One of the challenges in running cold simulations (ie those where the underlying team ratings and Venue Performance Values don’t change within a simulation replicate based on simulated results) is capturing the inherently increasing uncertainty about team ratings in future games.

We could ignore it entirely or, instead, attempt to incorporate the time-varying nature of that uncertainty in some way. I have chosen to follow the latter course in both my home and away simulations and my finals simulations. (For details about the methodology, see this blog post.)

Specifically, I’ve assumed that the standard deviation of teams’ offensive and defensive ratings is equal to 4.5 times the square root of the time between their latest rating and the date of the match in question, measured in days. This results in some quite large standard deviations for moderately distant games.

Applying that methodology to 10,000 of the 50,000 home and away season simulation replicates yields the following chart showing teams’ Finals fate overall and as a function of their ultimate ladder position at the end of the home and away season.

Overall, we see that the teams fall fairly naturally into five groups:

• Melbourne: just over 20% chances for the Flag and about 40% chances to make the Grand Final

• Geelong: just over 15% chances for the Flag and about 30% chances to make the Grand Final

• Brisbane Lions, Fremantle and Sydney: approximately 11-12% chances for the Flag and 22-24% chances to make the Grand Final

• Carlton, Collingwood, Richmond and Western Bulldogs: approximately 6-8% chances for the Flag and 12-16% chances to make the Grand Final

• St Kilda and Gold Coast: approximately 2% chances for the Flag and 3-4% chances to make the Grand Final

• Port Adelaide: less than 1% chance for the Flag and about 1.5% chance to make the Grand Final

• The Rest: all less than 0.5% chances to make the Finals and effectively zero chance of making the Grand Final

WEEK OF ELIMINATION IN FINALS

In this next chart we look at teams' chances for various Finals finishes, ignoring their home and away ladder positions (ie we focus solely on the heights of the bars in the previous chart). The numbers shown inside a cell are the percentage of simulation replicates (multiplied by 100) in which the specified team went out in the specified week of the Finals.

We see here that, if we define the season in terms of the five events listed above plus "Miss the Finals", the most-likely finishes for each team are estimated to be:

• Lose in a Preliminary Final: Melbourne, Geelong, Brisbane Lions, and Sydney (just)

• Lose in a Semi Final: Fremantle (just)

• Lose in an Elimination Final: Carlton, Collingwood, Richmond, and Western Bulldogs

• Miss the Finals: all other teams

GRAND FINAL PAIRINGS 

In this final chart we look at all of the Grand Final pairings that occurred in at least one of the simulation replicates. The numbers shown inside a cell are the percentage of simulation replicates (multiplied by 100) in which the team named in the row defeated the team named in the column in the Grand Final.

We see that the most common Grand Final has Melbourne defeating Geelong. This occurred in just over 4% of replicates. The opposite result - Geelong defeating Melbourne - occurred in another 3.4% of replicates.

(Note that zeroes in the chart represent pairings that did occur at least once but in less than 0.05% of replicates.)

MODELLING RATING VARIABILITY: AN ALTERNATIVE

Let’s explore the time-varying nature of team MoSHBODS ratings across the period from 2000 to now, by plotting the change in offensive rating and defensive rating for the same team X rounds apart.

(So, for example, the chart for X=3 has all the rating changes for teams between Rounds 1 and 4, Rounds 2 and 5, and so on.)

From this we can see that:

• The variability in changes in both offensive and defensive ratings does vary with time

• The changes are positively correlated

To investigate the shapes of the distributions we’ll next do density plots of each component, separately.

These all look somewhat Normal-like in shape, and a further analysis of their kurtosis suggests that they’re slightly leptokurtic (kurtosis around 3.5 to 4) for smaller Round gaps, but probably not too different from Normal for us to simulate on the basis of that being what they are.

Based on those charts (and analysing based on the day gap rather than the coarser, Round gap) we find that:

• For any given time D days ahead from a given point in the season, ratings variability roughly follows a Normal distribution with zero mean and a standard deviation equal to D to the Power of about 0.41

• Defensive and Offensive Rating changes are positively correlated and their covariance is best modelled as 0.158 x D. The covariance of the component rating changes is, I think, a quite important omission in my original approach.

Applying this methodology to both the home and away season simulations and then to 10,000 of those for the purposes of simulating the Finals yields the following, alternative outputs.

Perhaps the most striking feature here is that fewer teams are rated 50:50 chances in Grand Finals. Melbourne, for example, who won the Grand Final in 21% of replicates, and lost it in 17% of replicates under the methodology above, now win the Grand Final in 29% of replicates, and lose it in 19% of replicates.

Adding lots of variability to the ratings tends to make games in the moderately distant future closer to 50:50 propositions, which tends to reduce the probabilities associated with positive outcomes for stronger teams.

Conversely, less strong teams do less well, as we can see here for Gold Coast, St Kilda, and Port Adelaide, whose chances of making the Grand Final diminish substantially, albeit from a fairly low base.

Overall, the team-by-team main differences are:

• Melbourne’s Flag chances increase by about 30% (from 21% to 28.5%)

• Geelong’s Flag chances increase by about 10% (from 15.4% to 17%)

• Brisbane Lions’ Flag chances decrease by about 20% (from 11.9% to 9.5%)

• Sydney’s Flag chances increase by about 20% (from 10.8% to 13%)

• Fremantle’s Flag chances decrease by about 25% (from 10.7% to 7.9%)

• Carlton’s Flag chances decrease by about 50% (from 7.5% to 3.7%)

• Collingwood’s Flag chances decrease by about 20% (from 6.4% to 5%)

• Western Bulldogs’ Flag chances increase by about 25% (from 5.6% to 7%), partly because they are so highly rated by MoSHBODS

• Richmond are slightly more likely to go out in a Semi Final or Preliminary Final

• Gold Coast are slightly more likely to go out in an Elimination Final or Semi Final

• St Kilda, Port Adelaide, Hawthorn and GWS are all less likely to play Finals at all

For completeness’ sake, here are the Grand Final matchup numbers under this alternative methodology:

The Melbourne v Geelong Grand Final now turns up in over 10% of replicates, which is about one-third as much again as under the earlier methodology.

THE WAY AHEAD

I’ve no plan to swap exclusively to the alternative method of modelling rating variability, although I will be keeping an eye on both approaches for the remainder of the season.

My sense is that this new, alternative approach is superior, but I’d like to spend some time in the off-season making some more direct comparisons of them.