Why Sydney Won't Finish Fourth

As the ladder now stands, Sydney trail the Dogs by 4 competition points but they have a significantly inferior percentage. The Dogs have scored 2,067 points and conceded 1,656, giving them a percentage of 124.8, while Sydney have scored 1,911 points and conceded 1,795, giving them a percentage of 106.5, some 18.3 percentage points lower.

If Sydney were to win next week against the Lions, and the Dogs were to roll over and play dead against the Dons, then fourth place would be awarded to the team with the better percentage. Barring something apocalyptic, that'll be the Dogs.

Here's why. A few blogs back I noted that you could calculate the change in a team's percentage resulting from the outcome of a single game by using the following expression:

(1) Change in Percentage = (%S - %C)/(1 + %C) * Old Percentage

where %S = the points scored by the team in the current game as a percentage of the points it had already scored in the season,

and %C = the points conceded by the team in the current game as a percentage of the points it had already conceded in the season.

Now at this stage of the season, a big win or loss for a team will be one where the difference between %S and %C is in the 6-8% range, bearing in mind that a single game now represents about 1/20th or 5% of the season, so a 'typical' %S or %C would be about 5%. Scoring twice as many points as 'expected' then would give a %S of 10%, and conceding half as many as 'expected' would give a %C of 2.5%, a difference of 7.5%.

Okay, so consider a big loss for the Dogs, say 30-150. That gives a (%S - %C) of around -7.5%, which (1) tells us means the Dogs' percentage will change by about -7.5%/1.1 x 1.25, which is 8.5 percentage points. That drops the Dogs' percentage to about 116.

Next consider a big win for the Swans, again say 150-30. For them, that's a (%S - %C) of 6%, which gives them a percentage boost of 6%/1.02 x 1.06, which is about 6 percentage points. That lifts their percentage to about 112.5, still 3.5 percentage points short of the Dogs'.

To completely close the gap, Sydney needs its percentage change plus the Dogs' to exceed 18.3 percentage points, the percentage chasm it currently faces. Using this fact and the expression in (1) above for both teams, you can derive the fact that, to lift its percentage above the Dogs', Sydney needs the following to be true:

Sydney's (%S - %C) > 18.3% - 1.15 times the Dogs' (%S - %C)

Now my worst case 30-150 loss for the Dogs gives them a (%S - %C) of -7.6%. That means Sydney needs its (%S - %C) to be about 9.5%. So even if Sydney were to concede no points at all to the Lions - making %C equal to 0 - they'd need to score about 180 points to achieve this.

More generally still, Sydney need the sum of their victory margin and the Dogs' margin of defeat to be around 300 points if they're to grab fourth.

Sydney won't finish fourth.

Playing the Percentages

It seems very likely that this season, some ladder positions will be decided on percentage, so I thought it might be helpful to give you an heuristic for estimating the effect of a game result on a team's percentage.

A little maths produces the following exact result for the change in a team's percentage:

(1) New Percentage = Old Percentage + (%S - %C)/(1 + %C) * Old Percentage

where

%S = the points scored by the team in the game in question as a percentage of the points it has scored all season, excluding this game, and

%C = the points conceded by the team in the game in question as a percentage of the points it has conceded all season, excluding this game.

(In passing, I'll note that this equation makes it obvious that the only way for a team to increase its percentage on the basis of a single result is for %S to be greater than %C or, equivalently, for %S/%C to be greater than 1. Put another way, the team's percentage in the most current game needs to exceed its pre-game percentage.

This equation also puts a practical cap on the extent to which a team's percentage can alter based on the result of any one game at this stage of the season. For a team with a high percentage the term (%S - %C) will rarely exceed 5%, so a team with, for example, an existing percentage of 140 will find it hard to move that percentage by more than about 7 percentage points. Alternatively, a team with an existing percentage of just 70, which might at the extremes produce a (%S - %C) of 7%, will find it hard to move its percentage by more than about 5 percentage points in any one game.)

As an example of the use of equation (1) consider Sydney, who have scored 1,701 points this season and conceded 1,638, giving them a 103.8 percentage. If we assume, since this is Round 20, that they'll rack up a score this week that's about 5% of what they've previously scored all season and that they'll concede about 4%, then the formula tells us that their percentage will change by (5% - 4%)/(104%) * 103.8 = 1 percentage point.

Now 5% x 1,701 is about 85, and 4% x 1,638 is about 66, so we've implicitly assumed an 85-66 victory by the Swans in the previous paragraph. Recalculating Sydney's percentage the long way we get (1,701+85)/(1,638+66), which gives a 104.8 percentage and is, indeed, a 1 percentage point increase.

So we know that the formula works, which is nice, but not especially helpful.

To make equation (1) more helpful, we need firstly note that at this stage of the season the points that a team concedes in a game are unlikely to be a large proportion of the points they've already conceded so far in the entire season. So the (1+C%) in equation (1) is going to be very close to 1. That allows us to rewrite the equation as:

(2) Change in Percentage = (%S - %C) * Old Percentage

Now this equation makes it a little easier to play some what-if games.

For example we can ask what it would take for Sydney, who are currently equal with Carlton on competition points, to lift their percentage above Carlton's this weekend. Sydney's percentage stands now at 103.8 and Carlton's at 107.0, so Sydney needs a 3.2 percentage point lift.

Using a rearranged version of Equation (2) we know that achieving a lift of 3.2 percentage points from a current percentage of 103.8 requires that (%S - %C) be greater than 3.2/103.8, or about 3%. Now, if we assume that Sydney will concede points roughly equal to its season-long average then %C will be 1/19 or a bit over 5%.

So, to get the necessary lift in percentage, Sydney will need %S to be a bit over 5% + 3%, or 8%. To turn that into an actual score we take 8% x 1,701 (the number of points Sydney has scored in the season so far), which gives us a score of about 136. That's how many points Sydney will need to score to lift its percentage to around 107, assuming that its opponent this week (Fremantle) scores 5% x 1,638, which is approximately 82 points.

Within reasonable limits you can generalise this and say that Sydney needs to beat Fremantle by 54 points or more to lift its percentage to 107, regardless of the number of points Freo score. In reality, as Fremantle's score increase - and so %C rises - the margin of victory required by Sydney also rises, but only by a few points. A 60-point margin of victory will be enough to lift Sydney's percentage over Carlton's even in the unlikely event that the score in the Sydney v Freo game is as high as 170-110.

Okay, let's do one more what-if, this one a bit more complex.

What would it take for Melbourne to grab 8th spot this weekend? Well the Roos and Hawthorn would need to lose and the combined effect of Hawthorn's loss and Melbourne's win would need to drag Melbourne's percentage above Hawthorn's. Conveniently for us, Hawthorn and Melbourne meet this weekend. Even more conveniently, their respective points for and points against are all quite close: Hawthorn's scored 1,692 points and conceded 1,635; Melbourne's scored 1,599 and conceded 1,647.

The beauty of this fact is that, for both teams, in equation (2) Old Percentage is approximately 1 and, for any score, Hawthorn's %S will be approximately Melbourne's %C and vice versa. This means that any increase in percentage achieved by either team will be mirrored by an equivalent decrease in the percentage of the other.

All Melbourne needs do then to lift its percentage above Hawthorn's is to lift its percentage by one half the current difference. Melbourne's percentage stands at 97.1 and Hawthorn's at 103.5, so the difference is 6.4 and the target for Melbourne is an increase of 3.2 percentage points.

Melbourne then needs (%S-%C) to be a bit bigger than 3%. Since the divisors for both %S and %C are about the same we can re-express this by saying that Melbourne's margin of victory needs to be around 3% of the points it's conceded so far this season, which is 3% of 1,647 or around 50 points. Let's add on a few points to account for the fact that we need the margin to be a little over 3% and call the required margin 53 points.

So how good is our approximation? Well if Melbourne wins 123-70, Hawthorn's new percentage would be (1,692+70)/(1,635+123) = 1.002, and Melbourne's would be (1,599+123)/(1,647+70) = 1.003. Score 1 for the approximation. If, instead, it were a high-scoring game and Melbourne won 163-110, then Hawthorn's new percentage would be (1,692+110)/(1,635+163) = 1.002, and Melbourne's would be (1,599+163)/(1,647+100) = 1.003. So that works too.

In summary, a victory by the Dees over the Hawks by around 9-goals or more would, assuming the Roos lose to West Coast, propel Melbourne into the eight - not a confluence of events I'd be willing to wager large sums on, but a mathematical possibility nonetheless.

The Differential Difference

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.