from Amazon
Monday, July 24, 2006
62 to go
Would you prefer to be a worse team, but by good luck be ahead in the standings, or a better team, but by bad luck be behind in the standings?
Let’s start off with a simple simulation, where 100 games have already been played, and there is 62 to go. We have a four-team division, and all four teams are equals, and are currently at .500. Running 10,000 simulations, our four totally equal teams won or tied for the lead this often:
Team1 Team2 Team3 Team4
28.6% 28.4% 28.9% 28.1%
As you can see, even with 10,000 simulations, things don’t even out. But, the results are pretty close.
Now, let’s have some fun. Let’s make each team a true .500 team, but by good and bad luck, the current standings show:
Team 1 --
Team 2 -1
Team 3 -2
Team 4 -3
It’s a tight group. How badly off is team 4? 3 behind the lead, 62 games to go, but two other teams in the way.
Team1 Team2 Team3 Team4
42.9% 31.7% 22.8% 15.7%
Quite a difference! Essentially, each team has 70% the chance of the team above it. All because of a single game.
So far, we’ve assumed that all four teams are in fact equals in talent, and the standings were just a matter of a break here and there. Let’s take the flip-side: let’s make all four teams tied in the standings after 100 games, but make them unequals in talent. What kind of talent level does each team have to have to match the above percentages?
If I give the four teams this true talent level:
Team1 Team2 Team3 Team4
0.545 0.530 0.515 0.500
And start them all at .500, this is how often they’ll win or tie for the division lead:
Team1 Team2 Team3 Team4
42.0% 31.7% 23.3% 16.3%
And finally, let’s merge the two. Let’s have these true talent teams, with the following games behind the leader:
Team1 Team2 Team3 Team4
0.550 0.533 0.517 0.500
-3.......-2.......-1.......0
Team1 Team2 Team3 Team4
28.2% 28.8% 28.4% 28.8%
What this shows is that a really good team (.550 compared to a .500), but who find themselves three games behind that lucky team, has an even shot at the title. This is kind of like when you run against your big brother, and he gives you a 10-metre head start.