There have been quite a few overtime games in the NBA playoffs this year.  We have had one in the finals already and in an earlier series between the Bulls and Celtics, 4 out of 7 games went into overtime, with one game in double overtime and one game in triple overtime!

How often should we expect a basketball game to end tied after 48 minutes of play?  At first glance it would seem pretty rare.  If you look at the distribution of points scored by the home teams and by the visiting teams separately, they look pretty close to a normal distribution with a large variance.  If we made the crude hypothesis that the two distributions were statistically independent, then ties would indeed be very rare:  2.29% of all games would reach overtime.

But the scoring is not independent of course.  Similar to a marathon, the amount of effort expended is different for the team currently in front versus the team trailing and this amount of effort also depends on the current point differential.  But such strategy should have only a small effect on the probability of ties.  The team ahead optimally slows down to conserve effort, balancing this against the increased chance that the score will tighten.  Also, conservation of effort by itself should generally compress point differentials, raising not just the frequency of ties, but also the frequency of games decided by one or two points.

But overtime is almost 3 times more frequent than this:  6.26% of all NBA games are tied at the end of regulation play.  And games decided by just a few points are surprisingly rare:  It is more likely to have a tie than for the game to be decided by two points, and a tie is more than twice as likely as a one-point difference.  These statistics are quite dramatic when you see them visually.

Here is a frequency histogram of the difference in points between the home team and visiting team at the end of regulation play.  These are data from all NBA games 1997-2009.  A positive number means that the home team won, a zero means that the game was tied and therefore went into overtime.  Notice the massive spike at zero.


(There is also more mass on the positive end.  This is the well-known home team bias.)

What explains this? A star PhD student at Northwestern, Toomas Hinnosaar, and I have been thinking about this. Our focus in on the dynamics and strategy at the end of the game.  To give you some ideas, Toomas created the following striking video. It shows the evolution of the point differential in the last 40 seconds of the fourth quarter.  At the beginning, the distribution looks close to normal.  This is what the crude hypothesis above would predict.  Watch how the spike emerges in such a short period of time.

By contrast, here is the same animation at the end of halftime.  Nothing unusual.