Should tennis players, especially highly talented professionals, use their first serve as a second serve? As any tennis follower knows, first serves are usually much more effective than second serves, but they are also less reliable. Unless a player is afflicted with the yips (inability to get any serve in play), second serves are close to 100% reliable, but they are also much more likely to be returned by the server's opponent. Therefore, the liklihood of winning a point following a second serve is less than the percentage following a first serve that is successfully put in play.
Since the first of the two service attempts is the same in either case, the question comes down to, does the very reliable but less effective second serve produce a positive result on average compared the less reliable but more effective first serve?
One would think the answer is yes to the more reliable traditional second serve, because most players adopt this path. But, there are some players known to occasionally blast their second serve at first serve pace. By way of example, Alexander Zverev, Danil Medvedev and Nick Kyrgios do so at times. However, as a systematic strategy it seems very few have taken the possibility seriously. And yet, has anyone actually sat down and figured out the logic, one way or the other? One has to run the numbers to find out.
Let us examine the math, which turns out to be quite straightforward, indeed rather obvious, to see under what conditions hitting two first serves is recommended.
Let a = total point winning percentage as server for traditional strategy of using more reliable second serve when the first serve is a fault
Let b = total point winning percentage as server for strategy of hitting second serve as a first serve
s1 = percentage of first serves put into play
p1 = percentage of points won when first serve is good
s2 = percentage of second serves put in play and not double faulted
q2 = percentage of points won when second serve is good
p2 = total percentage of points won with second serve technique
a = s1p1 + (1- s1) s2q2
b = s1p1 + (1- s1) s1p1
But second serves are usually very reliable, so s2 ≈ 1 is a good approximation, in which case
s2 q2 ≈ q2 ≈ p2
This is consistent with the way professional tennis associations report second serve statistics, i.e. only percentage of points won, with double faults included in points lost on second serve. This, in turn, means that one can validly simplify the traditional serve strategy as follows
a = s1p1 + (1- s1) p2
We can now determine the crossover point where hitting a second serve with first serve technique yields the same total points won percentage as hitting the more traditional second serve, i.e. the point where
a = b
s1p1 + (1- s1) p2 = s1p1 + (1- s1) s1p1
Simplify by eliminating duplicated terms (subtract s1p1 from both sides, then divide both by (1- s1) )
p2 = s1p1
Finally, for a positive payoff from hitting a second serve opportunity as a first serve:
This last formula is easy to calculate for specific examples. The ATP (Association of Tennis Professionals, the men's professional tour level tennis organization) reports all three of the above statistics for ATP players, with data available going back to the beginning of the open tennis era and the founding of the ATP. Let's pick a few of the top players and see how they would fare with this approach. We'll use career stats for the top eight ATP players as of 31 August 2020. Upper case bold lables indicates ATP player statistics. Lower case italics labels are calculated values.
As an example of the break even situation, imaginary Player X has very good first serve statistics but poor second serve performance. Player Y has an even worse second serve, in which case it is better to hit two first serves. The data for some players is not available to hundredths of a percent, e.g. Tsitsipas and Berrittini, and Zverev for second serve win percentage. This is because the ATP web site only lists the top 200 players in each category, and the only data available for others is from individual player profiles, which is reported to the nearest percent.
From the data, it is apparent that the top players win enough points on their second serve so that hitting traditional high reliability second serves is the better strategy, and it is no surprise that they do so -- even without having ever explicitly analyzed the alternative. There are other factors at play as well. On exceptionally important points, does a player really want to risk hitting a big serve on the second serve and missing, thus double faulting and losing the point without ever putting the ball in play? One suspects that, despite any numbers to the contrary, the risk/reward tradeoff on, for example, a set point against might lead a player to opt for the higher reliability second serve.
And, then there's the very real fact that no player serves or wins points at a constant rate. Players are always trying to improve, which leads to change. And, players are streaky, serving better or worse than average across the years and in tournaments, matches, sets, and even within games -- especially as things get tight near the end of a closely contested set or match.
So in the end, very few players adopt the first serve on second serve opportunities as a fixed strategy. The only player currently on tour known for this practice is Maxime Cressy, and as of the date of this analysis he's ranked 168 on the ATP tour. Unfortunately, the ATP web site does not list statistics for Cressy, so it is difficult to know whether this is the right approach for him. Perhaps he's adopted the approach because he serves-and-volleys on every point, and he believes hitting a first serve on second serve opportunities gives him a better chance of avoiding being passed on the return of a weaker second serve as he is rushing the net.
For now, it seems likely that the practice of hitting another first serve if the first one is a fault will be only an occasional practice, used by a few players as a surprise tactic or during patches of very poor second serve performance.
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