Safe leads and lead changes in competitive team sports

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 semiprofessional team sports, and are more accurate than popular heuristics currently used in sports analytics.

Figure 1

Evolution of the score difference in a typical NBA game: the Denver Nuggets vs the Chicago Bulls on 26 November 2010. Dots indicate the four lead changes in the game. The Nuggets led for 2601 out of 2880 total seconds and won the game by a score of 98–97.

Figure 2

Distribution of the average number of lead changes per game in professional basketball.

Figure 3

The distribution of the time that a given team holds the lead, $\mathcal{O}\left(t\right)$. The homogeneous Poisson process data is virtually indistinguishable from the arcsine law, while the inhomogeneous Poisson process roughly accounts for the end-of-quarter enhanced scoring rate.

Figure 4

Schematic score evolution in a game of time $T$. The subsequent trajectory after the last lead change must always be positive (solid) or always negative (dashed).

Figure 5

Upper: Empirical probability that a scoring event occurs at time $t$, with the game-average scoring rate shown as a horizontal line. The data is aggregated in bins of 10 sec. each; the same binning is used in Fig. 14. Lower: Distribution of times $\mathcal{L}\left(t\right)$ for the last lead change.

Figure 6

The distribution of time for the last lead change, $\mathcal{L}\left(f\right)$, as a function of the fraction of steps $f$ for a 94-step random walk with persistence parameter $p=0.36$ as in the NBA ($\circ$) and $p=0.25$ corresponding to stronger persistence ($△$). The smooth curve is the arcsine law for $p=0.5$ (no antipersistence).

Figure 7

The distribution $\mathcal{L}\left(t\right)$ for nonzero bias [Eq. (12)]. The diffusion coefficient is the empirical value $D=0.0391$, and bias values are $v=0.002, v=0.004$, and $v=0.008$ (increasingly asymmetric curves). The central value of $v$ roughly corresponds to average NBA game-scoring bias if diffusion is neglected.

Figure 8

The maximal lead (which could be positive or negative) occurs at time $t$.

Figure 9

Distribution of times $\mathcal{M}\left(t\right)$ for the maximal lead.

Figure 10

One team leads by $L$ points when a time $\tau$ is left in the game.

Figure 11

Probability that a lead is safe versus the dimensionless lead size $z=L/\sqrt{4D\tau }$ for NBA games, showing the prediction from Eq. (15), the empirical data, and the mean prediction for Bill James' well-known “safe lead” heuristic.

Figure 12

Probability that a lead is safe versus $z=L/\sqrt{4D\tau }$ for (a) the stronger team is leading for $\text{Pe}=\frac{1}{5}, \frac{1}{2}$, and 1 (progressively flatter curves), and (b) the weaker team is leading for $\text{Pe}=-\frac{2}{5}, -\frac{4}{5}$, and $-\frac{6}{5}$ (progressively shifting to the right). The case $\text{Pe}=0$ is also shown for comparison.

Figure 13

Distribution of the average number of lead changes per game, for CFB, NFL, and NHL, showing the prediction of Eq. (3), the empirical data, and the results of a simulation in which scoring events occur by a Poisson process with the game-specific scoring rate.

Figure 14

Upper: Empirical probability that a scoring event occurs at time $t$, with the game-average scoring rate shown as a horizontal line, for games of CFB, NFL, and NHL. Lower: Distribution of times $\mathcal{L}\left(t\right)$ for the last lead change.

Figure 15

Distribution of times $\mathcal{M}\left(t\right)$ for the maximal lead, for games of CFB, NFL, and NHL.

Figure 16

Probability that a lead is safe, for CFB, NFL, and NHL, versus the dimensionless lead $z=L/\sqrt{4D\tau }$. Each figure shows the prediction from Eq. (15) and the corresponding empirical pattern.