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Safe Leads and Lead Changes in Competitive Team Sports

In this talk, we will investigate the time evolution of lead changes within individual games of competitive team sports. Exploiting ideas from the
theory of random walks, we first show that the number of lead changes within a single game follows a Gaussian distribution. We then 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. Finally, we derive a simple formula for the probability that a given lead is "safe", i.e., will not be
overturned by the end of the game, as a function of its size L and game time t. Using comprehensive data on more than 1.25 million
scoring events in roughly 40,000 games across four professional or semi-professional team sports, we find that our
predictions are generally in excellent agreement with real data, and are more accurate than popular heuristics currently
used in sports analytics.

Joint work with Marina Kogan (Colorado) and Sid Redner (Santa Fe)