Soft Parameter
As you may recall, the ATP revised its points table at the start of 2009, reducing the values for results less than titles.

It might be worth explaining how a points table is computed for a single-elimination event. And, to do that, it might be clearest to examine the two extreme cases.

At a single-elimination tournament, you eliminate half the players in each round. That, theoretically, means that in each round you knock out the weakest half of the players. That, it could be argued, means that each opponent is twice as difficult as the round before. So each win should be worth twice as much as the round before. So if a win is worth 1000 (as it is at Miami), then a final should be worth 500, a semifinal 250, a quarterfinal 125, etc.

This can actually be expressed as a number, a ratio between rounds. The author calls the actual points table an approximation to a "base curve" -- there is a formula (technically a geometric series) which the points awards roughly follow. But it's only a rough approximation because the numbers generally get rounded off slightly and so don't exactly follow the curve. Hence "base curve" for the formula to which the points table approximates. A base curve has a parameter, a number by which you multiply the point value for a particular round to get the value for the previous round. In the case above, the base curve parameter is 0.5.

At the opposite extreme is the situation in which you regard all wins as equal -- you assume that it's just as easy to beat #1 in the final as to beat #80 in the first round. Since there are seven rounds at Miami, that would mean that if a title is worth 1000, then a final is worth 857.(1000 minus 143, which is 1/7 of 1000). A semifinal is worth 714.(857 minus 143). And so forth. The base curve parameter in this case is effectively 1.0.

These are the extremes. Any reasonable points table .will have a base curve parameter somewhere between 0.5 and 1.0.

So where do the Tours stand? Recall that the current ATP points table for a 1000 point event like Miami is as follows:

Title: 1000
Final: 600 (ratio to preceding: 0.6)
Semifinal: 360 (ratio to the preceding: 0.6)
Quarterfinal: 180 (ratio to the preceding: 0.5)
Round of 16: 90 (ratio to the preceding: 0.5)
Round of 32: 45 (ratio to the preceding: 0.5)

As we said, the numbers in the actual point table don't quite fit an exact curve. It turns out that the base curve parameter for the ATP is roughly 0.57. If it were exactly .57, the points table would look like this:

Title: 1000
Final: 570
Semifinal: 325
Quarterfinal: 185
Round of 16: 106
Round of 32: 60

Now let's look at the WTA numbers:

:Title: 1000
Final: 700 (ratio to the preceding: 0.7)
Semifinal: 450 (ratio to the preceding: 0.64)
Quarterfinal: 250 (ratio to the preceding: 0.56)
Round of Sixteen: 140 (ratio to the preceding: 0.56)
Round of 32: 80 (ratio to the preceding: 0.57)

Again, we don't have a perfect curve. But the average parameter is about .64. If we took that as the parameter, we would get:

Title: 1000
Final: 640
Semifinal: 410
Quarterfinal: 262
Round of 16: 168
Round of 32: 107

So which is "right"? That is, which curve best represents the difficulty of winning in a particular round? There is no exact answer (unless you can tell us just how much harder it is to beat Roger Federer than Michael Berrer, anyway). But it's worth noting how big a difference this makes. At Miami, both the ATP and WTA have 96 players earning main draw points. But if we assume all seeds win their opening matches, and ignore qualifying points, then the WTA will hand out 8880 points at Miami. The ATP will hand out only 6000. That's a big differrence -- much bigger than the difference in the base curve parameter. This is the power of geometric progression. (Another illustration of that is shown by taking the point awards for the earlier rounds. In the final, the WTA awards 117% of what the ATP awards. But in the Round of 32, the WTA awards 178% percent of what the ATP awards!)

It is frankly hard to believe that #1 Roger Federer is twice as good as #2 Novak Djokovic, and that Djokovic is twice as good as #4 Rafael Nadal, and that Nadal is twice as good as #8 Andy Roddick (meaning, e.g., that Federer is four times as good as Nadal, and eight times as good as Roddick). Indeed, it's hard to believe that Federer is 1.75 times as good as Djokovic, which is the curve parameter of .57 implies. If Federer were that good, he would never lose. So a base curve parameter of 0.5 is absurd. A parameter of 1.0 is also absurd -- Federer may not be eight times as good as Roddick, but he is surely somewhat better. The author's feeling is that the parameter should be about 0.7. The WTA is a little below that. The ATP is far below that. The effects of that will probably eventually be measured in the volatility of the rankings. Unfortunately, we can't test that yet -- 2009 was a transition year, so it's no test. We won't be able to do a true test until the end of 2010.



The Real #2
This week, Caroline Wozniacki passed Dinara Safina to become the #2-ranked female tennis player. This prompted two questions: "Does she deserve it?" and "Who else is there?"

The answer to the first question is surely, "It depends on what you mean." We thought, for purposes of comparison, that we'd take a small sample of year-end #2 players and see what they had at the time. We decided to take a four-year increment. So we took the year-end #2 for 2006, 2002, 1998, and 1994.

2006 #2: Maria Sharapova. Her accomplishments: U. S. Open win, Australian Open and Wimbledon semifinals; 20-3 Slam record; 5 titles (Indian Wells, San Diego, U. S. Open, Zurich, Linz).

2002 #2: Venus Williams. Her accomplishments: finals at Roland Garros, Wimbledon, U. S. Open; 22-4 Slam record; 7 titles (Gold Coast, Paris Indoor, Antwerp, Amelia Island, Stanford, San Diego, New Haven).

1998 #2: Martina Hingis. Her accomplishments: Australian Open win; semifinals at other three Slams; 22-3 Slam record; 5 titles (Australian Open, Indian Wells, Hamburg, Rome, year-end Championships) [also won the doubles Grand Slam].

1994 #2: Arantxa Sanchez-Vicario. Her accomplishments: Roland Garros and U. S. Open wins, final at Australian Open; 23-2 Slam record; 8 titles (Amelia Island, Barcelona, Hamburg, Roland Garros, Canadian Open, U. S. Open, Princess Cup, Oakland)

We didn't stack this; we just decided that 15 years and four #2s was as much as we wanted to research, and that's what came out. To that we compare Wozniacki's results in the year up to and including Indian Wells 2010:

Wozniacki's accomplishments: U. S. Open final, 14-4 Slam record, 3 titles (Ponte Vedra Beach, Eastbourne, New Haven)

Thus Wozniacki has fewer titles than any #2 player we checked, and worse Slam results. Historically, it's pretty clear that Wozniacki stands below the "typical" #2 players.

Of course, she might improve on that. In any case, we aren't looking for the all-time #2 player. We're looking for the #2 player right now. This is why we ask the second question, "Who else is there?" That is, is there anyone who has a better right to the #2 ranking?

For this, of course, we don't want to use the WTA rankings, which should be called the "Player Punishment System," not "the rankings." A ranking system should be designed to determine who is the best player -- and Wozniacki is not ranked #2 because she is the second-best player, she is #2 because she is a top player (we aren't denying that) who happens to be healthy enough to play all the time. When you consider that Serena Williams, Dinara Safina, and Maria Sharapova have all missed significant time to injury, and Kim Clijsters has only been un-retired for half a year, and Justine Henin for only three months (and already has missed an event due to injury), it's clear that Wozniacki's biggest asset in the WTA rankings is her stamina.

To put this in perspective: Wozniacki now has 24 events (actual events, not nominal WTA events). Of the other players in the Top Ten, Agnieszka Radwanska has 22. Jelena Jankovic has 21. Dementieva and Stosur have 20. The other five have 19 or fewer events. Thus, additive rankings being additive rankings, Wozniacki has a big advantage just from how much she has played.

So we decided to take fourteen players who are reasonable candidates for #2 (the WTA's top 11, plus Clijsters, Sharapova, and Henin), and rank them under some alternate systems. In each of these categories, we will list the top five of our players, and plus Wozniacki if she is not Top Five.

Winning Percentage

Just what it sounds like: Percentage of matches won.

1. Clijsters........81%
2. Williams, S......80%
3. Henin............79%
4. Sharapova........77%
5. Safina...........77%
7. Wozniacki........73%

Even allowing that three of our top five have played limited schedules, we note that Wozniacki is only #4 among players who have played a whole year.

Winning Percentage, .Minimum 16 Events

To deal with all those players who haven't played enough, we require 16 events -- and add losses until they have 16.

1. Safina...........74%
2. Dementieva.......74%
3. Wozniacki........73%
4. Williams, S......73%
5. Williams, V......72%

This will be the best result Wozniacki produces under any system we could come up with on short notice. But note that she still isn't #2.

Winning Percentage, Premier Events

Lest the percentages above be biased by playing at small events, we calculate wins and losses based on taking only the Premier events (plus Slams and Championships)

1. Williams, S......81%
2. Clijsters........79%
3. Henin............78%
4. Safina...........75%
5. Sharapova........74%
7. Wozniacki........72%

No surprise to see Serena atop this one! But, again, Wozniacki is only #4 even among players who have played a full year.

Points Per Tournament

Just what it sounds like: Total WTA points earned (including those not counted toward a player's Best 16) divided by total actual events played in the last year.

1. Williams, S......618
2. Henin............550
3. Clijsters........409
4. Safina...........367
5. Williams, V......352
7. Wozniacki........291

Wozniacki really loves that #7 spot -- although, in this case, she isn't even #4 among players who have a full schedule; she turned out to be #5.

Points Per Tournament, Minimum 16 events

Same as above, but with a minimum divisor of 16 events. That is, if a player has fewer than 16 events, we divide her point total by 16 anyway..

1. Williams, S......540
2. Safina...........367
3. Williams, V......352
4. Kuznetsova.......330
5. Wozniacki........291

Note that Wozniacki is actually worse in points per tournament rankings than in won/lost. That probably says something about the tiers of the events she is playing.

Quality Points per Tournament

Our favorite for estimating future results. Based, of course, on the quality points the WTA no longer awards.

1. Henin............119
2. Williams, S......101
3. Clijsters.........99
4. Kuznetsova........66
5. Williams, V.......64
6. Azarenka..........61
7. Sharapova.........53
8. Safina............51
9. Wozniacki.........48

Wozniacki's worst result yet. This would seem to imply that she will fall before she can rise.

Modernized Divisor

The descendent of the pre-1997 divisor rankings, taking into account point inflation and the 16 event minimum. We take round points, add double quality points, and divide by 16 or the number of events, whichever is greater.

1. Williams, S......819
2. Williams, V......480
3. Safina...........468
4. Kuznetsova.......461
5. Azarenka.........413
6. Wozniacki........387

Well, at least Wozniacki isn't #7 in this one....

Declining Divisor

This is, in the author's opinion, the best ranking system we can make based just on points and events. We take total WTA round points, add four times the quality points (doubling once because of point inflation, then doubling again because quality points are a better measure of tournament strength than WTA tier), and use a minimum divisor of 16; for players with more than 16 events, we subtract a third of an event for each event past #16..

1. Williams, S......893
2. Kuznetsova.......615
3. Williams, V......609
4. Safina...........582
5. Azarenka.........546
6. Wozniacki........544

And so, to our shock and amazement, it appear that Svetlana Kuznetsova "ought" to be #2. At least until Miami. Certainly it doesn't appear that Wozniacki is the second-best player out there; she was not #2 in any system.

At least there isn't much doubt about who is #1. In every statistic, Serena Williams led the players who had played a full year. Often by a very wide margin (note her modernized divisor score, which is almost twice that of the #2 player. And her declining divisor score is more than 40% above the #2.)

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