The Summer of Jeff

Win Probability and Volatility in Tennis

Posted in tennis by Jeff on December 30, 2010

Last week, I presented three win expectancy graphs of the same up-and-down tennis match.  Tennis players and fans have an intuitive sense of the most “important” points in a match, and we can quantify that as well.

Again we can mimic win probability graphs from baseball–this time, I’ve added a measure of “volatility” for each point in the match.  Volatility is defined the difference in win probability between two hypotheticals: that the server wins the next point, and that the returner wins the next point.  In other words, if the score is 30-15, volatility is the difference between the WP at 40-15 and at 30-30.

For today, we’re going to stick with the Roddick-Pavel Davis Cup five-setter from 2006.  Here’s a recap of that match, if you want a better idea of what you’re looking at.

The green line is Roddick’s win probability, the purple line is volatility, and the spaces demarcate each set.  These numbers follow from the assumptions that the players are equal and that the server wins 65% of points.

In many ways, tennis is simpler than other sports for which you can show win probability and volatility.  Note the even ups-and-downs toward the end of the first set–that’s a deuce game where Pavel won each deuce-court point, but took a few tries to nail down the game.  The win probability at deuce doesn’t change within the game, so a long deuce game, such as the last game of this match, involves a lot of predictable zig-zagging.

Also predictable are the high volatility peaks in the final set.  A break of serve doesn’t count for much in the second set of a five-set match, but it makes a huge difference in the decider.

The peak volatility (26.5%) came in the third game of the final set, with Roddick serving 1-1, 30-40.  Had he won the point and reached deuce, his win probability would’ve gotten near 50%, as he would be likely to hold serve to 2-1.  But if Pavel won, the break knocked Roddick’s chances down to 21.7%, down a break in the fifth.  A similar moment came at 30-40 in the previous game when Pavel faced the same problem and a 24.2% volatility.

The volatility plummeted as Pavel kept winning to 5-1, but once Roddick crept back, it crossed 20% three more times, all at deuce in Pavel’s 5-4 service game.

I’ll be interested to see how the volatility graph looks in a variety of other situations–particularly deciding set tiebreaks in which the ultimate winner fights off multiple match points.

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Tennis Win Expectancy Graphs!

Posted in tennis by Jeff on December 23, 2010

It’s time to toss a lot of this year’s work into the blender and see what comes out.

In the last few months, I’ve calculated win expectancy charts for different parts of tennis matches.  Here’s the data for a single game, a tiebreak, and a set.  All of these are adjustable for assumed serving and returning skill levels, so we can can look at different win expectancies when the players are big servers, counterpunchers, or anything in between.

There’s not much tennis play-by-play out there, so the applications are lacking at this time.  But I’m working on it.

2006 Davis Cup: Roddick vs. Pavel

It’s February 2006, the first round of Davis Cup World Group play.  The USA has drawn Romania, and they are playing at home.  Opening the tie is the USA #1 (and world #3) Andy Roddick against the Romanian Andrei Pavel.

While Pavel broke in to the top 20 for a small chunk of his career, he was never a player of Roddick’s caliber.  At the time of this match, he was ranked #82 in the world.  Using only ranking points, Roddick had about an 87% chance of winning.  Add in the home-field, home-surface advantage, and on paper, it was even more lopsided.

A roller-coaster match

Here’s a quick summary of the match, so you know what you’re looking at when we get to the graphs.

First Set: Roddick and Pavel trade holds to 6-6, then Roddick easily takes the tiebreak, 7-2.

Second Set: Roddick cruises, breaking twice (at 1-1 and 2-4), to win the set 6-2.

Third Set: Pavel breaks Roddick at 0-1, but Roddick breaks back at 4-2.  They reach another tiebreak.  They trade early mini-breaks, but at 8-8, Roddick loses a point on serve, and Pavel holds to take the tiebreak 10-8.

Fourth Set: Pavel keeps the pressure on, breaking Roddick at love in the first game.  He breaks in the 5th game as well, winning this set 6-2 to even the match.

Fifth Set: This is almost unbelievable.  Roddick gets broken in the first game.  Pavel loses serve as well.  Roddick gets broken again.  Pavel reels off the next three games to reach 5-1, but Roddick fights back to 4-5.  Finally, in a ten-point game (the second longest of the match), Pavel holds to win the set 6-4, and the match.

The graphs!

Any WinEx graph varies depending on your assumptions.  Let’s start with what I’ll call the “baseball model.”  In it, we assume that each player has an equal chance of winning every point, serve or return.  (In professional men’s tennis, that’s false.  But as in baseball, it’s a convenient fiction.)

All of these graphs are from Roddick’s perspective.  The little gaps in the line separate each set.  Unsurprisingly, the most drama is pictured at the beginning and end of the final set.  The flip-flopping breaks are obvious at the beginning of the fifth set, and the hard-fought final game shows how close Roddick was to evening the score before finally losing.

Picturing the server advantage

Let’s change our assumptions to get closer to reality.  On hard courts, the average player wins about 64% of points on serve.  This is 2006 Roddick we’re talking about, so I edged it up to 65%.  Note that I’m altering the assumption for both players — this graph is based on the fiction that all players are equal, but the more truthful parameter that they are more likely to win points on serve.

Still looks about the same.  The extremes are a bit more pronounced in the fifth set — still, Roddick created some drama, but the lower win expectancy in the final game reflects the fact that Pavel had the ball on his racquet.

Approximating skill level

I don’t yet have granular stats for 2005 or 2006, so I don’t know exactly what percentage of serve and return points Roddick and Pavel were winning back then.  However, I do have their rankings, which generated the estimate that Roddick had about an 87% chance of winning.  We can argue about the effect of the Davis Cup atmosphere, the fact that Pavel was selected for the team, and so on, but it’s clear that Roddick had a big edge, at least on paper.

To reflect that edge, I changed the assumptions that each player would win 65% of points on serve.  Now, we figure that Roddick wins 69% of points on serve while Pavel wins 62%.  That spits out a win percentage of about 86% for Roddick, which is obvious in the graph, especially in the first set!

Roddick was up 3-0 in that 3rd set tiebreak.  In this model, he had a 99.0% chance at that point of winning the match!  What I find noteworthy is how, when a player is so heavily favored, the underdog can hack away at the advantage just by hanging in there.  It’s intuitively true, but still interesting to see in practice.  While Roddick started the match at 86%, Pavel got the edge down to 76% toward the end of the first set.

All together now

Here’s a look at all three models.  The red line is the 50/50 “baseball model,” the green line is the 65/65 model, and the blue line is the 69/62 skill-based model.

There’s a lot of great stuff here.  In future posts, I’ll get into more of it.

Greinke to the Brewers

Posted in baseball analysis by Jeff on December 20, 2010

The evidence would suggest that Doug Melvin is a very single-minded man.

One year, he was all about relief pitching.  Another, focused on defense.  Yet another, building starting pitching depth.

This year, he has focused on high-quality starting pitching to the exclusion of everything else.  First, he swapped 2008 first round pick Brett Lawrie for Blue Jays starter Shaun Marcum.  Now, he’s bet the farm on Zack Greinke, giving up four youngsters in a deal for the 2009 Cy Young Award winner.

One thing is clear: The Brewers rotation looked dreadful a few weeks ago, and now it looks very solid.  Greinke, Marcum, and Yovani Gallardo comprise a 1-2-3 that any team (except the Phillies) would desire, and adding Randy Wolf to that threesome makes it look even better.

Of course, the wisdom of a deal isn’t just in what the get.  The real question is, did the Brewers give up too much to increase their odds of winning in 2011?

The end of the Prince Fielder era

I’ve long been surprised that no one blew away Melvin with a trade offer for Prince.  For whatever reason, it hasn’t happened, and Fielder is still in the fold … at least for a few more months.  The team will look very different without him, and it’s all but certain that the first baseman will be elsewhere in 2012.

If the Brewers are going to win with Prince, it’s going to have to be in 2011.  And if the Crew is to win in 2012 and beyond without Prince, the team will have to rely less on a deadly 3-4 combination.

At the risk of stating the obvious: The trades make the Brewers better in 2011.  The difference between 400+ innings of Greinke and Marcum and 400+ innings of whoever the hell else Melvin would have dug up will be huge.

Some of the discarded players–Escobar, Cain, and maybe Jeffress–were part of the Crew’s 2011 plans, but none were as crucial as the starting pitching gained in exchange.  Escobar should continue to improve at the major league level, but there’s no guarantee that’s going to happen soon, suggesting that his .614 2010 OPS might be better as somebody else’s problem.

Cain excelled in an extended audition last year, but we may have seen the best he’ll ever produce.  He’s turning 25 in April, making it possible he’s a late bloomer, or more likely, that he’ll have a few league-average seasons before becoming a part-timer in his 30’s.  Jeffress has always had huge potential, but can only be viewed as a reliever at this point, and one with serious control issues.

The 2011 Brewers

In other words, there’s little short-term tradeoff.  I’m not convinced that the Greinke-Marcum duo makes the Brewers favorites in the Central next year, but you could certainly make the argument.

The biggest problems with the 2011 squad are the ones that existed before the deals.  A month ago, we wondered whether Escobar and some combination of Cain and Carlos Gomez would provide anything better than replacement-level offense.  Now the focus is on the same positions, but with less hope.  Now both spots are up for grabs this spring, with Yuniesky Betancourt and Gomez presumably in the lead.

I think it’s safe to assume that at least one of those two positions will end up being dreadful.  Maybe both.  I also think the Brewers should be happy with that result.  I’d rather have two studs and two replacement level players than four mediocre to average players.

Frankly, it’s tough to imagine Doug Melvin making any other (realistic) pair of moves to better boost his team’s chances for the 2011 playoffs.  If you don’t like the deal, it can only be because you’re concerned he traded away too much of the future.

2012 and beyond

Did he?  As is always the case in these sorts of deals, it will be at least seven or eight years before we know the whole story.

Let’s take a specific look at 2012.  As mentioned, the Brewers assume they’ll be without Fielder.  They will, however, still have a substantial core–just about everybody else in the 2011 lineup except for Fielder and Rickie Weeks, and the same top four in the rotation.  Slot Mat Gamel into first base, assume you’ve got a bit of money to play with after Prince’s departure and everybody else’s raise, and that still looks like a pretty good team.

To find the possibility of a serious downside, you have to look further into the future.  Sure, it would be nice to have Lawrie playing second base as soon as Weeks departs, but it’s very possible Lawrie will never play a major league inning as a second baseman.

It’s fun to wishcast a 2013 or 2014 Brewers team with Escobar at short, Cain in center, and Lawrie somewhere.  Maybe Jeffress closing and Odorizzi enjoying a successful rookie year in the rotation.  Realistically, though, Escobar and Cain may never be reliably league-average hitters; Lawrie may end up stuck in a corner; Jeffress could just as easily flame out as last a full season as an 8th-inning guy, and Odorizzi hasn’t yet pitched above single-A.

The combined packages for Greinke and Marcum are better than what Milwaukee sent to Cleveland for CC Sabathia, but not hugely so.  Here, we get two starters for two years each, instead of one for half a season.  Just because that deal hasn’t panned out for Cleveland doesn’t mean these won’t for the Jays or Royals.  But the playoff run in 2008 showed us just how unimportant prospects are in September–unless you swap them for someone valuable.

Without these deals, the Brewers wouldn’t have made the playoffs next year.  It’s possible they wouldn’t have broken .500.  Now, they are contenders in 2011, and it will only take one or two solid moves to make them contenders again in 2012.  Literally or figuratively, “the entire farm system” might just be worth it.

A New Low for SEO

Posted in business by Jeff on December 15, 2010

I got the following email today:

Hi, I am Your Name with WebSponsors. My company represents a leading provider of online MBA material.

They would like to purchase a simple, unobtrusive textual advertisement to go on the bottom of your webpage (http://www.gmathacks.com/resources/resource-review-the-mba-application-roadmap.html). It would say something like…

“MBA courses” or “MBA materials online” — with a link to our client’s site.

We are willing to pay $60 via PayPal immediately for this advertisement.

Please let me know if you are interested. Thank you for your time and consideration.

Your Name
Your Email

I get these once every week or two, and I’m amazed how many have obviously fake names.  But this is the first time someone didn’t include a name at all.

Best of all, whoever didn’t fill out the necessary form missed the return email address, as well.  In the email header, the “From” address is “Your Name.”   To find evidence of a real person in all this, you have to dig further in the header:

Return-Path: <webspons@vcg.vcgaffiliate.com>

By now, it’s a cliche to say, “These people want something from me, and they can’t even…”  But to me, this is a noteworthy step down from anything I’ve seen before.  I’m very glad my primary source of income is not in any way related to web advertising or affiliate links.

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ATP Total Points Won

Posted in tennis by Jeff on December 10, 2010

I’m always amused when a tennis player wins a match despite winning fewer than half of points played.  At the highest levels, tennis is often a game of small margins, where a few points in a tiebreak can make or break a match.

So I delved into the 2010 numbers to see what they would tell us about these unusual matches.

As a reference point, match winners take an average of 55.6% of total points.  Of the 2581 ATP main draw matches from this year in my database (excluding those won by retirement), 98 (3.8%) were won by the player who lost more than half of the points played.  Another 29 (1.1%) saw the points split evenly down the middle.

The most extreme 5% of matches in the other direction are those where the winner takes more than 64% of points.  Surprisingly, there are quite a few late-round matches in that batch.  In fact, the 13th most lopsided match of the season was the Brasil Open final, in which Juan Carlos Ferrero demolished Lukas Kubot while winning 70% of points.  Two other finals were among this most extreme 5%, including Rafael Nadal’s defeat of Fernando Verdasco at Monte Carlo.  That day, he won 64.3% of points.

Also of interest: the two most lopsided Grand Slam matches were Robin Soderling’s 1st and 2nd round takedowns of Laurent Recourdec and Taylor Dent.  Roland Garros shows up several times among the most lopsided 5%, but the US Open does not at all, and Wimbledon does only once: John Isner’s post-Mahut loss at the hands of Thiemo de Bakker.

Back to the wacky ones

What about the sub-50% winners?

It seems like every time I write about tennis, Isner’s name comes up.  Of the 98 matches in which the loser in points won the match, five names show up as winners three times or more.  Winning three of these close matches each are Stanislas Wawrinka, Nicholas Almagro, Mikhail Youhzny, and Fabio Fognini.  Isner amassed eight.  That’s more than 20% of the matches he won!

On the other side of the ball, Nadal, Juan Monaco, Michael Russell, and Richard Gasquet each lost three of these matches, while de Bakker and Marcos Baghdatis each lost four.  Ouch.

This sort of match only happened twice at the US Open and the Australian Open, but it occurred four times at the French, and six times at Wimbledon, including the marathon Isner-Mahut match.  Equal to Wimbledon was Indian Wells, with six such matches despite 32 fewer total contests.

Three finals are among these matches, as well: Bastad (Almagro over Soderling), Doha (Davydenko over Nadal), and Santiago (Bellucci over Monaco).

To wrap things up, here are the 20 matches from this season where the point total was most lopsided…in favor of the loser:

Tourney         Rd    Winner              Loser                   Pts  Won     W%  
Miami           R64   John Isner          Michael Russell         224  105  46.9%  
Indian Wells    R16   Tommy Robredo       Marcos Baghdatis        187   88  47.1%  
Bangkok         R16   Daniel Brands       Thiemo De Bakker        203   96  47.3%  
Santiago        R32   Joao Souza          Simon Greul             194   92  47.4%  
Queens Club     R64   Rajeev Ram          Karol Beck              162   77  47.5%  
Santiago        R32   David Marrero       Juan Martin Aranguren   225  107  47.6%  
Madrid          R32   John Isner          Santiago Giraldo        185   88  47.6%  
Vienna          R32   Pablo Cuevas        Thiemo De Bakker        176   84  47.7%  
Acupulco        R32   Pablo Cuevas        Marcos Daniel           205   98  47.8%  
Marseilles      R16   Gael Monfils        Andreas Seppi           207   99  47.8%  
Chennai         S     Stanislas Wawrinka  Dudi Sela               193   93  48.2%  
Washington DC   R64   Horacio Zeballos    Michal Przysiezny       203   98  48.3%  
Montpelier      R32   Frederico Gil       Edouard Roger Vasselin  205   99  48.3%  
US Open         R128  Arnaud Clement      Marcos Baghdatis        265  128  48.3%  
Miami           R128  Olivier Rochus      Richard Gasquet         180   87  48.3%  
Indian Wells    S     Ivan Ljubicic       Rafael Nadal            190   92  48.4%  
Doha            F     Nikolay Davydenko   Rafael Nadal            190   92  48.4%  
Atlanta         R16   John Isner          Gilles Muller           227  110  48.5%  
Shangai         R64   John Isner          Lukasz Kubot            229  111  48.5%  
St. Petersburg  S     Mikhail Youzhny     Dmitry Tursunov         233  113  48.5% 

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Server Dominance by Surface

Posted in tennis by Jeff on December 8, 2010

I’m starting to look at some ATP player stats broken down by surface.  We assume, correctly, that the faster the surface, the more the strength of the server dictates the outcome of the point.  Some players are much better on faster or slower surfaces.

My database of 2010 ATP-level matches should consist of all main-draw matches, though it doesn’t include the tour finals.  Taking all of those matches into account, here is a general measure of server dominance: percentage of service points won:

  • Clay: 61.5%
  • Hard: 63.7% [includes both indoor and outdoor]
  • Grass: 65.9%

It may be interesting in the future to look at this rate for specific tournaments.  I think the conventional wisdom is that the Australian Open surface is a bit slow for a hard court, while the U.S. Open surface is on the fast side.

For individual players, the grass numbers aren’t very meaningful, since there just aren’t very many matches played on grass.  But most players rack up a fair amount of competition on both clay and hard courts.  I took the 40 or so players with the most 2010 matches, and show their percentage of service points won on each surface:

Player                  n(C)  sv(C)  n(H)  sv(H)  n(G)  sv(G)  sv(H) - sv(C)  
Juan Carlos Ferrero       35  65.9%    10  62.1%     2  63.0%          -3.8%  
Potito Starace            30  63.9%    17  60.4%     1  57.4%          -3.6%  
Tomas Berdych             17  69.0%    41  67.4%     7  73.1%          -1.5%  
Nicolas Almagro           37  67.2%    27  66.1%     3  59.9%          -1.1%  
Juan Monaco               25  62.6%    24  61.5%     0                 -1.1%  
Stanislas Wawrinka        21  66.6%    30  65.5%     1  65.9%          -1.1%  
Juan Ignacio Chela        33  60.2%    18  59.3%     1  67.7%          -0.9%  
Ernests Gulbis            16  67.1%    33  66.4%     0                 -0.7%  
Robin Soderling           20  67.3%    47  66.9%     5  71.1%          -0.4%  
Andreas Seppi             24  62.0%    24  62.0%     4  62.7%           0.0%  
Denis Istomin             12  63.5%    36  63.6%    10  68.7%           0.1%  
Guillermo Garcia Lopez    20  63.0%    27  63.1%     7  63.9%           0.2%  
David Ferrer              36  64.4%    38  65.2%     4  67.5%           0.8%  
Fernando Verdasco         29  63.2%    34  64.2%     1  66.4%           1.0%  
Thiemo De Bakker          11  65.0%    32  66.2%     6  70.4%           1.1%  
Sam Querrey               16  65.7%    33  67.2%    11  70.7%           1.5%  
Mikhail Youzhny           17  63.3%    38  64.8%     3  60.6%           1.5%  
Jurgen Melzer             25  64.0%    41  65.6%     6  67.4%           1.6%  
Rafael Nadal              22  68.7%    44  70.3%    10  72.8%           1.7%  
Jeremy Chardy             21  60.9%    29  62.6%     5  67.5%           1.8%  
Marin Cilic               14  64.1%    42  65.9%     3  66.3%           1.8%  
Victor Hanescu            26  64.0%    18  65.8%     3  66.6%           1.8%  
Roger Federer             14  68.2%    49  70.0%    10  70.4%           1.9%  
Albert Montanes           34  62.2%    24  64.0%     3  68.9%           1.9%  
Richard Gasquet           24  65.1%    34  67.0%     2  73.3%           1.9%  
Mardy Fish                 4  64.2%    35  66.1%    13  74.9%           1.9%  
Thomaz Bellucci           29  61.5%    23  63.6%     3  64.4%           2.1%  
Marcos Baghdatis          14  62.9%    51  65.2%     3  57.3%           2.2%  
Feliciano Lopez           11  63.0%    30  65.9%     8  68.3%           2.8%  
Viktor Troicki            14  61.9%    43  64.8%     5  66.3%           2.9%  
Philipp Kohlschreiber     19  62.8%    34  66.0%     6  70.6%           3.2%  
John Isner                16  66.6%    42  70.0%     2  72.1%           3.5%  
Nikolay Davydenko          5  60.8%    38  64.5%     4  67.9%           3.8%  
Novak Djokovic            14  60.6%    46  64.5%     8  67.8%           3.9%  
Benjamin Becker            8  60.3%    40  64.4%    10  71.9%           4.1%  
Gael Monfils              12  61.5%    45  65.7%     4  67.2%           4.2%  
Andy Murray               10  61.1%    42  65.9%     8  74.1%           4.8%  
Jarkko Nieminen           13  60.4%    37  65.9%     4  61.5%           5.5%  
Michael Berrer             9  58.5%    38  64.4%     1  61.4%           5.9%  
Sergiy Stakhovsky          9  57.2%    32  63.3%     7  66.1%           6.1%  
Michael Llodra             5  58.7%    32  66.1%    11  73.0%           7.4%  
Andy Roddick               3  63.3%    54  73.0%     6  73.9%           9.6%

The columns n(C), n(H), and n(G) show the number of matches on each surface.  I sorted by the final column, the difference between sv(C) and sv(H), to highlight the surprising number of players who do not win more service points on hard courts.

Race Report: Joe Kleinerman 10K

Posted in running by Jeff on December 7, 2010

Sunday morning, I ran my first 10K, the NYRR Joe Kleinerman Classic in Central Park.  I haven’t been able to run too much in the last month or so, but I was excited to try the “short” distance for the first time.

The course was the standard Central Park loop, this time starting near East 102nd Street and heading counterclockwise.  I was amazed to see that nearly 5,000 runners turned out at 8 A.M. on a Sunday in roughly 28 degree weather.  I don’t think I’ve ever seen so many pairs of running tights in my life.

Anyway, I was shooting for a sub-45:00 time, recognizing that the goal might be a little far-fetched.  After all, my last half-marathon time translates to a 48:00 10K.  I knew I could do better than that, but by how much?

The first mile or so was frustrating–because of that disappointing half-marathon, NYRR didn’t give me credit for being very fast, and I had to start in the fourth corral.  Thus, I spent the entire race passing people.  So my first mile was the slowest–by far.  I’m slow to warm up, so I probably wouldn’t have run it any faster than 7:30, but that still would’ve made quite a difference.  After that first mile, I still had to do a bit of dodging, but it didn’t slow me down much.

Here are my mile-by-mile splits:

  1. 7:58
  2. 7:20
  3. 6:57
  4. 7:04
  5. 7:11
  6. 6:52
  7. 1:38 [0.26 miles – 6:26 pace]

Miles 1, 2, and 5 are the net uphills.  They average (according to my Garmin, which occasionally has wonky things to say about elevation) about 110 feet ascending and 80 feet descending.

Add it all together and you get … 45:02.  As you can see, I pushed pretty hard for the last mile and change, but it wasn’t quite enough to come in under 45:00.

For the first time, though, I finished a race feeling like I had given my best possible effort.  Maybe that’s easier to do in shorter races, since there aren’t as many miles to save yourself for.  In any event, I’m very happy with how things went.

Next up, the Ted Corbitt 15K, again in Central Park, 12 days from today!

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First Server Advantage? – Now With Data!

Posted in tennis by Jeff on December 7, 2010

Last week I wrongly concluded that the first server in a set had a structural advantage.  In fact, both the server and returner of the first game begin each set with an equal chance of winning–at least, if the players themselves are of equal skill levels.

But what about in practice?

If the conventional wisdom is to be believed, the first server does have a psychological advantage.  If the set reaches 4-4, the first server can hold for 5-4, then the second server must hold simply to stay alive.  If he does, the process repeats itself at 5-5 and 6-5.  In other words, in a tight set, the pressure is disproportionately on the second server.

What the data says

While we have final scores for every ATP match for the last several years, we don’t always know who served first.  However, we do have some match-by-match statistics, including the number of service games for each player.   When the number of service games is equal, it’s anybody’s guess who served first.  But if one player served more games than the other, he must have served first.  As it turns out, about half of all matches have an odd total number of games.

(Is this a biased sample?  It could be, but it’s not clear to me that it is.  If we were only looking at individual sets, it would be biased–6-3’s are more likely to favor the server, for instance–but at the match level, many contests have one set with an odd number of games and one or two with an even number of games, resulting in a score like 7-6(4), 3-6, 6-4.  That said, the possibility of a bias can’t be ignored, as we’ll see.)

Let’s dive in.  I have stats for 2674 ATP-level matches from 2010.  1316 of those had an odd number of service games, so we can analyze those.  That subset of matches gives us a total of 3464 sets, of which 53.2% were won by the player who served first in that set.  That’s a substantial edge.

I decided to run the same test on the 2010 Challenger tour.  There, we have 4695 matches, 2255 of which are usable for this purpose, giving us 5274 sets.  You couldn’t ask for anything much more consistent: At this level, the first server won 53.0% of sets.

What about deciders?

If we’re talking about pressure, let’s turn to where there’s some real pressure.  Sure, the returner is playing catch-up in every set, but it only really matters in a deciding set.

At the ATP level, the available sample shrinks to 493 matches, of which the first server in the deciding set won 51.7%.  Surprisingly, among Challengers, the first server in the final set won only 48.8% of the 797 relevant matches.

These results, especially the Challenger numbers, are far from intuitive.  It turns out that the first server has the most dramatic edge in the first set.  In the Challenger dataset, the first server won the first set of a match 61.6% of the time!  For the ATP-level matches, the edge goes up to 64.6%.  At this point, I have to suspect that the data is biased.  A 6-3 first set makes a match more likely to show up in this dataset, and the first server is much more likely to win 6-3.

What about “tight” sets?

Here’s another possibility: The first server’s supposed edge really doesn’t kick in until the set gets to about 4-4.  There may be a bit of a psychological advantage to serving first, but if someone is winning 6-1 or 6-2, it’s for reasons other than the subtle gain conferred by a coin toss.

If we limit ourselves to sets with 10 games or more, we get another wacky result.  In ATP matches, the player who serves first wins a “tight” set only 44.0% of the time.  In the Challenger dataset, it’s 44.1%.  Again consistent, but far from intuitive!  I would have expected the opposite.

Aside from the plausible aggregate numbers (53% of sets going to the initial server), it’s tough to explain a lot of this.  And I’m not sure it’s worth the effort, given the possibility that the dataset is biased.  Without better data, it may not be possible to draw any solid conclusions on the advantage gained by serving first.

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Serving Against Markov

Posted in tennis by Jeff on December 5, 2010

The last few days I’ve presented a lot of theory. My win expectancy tables are based on what you might call a “video game” model of tennis–players are presumed to perform at the same level on every single point. In a one-on-one sport played by humans, I think it’s safe to assume that that’s not the case.

Some of the specific assumptions are more easily tested than others. Let’s start with one of the easy ones.

Starting from the beginning of this project a few days ago, I used Markov chains to translate service points won to service games won.  There’s obviously a correlation, but does my model reflect reality?

There are plenty of reasons to suspect that it may not.  First off, it doesn’t consider the differences between the deuce and ad courts.  Some players are stronger servers (or returners) on one side or the other.  Second, anyone who has even played or watched tennis can tell you that good serving isn’t always consistent.  Even a server who is perfectly consistent isn’t going to put up the same numbers every match–the surface, the weather, and most of all his opponent will have something to say about that.

Tennis isn’t awash with numbers, but this is a case where we are able to use ATP statistics for a quick reality check.

Testing the model

I took the top 50 players in the most recent ATP rankings.  In the table below, you’ll see each man’s percentage of service games played in 2010, his percentage of service points won, and his number of service games won.  Next, I show the percentage of service games the Markov model predicts he would win, based on his percentage of service points won.

The final columns show two ways of measuring the results.  The first of the two is the raw difference between the number of service games he actually won and the number that my model predicts he would win.  For 40 of the 50 players, this is negative!  Clearly, the model is not quite an accurate representation of reality, at least at the aggregate level.

To show something a little different, the final column adjusts those numbers so that they sum to zero.  You might think of this like wins above or below a Pythagorean expectation for a baseball team.  As in the baseball analogue, we might be tempted into speculating that this plus/minus rating represents some kind of clutch skill.  We’ll come back to that in a bit.

Here are the numbers:

First         Last             Sv Gms  Pts Won  Gms Won    Pred Gms    Diff  Diff+  
John          Isner               916     0.69      0.9        0.89      11     19  
Marin         Cilic               769     0.65     0.84        0.83       8     15  
Novak         Djokovic            907     0.64     0.82        0.81       7     15  
Ivan          Ljubicic            563     0.65     0.84        0.83       6     11  
Albert        Montanes            724     0.63      0.8        0.79       4     10  
Jurgen        Melzer              914     0.65     0.83        0.83       0      8  
Rafael        Nadal              1001      0.7      0.9        0.90      -1      8  
Feliciano     Lopez               581     0.66     0.85        0.85       2      8  
Robin         Soderling           942     0.67     0.86        0.86      -1      7  
Gael          Monfils             793     0.65     0.83        0.83       0      7  

Sam           Querrey             810     0.67     0.86        0.86      -1      6  
Andrey        Golubev             440     0.62     0.78        0.78       2      6  
Mikhail       Youzhny             771     0.64     0.81        0.81      -2      5  
Jarkko        Nieminen            669     0.64     0.81        0.81      -2      4  
Tomas         Berdych             864     0.68     0.87        0.88      -4      3  
Nicolas       Almagro             898     0.66     0.84        0.85      -5      3  
Andy          Murray              791     0.66     0.84        0.85      -5      2  
Potito        Starace             541     0.63     0.79        0.79      -3      2  
Richard       Gasquet             724     0.66     0.84        0.85      -4      2  
Stanislas     Wawrinka            641     0.66     0.84        0.85      -4      2  

Juan Ignacio  Chela               593      0.6     0.73        0.74      -3      2  
Sergiy        Stakhovsky          588     0.62     0.77        0.78      -3      2  
Yen Hsun      Lu                  425     0.62     0.77        0.78      -2      1  
Janko         Tipsarevic          515     0.65     0.82        0.83      -5     -1  
Nikolay       Davydenko           560     0.65     0.82        0.83      -5     -1  
Thiemo        De Bakker           620     0.67     0.85        0.86      -7     -1  
Florian       Mayer               474     0.64      0.8        0.81      -6     -2  
Roger         Federer             980      0.7     0.89        0.90     -11     -2  
David         Nalbandian          387     0.63     0.78        0.79      -6     -2  
Viktor        Troicki             721     0.64      0.8        0.81      -9     -3  

Marcos        Baghdatis           823     0.64      0.8        0.81     -10     -3  
Fernando      Verdasco            829     0.64      0.8        0.81     -10     -3  
Alexandr      Dolgopolov          561     0.63     0.78        0.79      -8     -3  
Andy          Roddick             868     0.72     0.91        0.92     -11     -4  
Jeremy        Chardy              630     0.63     0.78        0.79      -9     -4  
Jo Wilfried   Tsonga              607     0.68     0.86        0.88      -9     -4  
Guillermo     Garcia Lopez        662     0.63     0.78        0.79     -10     -4  
Julien        Benneteau           561     0.62     0.76        0.78      -9     -4  
Juan          Monaco              584     0.62     0.76        0.78      -9     -4  
Thomaz        Bellucci            706     0.63     0.78        0.79     -10     -4  

Philipp       Kohlschreiber       723     0.66     0.83        0.85     -11     -5  
Juan Carlos   Ferrero             536     0.65     0.81        0.83     -11     -6  
Ernests       Gulbis              542     0.67     0.84        0.86     -11     -7  
Marcel        Granollers          508     0.61     0.73        0.76     -13     -9  
Gilles        Simon               462     0.65      0.8        0.83     -14    -10  
David         Ferrer              950     0.65     0.81        0.83     -19    -10  
Mardy         Fish                647     0.68     0.85        0.88     -16    -11  
Michael       Llodra              551     0.67     0.83        0.86     -17    -12  
Tommy         Robredo             512     0.64     0.78        0.81     -17    -12  
Denis         Istomin             720     0.65      0.8        0.83     -21    -15

Any list with John Isner at the top or bottom is going to lend itself to some breezy conclusions, but a more thorough look tells us that strong and weak servers are scattered throughout the list.

Let’s return to the issue of consistency, and see how much it might account for these differences.

Impossible consistency

Imagine a hypothetical player who, on average, is a middle-of-the-pack server, winning 65% of his service points.  Whether because of his own inconsistency, or because of the variety in environments and opponents, he wins 60% of his service points half the time, and 70% of his service points the other half.

A perfectly consistent 65 percenter, according to the model, will win 83% of service games.  But the half-60/half-70 server will win only 81.9% of service games.  For a player who compiles 800 service games, that’s nine fewer winning games than the algorithm predicts from his aggregate numbers.

A fair amount of variance could be due solely to differences between opponents.  David Ferrer won 42% of return points this year, while Isner won only 29%.  Factor in higher success rates on hard courts or in warmer weather, and it’s a wonder that so many players come close to winning as many service games as are expected of them!

We may be able to get closer to the bottom of this by looking at single-match results.  Roger Federer probably wins different percentages of service points against Ferrer than he does against Isner, but perhaps when we adjust each match for the opponent’s return skills, we’ll discover that he regularly outperforms the model.

Strategy and clutch

One factor working in the other direction is that of strategy.  As we saw in the single-game probability table, a reasonably good server at 30-0, 40-0, or 40-15 has little to fear.  Nobody is tanking points with those scores, but some players may take the opportunity to try something riskier than usual, perhaps to keep his opponent guessing later in the match.

With sufficient point-by-point data, we could test this hypothesis, but without it, we can only speculate that servers may be less likely to win these specific points when the game is already heavily in their favor.  If this is frequently the case, players should be winning more service games than the model predicts.

Finally, back to clutch.  Because there are so many other factors at play, it would be foolish to point to the rightmost column in the table above and call it anything in particular, let alone “clutch.”  Among other problems, we don’t even know whether outperforming the model is a repeatable skill.  I’d love to show you the tables from previous years, but the necessary data collection is going to take some time.

7-point Tiebreak Win Expectancy Tables

Posted in tennis by Jeff on December 4, 2010

If you’re joining in progress, here’s an intro, along with single set tables and single game tables.

A Markov model of a tiebreak gets a little tricky–depending on where you are in the breaker, the remaining points could be evenly split between the two servers, off by one, or even off by two.  But that’s my problem…below you’ll find the win expectancy at every possible score in a tiebreak for each of three different types of players/surfaces.

The three sets of percentages are for three types of evenly matched players.  “0.6” refers to a pair of players who win 60% of service points, which could model counterpunchers or stronger servers on a slower surface.  0.65 is probably about average for a hard court match, and 0.7 is a battle between two dominant servers.  The table quantifies what we already know: The more effective the servers, the more valuable each mini-break.

To read the table, start with the first three columns of any row. The second row, for instance, shows percentages for a player who is serving (“s”) at 0-1. In the 0.60 model, he has a 40.8% chance of winning the tiebreak.

              0.60            0.65            0.70         
              p(W)   p(L)     p(W)   p(L)     p(W)   p(L)  
0  0  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
0  1    s    40.8%  59.2%    41.8%  58.2%    42.7%  57.3%  
0  1    r    36.2%  63.8%    34.7%  65.3%    32.9%  67.1%  
0  2  s/r    27.0%  73.0%    26.5%  73.5%    25.6%  74.4%  
0  3    s    18.4%  81.6%    18.6%  81.4%    18.6%  81.4%  
0  3    r    15.1%  84.9%    13.7%  86.3%    12.1%  87.9%  
0  4  s/r     8.5%  91.5%     8.0%  92.0%     7.2%  92.8%  
0  5    s     3.8%  96.2%     3.7%  96.3%     3.5%  96.5%  
0  5    r     2.8%  97.2%     2.3%  97.7%     1.8%  98.2%  
0  6  s/r     0.7%  99.3%     0.6%  99.4%     0.5%  99.5%  

1  0    r    59.2%  40.8%    58.2%  41.8%    57.3%  42.7%  
1  0    s    63.8%  36.2%    65.3%  34.7%    67.1%  32.9%  
1  1  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
1  2    s    40.0%  60.0%    41.0%  59.0%    42.0%  58.0%  
1  2    r    35.0%  65.0%    33.3%  66.7%    31.4%  68.6%  
1  3  s/r    24.9%  75.1%    24.3%  75.7%    23.4%  76.6%  
1  4    s    15.7%  84.3%    15.9%  84.1%    15.8%  84.2%  
1  4    r    12.4%  87.6%    11.1%  88.9%     9.5%  90.5%  
1  5  s/r     5.9%  94.1%     5.4%  94.6%     4.8%  95.2%  
1  6    s     1.7%  98.3%     1.7%  98.3%     1.5%  98.5%  
1  6    r     1.2%  98.8%     0.9%  99.1%     0.7%  99.3%  

2  0  s/r    73.0%  27.0%    73.5%  26.5%    74.4%  25.6%  
2  1    r    60.0%  40.0%    59.0%  41.0%    58.0%  42.0%  
2  1    s    65.0%  35.0%    66.7%  33.3%    68.6%  31.4%  
2  2  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
2  3    s    38.9%  61.1%    40.0%  60.0%    41.2%  58.8%  
2  3    r    33.3%  66.7%    31.5%  68.5%    29.4%  70.6%  
2  4  s/r    22.2%  77.8%    21.5%  78.5%    20.6%  79.4%  
2  5    s    12.1%  87.9%    12.3%  87.7%    12.3%  87.7%  
2  5    r     9.0%  91.0%     7.8%  92.2%     6.5%  93.5%  
2  6  s/r     2.9%  97.1%     2.6%  97.4%     2.2%  97.8%  

3  0    r    81.6%  18.4%    81.4%  18.6%    81.4%  18.6%  
3  0    s    84.9%  15.1%    86.3%  13.7%    87.9%  12.1%  
3  1  s/r    75.1%  24.9%    75.7%  24.3%    76.6%  23.4%  
3  2    r    61.1%  38.9%    60.0%  40.0%    58.8%  41.2%  
3  2    s    66.7%  33.3%    68.5%  31.5%    70.6%  29.4%  
3  3  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
3  4    s    37.3%  62.7%    38.7%  61.3%    40.0%  60.0%  
3  4    r    30.9%  69.1%    28.9%  71.1%    26.6%  73.4%  
3  5  s/r    18.2%  81.8%    17.6%  82.4%    16.6%  83.4%  
3  6    s     7.2%  92.8%     7.4%  92.6%     7.4%  92.7%  
3  6    r     4.8%  95.2%     4.0%  96.0%     3.2%  96.9%  

4  0  s/r    91.5%   8.5%    92.0%   8.0%    92.8%   7.2%  
4  1    r    84.3%  15.7%    84.1%  15.9%    84.2%  15.8%  
4  1    s    87.6%  12.4%    88.9%  11.1%    90.5%   9.5%  
4  2  s/r    77.8%  22.2%    78.5%  21.5%    79.4%  20.6%  
4  3    r    62.7%  37.3%    61.3%  38.7%    60.0%  40.0%  
4  3    s    69.1%  30.9%    71.1%  28.9%    73.4%  26.6%  
4  4  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
4  5    s    34.8%  65.2%    36.5%  63.5%    38.2%  61.9%  
4  5    r    27.2%  72.8%    24.9%  75.1%    22.4%  77.7%  
4  6  s/r    12.0%  88.0%    11.4%  88.6%    10.5%  89.5%  

5  0    r    96.2%   3.8%    96.3%   3.7%    96.5%   3.5%  
5  0    s    97.2%   2.8%    97.7%   2.3%    98.2%   1.8%  
5  1  s/r    94.1%   5.9%    94.6%   5.4%    95.2%   4.8%  
5  2    r    87.9%  12.1%    87.7%  12.3%    87.7%  12.3%  
5  2    s    91.0%   9.0%    92.2%   7.8%    93.5%   6.5%  
5  3  s/r    81.8%  18.2%    82.4%  17.6%    83.4%  16.6%  
5  4    r    65.2%  34.8%    63.5%  36.5%    61.9%  38.2%  
5  4    s    72.8%  27.2%    75.1%  24.9%    77.7%  22.4%  
5  5  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
5  6    s    30.0%  70.0%    32.5%  67.5%    35.0%  65.0%  
5  6    r    20.0%  80.0%    17.5%  82.5%    15.0%  85.0%  

6  0  s/r    99.3%   0.7%    99.4%   0.6%    99.5%   0.5%  
6  1    r    98.3%   1.7%    98.3%   1.7%    98.5%   1.5%  
6  1    s    98.8%   1.2%    99.1%   0.9%    99.3%   0.7%  
6  2  s/r    97.1%   2.9%    97.4%   2.6%    97.8%   2.2%  
6  3    r    92.8%   7.2%    92.6%   7.4%    92.7%   7.4%  
6  3    s    95.2%   4.8%    96.0%   4.0%    96.9%   3.2%  
6  4  s/r    88.0%  12.0%    88.6%  11.4%    89.5%  10.5%  
6  5    r    70.0%  30.0%    67.5%  32.5%    65.0%  35.0%  
6  5    s    80.0%  20.0%    82.5%  17.5%    85.0%  15.0%  
6  6  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%

Of course, a tiebreak can continue past 6-6. Any tie (e.g. 9-9) means that each player has a 50% chance of winning. Any other score (e.g. 8-9, 11-10) has the same probabilities as 6-5 or 5-6.

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