The latest round of simulations has one of Hawthorn, West Coast or Fremantle taking out the Minor Premiership in all but a handful of replicates, and Hawthorn collecting that prize about half of the time. I find it hard to imagine a time in recent history where a team 3rd on the ladder so late in a season is such an overwhelming favourite for this honour.

That assessment makes the Hawks seem extraordinary value at their current TAB price of $3 (though readers are cautioned to consider my wagering performance so far this season before contemplating any hasty action based on this or any other probabilistic assessment appearing in this blog - the chart in the "Latest Information" section in the sidebar makes for sobering review).

Apparent value is also evident in the Top 4 prices being offered for the Tigers, Dogs and Pies - especially in the $21 being dangled for the Dogs, who the simulations assess as being about 10% chances for a Top 4 finish.

The Dogs are also value at $1.55 for a Top 8 berth, while the Roos and the Cats carry an attractive, though not as large, edge in their prices for Missing the 8.

The inputs for the simulations, shown again this week as probabilities, appear in the table at left. The differences compared to last week are generally small, though I was amused to see that the ChiPS Rating System (from which these inputs are derived) now assesses the Hawks as 99.7% chances against the Lions in Round 22, and 98.8% chances against the Blues in Round 23. Those near-certain probabilities are equivalent to making the Hawks 100-point and 81-point favourites in these matchups, respectively. Somehow I doubt we'll ever see a 99.5 point line market, but who knows?

To the chart now known as "The Dinosaur Chart" then, where we see, as expected, a general narrowing in the set of available finishes for most teams, though still a smorgasbord of possibilities for a few teams that have non-trivial chances of ending the season in as many as nine different positions.

Adelaide probably has the richest menu, and certainly the most symmetric, though the Pies, Cats, Giants, Roos, Tigers and Dogs also have buffet-like opportunities.

Viewed, instead, as a heatmap, this same data reveals a natural clustering of the teams into four groups much as we saw last week, though it appears that the Swans' and Port Adelaide's position within their cluster is more certain than that of the other teams in the same group.

(Speaking, as I did earlier, of amusing, it was fun to see that the Dogs took out the Minor Premiership in exactly 1 of the 100,000 simulations. The bar for that outcome in the Dinosaur Chart above is so small that it can't be rendered visible, and appears here in the heatmap merely as a white block with a rounded 0 in it.)

TOP 2'S, 4'S AND 8'S

A Hawthorn / West Coast ordering is, comfortably, the most-likely 1-2 home-and-away season finish at the moment, this result appearing in over 30% of replicates. Next most common is a Fremantle / Hawthorn finish, which emerged almost 20% of the time.

Combined, a {Hawthorn, West Coast} pairing cropped up, in either order, over 45% of the time, and a {Fremantle, Hawthorn} pairing arose over 36% of the time. 

The Top 9 pairings all involve Hawthorn, West Coast, Fremantle and Sydney, and it's not until we get to the 10th most-common pairing that Richmond makes an entrance. Even then, it's a very quiet one, representing just 0.01% of replicates.

Richmond also waits until the sixth-most popular Top 4 before making an appearance, there being part of a quartet with probability of just 2.8%. Far more common is the quartet of Hawthorn / West Coast / Fremantle / Sydney, which transpires, in that order, in about 15% of all replicates.

A little over two-thirds as common is the Fremantle / Hawthorn / West Coast / Sydney ordering, it the only other quartet to appear in more than 10% of all replicates.

The Western Bulldogs appear in just one of the 10 most-common Top 4s, finishing fourth in a Hawthorn / West Coast / Fremantle / Western Bulldogs grouping that arose in just 2.8% of replicates.

Finally, it's salutary to note that, even this late in the season, the most likely Top 8 still warrants fair odds of over 360/1. That most likely ordering goes Hawthorn / West Coast / Fremantle / Sydney / Richmond / Kangaroos / Western Bulldogs / Adelaide, and it occurred in just 274 of the 100,000 simulations.

Amongst the most-common Top 8s, Hawthorn lies 1st, West Coast 2nd, and Fremantle 3rd in all but two of them. Sydney lies 4th in them all. Still, as I've been saying for the past few weeks, the probabilities here are so small that we should be cautious about what they mean for the true likelihood of any of them occurring in reality.

DEPENDENCIES IN FINALS APPEARANCES

Next, let's use 10,000 replicates to estimate the chances than any particular team with Finals aspirations might rue or applaud the inclusion or exclusion of another team in the 8.

We find, using the difference between a team's likelihood of making or missing the 8 when some other team is assumed to make or miss the 8, that:

• Adelaide's and Collingwood's fate is most tied to Richmond's
• Geelong's and GWS's fate is most tied to Adelaide's
• The Roos' and Port's fate is most tied to the Dogs'
• Richmond's fate is most tied to Adelaide's and Collingwood's
• The Dogs' fate is most tied to Port Adelaide's

However, now that teams have widely differing base probabilities of making the 8, the raw difference measure I've used for that bulleted list above might be a little deceptive. Whilst, for example, the Dogs could be 43% chances of making the 8 if Port make the 8, and 82% chances if Port miss the 8, because Port are now so unlikely to make the 8, this difference of 39% points somewhat overstates Port's real influence on the Dogs' fate. 

A better measure would be one that takes the likelihood of the relevant outcome into account, and it's such a measure that I've now used to assess what I'm calling the "Weighted Impact Index" of each of the remaining games (alongside the "Raw Impact Index", which is calculated as I explained last week). For those of you interested in the technical aspects of this Index - and it is more complex - I've included a section at the end of this post explaining how it's calculated, but the main thing to understand is that the measure provides a better indication of the likely impact of a game on the chances of the Finals aspirants, rather than a measure of the absolute impact when we compare the implications of a home team win with those of a home team loss ignoring how differently likely those two outcomes might be.

This new measure reduces, most of all, the assessed impact of those games where the outcome is nearer certain, and tends to increase the impact of those games where the outcome is more a 50:50 proposition. A perfect example of the need for such a Weighted Index is the Round 22 Hawks v Lions contest, where the Hawks are assessed as near 100% chances. Though it might be the case that, as it is, there's a 21% point difference between GWS' chances of making or missing the 8 depending on the result of that game, in truth the impact of that game on GWS' chances (and everyone else's) is almost certainly going to be near zero.

What we find then, looking at the Weighted (or, as it turns out, also the Raw) view, is that the Round 22 Cats v Pies game is now the most important of the remaining games in the context of the Final 8, and that next week's Crows v Tigers game is next-most important.

The Round 21 Pies v Tigers game is third-most important taking the Weighted view, and the Round 20 Dons v Crows game is fourth-most important.

Looking across the entirety of each remaining round, Round 21 is currently looking most pivotal as it includes three of the six most-important games as assessed at the present time.

WEIGHTED IMPACT INDEX CALCULATION

The Weighted Impact Index for each game is calculated as the sum of the expected absolute difference between all of the Finals aspirants' chances of making the Final 8 dependent on the outcome of that game relative to that team's unconditional chances.

So, for example, if a team is a 20% chance of making the 8 if the Home team wins and a 40% chance if the Home team loses (which, under the Raw Index, would have this team contributing 20% to the Impact measure of this game), and the probability of a Home team victory is 70%, then the contribution of the impact on this team to the Weighted Impact for this game will be given by: 70% x abs(20% - (20% x 70% + 40% x 30%)) + 30% x abs(40% - (20% x 70% + 40% x 30%)) = 8.4%.

The logic is that there's a 70% chance this team's probability will be 6% different from (lower than) the unconditional expectation, and a 30% chance that it'll be 14% different from (higher than) the unconditional expectation, the weighted sum of which is 8.4%. This is equal to the expected absolute change in this team's Finals chances once the result is known relative to their pre-game unconditional chances.

To get a feel for how this measure better reflects the true likely impact of a game's result where one team is an overwhelming favourite, consider another example where, again, the team is a 20% chance of making the 8 if the Home team wins and a 40% chance if the Home team loses, but now assume that the probability of a Home team win is only 5%.

The calculation now is: 5% x abs(20% - (20% x 5% + 40% x 95%)) + 95% x abs(40% - (20% x 5% + 40% x 95%)), which is just 1.9%, reflecting the fact that this game is much less likely to have a significant effect on the team's chances of making the 8.

The final Weighted Impact Index for a game is merely the sum of eight such calculations, one for each of the Finals aspirants.