Heavy Topspin

Italian translation at settesei.it

Taking the next step beyond yesterday’s post about servers’ success rates in the deuce and ad courts, along with each in-game score, let’s look at the tendencies of righties and lefties.

As I speculated yesterday, righties are more successful in the deuce court (64.0% to 62.3% of points won), while lefties are better serving to the ad court (63.0% to 62.3%).  The difference for lefties is a little more dramatic (62.8% to 61.8%) if we remove Rafael Nadal from the sample.

(In all of the numbers today, I’ll present lefties in two forms: with Nadal, and without Nadal.  While Rafa is just one player, he makes up nearly one-third of the service points played by lefties in the dataset we’re working with of 2011 grand slam men’s singles matches tracked by Pointstream.  As we’ll see, Nadal appears to have some tendencies that separate him from the lefty pack.)

This would seem to give lefties a bit of a strategic edge; more than three-quarters of break points are played in the ad court, including all of the break points (30-40, 40-AD) that bring the server back to even.  If lefties are more likely to win those points, they would seem to be more likely to fend off such threats.  Of course, it might cut both ways: A weakness in the deuce court may lead to more break points needing to be fended off.

Oddly enough, lefties do not seem to employ their advantage at the most common break point score, 30-40.  Both righties and lefties win 30-40 points at about 6% less frequently than they win points in general.  The most marked difference is at 40-AD, where righties win 10% fewer points than average, but lefties win only 3% fewer points than average.  Rafa accounts for a big part of that difference, probably attesting to his mental strength.  Without Rafa, lefties win 6% fewer points than average at 40-AD; still quite a bit better than righties.

(Remember, there’s a bias inherent in this approach.  If a server reaches 40-AD, the score itself reflects a disadvantage.  Federer and Isner don’t serve many 40-AD points, precisely because their serves are so dominant.  Verdasco, Fognini, and whoever is playingDjokovic serve more 30-40 and 40-AD points, meaning that the sample of 40-AD is disproportionately full of men serving against the odds.)

There’s much more to do here, but in the meantime, these broad differences between righties and lefties give us plenty to think about.

The table below shows all the numbers described above.  The columns with numbers between 0.85 and 1.1 indicates, for each type of server (righty, lefty, etc.), how their performance at a specific point compares to their overall rate.  That allows us to better compare righties (winners of 63.1% of service points) with lefties-minus-Nadal (winners of 62.3%).

SCORE    WIN%       RH               LH            LH-xRN          
ALL     62.9%    63.1%            62.7%             62.3%          
DC CT   63.5%    64.0%   1.01     62.3%   0.99      61.8%   0.99   
AD CT   62.3%    62.3%   0.99     63.0%   1.00      62.8%   1.01   

SCORE    WIN%       RH               LH            LH-xRN          
g0-0    63.4%    63.5%   1.01     63.5%   1.01      63.1%   1.01   
g0-15   60.6%    60.8%   0.96     59.5%   0.95      59.5%   0.96   
g0-30   62.0%    62.6%   0.99     59.4%   0.95      60.6%   0.97   
g0-40   54.8%    54.1%   0.86     56.3%   0.90      54.6%   0.88   

g15-0   63.8%    63.7%   1.01     65.3%   1.04      64.8%   1.04   
g15-15  63.4%    63.8%   1.01     59.5%   0.95      59.6%   0.96   
g15-30  60.1%    60.2%   0.95     60.6%   0.97      61.4%   0.99   
g15-40  61.2%    61.5%   0.98     58.2%   0.93      58.3%   0.94   

SCORE    WIN%       RH               LH            LH-xRN          
g30-0   64.9%    65.0%   1.03     64.1%   1.02      63.0%   1.01   
g30-15  62.7%    62.5%   0.99     65.7%   1.05      65.7%   1.05   
g30-30  64.0%    64.3%   1.02     62.5%   1.00      60.9%   0.98   
g30-40  59.3%    59.2%   0.94     58.6%   0.94      59.3%   0.95   

g40-0   67.1%    66.8%   1.06     67.0%   1.07      65.1%   1.05   
g40-15  65.6%    66.0%   1.05     62.5%   1.00      61.6%   0.99   
g40-30  63.7%    63.6%   1.01     67.0%   1.07      65.6%   1.05   
g40-40  61.6%    61.9%   0.98     61.4%   0.98      62.2%   1.00   
g40-AD  57.8%    57.0%   0.90     60.5%   0.97      58.7%   0.94   
gAD-40  62.3%    62.4%   0.99     59.7%   0.95      61.4%   0.99

Italian translation at settesei.it

If tennis players were machines, each player would have the same probability of winning every point.  Winning the point at 40-15 would be equally likely as winning the point at 15-40.  It seems a safe bet that this isn’t the case, and today I’m going to start talking about the difference, and why it exists.

To begin with, let’s look at the outcome of every grand slam men’s singles point in 2011, sorted by the score before the point was played.  (I’ll explain some of this in a minute.)

SCORE     PTS    WON   WIN%   REL  
g0-0    10757   6820  63.4%  1.00  
g0-15    3941   2390  60.6%  0.97  
g0-30    1552    963  62.0%  0.98  
g0-40     591    324  54.8%  0.88 

g15-0    6823   4356  63.8%  1.02  
g15-15   4858   3081  63.4%  1.00  
g15-30   2741   1648  60.1%  0.97  
g15-40   1416    866  61.2%  0.96  

SCORE     PTS    WON   WIN%   REL  
g30-0    4355   2826  64.9%  1.02  
g30-15   4609   2890  62.7%  1.01  
g30-30   3366   2155  64.0%  1.01  
g30-40   2080   1234  59.3%  0.95 

g40-0    2824   1895  67.1%  1.08  
g40-15   3819   2507  65.6%  1.03  
g40-30   3466   2209  63.7%  1.02  
g40-40   4556   2806  61.6%  0.97 

g40-AD   1749   1011  57.8%  0.93  
gAD-40   2806   1748  62.3%  1.00  

SCORE     PTS    WON   WIN%        
ALL     66309  41729  62.9%        
DC CT   34679  22024  63.5%        
AD CT   31630  19705  62.3%

One thing that sticks out is that as players get closer to winning a game (30-0, 40-0), they are more likely to win the next point.  When facing (or approaching) break point, they have less success.

Much of that (and maybe all of it) is simply the bias of the sample.  If a player reaches 40-0, he’s more likely to be a player who is dominant on serve, or facing a returner who hasn’t found the range.  A disproportionate number of 40-0 points are served by players who are better-than-average servers.  Similarly, a disproportionate number of 0-40 points are served by players without dominant service games … or served against Novak Djokovic.

Deuce and ad courts

A more useful finding is that players win more points in the deuce court.  In this sample, the server won 63.5% of points in the deuce court and 62.3% of points in the ad court.  This may be because right-handers (who make up about 85% of this sample) are more successful when serving across their body, but I haven’t tested that yet.

(If it is true that players are better serving across their body, then the difference is even more stark.  Assuming that righties and lefties have the same difference in success rates, the “serve across your body” success rate–deuce for righties, ad for lefties–should be about 63.8%, while the “serve away from your body” rate–ad for righties, deuce for lefties–should be 62.1%.)

Thus, the difference between success rates at 0-0 and 0-15 isn’t as extreme as it looks at first; some of the 0-15 winning percentage is due to the difficulty of serving to the ad court.  That’s the purpose of the ‘REL’ column, which shows how the winning percentage on that point relates to the average winning percentage in the relevant court.

If this difference is universally true, it would require a change in win probability tables.  For instance, when the returner reaches break point–which is more often in the ad court, at 30-40 or 40-AD–his chance of winning the game is a percentage point or two higher than previously estimated.  As long as he’s playing a right-hander, anyway.

There’s plenty more to investigate here.  To determine whether players really raise or drop their performance levels (for instance, raising their game against break point, or taking it easy at 40-0), we’ll need to switch to a player-by-player basis, to reduce the skewing effect of dropping every player in the same bucket.

A couple of months ago I presented some research that showed that, in the average men’s grand slam match, the server’s advantage was neutralized somewhere between the 4th and 8th shot.

That research left a major question unanswered: How do the results differ between first and second serves?  Some second serves are hardly better than rallying shots, so it stands to reason that the server’s advantage is neutralized even faster on the second serve.

Using all of the Pointstream-tracked men’s matches from this year’s grand slams, we have an enormous population of points, in which 63.2% of points were won by the server.  When the first serve went in, the percentage jumped to 71.7%.  If the first serve went out, the server’s chance of winning fell to 50.2%, then rose again to 56.1% if he landed his second serve.

On first serve points, the server’s advantage was not neutralized until at least the 8th shot of the rally, and perhaps not until the 9th or 10th.  On second serve points, however, the advantage was gone (or very nearly so) as soon as the returner got the ball back in play.

Below, find the exact percentages of service points won for rallies that reach various lengths.  In the table below, the ‘1’ row refers to points with at least one stroke (the serve) that went in.  A one-stroke rally is defined as an ace, service winner, or return error.  A two-stroke rally is defined as a point in which the return landed in but the server doesn’t get his second shot back in play.  Note that each individual percentage is biased in favor of the player (server or returner) who has the chance to put the point away; the point can be considered neutralized when the biased even out (e.g. 55, then 45, then 55, and so on).

Rally    All     1sts     2nds
0      63.2%
1      66.1%    71.7%    56.1%
2      50.3%    53.7%    45.6%
3      59.9%    63.6%    54.9%
4      46.4%    47.8%    44.6%
5      58.1%    60.3%    55.6%
6      44.9%    45.9%    43.9%
7      57.0%    58.2%    55.6%
8      44.5%    45.1%    43.9%
9      55.9%    56.4%    55.5%
10     44.1%    44.1%    44.0%
11     55.8%    55.9%    55.8%
12     43.2%    43.1%    43.3%
13     55.6%    55.6%    55.5%
14     43.5%    43.5%    43.6%
15     55.5%    55.3%    55.6%

Italian translation at settesei.it

As promised the other day, there’s a lot we can do with point-by-point and win probability stats for over 600 grand slam matches.

I’ve beefed up those pages a bit by borrowing some ideas from Brian Burke at Advanced NFL Stats.  He invented a couple of simple metrics using win probability stats to compare degrees of comebacks and the excitement level of (American) football games.

The concepts transfer to tennis quite nicely.  Comeback Factor identifies the odds against the winner at his lowest point.  I’ve defined it the same way Burke does for football: CF is the inverse of the winning player’s lowest win probability.  In the US Open Federer/Djokovic semifinal, Djokovic’s win probability was as low as 1.3%, or 0.013.  Thus, his comeback factor is 1/.013, or about 79.  That’s about as high a comeback factor as you’ll ever see.

On the other end, comeback factor cannot go below 2.0 — that’s the factor if the winning player’s WP never fell below 50%.  Matches in which the winner dominated are often very close to 2.0, as in the Murray/Nadal semifinal.  In that match, Nadal’s low point was facing a single break point at 2-3 in the first set; the comeback factor is 2.3.

A good way to think about comeback factor is this: “At his lowest point, the winning player faced odds of 1 in [CF].”

Excitement Index is a measure of volatility, or the average importance of each point in a match.  “Volatility” measures the importance of each individual point; EI is the average volatility over the course of a match.

(Burke sums the volatilities, reasoning that in football, a fast-paced game with many plays is itself exciting.  Since there is no clock in tennis [not exactly, anyway], it seems appropriate to average the volatilities.  Win probability already considers the excitement and importance of a deciding final set.)

At the moment, I’m calculating EI by multiplying the average volatility by 1000.  The Murray/Nadal match is 35 (not very exciting, though Murray fought back), the Djokovic/Federer match is 47 (more on that in a minute), while the 2nd rounder between Donald Young and Stanislas Wawrinka is 64.  I haven’t looked at all the matches yet, but EI should generally fall between 10 and 100, possibly exceeding 100 in rare instances like the Isner/Mahut marathon.

It seems like Djok/Fed should be higher, perhaps because we remember the excitement of the final set.  (And it may be that the final set should be weighted accordingly.)  But looking at the match log, there were an awful lot of quick games, which translate to relatively low volatility.  By contrast, Donald/Stan was more topsy-turvy throughout, as the players traded sets, then send volatility through the roof with a pair of breaks midway through the final set.

Both EI’s scaling and its exact definition are works in progress.  When I get a chance, I’ll do a survey of matches for which I have point-by-point data to further investigate both of these new (to tennis) metrics.

Win probability graphs and stats are now available for over 600 grand slam matches from 2011.  Thanks to IBM Pointstream from this year’s slams, there is a wealth of data available like never before.

Here’s the main menu.

Here’s a sample match: The US Open semifinal between Federer and Djokovic.

When I first started publishing tennis research, win probability was one of my focuses.  You can find earlier work here, which links to specific tables for games, sets, and tiebreaks.  I’ve also published much of the relevant code, which is written in Python.

Win probability represents the odds of each player winning after every point of the match, based on the score up to that point and which player is serving. It makes no assumptions about the specific skill levels of each players, but does assume that the server has an advantage, which varies based on surface and gender.  With every point, each player’s win probability goes up or down, and the degree to which it rises or falls is dependent on the importance of the point–at 4-1, 40-0, winning the point is nice, but losing the point just delays the inevitable; at 5-6 in a tiebreak, the potential change in win probability is huge.

To quantify that in the graphs, I show another metric: Volatility, which measures the importance of each point. It is equal to the difference in win probabilities between the server winning and losing the following point. 10 percent is exciting, 20 percent is crucial, and 30 percent is edge-of-your-seat stuff.

Assumptions

To produce these numbers, I needed to make several simplifying assumptions.  Some are more important than others; here are the big two:

• The players are equal.
• Each player’s ability does not vary from point to point.

The first of these is almost always false, and the second is probably false as well.  The first, however, makes things more interesting.  In most matches Novak Djokovic plays these days, he goes in with an 80-percent-or-better chance of winning.  If we graphed one of his matches starting at 85 percent, we’d usually get a very slowly ascending line.  Instead, by starting at 50 percent, we can see where he and his opponent had their biggest openings, and who took advantage.

(In this long-ago post, I showed a sample graph with an assumption similar to the 85 percent for Djokovic, and you can see some of what I mean.)

Assuming that the players are equal also sidesteps of messy question of how to quantify each player’s skill level on that day, on that surface, against that opponent.

The second big assumption ignores possibility real-world attributes like clutch performance and streakiness, along with more pedestrian considerations like some players’ stronger serving in the deuce or ad court.

Another long-ago article of mine suggests that servers are not absolutely consistent, possibly because of natural rises and falls in performance, also possibly because of risk-taking (or lack of concentration) in low-pressure situations.  One of the most interesting directions for research with these stats is into this inconsistency: We need to figure out whether some players are more consistent than others, whether “clutch” exists in tennis, and much more.

One more set of assumptions regards the server’s advantage.  Since these graphs only encompass the four grand slams, I set the server’s win percentage for each tournament.  The numbers I used for men are: 63% in Australia, 61% at the French, 66% at Wimbledon, and 64% at the U.S. Open.  I used percentages two points lower for women at each event.

More on Win Probability

There’s very little out there on win probability and volatility in tennis.  I wasn’t the first person to work out the probability of winning a game, a set, or a match from a given score, but as far as I know, I’m the only person publishing graphs like this.  Much of the problem is the limited availability of play-by-play descriptions for professional tennis.

That problem doesn’t apply to baseball, where win probability has thrived for years.  Here’s a good intro to win probability stats in baseball, and fangraphs.com is known for its single-game graphs–for instance, here’s tonight’s’s Brewers game.  In many ways, win probability is more interesting in baseball than in tennis.  In tennis, there are only two possible outcomes of each point, while in baseball, there are several possible outcomes of each at-bat.

Enjoy the graphs and stats!