# What Proportion of Close Games Should the Better Team Win?

/These week there's been a lot of talk about Hawthorn and their ability to "win the close ones", one narrative being that they are somehow able to do this more often than they "should" given that they're 5 and 0 in games finishing with a margin of under a goal this season.

The first thing to point out is that nothing can usefully be concluded based on a sample of 5, so I won't be saying anything specific about Hawthorn, but I think the more interesting question is to ask how you might determine how many close games a team "should" win.

Implicit in many of the discussions is that teams, regardless of their relative abilities, all become 50:50 propositions in close games. I think that position is logically untenable. If a better side was expected to record, say, 60% of the scoring opportunities at the time the game started then, field position notwithstanding they must surely be expected to generate a better than 50% share of any remaining opportunities in the back end of a close contest.

My initial assumption was that, assuming level scores, the two team's chances with any fraction of the game remaining should be the same as they were at the start. That assumption, on reflection, is also wrong. The original assessments of the two teams' chances were based on the total set of scoring shots expected to be generated during the entire course of the game but, as time reduces and the number of likely remaining scoring shots diminishes, it stand to reason that the weaker team has a higher likelihood, by chance alone, to generate the majority of the remaining scoring shots and therefore win.

Imagine, for example that we placed 100 balls in a bowl, 55 green and 45 blue from which we made 50 draws at random with replacement. The likelihood of drawing more blue than green balls - of blue "winning", if you like - in this case is about 19%. But, if we only make 5 draws instead, the probability climbs to 41%.

So, the better team should win close games more than 50% of the time but less often than its pre-game probability would have suggested.

One way of coming up with a figure for the proportion they should win is to proceed as follows.

We've found in previous analyses (for example, this empirical one from 2009 and this theoretical one from 2014) that the final margin of a V/AFL game can be modelled as a Normal random variate with mean equal to the pre-game expected margin and standard deviation of about 36.

That assumption can be used to model the game margin distributions for teams of varying levels of pre-game superiority and to ask of the resulting distribution: of all games that finished with a margin of X points or less (as a win or a loss for the superior team), in what proportion does the superior team win?

The answer to that question is summarised in the chart below.

Each line in the chart relates to a team with some fixed level of pre-game superioirity. The bottom line is a team that is 0 points better than its opponent (ie an equal-favourite), the next to a team that is assessed as a 6-point favourite, and so on in 6-point increments. The top line relates to a 60-point pre-game favourite.

As we move from left to right across a line we increase the range of victory margins that we are considering. We can read the proportion of games that the superior team is expected to win for any given victory margin from the vertical axis.

So, for example, a team that was a 6-point pre-game favourite would be expected to win about 53% of games that finished with a margin of 18 points or less (for either side).

What we see then is that, as suggested, superior teams are expected to win more than one-half of games finishing with a margin within any nominated range, but also that the probability doesn't stray all that far from 50% all that quickly for teams that are only slightly superior. For example, even a 24-point better side will win only about 57% of games won by 18 points or fewer.

Vastly superior sides, however, win much larger proportions of close games. For example, a 48-point better team will win 70% of games decided by 24 points or fewer.

Armed with this chart it would be possible to go back for any team and use their pre-game bookmaker handicaps in a large enough sample of games that finished as narrow wins or losses, and assess the extent to which they have under- or over-performed in those games.

I'll leave that, in fine professorial style, as an exercise for the reader.