# 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.

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