Game story space of professional sports: Australian rules football

Sports are spontaneous generators of stories. Through skill and chance, the script of each game is dynamically written in real time by players acting out possible trajectories allowed by a sport's rules. By properly characterizing a given sport's ecology of “game stories,” we are able to capture the sport's capacity for unfolding interesting narratives, in part by contrasting them with random walks. Here we explore the game story space afforded by a data set of 1310 Australian Football League (AFL) score lines. We find that AFL games exhibit a continuous spectrum of stories rather than distinct clusters. We show how coarse graining reveals identifiable motifs ranging from last-minute comeback wins to one-sided blowouts. Through an extensive comparison with biased random walks, we show that real AFL games deliver a broader array of motifs than null models, and we provide consequent insights into the narrative appeal of real games.

Figure 1

Representative “game story” (or score differential “worm”) for an example AFL contest held between Geelong and Hawthorn on April 21, 2014. Individual scores are either goals (six points) or behinds (one point). Geelong won by 19 with a final score line of 106 (15 goals, 16 behinds) to 87 (12 goals, 15 behinds).

Figure 2

Conditional probability of scoring the next goal or behind given a particular lead size. Bins are in six-point blocks with the extreme leads collapsed: $< -72, -71$ to $-66,...,-6$ to $-1$, 1 to 6, 7 to $12,...,>72$ . As for most sports, the probability of scoring next increases approximately linearly as a function of current lead size.

Figure 3

Conditional probability of winning given a lead of size $\ell$ at the end of the first three quarters (a–c) and with 10 min to go in the game (d). Bins comprise the aggregate of every six points as in Fig. 2. The dark blue curve is a sigmoid function of the form ${[1+e^{-k(\ell -{\ell }_0)}]}^{-1}$ where $k$ and ${\ell }_0$ are fit parameters determined via standard optimization using the Python function scipy.optimize.curve_fit (Note that ${\ell }_0$ should be 0 by construction.) As a game progresses, the threshold for likely victory (winning probability 0.90, upper red lines) decreases as expected, as does a threshold for a close game (probability of 0.60, lower red line). The slope of the sigmoid curve increases as the game time progresses showing the evident greater impact of each point. We note that the missing data in panel (a) are a real feature of the specific 1310 games in our data set.

Figure 4

Skill bias $\rho$ represents a team's relative ability to score against another team and is estimated a posteriori by the fraction of scoring events made by each team Eq. (1). (a, b) Kolmogorov-Smirnov test $D$ statistic and associated $p$-value comparing the observed output skill bias distribution produced by a presumed input skill distribution $f$ with that observed for all AFL games in our data set, where $f$ is Gaussian with mean 0.5 and its standard deviation $\sigma$ is the variable of interest. For each value of $\sigma$, we created 1000 biased random walks with the bias $\rho$ drawn from the corresponding normal distribution. Each game's number of events was drawn from a distribution of the number of events in real AFL games (see text). Panel (b) is an expanded version of the shaded region in panel (a) with finer sampling. The black vertical dashed lines bound a plateau in the measured $p$-values, and the blue vertical dashed line is our estimated best fit of $\sigma \simeq 0.088$. We compare the resulting observed bias distribution with that of Ref. [12] in Fig. 5.

Figure 5

Comparison of the observed AFL skill bias distribution [balance of scoring events $\rho$ given in Eq. (1), dashed blue curve] with that produced by two approaches: (1) We draw $\rho$ from a normal distribution using the best candidate $\sigma$ value with mean 0.50 as determined via Fig. 4 (solid red curve) and (2) we choose $\rho$ from the complete list of observed biases from the AFL (hashed green curve, the replication method of Ref. [12]). For the real and the two simulated distributions, both $\rho$ and $1-\rho$ are included for symmetry. The fitted $\sigma$ approach produces a more accurate estimate of the observed biases, particularly for competitive matches ($\rho$ close to 0.50) and one-sided affairs. Inset: Upper half of the distributions plotted on a semi-logarithmic scale (base 10) revealing that the replication method of Ref. [12] also overproduces extreme biases, as compared to the AFL and our proposed correction using a numerically determined $\sigma$.

Figure 6

Variance in the instantaneous margin as a function of event number for real AFL games (solid red curve) and biased random walks as described in Sec. 3, (solid blue curve). We perform fits in logarithmic space using standard least squares regression (solid black curve for real games, dashed black for the null model). The biased random walks satisfactorily reproduce the observed scaling of variance. It thus appears that AFL games stories do not exhibit inherently superdiffusive behavior but rather result from imbalances between opposing teams.

Figure 7

Top 10 pairwise neighbors as determined by the distance measure between each game described by Eq. (2). In all examples, dark gray curves denote the game story. For the shorter game of each pair, horizontal solid blue lines show how we hold the final score constant to equalize lengths of games.

Figure 8

Heat maps for (a) the pairwise distances between games unsorted on a ring; (b) the same distances after games have been reordered on the ring so as to minimize the cost function given in Eq. (3); (c) the same as panel (b) but with game indices cycled to make the continuous spectrum of games evident. We include only every 20th game for clarity and note that such shuffling is usually performed for entities on a line rather than a ring. The games at the end of the spectrum are most dissimilar and correspond to runaway victories and comebacks (see also Fig. 9).

Figure 9

Heat matrix for the pairwise distances between games, subsampled by a factor of 20 as per Fig. 8. A noticeable split is visible between the blowout games (first six clusters) and the comeback victories (last three clusters). We plot dendrograms along both the top and left edges of the matrix, and as explained in Sec. 5c, the boxed numbers reference the 18 motifs found when the average intracluster distance is set to 11 points. These 18 motifs are variously displayed in Figs. 12 and 13.

Figure 10

Average intracluster distance $\langle \rho \rangle$ as a function of cluster number $N$. Vertical red lines mark the first occurrence in which the average of the intracluster distance of the $N$ motif clusters had a value below 12, 11, 10, 9, and 8 (red text beside each line) points respectively. The next cut for seven points gives 343 motifs.

Figure 11

Histograms of the number of motifs produced by 100 ensembles of 1310 games using the random walk null model and evaluating at 11- and nine-point cutoffs (a, b) as described in Sec. 5b. For real games, we obtain by comparison 18 and 71 motifs [vertical red lines in panels (a) and (b)], which exceeds all 100 motif numbers in both cases and indicates AFL game stories are more diverse than our null model would suggest.

Figure 12

Eighteen game motifs as determined by performing hierarchical clustering analysis and finding when the average intracluster game distance $\langle \rho \rangle$ first drops below 11 points. In each panel the main curves are the motifs, the average of all game stories (shown as light gray curves in background) within each cluster, and we arrange clusters in order of the motif winning margin. All motifs are shown with the same axis limits. Numbers of games within each cluster are indicated in the bottom right corner of each panel along with the average number of the nearest biased random walk games (normalized per 1310). Motif colors (online version) correspond to relative abundance of real versus random game ratio $R$ as red: $R\ge 1.1$; gray: $0.9<R<1.1$; and blue: $R\le 0.9$. See Fig. 13 for the same motifs reordered by real game to random ratio.

Figure 13

Real game motifs for an 11 point cut off as per Fig. 12 but reordered according to decreasing ratio of adjacent real to biased random games, $R$, and with closest biased random walk rather than real game stories plotted underneath in light gray. See the caption for Fig. 12 for more details.

Figure 14

Seventy-one distinct game motifs as determined by hierarchical clustering analysis with a threshold of nine points, the fourth cutoff shown in Fig. 10 and described in Sec. 5. Motifs are ordered by their final margin, highest to lowest, and real game stories are shown in the background of each motif. Cutoffs for motif colors red, gray, and blue correspond to real-to-random ratios 1.1 and 0.9, and the top number indicates motif rank according to final margin (see online version). The same process applied to the biased random walk model for our 100 simulations typically yields only 45 to 50 motifs (see Fig. 11). We discuss the 10 highlighted motifs in the main text and note that we have allowed the vertical axis limits to vary.

Figure 15

Motifs (red curves) from Fig. 14 rearranged in order of descending ratio of the number of real games to the number of adjacent biased random walk games, as described in Sec. 5c, and adjacent real game stories are shown in gray for each motif.

Figure 16

Final margin of motifs as a function of real-to-random ratio $R$ for real AFL games at the 18 and 71 motif levels, panels (a) and (b), respectively, with linear fits. On the right of each plot, extreme blowout motifs ending in high margins have no or relatively few adjacent random walks (red, $R\ge 1.1$). On the left, game stories are more well represented by random walks (blue, $R\le 0.9$). There is considerable variation, however, particularly in the 71 motif case, and we certainly see some close finishes with $R\ge 1$ [e.g., the massive comeback, motif no. 71, Fig. 15].

Figure 17

Illustration of our prediction method given in Eq. (6). We start with a game story $g_{obs}$ (red curve, see online version) for which we know up until, for this example, 60 min ($t=3600$). We find the $N=50$ closest game stories based on matching over the time period 45 to 60 min (memory $M=15$) and show these as gray curves. We indicate the average final score $F(g_{obs},t/60,M,N)$ for these analog games with the horizontal blue curve.

Figure 18

Fraction of games correctly predicted using the average final score of $N$ analog games, with adjacency evaluated over the last $M$ min at the three game times of (a) 30, (b) 60, and (c) 90 min. Increasing the number of analogs provides the strongest benefit for prediction while increasing memory either degrades or does not improve performance. Because prediction improves as a game is played out, the color bars cover the same span of accuracy (0.06) but with range translated appropriately.

Figure 19

Prediction accuracy using the described game shape comparison model using $N=50$ analogs and a memories of $M=1$ (blue curve) and $M=10$ (green curve), compared with the naive model of assuming that the current leader will ultimately win (red curve).