The Summer of Jeff

Advantage: First Server

Posted in tennis by Jeff on December 1, 2010

Update, 12/2: The conclusions reached below are flat-out wrong.  See my 5:16 PM comment for details.  I’m leaving this up, unedited for now, since I think many of the assumptions and thought processes are of value.  But, long story short, the initial server in a set does not hold any statistical advantage.

I’ve been working up a Markovian analysis of tennis matches. I’ve still got a ways to go before I can turn it into something like a full win expectancy chart, but in the process, I’ve stumbled across something very interesting.

The player who serves first has a measurable advantage. The assumptions are minimal: As far as I can tell, the conclusion holds whenever the players win more service points than return points (except at the extreme, as we’ll see below), and in professional tennis, that is virtually always the case.

In fact, the advantage is often two-fold:

  • The initial server is more likely to win the set.
  • The initial server retains a very good chance of serving first in the next set.

Let’s go deeper.

Assuming that players are more likely to win points on serve than against serve, it’s fairly intuitive to see how the field is tilted toward the first server.  Most possible outcomes (6-0, 6-2, 6-4, tiebreak) give each player an equal number of service games.  But while 6-1 and 6-3 give the initial server more service games, there is nothing (at least within that set) that serves as a balance.

Thus, the initial server will, in the aggregate, serve more than his opponent.  More serving equals more winning.

Let’s assume we have two equal players who, given their skills, are each likely to win 63% of points on serve.  That translates into about 80% of service games.  Incidentally, I’m using .63 because that’s about the number in a hypothetical matchup between David Ferrer and Robin Soderling.  63%/80% also turns out to be very close to the ATP average.

Here’s how the probabilities break down:

  • 6-0: 0.004
  • 6-1: 0.043
  • 6-2: 0.072 This was my mistake; it should be ~0.04.
  • 6-3: 0.186
  • 6-4: 0.068
  • 7-5: 0.052
  • 7-6: 0.106
  • 6-7: 0.106
  • 5-7: 0.052
  • 4-6: 0.206
  • 3-6: 0.048
  • 2-6: 0.042
  • 1-6: 0.011
  • 0-6: 0.004


  • Wins: 0.53
  • Loses: 0.47
  • Reaches 5-5: 0.316
  • Reaches 6-6: 0.213

Once the set reaches 4-4, the returner has won himself an even playing field.  Thus, it’s not surprising to discover that the more dominant the servers, the less the advantage.  If we increase the odds of winning service points to 67% (about 86% of service games), the first servers advantage decreases to 52.3%.  At 70% of service points (90% of games), the edge falls to 51.6%.

It turns out that the first-server advantage is greatest when the probability of winning service points is about 61%.  As we’ve seen, it decreases as it gets higher, and as you might expect, it decreases as the probability of winning service points approaches 50%.  At 50%, by definition, there’s no advantage to being a server, whether we’re talking point, game, or set.

The next set

The possibility of the scores 6-3 and 6-1 are what give the initial server his advantage, but they hold a disadvantage as well.  If you win the first set 6-3, your opponent will serve first in the second set, thus gaining the advantage for himself.

To be more precise, there are six scenarios in which the initial server does not serve first in the subsequent set: 6-1, 6-3, 7-6, 6-7, 3-6, and 1-6.  Given evenly-matched players, 6-3, 7-6, and 6-7 are relatively common.

Are they common enough to counteract the first-server advantage?  Usually, no.  Here are six types of players, from the dominant server (0.7) to the average matchup (0.6, 0.63), to the weak-serving counterpunchers (0.54), along with the probability that they serve first in the following set:

  • 0.70: 0.39
  • 0.67: 0.45
  • 0.63: 0.50
  • 0.60: 0.53
  • 0.57: 0.55
  • 0.54: 0.56

To me, this is the shocker.  In the Ferrer-Soderling matchup, the player who wins the coin toss gets a big edge in the first set, and loses nothing.  For return-dominated matches, the advantage is compounded in subsequent sets!

The only case that looks “fair” is that of the dominant servers.  When, say, John Isner and Sam Querrey are going at it, the first-server advantage is about as tiny as it gets in the first set, and the initial server is more likely to begin the second set as the returner.

The coin toss and the match

Of course, there’s only one stat that matters to the player, and that’s the W.  It’s clear that in some matchups, the initial server begins with an edge in his favor, but how big is it?

In a best-of-three-set match, the initial server almost always has the advantage.  In the Ferrer-Soderling example, the first server has a 51.6% chance of winning the match.  Here’s the breakdown using the same categories–0.7 represents dominant servers, down to .54, the extreme counterpunchers:

  • 0.70: 0.503
  • 0.67: 0.508
  • 0.63: 0.516
  • 0.60: 0.520
  • 0.57: 0.518
  • 0.54: 0.512

The break-even point is roughly 72% of service points won.  Above that, the initial server has a tiny disadvantage.  That’s not unheard of–I was able to find a match between Isner and Kevin Anderson where they won 73% of service points between them.

For everybody else, it’s safe to say that in the vast majority of ATP matches (and an even more staggering majority of WTA contests), the player who wins the coin toss and elects to serve has a distinct advantage.

15 Responses

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  1. Phil Birnbaum said, on December 1, 2010 at 5:53 pm

    Suppose you’re leading 5-1 and serving. Is it worthwhile to deliberately lose a game, so that you win 6-2 instead of 6-1 and gain first service in the next set?

    • e poc said, on December 1, 2010 at 6:24 pm

      You’d have to be extremely confident that you could break your opponents serve in the 5-2 game. So, probably, no.

      • Phil Birnbaum said, on December 2, 2010 at 4:58 am

        Well, you have to be extremely confident that you won’t lose the set. Otherwise, the worst case is that you break even — like if you win 6-3.

        But, point taken.

    • Jeff said, on December 2, 2010 at 2:04 pm

      Pretty soon, I should be able to give a precise answer to that, when I work out the full WinEx charts.

      But some back-of-the-envelope, assuming each server has an 80% chance of winning each service game:
      – at 5-1 serving, the server has a ~99% chance of winning the set. ~93% of the time, he’ll end up serving 2nd the next set, since he’ll win 6-2 or 6-4 (or more rarely, 7-5).
      – if he tanks, he’s then returning at 5-2. Now his chance of winning the set is approximately 93%. That 93 breaks down as 26/67 serving 1st/2nd in the next set.

      So…it might be close if, by tanking, you were guaranteed to get the 1st serve in the following set. But you don’t.

      Incidentally, say you tank on 5-1, try to break at 5-2, but if you don’t, tank *again* at 5-3. Then you try to win. This drastically lowers your chance of winning, but increases chances of serving first in the following set:
      – win, serve 1st: 58%
      – win, sv 2nd: 10%
      – lose, sv 1st: 22%
      – lose, sv 2nd: 10%

      Fun stuff.

  2. Dick Whitman said, on December 1, 2010 at 6:03 pm

    Jeff, I’ve been looking into this type of research specifically for tennis for a while now. Glad someone else is, too. Look forward to seeing more!

    • Jeff said, on December 2, 2010 at 1:51 pm

      Thanks, glad to hear I’m not alone.

  3. Seth Burn said, on December 1, 2010 at 7:28 pm

    Have you tried controlling for quality of players? I just want to make sure that this effect is coming coming from different quality players having different preferences.

    • Jeff said, on December 2, 2010 at 1:51 pm


      Everything in the post is about evenly-matched players with identical ‘preferences.’

  4. Xeifrank said, on December 1, 2010 at 9:06 pm

    Do you see this same effect in men’s/women’s/mixed doubles and women’s singles?

    Intuitively this doesn’t make sense as both singles players have the same number of service games in a set unless one of the players gets broke. And it shouldn’t make any difference whether or not holding serve in an odd or an even numbered game is more difficult to do or not. In a set with no breaks, the person who serves second will have the added pressure of serving at 4-5 and 5-6 to stay in the set.

    If the empirical data is showing that it is more difficult to win a set if you serve 2nd, is it showing a particular set score or set of set scores that are more difficult than average to win?
    vr, Xeifrank

    • Jeff said, on December 2, 2010 at 1:50 pm


      There’s no empirical data behind this. This is all theoretical. Pressure has nothing to do with it.

  5. Jeff said, on December 2, 2010 at 5:16 pm

    Well, this was fun while it lasted.

    I had myself convinced that the first-server advantage made intuitive sense, but it doesn’t.

    A typo in my code was letting the first server serve for the set at 5-1 instead of returning at 1-5, so the odds of winning 6-2 are inflated.

    So the odds of the first server winning the set against an equal opponent are … drum roll … 50%.

    The good news is that I’m now close to a full win expectancy chart at a game-by-game level, maybe even typo-free.

    • Seth Burn said, on December 2, 2010 at 7:00 pm

      You still need to fix the other 2-6 cell, presumably it should be .072. At that point it sums up to 1 again.

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