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 $x$ be the probability that it goes in and the point is lost and let $y$ be the probability that it goes in and the point is won.  Then $x + y \leq 1$ and $1 - (x +y)$ is the probability of a fault (the serve goes out.) The arsenal of serves is just the set of pairs $(x,y)$ that a server can muster.

Your second serve should be the one that has the highest $y$ 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 $y$ 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 $y$ that matters.  Because not all ways of not-winning are equivalent.  You have that second serve $y$ 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 $y$ than your second serve (it can’t be higher because your second serve maximizes $y$) in return for a lower $x$.  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 $x$ to zero with no sacrifice in winning percentage $y$.  At the other extreme you wouldn’t want to reduce $y$ to zero.

But at the margin, if you can reduce $x$ at the cost of a comparatively small reduction in $y$ 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

1. The winning percentage on first serves should be lower than on second serves.
2. First serves go in less often than second serves.
3. 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.