How Long Does the Server’s Advantage Last?

In professional tennis, it’s a given that the server has an advantage.  The size of that advantage depends on the abilities of the two players and the surface, but especially in men’s tennis, it’s a sizable edge.  On average, a server in an ATP match starts a point with a roughly 65% chance of winning.

But how long does it last?  It seems that, at some stage in the rally, the server’s advantage has disappeared.  Four or five strokes in, the server may still be benefiting from an off-balance return.  But by ten strokes, one would assume that the rally is neutral–that the advantage conferred by serving has evaporated.

As usual with tennis analysis, one question begets several more.  Does the server’s advantage last longer on faster surfaces?  Do women settle into “neutral” rallies sooner than men do?  Do dominating players, like this year’s edition of Novak Djokovic, take away the server’s advantage faster than the average player?

Using the rally counts provided by Pointstream at the last three majors, we can start to answer these questions.

Neutralizing the serve

The first step is to take all the matches we have rally-count data for, and average them out.  Then, for each point length, we calculate the odds that the server wins a point of at least that length.  So, for instance, we look at all points of five shots or more, and figure out how many of those the server wins.

Each one of these numbers is biased, because a rally of exactly five-strokes is, by definition, won by the server.  The server either hits a winner on his third shot (the fifth overall), or the returner makes an error attempting to hit his own third shot (the sixth overall).  Thus, if we look at all points of at least five strokes, the exactly five-stroke rallies virtually guarantee that the server will have the advantage.

However, the same reasoning shows us that a six-stroke rally will be biased in favor of the returner.  When we do the math for at-least-five, at-least-six, at-least-seven, and so on, we’ll see a yo-yo effect.  When the biases have equal effect, that means the serve is neutralized.

Here are the results for the approximately 150 grand slam matches with Pointstream data so far this year:

At least…  Win%  Notes                          
0          63%   before point begins            
1          66%   if serve goes in               
2          50%   if serve is returned           
3          60%   if server makes second shot    
4          46%   if returner makes second shot  
5          58%                                  
6          45%                                  
7          57%                                  
8          44%                                  
9          56%                                  
10         44%                                  
11         56%                                  
12         43%                                  
13         56%                                  
14         43%                                  
15         56%

In the table, “Win%” refers to the server’s chance of winning the point.  The biases even out somewhere between the 4th and 8th shot, meaning that in that zone, the server’s advantage is neutralized.

While the server retains the advantage at least until the fourth shot, it is interesting to see how quickly it decays.  Dropping from 66% upon making a serve to 56% once the advantage is neutralized, it loses more than half the difference between the first and third shots.  Thus, the returner doesn’t negate the server’s advantage simply by getting the ball back in play, but he does take a large step toward doing so.

Does surface matter?

As usual, it sure does.  The numbers for the Australian and French Opens are similar, and since they make up 2/3 of the data set, they are close to the aggregate numbers shown above.  But Wimbledon, as is so often the case, seems to play by a different set of rules:

At least…    Wimby    Austr    French  
0            66%        62%       62%  
1            68%        64%       67%  
2            52%        50%       48%  
3            62%        59%       58%  
4            48%        46%       45%  
5            61%        57%       57%  
6            47%        44%       44%  
7            61%        55%       56%  
8            47%        44%       44%  
9            59%        55%       54%  
10           47%        43%       43%  
11           60%        54%       55%  
12           46%        43%       43%  
13           59%        55%       55%  
14           43%        45%       42%  
15           56%        54%       56%

The biases don’t balance out until the very bottom–at 14 or more shots!  That’s only about 3% of points.  I’m not sure how to explain this, except perhaps psychologically, that on grass (considered the best surface for servers), players are less successful in return games simply because that’s what they expect to happen.  Regardless of surface, I can’t understand why else the server’s advantage would persist into double-digit shot counts.

What about the ladies?

WTA players (on average) start each service point with a smaller advantage than their male counterparts, and as it turns out, that advantage evaporates more quickly.

We saw a moment ago that, by putting the return in play, an ATP returner gives himself a 50% chance of winning the point–at least until his opponent hits another shot.  Women, however, knock the server’s winning percentage down to 47% by making the return.

The returner clearly neutralizes the point by the fourth stroke overall, and–here’s the good part–takes over a slight advantage herself by her third shot, the sixth stroke overall.  By making that sixth shot, the returner has a 57% chance of winning the point, while the server will never reach 57% again.  The advantage is only a percentage or two, but from the sixth stroke on, the returner has the edge.

Finally, presenting Novak Djokovic

Pointstream has tracked 17 of Djokovic’s slam matches this year, giving us a good set of data to work with.  When a man is having a season like this one–in large part because of his return game–it’s fascinating to see how comprehensively he is outplaying his opponents.

In the same terms as the tables above, here are Djokovic’s serve and return points across those 17 matches.  The return points are shown with the server’s winning percentages:

At least…    ND Sv    ND Ret  
0            70%         57%  
1            72%         60%  
2            59%         42%  
3            68%         50%  
4            56%         39%  
5            68%         49%  
6            57%         38%  
7            68%         47%  
8            58%         38%  
9            68%         45%  
10           55%         34%  
11           65%         44%  
12           55%         33%  
13           66%         38%  
14           52%         29%  
15           65%         40%

After seeing the averages above, you might reasonably conclude that these numbers are out of this world.  Even with the bias of 4-, 6-, and 8-stroke rallies, as discussed above, Djokovic still maintains an edge.  For everyone else, once fifth or sixth shot is struck, the point is a 50/50 proposition.  For Novak, it’s at least 60/40 in his favor.

The amazing stats are on his return.  When he gets his return back in play, he’s more than likely to win the point.  That may not surprise anyone who has watched Djokovic play this year, but consider how remarkable that is in the context of modern men’s tennis.  By the 8th stroke or so, he’s back to the 60/40 odds of the service points that turn into longer rallies.

Thanks to Carl Bialik for suggesting this topic.