Imagine that the TAB Bookmaker was exceptionally good at his job and that his assessments of every game this season were what a statistician would label an "unbiased estimator" of the final result. In other words, imagine that the handicap he set in every game was an accurate reflection of the true expected margin that we would have seen in that game had it been played a substantial number of times to reduce the effects of on-the-day randomness.
Further, imagine that the approach I've been using to model those in-game random elements to simulate the results of the games in the remainder of the season (as described in this post) is an adequate and faithful way of encapsulating this randomness.
Granted those conceits it's possible to simulate the season so far, using the (negative of the) TAB Bookmaker's pre-game line market handicap for each game as his estimate of the likely outcome, to explore how the competition ladder might look instead right now after 20 rounds, had this Bookmaker been right and had the random elements played out in the different ways they might consistent with the way I've modelled them, across, say, 100,000 simulation replicates.
If we do that we might then describe the outcomes of that exercise in some of the now-familiar outputs from the Finalist simulations - for example the table at right showing the probabilities that, at this point in the season, a team might be lying 1st, in the Top 4, in the Top 8, or last.
It reveals that, in over half of the replicates, Hawthorn would now lead the competition and that in just 21% would Fremantle do so, as they do in the version of the "simulation" that we're experiencing. Also revealed in the fact that there's a version of reality, where the results of games were completely consistent with the TAB Bookmaker's views and where randomness had chanced its way onto the stage exactly as we've empirically modelled it, but where the Lions currently lead the competition. Granted, it happened in only 3 of the 100,000 replicates, but it is conceivable, such is the nature and size of the random elements of football when played out across a 170-game canvas.
There are also versions of season 2015 - a bit over 1% of them - where the Dons currently sit in the Top 4 and even more versions (12%) where the Dons sit in the Top 8. And there are plausible if extreme versions of 2015 where the Swans or even Fremantle, currently sit in 18th.
One of the fascinating things about this exercise, I find, is how it makes you think about the relative contributions of luck and ability to the fortunes of the teams, and how much we tend to downplay the former and focus mainly on the latter.
The other interesting aspect is to look at how far from their expected or most likely ladder position each team currently sits. In this regard, Port Adelaide might be seen either as the most unlucky or most underperformed of the teams this season, the simulations suggesting their most-likely current position is 3rd despite the fact that they sit 12th on the ladder.
Conversely, the Western Bulldogs might be seen as the luckiest or most overachieving team, their actual ladder position of 4th contrasting with their most-likely ladder of 11th should they have performed in line with TAB Bookmaker expectations.
One commonsense check on the results is to order the teams based on the average TAB handicap that each has faced, and to compare this ordering with the average ordering in the simulations.
Those average handicaps appear in the table at right and show a very similar ordering of the teams. Port Adelaide, for example, sit 5th on this measure having, on average given their opponents 11.4 points start in the line market, and the Dogs sit 12th having enjoyed, on average, almost a 4 point start.
We can also, of course, created the usual Dinosaur Charts, though the longer-term nature of the simulations we're performing here, looking at 20 rather than, say, 4 or 5 rounds, makes for charts that are more wave-like in nature. One thing that is immediately apparent from this chart is how broad the range of feasible current ladder positions is for every team.
Another perspective on this same observation is provided by the heat map below, the vast pink swathes of which reflect that same broad spectrum of possibilities for almost every team.
In addition to looking at the possibilities for individual teams, we can also consider the most likely Top 2s and Top 4s at this point in the season, details of which are provided below.
Across the 100,000 replicates, about 1 in 5 saw the competition with a Hawthorn / Fremantle 1-2 pairing and about another 1 in 10 saw it with the same pairing but in the opposite order.
Sydney lies second to the Hawks in another 8% or replicates, while Port Adelaide sits Runner Up to the Hawks in another 7%.
None of the 10 most-common pairings includes the Fremantle / West Coast duo that we actually have in the current ladder (though West Coast does sit second behind Hawthorn in about 7% of replicates).
About 1 in 40 replicates had Port Adelaide in first and Hawthorn in second, which is yet more evidence of how different Port Adelaide's actual fate is to that which the TAB Bookmaker's opinions have implied.
All 10 of the most-common Top 4s at this stage of the season from the simulations had the Hawks lying in first, and the eight most-common of those had Fremantle sitting in second behind them.
Randomness being what it is though, and having 20 rounds to play itself out, even the most-common quartet occurred in only just over 1 replicate in 100.
Rampant randomness is even more apparent when we look at the Top 8s churned out by the simulations, no single one of which appeared in more than 8 replicates. So, even if we'd known about the entire 20 rounds at the start of the season what the TAB Bookmaker has come to know about each round just before it's taken place, and even if the competition had panned out exactly as he'd imagined in each round, we'd have had, at best about a 12,500/1 shot of seeing the Top 8 we most expected right now.
(For the record, that most-common - though it's not at all "common" and only barely "most" - ordering was Hawthorn / Fremantle / Kangaroos / Sydney / West Coast / Port Adelaide / Collingwood / Richmond).
You might, I suppose, consider all this simulation and discussion to be purely academic and esoteric and divorced from the reality of season 2015. But the results here reflect two realities: the beliefs that the TAB Bookmaker held just prior to each round, and the contribution that randomness makes to the outcome of every game of football, as near as I can empirically model it. And that, I think, makes these results, if not profound, then certainly worth reflecting upon.
If the same sets of inputs to a season can yield such a diversity of outputs we should acknowledge that the ladder we see today reflects a mixture of ability and luck - perhaps more of the latter than we sometimes admit - and that the plaudits we shower on the successful and the brickbats we toss at the apparent underachievers might sometimes be more than either deserve.