I've been part of some interesting conversations this week on Twitter about definitions of "close" and "blowout" games of Australian Rules.

Alternative definitions of "close" have mostly focussed on the final game margin, though some have also suggested that the margin at some earlier point in the game might also be included - say at Three-Quarter Time, or with a few minutes to play. Most defined a threshold for "close" in terms of points, most commonly 12, though a couple of people noted that, with the much lower total scores in earlier, VFL seasons (see chart are right), defining a threshold in terms of total score might also make sense if the goal is to make longer-term comparisons.

In discussions about the notion of a "blowout" debate was only ever about the the size of the final margin that should be use to demarcate such games. Suggestions ranged from a low of 8 goals to a high of 10 goals.

Consistent with the general thrust of both discussions, in the charts that follow I've tracked the proportion of games in each season where the final margin has been below some cutoff (in the "close" games chart) or above some cutoff (in the "blowouts" chart). Both charts use data from all games from Round 1 of 1897 to Round 15 of 2015.

The dotted lines on each chart provide a smoothed view of the underlying time series (technically, they're 5-season, uncentred moving averages) and broadly suggest that the recent trend is towards more "blowouts" and, arguably, slightly more "close" games.

It's interesting to speculate about why this might be the case - are the greater number of "blowouts", for example, being driven by inequities in financial resources that lead to greater variability in team abilities - but in this blog I want to consider just how far the current trends might be from some "ideal".

To do this I'm going to proceed by simulation using the same methodology as I've been using to create the simulations of the rest of the season, the most relevant feature of which for our current purposes is that game margins are assumed to follow a Normal distribution with a standard deviation of 36 about some mean based on the relative abilities of the teams.

To incorporate the aspects of a real (imbalanced) draw I'm going to simulate the entirety of the 2015 season varying only the assumed (constant) abilities of the 18 teams. The expected game margin in any game will be set equal to the difference in the assumed Ratings of the participating teams and we'll further assume that home ground advantage is equal to zero.

Four separate runs of 10,000 simulations of the entire 2015 season were made under the following scenarios:

1. Assuming that all teams were of equal ability (ie Rated 1,000)
2. Assuming that all teams had their current ChiPS Ratings
3. Assuming that all teams' Ratings were 25% nearer to 1000 than their current ChiPS Ratings (so, for example, a team Rated 1,030 would become Rated 1,022.5, and one Rated 950 would become Rated 962.5)
4. Assuming that all teams' Ratings were 50% nearer to 1000 than their current ChiPS Ratings

In all cases, teams are assumed to maintain the same Rating throughout a season.

The outputs for Scenario 1 provide us with a best-case outcome and will give an estimate of the proportion of "close" games and "blowouts" that we should expect in a perfectly balanced competition.

Scenario 2 will give estimates of "close" games and "blowouts" that might be expected in the current season, while Scenarios 3 and 4 will help us understand the extent to which interventions that moved us some way towards perfect balance might be expected to be reflected in the proportions of "close" and "blowout" games in a season like the current one.

The results of the simulations appear below, firstly for the proportion of "close" games expected in a season where we define "close" as finishing with a margin under 2 goals. These plots are called "violin plots" and, as Wikipedia describes, they are "similar to box plots, except that they also show the probability density of the data at different values". In the versions I've created, three dots are included to denote the 25th, 50th (ie median) and 75th percentiles of the simulation results (so, for example, only 25% of simulated seasons finished with a proportion of close games higher than the third dot).

So, given the assumptions I've made about the distribution and variability of game margins, if we repeatedly simulate the current season, further assuming that all teams are Rated equally throughout, we would expect about 25% of games to be "close". So far in 2015 we've had almost exactly this proportion of "close" games, as we did in 2014.

Earlier seasons saw fewer "close" games however, with only about 21% in 2013, 18% in 2012, and 21% in 2011. Doubtless, the introduction of two new teams contributed to this reduction, though it might also be that we've simply been fortunate so far this season (and last) to witness as many "close" games as we have, the simulation results based on the current distribution of team Ratings (the second violin) suggesting that proportions of "close" games nearer 20% might be more the norm. What might also be helping, of course, is that the distribution of home ground advantages across teams, which are entirely ignored in these simulations, could tend to reduce the average effective differences in team abilities.

The two rightmost violins give an indication of how reductions in the variability of team Ratings would help lift the expected proportion of "close" games.

Next then to "blowouts", the chart for which appears below and uses 10 goals or more as the margin defining such a game.

In an evenly-matched competition where the expected margin is, by definition, always zero, a 60 point victory represents a result about 1.7 standard deviations from the mean. Results so deviant are relatively rare, and we see on average only about 10% of games finish with such a margin in our "ideal" competition. 

This season, and the two that have gone before it, have produced about twice as many "blowouts" as this ideal, a statistic that is completely consistent with the results in the second set of simulations. In that sense then, the current imbalances in the abilities of teams are delivering exactly the proportion of "blowout" games we should expect.

While complete parity in team Ratings seems an unrealistic goal, we can see from the third set of simulations that a more modest goal of reducing the variability such that all teams are 25% nearer parity, would have a significant impact on reducing the proportion of expected "blowouts" in a season. To provide some context for what that goal implies, it would mean making the weakest teams about a goal better than they are now and curtailing the abilities of the strongest teams by about the same amount.

SUMMARY AND CONCLUSION

The inherent variability in Australian Rules football limits the proportion of close games that we can reasonably hope to see and also means that blowouts, even amongst evenly-matched teams, are inevitable.

Defining "close" games as those that finish with a margin under two goals, and "blowouts" as those that finish with a margin of 10 goals or more, simulations reveal that 25% "close" games and 10% "blowouts" are fundamental limits.

This season has produced "close" games at about this idealised rate, though the spread of team Ratings suggests that this might only be a temporary glut. The season's also produced "blowouts" at almost exactly the rate we'd expect given those same team Ratings.

Efforts to reduce the variability in underlying team abilities will serve to lift the proportion of "close" games nearer the theoretical maximum, and drop the proportion of "blowouts" nearer the theoretical minimum. Making weaker teams about one goal stronger, and stronger teams about one goal weaker would lift the expected proportion of "close" games by about 1 game per round (ie from 20% to 22%), and drop the expected proportion of "blowout" games by about the same amount (ie from 18% to 16%).

Exactly how that might be done is a topic for another blog (and probably another blogger).