How Important is the Seventh Game of the Set?

Italian translation at settesei.it

Few nuggets of tennis’s conventional wisdom are more standard than the notion that the seventh game of each set is particularly crucial. While it’s often difficult to pin down such a well-worn conceit, it seems to combine two separate beliefs:

1. If a set has reached 3-3, the pressure is starting to mount, and the server is less likely to hold serve.
2. The seventh game is somehow more important than its immediate effect on the score, perhaps because the winner gains momentum by taking such a pivotal game.

Let’s test both.

Holding at 3-3

Drawing on my database of over 11,000 ATP tour-level matches from the last few years, I found 11,421 sets that reached three-all. For each, I calculated the theoretical likelihood that the server would hold (based on his rate of service points won throughout the match) and his percentage of service games won in the match. If the conventional wisdom is true, the percentage of games won by the server at 3-3 should be noticeably lower.

It isn’t. Using the theoretical model, these servers should have held 80.5% of the time. Based on their success holding serve throughout these matches, they should have held 80.2% of the time. At three-all, they held serve 79.5% of the time. That’s lower, but not enough lower that a human would ever notice. The difference between 80.2% and 79.5% is roughly one extra break at 3-3 per Grand Slam. Not Grand Slam match–an entire tournament.

None of that 0.7% discrepancy can be explained by the effect of old balls [1]. Because new balls are introduced after the first seven games of each match, the server at three-all in the first set is always using old balls, which should, according to another bit of conventional wisdom, work against him. However, the difference between actual holds and predicted holds at 3-3 is slightly greater after the first set: 78.9% instead of the predicted 79.8%. Still, this difference is not enough to merit the weight we give to the seventh game.

The simple part of our work is done: Servers hold at three-all almost as often as they do at any other stage of a match.

Momentum from the seventh game

At 3-3, a set is close, and every game matters. This is especially true in men’s tennis, where breaks are hard to come by. Against many players, getting broken so late in the set is almost the same as losing the set.

However, the focus on the seventh game is a bit odd. It’s important, but not as important as serving at 3-4, or 4-4, or 4-5, or … you get the idea. The closer a game to the end of the set, the more important it is–theoretically, anyway. If 3-3 is really worth the hoopla, it must grant the winner some additional momentum.

To measure the effect of the seventh game, I took another look at that pool of 11,000-plus sets that reached three-all. For each set, I calculated the two probabilities–based on each player’s service points won throughout the match–that the server would win the set:

1. the 3-3 server’s chance of winning the set before the 3-3 game
2. his chance of winning the set after winning or losing the 3-3 game

In this sample of matches, the average server at three-all had a 48.1% chance of winning the set before the seventh game. The servers went on to win 49.4% of the sets [2].

In over 9,000 of our 3-3 sets, the server held at 3-3. These players had, on average, a 51.3% chance of winning the set before serving at 3-3, which rose to an average of a 57.3% chance after holding. In fact, they won the set 58.6% of the time.

In the other 2,300 of our sets, the server failed to hold. Before serving at three-all, these players had a 35.9% chance of winning the set, which fell to 12.6% after losing serve. These players went on to win the set 13.7% of the time. In all of these cases, the model slightly underestimates the likelihood that the server at 3-3 goes on to win the set.

There’s no evidence here for momentum. Players who hold serve at three-all are slightly more likely to win the set than the model predicts, but the difference is no greater than that between the model and reality before the 3-3 game. In any event, the difference is small, affecting barely one set in one hundred.

When a server is broken at three-all, the evidence directly contradicts the momentum hypothesis. Yes, the server is much less likely to win the set–but that’s because he just got broken! The same would be true if we studied servers at 3-4, 4-4, 4-5, or 5-5. Once we factor in the mathematical implications of getting broken in the seventh game, servers are slightly more likely to win the set than the model suggests. Certainly the break does not swing any momentum in the direction of the successful returner.

There you have it. Players hold serve about as often as usual at three-all (whether they’re serving with new balls or not), and winning or losing the seventh game doesn’t have any discernible momentum effect on the rest of the set [3]. Be sure to tell your friendly neighborhood tennis pundits.

Notes:

1. Using a more limited dataset, Magnus and Klaassen found that new balls did not result in more holds of serve.
2. It’s not entirely clear why these numbers aren’t 50%. My best guess is that underdogs are able to stay close early in sets, reaching 3-3 a bit more often than the model would predict. That’s a project for another day.
3. I ran the same tests against WTA, women’s ITF, Challenger, and Futures matches to see if the results would be different by gender or level. The ITF numbers are the reverse of most of the other groups, but overall, none of these subsets contradict anything I’ve generalized from the ATP numbers.

   LEVEL                   WTA    ITF  CHALL    FUT  
   Matches               11203  17143  18717  14052  
                                                     
   Hold%                 64.3%  54.9%  75.8%  69.9%  
   Hold at 3-3           63.4%  57.1%  74.6%  69.4%  
   Hold% (no 1st set)    63.9%  54.4%  75.4%  69.6%  
   Hold at 3-3 (no 1st)  64.0%  56.4%  73.6%  68.4%  
                                                     
   Prob at 3-3           49.2%  49.1%  47.8%  48.2%  
   Server set%           50.0%  49.4%  48.0%  48.7%  
   
   WIN at 3-3:                                       
   Prob at 3-3           54.6%  56.6%  51.8%  53.2%  
   Prob at 4-3           65.0%  69.2%  58.8%  61.5%  
   Set won%              65.8%  68.7%  58.9%  61.2%  
   
   LOSE at 3-3:                                      
   Prob at 3-3           40.0%  39.1%  36.1%  36.8%  
   Prob at 3-4           21.5%  24.2%  14.9%  17.8%  
   Set won%              22.8%  23.8%  16.1%  20.3%