The elusive final set – are tennis finals always close run contests?

Author: Thomas Fanshawe


The Wimbledon Championships.

Photo by Tim Schofield courtesy of wikimedia

At the 2013 Wimbledon Championships, the top two seeded players reached the final of the men’s singles competition, while two relative outsiders reached the final of the women’s. Many commentators predicted close, exciting finals and appeared surprised when both matches were decided in straight sets. Is this as unlikely as it first appeared? To answer this I examined the results from all Wimbledon singles finals to see what the statistics tell us.

The first Wimbledon Championships was held in 1877, when Spencer Gore beat William Marshall to become the first men’s singles champion. The basic rules have remained unchanged ever since, with matches contested over a maximum of five sets, and the match ending when either player wins three. In the women’s competition, first won by Maud Watson in 1884, only two sets are required for victory. A match is said to be won in ‘straight sets’ if the set score is 3-0 in a men’s match, or 2-0 in a women’s match.

The table below shows the results, in terms of the set score, in all Wimbledon singles finals up to and including 20131. Even though we might expect many finals to be contested by two strong players of approximately equal ability, we can immediately see that the straight set scoreline is by far the most common in both the men’s and women’s competitions. Why should this be the case?


Table 1: Frequency of set scores in Wimbledon singles finals.

Looking at the simpler women’s data first, there are a couple of assumptions to take into account. We might suppose that first, in a particular final, the stronger player has a certain probability of winning each set. We might also propose that the outcomes of the sets are independent, so the outcome of each set is unaffected by what has happened before. We call this the ‘independence model’. It is simple to calculate the probability that the match will end in a 2-0 scoreline2. The image below (Figure 1) shows how this probability (p) changes (in practice we consider only p>=0.5 as the probability refers to the stronger player).

What we can see from this is, except when the players are perfectly evenly matched (p=0.5), the match is more likely to end 2-0 than 2-1. This perhaps should not be a surprise: for a match to finish 2-1, the player who won the first set, likely to be the stronger player, must lose the second. Note also that the chance of a 2-1 score under the rules is less than the chance of a 2-1 score, were the players to play three sets regardless of the score after the first two.

So for a 2-1 scoreline to be more likely that a 2-0 scoreline, not only would the set outcomes need to be dependent, but winning a set would need to confer a disadvantage to a player in the next. One might dream up several explanations for this seemingly improbable scenario, such as the winning player becoming tired or complacent, the losing player becoming more determined, or even a perceived advantage of serving in the first game of the set.

However, in this data-set we can see no evidence for this. The table below shows the order of sets won by each player in the finals that ended with a 2-1 score, with ‘A’ representing a set won by the eventual winning player and ‘B’ a set won by the losing player. As the frequency of the ‘ABA’ and ‘BAA’ patterns are similar, there is no reason to suspect that set outcomes are not independent. Indeed, the independence model fits the data extremely well3.

Table 2: Order of sets won in Wimbledon singles finals. ‘A’ denotes a set won by the eventual winning player, and ‘B’ one won by the eventual losing player. (Freq.=Frequency)

This means in a typical women’s final, we expect the stronger player to have an 80% chance of winning each set, which does not seem to be a formula for a close-run contest. This probability may of course vary from year to year – for every Venus Williams versus Serena Williams, there is a Martina Navratilova versus Zina Garrison. However, as Figure 1 shows, even on occasions when the players are more equally matched it is irrational to predict a 2-1 scoreline.

For the men’s data, we can also use the same probability formula2. The graph (Figure 1) shows that the straight set scoreline is no longer always the most likely4. However, 3-1 is always at least as likely as 3-2. A 3-2 scoreline requires the score to be 2-1 after three sets, and for the losing player (usually the weaker player) to win the fourth set.

What all this data shows is there is no reasonable scenario in which a 2-1 (women) or 3-2 (men) scoreline is logically the most likely outcome. This is supported by both historical data and modelling.

To predict such a score exposes either a lack of understanding of probability or a desire to ratchet up the anticipation of a match in the hope that it will be more thrilling than it is likely to be. Straight sets victories are common even when players are evenly matched. The 2014 Wimbledon women’s singles final will probably be decided in straight sets.

References

  • 1. The men’s competition in 1931, when the final was not played because of injury, is excluded, and the 1911 final, which was stopped at 2-2 when a player retired, is regarded as a five-set match.
  • 2. Probability of the final set scoreline in terms of the probability p of winning a set, under the independence model.

    Women Prob(2-0) = p2 + (1-p)2
                 Prob(2-1) = 1 – Prob(2-0)

    Men      Prob(3-0) = p3 + (1-p)3
                 Prob(3-1) = 3p(1-p)[p2 + (1-p)2]
                 Prob(3-2) = 1 – Prob(3-0) – Prob(3-1)
  • 3. The value of p that fits the data best (the maximum likelihood estimate, or MLE) is 0.80, with 95% confidence interval 0.72 to 0.86. Figure 2 shows the likelihood function, reaching a maximum at p=0.80.
  • 4. For values of p less than 0.69, a 3-1 score is more likely than 3-0, and if p is less than 0.65, 3-2 is more likely than 3-0.
  • Figure 1 notes: Relationship between the probability of winning a set and the probability of the final set scoreline, under the independence model.
  • Figure 2 notes: Log-likelihood function for the probability of winning a set under the independence model.

  • Figure 2 also shows the likelihood function of p for the men’s data under the independence model. The MLE is p=0.74, with 95% confidence interval 0.68 to 0.79. However, this model fits the data quite poorly. The predicted number of 3-0, 3-1 and 3-2 scorelines are 53, 45 and 28 respectively, and so there are far fewer 3-1 scorelines than we would anticipate. The reasons for this are unclear, but in such a restricted data-set it is impractical to fit extensions to the independence model that remain interpretable.

    For example, a mixture model in which 81% of the finals are evenly matched, with p=0.5, and the remaining 19% are walkovers, with p=1, fits the data extremely well, but appears implausible. The order of sets won by each player (Table 2) does not reveal any obvious patterns of dependency between sets, although readers are invited to draw their own conclusions. To test whether the poor fit of the independence model in men’s matches was true in another setting, we also examined the results of the men’s Wimbledon semi-finals since 1968, the start of the professional era.

    Here, the MLE (p=0.77) under the independence model provides an excellent fit to the frequencies of 3, 4 and 5-set matches of 43, 31 and 18 respectively. It is also reasonable that the MLE for the semi-final should be slightly higher than the MLE for the final, as players in the semi-final would be expected to be less evenly matched. The reason for the relative dearth of 3-1 scores in finals therefore remains unclear.

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