by Antony Ugoni

From the outset I should declare my love of the format of the final 8. It has everything I could hope for in a finals system. That is

• Rewards for finishing close to the top of the eight. Teams in positions 1 to 4 earn either a week’s rest or a second week’s game against an inferior team.
• The possibility of elimination during every final played by teams finishing in the lower echelons of the top eight.
• Four weeks of finals football (just enough to keep me wanting for more).
• A bumper first weekend of finals with four games to watch with no overlaps in schedule.
• Elegant symmetry.

I am also acutely aware of football as a commercial venture. That is, if the AFL’s research showed that more finals, via the participation of more teams, would be readily accepted (and eagerly consumed) by the paying public then the eligibility for finals involvement may extend deeper than the top 8. It is the AFL commission’s duty to keep the league financially viable, and if more finals teams will bring further revenue then so be it.

The purist in me, however, has been considering eligibility of contending finals for some time. Consider Essendon in 2009 or Brisbane in 1997, both making it into the final 8 with 10 wins and 1 draw. For me, these are losing home and away seasons yet somehow still worthy enough to contend for the ultimate prize. Such seasons had me wondering the following

• What is the lowest number of wins a team can still sneak into the final 8 with?
• What number of wins will guarantee a team plays in the finals (assuming top 8)?
• Assuming we never want teams playing finals with anything less than 11 wins from 22 games, how many teams should be included in the finals series.

With next year seeing the admission of an 18th team into the AFL’s home and away season I decided to explore such questions. My first attempt at each of the questions was through the elegance of mathematics. The number of permutations of a full home and away season of results was way beyond my capability. In any given round there are 2 raised to the power of 9 (512) different ways wins and losses can be distributed (no wonder football tipping is tough). Take this number and raise it to the power of 22 rounds and you end up with a seriously big number. So I turned to pure computing grunt instead. That is, I simulated one million home and away seasons and collected the round 22 ladders of each. I have provided further notes around the methodology at the end of this article, but if you are interested in further discussion I can be reached via antonyugoni@hotmail.com.

The answers to these questions are as follows:

Lowest possible number of wins a team can win and still make it into the final eight?

I had hoped that 11 wins (a 50% winning record) would be the bare minimum a team would need to win to make it to finals football, but the seasons of 2009 and 1997 left me in doubt. After interrogating the 1,000,000 simulations I found that it is actually possible for a team to make it into the finals with a paltry 9 wins. Admittedly this happened on only one occasion…but it happened. Wins for eighth place ranged from 9 to 14 in the following proportions: 0.0% (9 wins, 1 in 1,000,000), 0.7% (10 wins, 6,666), 49.9% (11 wins, 498,924), 48.4% (12 wins, 483,539), 1.1% (13 wins, 10,863) and 0.0% (14 wins, 7). Interestingly the same simulation for 16 teams in a 22 round home and away season also yielded seasons where eighth place had 9 wins (occurring 9 times in 1,000,000 simulations).

Number of wins to guarantee a position in the top 8.

To “guarantee” a position in the top 8 means a team isn’t depending on superior percentage to elevate them above a team with the same number of wins. For example, in 93.1% of simulated ladders where the team in eighth position won 11 games, the team in ninth position also won 11 games. That is, in less than 7% of seasons teams will be assured of playing finals if they win no more than 11 games.

Trawling through the 1,000,000 ladders shows that every time a team in eighth position wins 14 games the team in ninth position has never equalled this number of wins. In brief, to guarantee a team plays finals footy they need to win 14 games.

A finals series that will only ever have participants with a minimum of 11 home and away wins.

When I saw that first place on the ladder could be achieved with only 12 wins (80 of 1,000,000 simulations) I immediately suspected a shallow pool of finals contenders, however I was pleasantly surprised to see that teams finishing as low as fifth in these simulations never won less than 11 games (in 7 simulations sixth placed teams won 10 games only). Thus there may still be a place for the old McIntyre Final 5 system the VFL/AFL first used in 1972 and continued to use through to 1990.

And so, what to from here? Probably nowhere really. The commercial reality is that we are unlikely to ever see less than eight team contending AFL finals. The implied trade-off for teams with small numbers of wins is they will undoubtedly have small chance of becoming premiers in that year, however the purist in me wants to see no reward for mediocrity.

For those interested in the methodology used for the 1,000,000 simulations, read on.

All 198 games of a season have been simulated under the assumptions listed below, with wins and losses tallied for each team. This was repeated 1,000,000 times.

Assumptions:

1. The competition is split into 3 groups of 6 teams each. Each team plays every other team within their group twice and every team in the other two groups once.
2. The probability of winning a match (any match) is 50%.
3. There are no drawn games.

Drawn games are excluded for two reasons. The first being my interest in the number of wins. That is the water cooler conversations at the beginning of the season concerned with number of wins (not wins AND draws) to make it into the final eight. Secondly, the added dimension of a possible third outcome added to each game made programming significantly more complicated than I was prepared to pursue.

1. Gigs says:

Love it Antony. Anyone who takes the time to simulate 1,000,000 footy seasons gets my vote!

2. Gigs says:

And although I’m sad you didn’t include draws in your simulations, your “water-cooler” observation is correct. Every time I start talking about the beauty of draws around the water-cooler, I end up very quickly being the only one there.

3. Peter Flynn says:

Nice piece of modelling Antony.

The basis of this article would be worthy of inclusion in a mathematics teacher journal/magazine and could be developed into an interesting student investigation.

I reckon state your assumptions at the start. I was wondering what they were throughout.

Being able to vary the value of p would be the obvious improvement in your simulation. How much difference would this make to your findings? Hmmmmm. Possibly not much.

4. lee donovan says:

Did my team freo do any good in any of those 1 million seasons or do I have to keep waiting

5. Gigs says:

Lee, I expect the Dockers would have won a few of those million, the Saints would’ve made quite a few GF’s but won none of them and that my team, the Dogs, would have had many heart-breaking PF losses but not made the big one.

Assuming the inclusion of already completed actual seasons, the “Ladder of Premierships” would read, at the top and bottom:

Carlton 149,123 premierships
Essendon 149,123 (those two still inseparable)
Gold Coast 120,123
GWS 98,425

Fremantle 926
North Melbourne 112
Footscray 1
St Kilda 1

6. Dips says:

P Flynn – I knew I’d find a comment from you on this page. Superb.

7. Salc says:

A man after my own heart – just love a good set of stats. After thinking about this a bit, to reach the extremes it would require that the top 7 teams be completely dominant and the top six only play each other once with the top 6 teams winning 22, 21, 20, 19, 18 and 17 with the 7th team registering 15 wins losing to the top team twice and then once to the the rest of the top 6.

This accounts for 132 wins of the 198 games which leaves 66 games to be won by the remaining 11 hack teams. If these are distributed evenly then it is possibly for the 8th team to finish on 6 wins and make the finals on percentage with the same number of wins as the wooden spooner! I am guessing tanking might not be such an issue in this season.

As for the guarantee – that is an extremity I can’t quite reach yet.

8. Phil of Glen Iris says:

Brilliant stuff Antony. For me, the biggest factor that distances the simulation from reality is the existing unfairness of the draw and the impact that this has on the likely ladder configuration. With this force inexorably making the big clubs bigger and the weak clubs weaker, it’s hard to envisage even 80 of 1,000,000 AFL seasons with the first placed team having only 12 wins. It would be a weakling Collingwood team indeed that managed only this number from its fan-friendly Friday nights, blockbusters, virtual MCG residency etc.

9. David Bergman says:

I love a man who loves footy stats! If you were my boss I’d not only talk footy with you at the water cooler but at my desk all day
Keep up the good work!!

10. Monica Bellucci says:

Great work Antony!

Just want to let you know that I find mathemeticians highly desirable.

I will be utilising your hotmail address and would love to get into further “discussion”.

X

ps just the thought of 1 million simulations drives me wild!

11. Pinko says:

So in a season where 8th spot finishes on less than 11-wins one would assume that 1st and possibly 2nd spot have had stand out seasons (not sure if your simulations support this). So, would it be a fair & best reward (every pun intended) for 1st to play 8th in the first week of the finals?

Also, after 1,000,000 seasons I am betting Michael Tuck holds the games record on 21,967,002.

12. Nancy C says:

Ant – stop stuffing around and go mow the lawns!

13. Walshy says:

Antony, great work and even greater reward that Monica Bellucci loves stats.
Maths is where it’s at.