In a pro basketball game, when is a lead safe?

In a pro basketball game, when is a lead safe? This analysis

“Safe Leads and Lead Changes in Competitive Team Sports,” Aaron Clauset [pictured here], M. Kogan, and S. Redner, arXiv:1503.03509v1 [stat.AP] 11 Mar 2015. (Thanks to Marcos Carreira for bringing this to our attention.) The authors, at the University of Colorado, the Santa Fe Institute, and Boston University, report:

“We investigate the time evolution of lead changes within individual games of competitive team sports. Exploiting ideas from the theory of random walks, the number of lead changes within a single game follows a Gaussian distribution. We show that the probability that the last lead change and the time of the largest lead size are governed by the same arcsine law, a bimodal distribution that diverges at the start and at the end of the game. We also determine the probability that a given lead is “safe” as a function of its size L and game time t. Our predictions generally agree with comprehensive data on more than 1.25 million scoring events in roughly 40,000 games across four professional or semi-professional team sports, and are more accurate than popular heuristics currently used in sports analytics….

The kicker is in their conclusion:

“Cynically, our results suggest that one should watch only the first few and last few minutes of a professional basketball game; the rest of the game is as predictable as watching repeated coin tossings. On the other hand, the high degree of unpredictability of events in the middle of a game may be precisely what makes these games so exciting for sports fans.”

Here’s further detail from the paper:

The paper also explores the same question as it plays out in other sports — American college football, American pro football, and professional hockey — where the answer is not so dismaying.