# Should a tennis player ever serve two first serves?

On a recent episode of The Tennis Podcast, presenter Catherine Whitaker was analysing Andy Murray’s shock exit from the 2017 Australian Open. A feature of the match was that Murray’s opponent, Mischa Zverev, had been eating Murray’s second serve for breakfast — Murray won only 36% of points behind his second serve in the match.

“When you’re winning that few points on second serve, why not just hit two first serves?” Catherine asked.

Good question. Would Andy Murray have actually been better off hitting two first serves in that match? Let’s do the maths.

Disclaimer: There’s a bit of algebra coming up. It’s not complicated but if it’s not your bag just scroll down to ‘TL;DR’.

Firstly let’s build the equation for proportion of service points won.

Let first serve percentage be ‘a’
Let percentage of points won behind first serve be ‘b’
Let percentage of points won behind second serve be ‘y’

(For simplicity all percentages here will be expressed as proportions in the equations, where 100% = 1, 50% = 0.5 and so on).

A player’s total proportion of points won on serve (call this W1) will be given by this equation:

W1 = ab + y(1-a)

In his match against Zverev, Murray got 66% of his first serves in, and won 65% of points behind his first serve. So we can plug these numbers in and get his overall % of points won on serve:

W1 = (0.66 x 0.65) + 0.36(1–0.66)
W1 = 0.429 + 0.1224
W1 = 0.5514

This tallies with what the match stats show: Murray won 55.14% of his service points in this match.

But what would his hypothetical % of service points won have been had he served two first serves?

We can build an equation in the same way. Let’s call his hypothetical win percentage for two first serves W2.

W2 = ab + ab(1-a)

Now plugging in Murray’s numbers:

W2 = (0.66 x 0.65) + [(0.66 x 0.65)(1–0.66)]
W2 = 0.429 + 0.14586
W2 = 0.57486

The maths shows that Murray would theoretically have increased his percentage of points won on serve from 55.1% to 57.5% by serving two first serves. In a sport with margins as fine as those in tennis, this is worth taking note of.

#### Generalising

We’ve plugged in the numbers for this specific match, but the equations can be simplified to give a simple rule to answer the question ‘when should a player serve two first serves?’.

We’re looking for instances where W2 would be greater than W1. So:

W1 < W2
ab + y(1-a) < ab + ab(1-a)
y(1-a) < ab(1-a)
y < ab

Basically, if a player’s win percentage on second serve (y) is dipping below his or her first serve percentage (a) multiplied by his or her win percentage behind first serve (b), then he or she would theoretically be better off serving two first serves.

Andy Murray is particularly prone to this phenomenon, as his first serve is strong and generally consistent while his second serve is still relatively weak, even though he upgraded it significantly last year. But he’s not unique. The stats reveal a significant number of matches where a player’s hypothetical W2 percentage would have exceeded their actual W1 percentage. In Novak Djokovic’s surprise loss to Kei Nishikori in the 2014 US Open, for example, Djokovic only won 37% of points behind his second serve. He could hypothetically have improved his overall win percentage on serve from 62% to 67% by serving two first serves.

Without going back through all the records, one suspects that a large number of Murray’s matches against top players pre-2016 would also have had better W2 scores than W1 scores.

TL;DR: Going purely by the stats, Andy Murray could theoretically have won more points on serve in his match against Zverev by serving two first serves. There are also many other matches where the stats show that a player could have increased his or her win percentage on serve by serving two first serves.

#### So why don’t players serve two first serves?

The clue is probably in the word ‘theoretically’. Tennis isn’t all about maths; there is complex psychology at play. Perhaps the prospect of double faulting on around 15% of your service points (as you would if you served two first serves) would be so alarming that your serve would go to pieces entirely. Or perhaps you would be handing a massive psychological advantage to your opponent, by essentially saying ‘I don’t think I can get the better of you in a rally’.

The truth is probably a combination of these and other factors. But we are in an era when the game’s best returners — Djokovic, Murray, Nadal, Nishikori — have turned the second serve into a significant disadvantage for their opponents (all these players win more than 55% of their return points when facing a second serve).

Maybe the psychology of ‘I’m serving therefore I have the upper hand’ is an outdated hangover from the days of faster courts and less athletic opponents, and players should be thinking differently about their serves. As Big Bill Tilden said, ‘Never change a winning game; always change a losing one’. Whatever the psychology of the situation, there is no getting around the fact that 36% on your second serve is emphatically not a winning game, and you would think that a player in that situation should be willing to try some pretty radical tactics.

In a sport where players sleep in oxygen tents and calibrate their food intake down to the last calorie to gain a few fractions of a percent advantage over their opponents, there may yet be some unrealised marginal gains out there for a player who is willing and able to overcome the psychological factors and cash in on a few extra service points by ditching their second serve.

You can listen to the post-Murray-match episode of The Tennis Podcast here:

And you can view a full breakdown of the match stats here:

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