## Uneven Divisions Produce Uneven Results

A number of sources have indicated that the NHL may be switching to a 4 division, 2 conference set-up starting in the 2012-2013 hockey season.  This would almost certainly results in two 8 team divisions and two 7 team divisions.  As a result, we've seen a lot of speculation about how playoff seeding could change.  Many people have argued that the NHL ought to go to a divisional playoff format, where the top 4 teams in each division make the playoffs and play other teams from their division before moving on to face teams from other divisions in later rounds of the playoffs.  I have argued against this idea on the grounds that teams in the 7 team divisions would have an unfair advantage in that they have a better chance of making the playoffs due to having less teams to make.

Whenever I make this claim, though, I find that a number of people disagree, arguing that there is no unfairness in this format.  To try to clarify this, I've run some simulations in order to see whether an advantage exists, and if so, what it looks like.  Follow me after the jump and we'll take a look at the results.

The actual mathematical advantage that teams in the 7 team division have is 12.5%; that is to say that teams in the 7 team division will, over a run of multiple seasons, make the playoffs on average 12.5% more frequently than teams from the 8 team divisions.  But that math isn't very convincing to some people, so instead I wrote some code to simulate some results to try to make the explanation more clear.

The first thing I did was write a program that would generate random playoff seedings for two divisions; Division A had 8 teams and Division B had 7.  I randomly assigned 4 teams from each division to the playoffs.  At least, that's what I intended to do, but I organised the code poorly and it didn't work.  So I threw out the code and started again, and my second attempt was much more successful (and much more efficient).  I then ran 10,000 seasons to see what kind of an advantage the teams in the 7 team divisions might rack up.  Of course, that was unnecessary, since the 12.5% advantage that those teams have isn't dependent on how many seasons you run the simulation for - it's always true.  One thing that did emerge over those 10,000 seasons was that on a long enough time scale, even the worst teams in the 7 team divisions will garner more playoff appearances than the best teams in the 8 team divisions.

The problem with that is that the NHL will never run 10,000 seasons, and the numbers begin to look meaningless once they get that large.  So I re-focussed, and instead I ran 10 batches of 10 seasons.  10 years is how long the NHL has had 30 teams, so that seemed like a good length of time.  I ran 10 simulations mostly because it's a nice simple number.  So what were the results?  I've uploaded a spreadsheet for anyone who wants to see the individual results, but here were the most important things I discovered:

14 teams in the large division (Division A) had 7 or more playoff appearances in a 10 year run (out of the 100 possible appearances).  By contrast, 22 teams in the small division (Division B) had 7 or more playoff appearances in a 10 year run, a very significant increase. That means teams in the small division were 57% more likely to have 7 or more playoff appearances in a 10 year run.

At the low end of the scale, 14 teams had 3 or fewer playoff appearances in a 10 year run.  At the other end, the same was true for just 6 teams in Division B.  That means teams in Division A were 133% more likely to have just 3 playoff appearances than teams in Division B were.

The last thing I looked at in these runs was the median number of playoff appearances in each division.  For Division A, it was 5; for Division B, it was 6.  That might not seem like much, but it is one more indication that teams in a 7 team division have an advantage over teams in an 8 team division.

So those are the results, and they seem fairly clear to me.  While it's impossible to predict what might happen in any one season, over a reasonable time span (such as one decade) the advantage that teams in a 7 team conference have over teams in an 8 team conference becomes clear.  The results are better at both the low end, the mid-range, and the high end.  The advantage is quite significant, and that is why divisional playoffs with uneven divisions is a bad idea.

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