Here is Sandeep’s post on the data discussed in the New York Times about winning percentages on first and second serves in tennis.There are a few players who win with higher frequency on either the first or second serve and this is a puzzle. Daniel Khaneman even gets drawn into it. (To be precise, we are calculating the probability she wins on the first serve and comparing that to the probability she wins conditional on getting to her second serve. At least that is the relevant comparison, this is not made clear in the article. Also I agree with Sandeep that the opponent must be taken into consideration but there is a lot we can say about the individual decision problem. See also Eilon Solan.)
And the question persists: would players have a better chance of winning the point, even after factoring in the sure rise in double faults, by going for it again on the second serve — in essence, hitting two first serves?
But this is the wrong way of phrasing the question and in fact by theory alone, without any data (and definitely no psychology), we can prove that most players do not want to hit two first serves.
One thing is crystal clear, your second serve should be your very best. To formalize this, let’s model the variety of serves in a given player’s arsenal. For our purposes it is enough to describe a serve by two numbers. Let be the probability that it goes in and the point is lost and let be the probability that it goes in and the point is won. Then and is the probability of a fault (the serve goes out.) The arsenal of serves is just the set of pairs that a server can muster.
Your second serve should be the one that has the highest among all of those in your arsenal. There should be no consideration of “playing it safe” or “staying in the point” or “not giving away free points” beyond the extent to which those factor into maximizing the probability of the serve going in and winning.
But it’s a jumped-to conclusion that this means your second serve should be as good as your first serve. Because your first serve should typically be worse!
On your first serve it’s not just that matters. Because not all ways of not-winning are equivalent. You have that second serve to fall back on so if you are going to not-win on your first serve , better that it come from a faulted first serve than a serve that goes in but loses the point. You want to leverage your second chance.
So, you want in your arsenal a serve which has a lower than your second serve (it can’t be higher because your second serve maximizes ) in return for a lower . That is, you want decisive serves and you are willing to fault more often to get them. Of course the rate of substitution matters. The best of all first serves would be one that simply lowers to zero with no sacrifice in winning percentage . At the other extreme you wouldn’t want to reduce to zero.
But at the margin, if you can reduce at the cost of a comparatively small reduction in you will do that. Most players can make this trade-off and this is exactly how first serves differ from second serves in practice. First serves are bombs that often go out, second serves are rarely aces.
So when Vanderbilt tennis coach Bill Tym says
“It’s an insidious disease of backing off the second serve after they miss the first serve,” said Tym, who thinks that players should simply make a tiny adjustment in their serves after missing rather than perform an alternate service motion meant mostly to get the ball in play. “They are at the mercy of their own making.”
he might be just thinking about it backwards. The second serve is their best serve, but nevertheless it is a “backing-off” from their first serve because their first serve is (intentionally) excessively risky.
Statistically, the implications of this strategy are
- The winning percentage on first serves should be lower than on second serves.
- First serves go in less often than second serves.
- Conditional on a serve going in, the winning percentage on the first serve should be higher than on second serves.
The second and third are certainly true in practice. And these refute the idea that the second serve should use the same technique as the first serve as suggested by the Vanderbilt coach. The first is true for most servers sampled in the NY Times piece.
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September 1, 2010 at 11:30 pm
Brian
Interesting…professional tennis players, believe it or not, swing their racquet faster on their second serves (doesn’t necessarily mean they’re hitting the ball harder). Swinging faster at a slight angle imparts topspin, which lets the ball clear the net by a greater margin, while allowing it to “dip” into the box.
So pros actually exert a greater amount of energy, and use faster head speed to hit a “safer” serve…
September 2, 2010 at 2:05 pm
Jonathan Weinstein
Jeff,
I don’t think anyone claimed that the serves should optimally be identical. They claimed that some players would do better if they served two first serves *compared to current performance*, i.e. they violate your #1, and indeed the numbers bear this out.
Also, there is a vocabulary gap; you have defined “best” serve as maximizing y, which of course is the optimal second serve, but this definitely isn’t how tennis people talk. “Best serve” in tennis-speak doesn’t have a simple definition I think, but it would mean something closer to “optimal first serve” than “optimal second serve.”
September 2, 2010 at 2:24 pm
jeff
Taking the data at face value, yes a few players have better records with first serves than second serves. But I doubt the face-value interpretation. For example, suppose that against weak opponents you don’t have to hit as many second serves as against strong opponents (say because the matches go longer and your first serve weakens.) Then your second serves look worse on average because they are used disporportionately against stronger players.
Anyway the main point of the post is that you should not be thinking of first serves as the unconstrained optimum and thinking about how to tweak that to get to a good second serve. The second serve is the unconstrained optimum and the first serve is a distortion of that.
September 2, 2010 at 9:12 pm
Identification Strategy For Second Serves « Cheap Talk
[…] 2, 2010 in Uncategorized | Tags: sport, statistics | by jeff The data on first vs. second serve win frequency cannot be taken at face value because of selection problems that bias against second serves. The […]
September 3, 2010 at 11:38 am
Jonathan Weinstein
No question you’re right, given a set of available (x,y) pairs, the way to solve it for us math people would be to first find the optimal 2nd serve (easy: max y), then backwards induct the optimal 1st serve. And maybe tennis people could gain by thinking this way also, but I have a feeling it’s natural for them to treat the serve that is closer to their physical max velocity as the baseline and then modify from there.
Interestingly, it is possible from existing statistics (though the numbers in the nytimes chart don’t suffice) to also work out whether any player would improve by hitting their current 2nd serve twice. Based on some rough calculations, it appears that almost everyone would do much worse this way. So we have, for many players, relative to the way they currently play:
1st serves are roughly optimal as 2nd serves (at least, close to as good as actual 2nd serves, or better)
2nd serves are *not* roughly optimal as 1st serves
What picture do we get from this? A portion of the frontier in (x,y)-space must have roughly constant y for a range of x, making those serves about equally good as second serves, but very different as first serves as x varies.
It is interesting that violations of Jeff’s #1 are just as consistent with not going for enough on 1st serve as with not going for enough on 2nd serve, but the article never mentioned this.
A final thought: While (x,y)-space is a natural way to look at things mathematically, players and coaches probably think more in terms of (p,w)-space, where p=getting it in and w=winning conditional on in. Of course these map to each other bijectively.
September 4, 2010 at 1:34 pm
jeff
agreed on all points. another thought: you hear announcers say “he’s lucky to get a look at a second serve.” which in theory, and for many servers in practice, is the wrong way of looking at it. because the second serve is more threatening.
but of course what they mean is “he’s lucky that the server has only one chance left.”
May 3, 2011 at 1:08 pm
What’s Wrong With This Logic? « Cheap Talk
[…] a server should win a larger percentage of second-serve points compared to first-serve points; that much we know. Partly that’s because a server optimally serves more faults (serves that land out) on […]
May 5, 2011 at 12:08 am
Pierre
A very late comment, since I just discovered this blog:
David Gale–yes, the mathematician–got this some 40 years ago in Mathematics Magazine, available at http://www.jstor.org/pss/2689074.
We discovered that after re-doing it with friends, which led us to abandon the project. But this was fun. Probably a nice thing to (try to) estimate is how rational players are, and how well they know themselves and opponents when serving. I’ve been told some people around have point-level data…
Also, I am not a big fan of the “effort budgeting” story between first and second, given that the serving effort seems negligible compared to the rest. On the contrary, when players get tired, they sometimes serve harder to economize on running at the risk of losing directly the point (ok, this is also consistent with the first comment). Effort budgeting is probably a different problem, requiring a broader perspective, such as the whole match level–think of slacking off in the 3rd set when leading by two.
May 5, 2011 at 9:12 am
jeff
thank you very much for the reference. i will be blogging again on this subject soon.
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