zero mean?
Well firstly we need to deal with the definition of the term HAM, for which there is - at least - two logical definitions.
The first definition, which is the one I usually use, is calculated from the Home Team perspective and is Home Team Score - Away Team Score + Home Team's Handicap (where the Handicap is negative if the Home Team is giving start and positive otherwise). Let's call this Home HAM.
As an example, if the Home Team wins 112 to 80 and was giving 20.5 points start, then Home HAM is 112-80-20.5 = +11.5 points, meaning that the Home Team won by 11.5 points on handicap.
The other approach defines HAM in terms of the Favourite Team and is Favourite Team Score - Underdog Team Score + Favourite Team's Handicap (where the Handicap is always negative as, by definition the Favourite Team is giving start). Let's call this Favourite HAM.
So, if the Favourite Team wins 82 to 75 and was giving 15.5 points start, then Favourite HAM is 82-75-15.5 = -7.5 points, meaning that the Favourite Team lost by 7.5 points on handicap.
Home HAM will be the same as Favourite HAM if the Home Team is Favourite. Otherwise Home HAM and Favourite HAM will have opposite signs.
There is one other definitional detail we need to deal with and that is which handicap to use. Each week a number of betting shops publish line markets and they often differ in the starts and the prices offered for each team. For this blog I'm going to use TAB Sportsbet's handicap markets.
TAB Sportsbet Handicap markets work by offering even money odds (less the vigorish) on both teams, with one team receiving start and the other offering that same start. The only exception to this is when the teams are fairly evenly matched in which case the start is fixed at 6.5 points and the prices varied away from even money as required. So, for example, we might see Essendon +6.5 points against Carlton but priced at $1.70 reflecting the fact that 6.5 points makes Essendon in the bookie's opinion more likely to win on handicap than to lose. Games such as this are problematic for the current analysis because the 'true' handicap is not 6.5 points but is instead something less than 6.5 points. Including these games would bias the analysis - and adjusting the start is too complex - so we'll exclude them.
So, the question now becomes is HAM Home, defined as above and using the TAB Sportsbet handicap and excluding games with 6.5 points start or fewer, normally distributed with zero mean? Similarly, is HAM Favourite so distributed?
We should expect HAM Home and HAM Favourite to have zero means because, if they don't it suggests that the Sportsbet bookie has a bias towards or against Home teams of Favourites. And, as we know, in gambling, bias is often financially exploitable.
There's no particular reason to believe that HAM Home and HAM Favourite should follow a normal distribution, however, apart from the startling ubiquity of that distribution across a range of phenomena.
Consider first the issue of zero means.
The following table provides information about Home HAMs for seasons 2006 to 2008 combined, for season 2009, and for seasons 2006 to 2009. I've isolated this season because, as we'll see, it's been a slightly unusual season for handicap betting.
Each row of this table aggregates the results for different ranges of Home Team handicaps. The first row looks at those games where the Home Team was offering start of 30.5 points or more. In these games, of which there were 53 across seasons 2006 to 2008, the average Home HAM was 1.1 and the standard deviation of the Home HAMs was 39.7. In season 2009 there have been 17 such games for which the average Home HAM has been 14.7 and the standard deviation of the Home HAMs has been 29.1.
The asterisk next to the 14.7 average denotes that this average is statistically significantly different from zero at the 10% level (using a two-tailed test). Looking at other rows you'll see there are a handful more asterisks, most notably two against the 12.5 to 17.5 points row for season 2009 denoting that the average Home HAM of 32.0 is significant at the 5% level (though it is based on only 8 games).
At the foot of the table you can see that the overall average Home HAM across seasons 2006 to 2008 was, as we expected approximately zero. Casting an eye down the column of standard deviations for these same seasons suggests that these are broadly independent of the Home Team handicap, though there is some weak evidence that larger absolute starts are associated with slightly larger standard deviations.
For season 2009, the story's a little different. The overall average is +8.4 points which, the asterisks tell us, is statistically significantly different from zero at the 5% level. The standard deviations are much smaller and, if anything, larger absolute margins seem to be associated with smaller standard deviations.
Combining all the seasons, the aberrations of 2009 are mostly washed out and we find an average Home HAM of just +1.6 points.
Next, consider Favourite HAMs, the data for which appears below:
The first thing to note about this table is the fact that none of the Favourite HAMs are significantly different from zero.
Overall, across seasons 2006 to 2008 the average Favourite HAM is just 0.1 point; in 2009 it's just -3.7 points.
In general there appears to be no systematic relationship between the start given by favourites and the standard deviation of the resulting Favourite HAMs.
Summarising:
Okay then, are Home HAMs and Favourite HAMs normally distributed?
Here's a histogram of Home HAMs:
And here's a histogram of Favourite HAMs:
There's nothing in either of those that argues strongly for the negative.
More formally, Shapiro-Wilks tests fail to reject the null hypothesis that both distributions are Normal.
Using this fact, I've drawn up a couple of tables that compare the observed frequency of various results with what we'd expect if the generating distributions were Normal.
Here's the one for Home HAMs:
There is a slight over-prediction of negative Home HAMs and a corresponding under-prediction of positive Home HAMs but, overall, the fit is good and the appropriate Chi-Squared test of Goodness of Fit is passed.
And, lastly, here's the one for Home Favourites:
In this case the fit is even better.
We conclude then that it seems reasonable to treat Home HAMs as being normally distributed with zero mean and a standard deviation of 37.7 points and to treat Favourite HAMs as being normally distributed with zero mean and, curiously, the same standard deviation. I should point out for any lurking pedant that I realise neither Home HAMs nor Favourite HAMs can strictly follow a normal distribution since Home HAMs and Favourite HAMs take on only discrete values. The issue really is: practically, how good is the approximation?
This conclusion of normality has important implications for detecting possible imbalances between the line and head-to-head markets for the same game. But, for now, enough.
It's been a while since we looked at teams' quarter-by-quarter performances. This blog looks to redress this deficiency.
(By the way, the Alternative Premierships data is available as a PDF download on the MAFL Stats website .)
The table below includes each teams' percentage by quarter and its win-draw-lose record by quarter as at the end of the 14th round:
(The comments in the right-hand column in some cases make comparisons to a team's performance after Round 7. This was the subject of an earlier blog.)
Geelong, St Kilda and, to a lesser extent, Adelaide, are the kings/queens of the 1st quarter. The Cats and the Saints have both won 11 of 14 first terms, whereas the Crows, despite recording an impressive 133 percentage, have won just 8 of 14, a record that surprisingly has been matched by the 11th-placed Hawks. The Hawks however, when bad have been very, very bad, and so have a 1st quarter percentage of just 89.
Second quarters have been the province of the ladder's top 3 teams. The Saints have the best percentage (176) but the Cats have the best win-draw-lose record (10-1-3). Carlton, though 7th on the ladder, have the 5th best percentage in 2nd quarters and the equal-2nd best win-draw-lose record.
St Kilda have also dominated in the 3rd quarter racking up a league-best percentage of 186 and a 10-0-4 win-draw-lose record. Geelong and Collingwood have also established 10-0-4 records in this quarter. The Lions, though managing only a 9-1-4 win-draw-lose record, have racked up the second-best percentage in the league for this quarter (160).
Final terms, which have been far less important this year than in seasons past, have been most dominated by St Kilda and the Bulldogs in terms of percentage, and by the Dogs and Carlton in terms of win-draw-lose records.
As you'd expect, the poorer teams have tended to do poorly across all terms, though some better-positioned teams have also had troublesome quarters.
For example, amongst those teams in the ladder's top 8 or thereabouts, the Lions, the Dons and Port have all generally failed to start well, recording sub-90 percentages and 50% or worse win-draw-lose performances.
The Dons and Sydney have both struggled in 2nd terms, winning no more than 5 of them and, in the Dons' case, also drawing one.
Adelaide and Port have found 3rd terms most disagreeable, winning only, respectively, 6 and 5 of them and in so doing producing percentages of around 75.
No top-ranked team has truly flopped in the final term, though the Lions' performance is conspicuous because it has resulted in a sub-100 percentage and a 6-0-8 win-draw-lose record.
Finally, in terms of quarters won, Geelong leads on 39 followed by the Saints on 38. There's then a gap back to the Dogs and the Pies on 32.5, and then Carlton, somewhat surprisingly given its ladder position, on 32. Melbourne have only the 3rd worst performance in terms of total quarters won. They're on 19.5, ahead of Richmond on 19 and the Roos on just 16.5. That means, in an average game, the Roos can be expected to win just 1.2 quarters. Eleven of the 16.5 quarters won have come in the first half of games so, to date anyway, Roos supporters could comfortably leave at the main change without much risk of missing a winning Roos quarter or half.
In a famous - some might say, infamous - paper by Wolfers he analysed the results of 44,120 NCAA Division I basketball games on which public betting was possible, looking for signs of "point shaving".
Point shaving occurs when a favoured team plays well enough to win, but deliberately not quite well enough to cover the spread. In his first paragraph he states: "Initial evidence suggests that point shaving may be quite widespread". Unsurprisingly, such a conclusion created considerable alarm and led, amongst a slew of furious rebuttals, to a paper by sabermetrician Phil Birnbaum refuting Wolfers' claim. This, in turn, led to a counter-rebuttal by Wolfers.
Wolfers' claim is based on a simple finding: in the games that he looked at, strong favourites - which he defines as those giving more than 12 points start - narrowly fail to cover the spread significantly more often than they narrowly cover the spread. The "significance" of the difference is in a statistical sense and relies on the assumption that the handicap-adjusted victory margin for favourites has a zero mean, normal distribution.
He excludes narrow favourites from his analysis on the basis that, since they give relatively little start, there's too great a risk that an attempt at point-shaving will cascade into a loss not just on handicap but outright. Point-shavers, he contends, are happy to facilitate a loss on handicap but not at the risk of missing out on the competition points altogether and of heightening the levels of suspicion about the outcome generally.
I have collected over three-and-a-half seasons of TAB Sporsbet handicapping data and results, so I thought I'd perform a Wolfers style analysis on it. From the outset I should note that one major drawback of performing this analysis on the AFL is that there are multiple line markets on AFL games and they regularly offer different points start. So, any conclusions we draw will be relevant only in the context of the starts offered by TAB Sportsbet. A "narrow shaving" if you will.
In adapting Wolfers' approach to AFL I have defined a "strong favourite" as a team giving more than 2 goals start though, from a point-shaving perspective, the conclusion is the same if we define it more restrictively. Also, I've defined "narrow victory" with respect to the handicap as one by less than 6 points. With these definitions, the key numbers in the table below are those in the box shaded grey.
These numbers tell us that there have been 27(13+4+10) games in which the favourite has given 12.5 points or more start and has won, by has won narrowly by enough to cover the spread. As well, there have been 24(11+7+6) games in which the favourite has given 12.5 points or more start and has won, but has narrowly not won by enough to cover the spread. In this admittedly small sample of just 51 games, there is then no statistical evidence at all of any point-shaving going on. In truth if there was any such behaviour occurring it would need to be near-endemic to show up in a sample this small lest it be washed out by the underlying variability.
So, no smoking gun there - not even a faint whiff of gunpowder ...
The table does, however, offer one intriguing insight, albeit that it only whispers it.
The final column contains the percentage of the time that favourites have managed to cover the spread for the given range of handicaps. So, for example, favourites giving 6.5 points start have covered the spread 53% of the time. Bear in mind that these percentages should be about 50%, give or take some statistically variability, lest they be financially exploitable.
It's the next percentage down that's the tantalising one. Favourites giving 7.5 to 11.5 points start have, over the period 2006 to Round 13 of 2009, covered the spread only 41% of the time. That percentage is statistically significantly different from 50% at roughly the 5% level (using a two-tailed test in case you were wondering). If this failure to cover continues at this rate into the future, that's a seriously exploitable discrepancy.
To check if what we've found is merely a single-year phenomenon, let's take a look at the year-by-year data. In 2006, 7.5-to 11.5-point favourites covered on only 12 of 35 occasions (34%). In 2007, they covered in 17 of 38 (45%), while in 2008 they covered in 12 of 28 (43%). This year, to date they've covered in 6 of 15 (40%). So there's a thread of consistency there. Worth keeping an eye on, I'd say.
Another striking feature of this final column is how the percentage of time that the favourites cover tends to increase with the size of the start offered and only crosses 50% for the uppermost category, suggesting perhaps a reticence on the part of TAB Sportsbet to offer appropriately large starts for very strong favourites. Note though that the discrepancy for the 24.5 points or more category is not statistically significant.
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