ANGELIQUE KERBER’S reign at the top of the world rankings for women’s tennis came to an end on July 10th, when Garbiñe Muguruza (right) of Spain knocked her out of Wimbledon in the fourth round. Although Ms Kerber had not yet won a title in 2017, the numbers suggested she was a modest favourite in the match: according to Elo, a statistical rating system that evaluates players based on their performances and the quality of their opponents, she was both the better player overall and was superior specifically on grass courts, the surface of Britain’s grand slam. A forecast based on the model put Ms Kerber’s chances of advancing at 66.9%.
Bettors, however, saw things differently. Entering the match, it was Ms Muguruza who was the bookmakers’ favourite, priced at an estimated probability of victory around 62.5%. Ms Kerber’s season-long slump surely played a part in how the wagering crowd valued her chances, but it can’t explain the entire discrepancy: Ms Muguruza’s season has been only slightly better. A more likely explanation for the market’s preference for the lower-ranked Spaniard was the history between the two players. In recent years, Ms Muguruza has clearly had the better of Ms Kerber, winning each of their past four meetings dating back to the 2015 French Open.
Commentators often assert that tennis is a game of matchups. At one level, the statement is a truism. But an implicit corollary of that claim is that certain players, or individuals with a particular style of play, are likely to fare better or worse against specific opponents with whom they “match up” unusually well or poorly than they would when playing a generic rival of equivalent skill. If true, you could produce more accurate predictions for the results of such matches by blending a standard comparison of the players’ overall abilities with a dose of their past head-to-head record—or, at the extreme, discarding the former entirely.
There is no shortage of anecdotal cases to suggest that “small data”—a modestly-sized sample of highly relevant observations, which involve both of the players in the match you want to forecast—might be more informative than ratings based on entire seasons or careers. Even though the latter are derived from much larger samples of matches, they require you to extrapolate players’ performances against a wide range of opponents to their specific matchups against a unique adversary. Ms Kerber’s most recent loss certainly seems to support the theory, as do a few famous rivalries. Serena Williams has won a whopping 18 consecutive matches against Maria Sharapova, who is herself among the all-time greats. Before Roger Federer retooled his backhand to better handle Rafael Nadal’s ferocious topspin, Mr Nadal won 23 of 34 career meetings, more often than not as the lower-ranked of the two players.
In general, these two sources of information tend to point in the same direction. Across 9,000 women’s matches since 1990 in which the competitors had faced each other at least three times and one of player won a majority of the meetings, the head-to-head record pointed in the same direction as the players’ Elo ratings two-thirds of the time. The 3,000-strong subset of contests in which the head-to-head and Elo ratings disagree does give some credence to the “matchups” theory: the player with the edge in previous meetings but a lower Elo score emerged triumphant 60% of the time. However, when we combine Elo ratings and head-to-head records into a joint forecast, the former carry far more weight. The recipe that yields the best predictions weights the overall Elo score at around 50 past head-to-head matchups. In theory, that means that if two players had already faced off 100 times, the forecast would consist of two-thirds their previous record against each other and one-third the Elo projection. However, in practice, only a handful of pairs of active players have faced off even 10 times. As a result, Elo will almost always make up at least five-sixths of the blend, calling into question whether incorporating the head-to-head records at all is worth the trouble.
But if the markets were wrong to place so much weight on Ms Kerber’s recent struggles against Ms Muguruza—Ms Kerber did win the first three of the pair’s eight matches—why does she keep falling to the Spaniard? The first suspect is simple chance: even if Ms Kerber were a 66.9% favorite in five consecutive matches, she would still have had a one-in-250 chance of losing the lot. Such odds might seem slim at first. But given the large volume of matches played every year, it is highly probable that at least one player would happen to suffer a long losing streak against a specific, inferior rival simply by virtue of random variation.
Nonetheless, just because any long run of coin flips will include extended streaks of heads or tails does not preclude the possibility that Ms Kerber really does match up poorly against Ms Muguruza. To explore such a hypothesis, we need a bigger dataset than a handful of head-to-head matchups, but one that is still limited to players whose games are roughly similar to those of the competitors in question. For example, Ms Kerber is a resourceful, defensive player, who succeeds largely by outlasting and out-thinking her opponents. In contrast, Ms Muguruza is more aggressive, seeking to end points quickly with her powerful serve and groundstrokes aimed for the corners. Such differences can be measured statistically: according to a metric called Aggression Score (AS), which approximates the frequency with which a player attempts to end points, Ms Kerber ranks in around the 25th percentile on the women′s tour, whereas Ms Muguruza winds up near the 65th.
Sure enough, it is opponents of Ms Muguruza’s type—the third quartile of AS, the cohort of players who are aggressive but not extremely so—that have given Ms Kerber trouble. Since the beginning of last season, the German has feasted on her fellow passive players, winning 65% of matches against opponents in her own AS quartile. She has also withstood the ball-bashing of the opposite extreme, prevailing in 69% of her contests against the most aggressive rivals. However, against players in Ms Muguruza’s category, she has won a mere 40%. Of the women currently ranked in the top ten, Ms Kerber holds the worst record against third-quartile opponents. (This discrepancy is not because players with Ms Muguruza’s approach tend to be better overall: the average Elo score is roughly the same for each of the four quartiles, both for the tour as a whole and specifically among Ms Kerber’s opponents.)
This suggests that Ms Kerber’s run of poor results against her Spanish rival are probably more than just a fluke. Instead, it seems to represent a tactical weakness, one which can be measured using statistics on playing style. Because this breed of tennis analysis relies on data at the shot-by-shot level, which is not available for the majority of tour-level matches, it remains in its infancy. But as the state of the art advances, it will become easier to identify the stylistic characteristics that seem likely to justify the cliché that tennis is a game of matchups. And for Ms Kerber, better analytics will point to the gaps in her game that make her so vulnerable against a certain type of opponent—and perhaps even enable her to end her five-match losing streak against Ms Muguruza.