A rudimentary NFL season simulation
Following a post by Tango a couple of weeks ago on the playoff systems of the various sports, I thought I'd try writing a simulation. This is an update of that work in progress. Actually, I've only started on the NFL, and I haven't even done playoffs yet, just the regular season. But I thought I'd at least share what I've got so far.
In the simulation, each of the 32 teams was assigned a "true talent," from a normal distribution with mean .500 and standard deviation .143. No team was allowed to have talent higher than .900 or lower than .100; if they did, they were moved to .900 or .100. Then, all 32 teams were moved the same amount (arithmetically) in the same direction to get the overall talent to average exactly .500. (I think this method actually reduces the expected SD below .143, but I didn't bother fixing that.)
The 16-game schedule is random, instead of unbalanced (with the restriction that a team can't face any another team more than twice). There are no tie games. There is no home field advantage (although that would be easy to add in). The chance of winning each game is determined by the log5 method. There are no ties in games. Ties in the standings (division or wild card) are broken randomly.
As I said, I stopped there for now; haven't done playoffs yet. That's the next step, along with home field advantage.
Anyway, here are some results. Each result is out of 100,000 seasons. Every result came from a different run of the simulation. Results varied a fair bit per run, but I think everything is reasonably typical.
I checked for all teams out of 3,200,000 (32 teams, 100,000 seasons) that finished more than 8 games above or below their talent. That's hard to do, obviously. Also, the worse or better you are, the harder it is. It's (relatively) easier for an 8-8 team to go 16-0 than for a 3-13 team to go 11-5. Amplifying that is the fact that there are a lot more 8-8 teams than 3-13 teams. However, offsetting that, a little bit, is the fact that the 3-13 team can also go 12-4 or 13-3 or better.
In any case ... there were 43 cases where a team differed from its talent by 8 games or more. Of those, 26 were teams that outperformed, and 17 were teams that underperformed.
The biggest differential was in season 98,534, where the Broncos a team that had talent of 4.56 wins (out of 16), but went 14-2, for a differential of 9.44 games. That was the only team with a differential of 9 or more. Part of the reason it did so well was that it faced inferior opponents. You'd expect any given team's opponents to average 8.00 games of talent. But in that season, the Broncos' opponents' talent was only 7.45 games. Not a huge difference, but still.
Actually, when it comes to extreme events, a small difference in opponents makes a big difference in probability. Of the 43 teams in the sample, 38 of them had records that went in the direction "aided" by the opposition (in the sense that the underperforming teams played better-than-expected opponents, and vice versa). That's 38-5 in favor.
The worst team in the sample was the season 63,924 Jets, a 3.03 team that went 12-4 (playing 7.11-win opponents). The best team in the sample was a Bucs team that was expected to win 12.04 games, but instead went 4-12 (playing 8.58-win opponents).
I also took a look at teams that went 0-16.
Those results probably aren't as realistic, because they're heavily dependent on the shape of the tail of the talent distribution ... and we really don't know what that is. Recall that we chose a normal distribution that gets truncated at .100 (1.6 wins). Both those choices -- normal, and truncated -- are arbitrary and probably not close enough to real life. (Also, teams could drop below .100 in talent from the adjustment that sets all league-years to .500.)
In addition, the other shortcuts in the simulation probably skew the results too. The mainstream results are probably right, but the extremes are extremely sensitive to some of the assumptions.
With those caveats: there were 5,663 of those 0-16 teams out of 3.2 million, and their average talent was .181, which is just under 3-13. I suspect the talent of actual flesh-and-blood 0-16 teams is higher than that, but I really don't know.
16-0 should be exactly symmetrical, so I won't show that separately.
I checked for four-way ties where every team has the same record. That happened 878 times out of 800,000, or about once every century.
There were five seasons out of 100,000 where two divisions had a four-way tie. Actually, that might be a little high ... the test runs had only 1 or 2 such seasons.
Anyway, before I start on the playoffs, and repeating this for other sports leagues, I'm looking for feedback on what I've got so far. Any suggestions?
And, if you want me to run the sim and check for something in particular, let me know in the comments. It's real easy to add a couple of lines of code to check for something specific.