The formula is less complex than it seems, and is given below:
Tier 1 and 2 Main Draw: 1.4 * wins/losses Tier 3/4/5 Main Draw: 1.3 * wins/losses Tier 1/2 Qualifying: 1.2 * wins/losses Tier 3/4/5 Qualifying: 1.1 * wins/losses $75k/$50K Challenger Main Draw: 1.1 * wins/losses $25K Challenger Main Draws: 1.0 * wins/losses $75K/$50K Challenger Qualifying: 0.9 * wins/losses $25K Challenger Qualifying: 0.8 * wins/losses $10K Challenger Main Draw: 0.7 * wins/lossesAdd in any penalty losses (see below).
Penalty losses are calculated in the following manner:
There is one exceptions to the rule on penalty losses: Players who fail to win a single match aren't given penalty losses. They're punished enough for failing to win any matches.
I should point out here that I'm not really the one who determined the
formula. I've basically cribbed a formula devised by mathematics professor
Dr. Arpad Elo several decades ago for rating the relative strengths of chess
players. Elo's system works well enough for chess that it's the standard
for chess ratings around the world. The system has also been used for
official rankings of Scrabble players. Doing a search on "Arpad Elo" on any
search engine should provide you far more information than I can.
Note: In chess and other pursuits, Elo's system is used
not just to determine the original ratings (often called "performance
ratings"), but also to predict future results -- that is, given a round-robin
tournament with a certain strength of opposition, you can determine how many
matches a particular player is supposed to win. Due to the different surfaces
and the single-elimination nature of tennis tournaments, the ratings can't
really be used for predictive purposes. Only the "performance rating" aspect
is being used.
If you're wondering how I came up with the original ratings, I started by entering all of the WTA Tour results from 1999 into a spreadsheet, and looking at three things: a player's winning percentage, the mean of her opponents' winning percentages, and the mean of the means of their opponents' winning percentages. (Basically, it's an approximation of the Elo formula I've described above; for those of you who follow American college basketball, this is also fairly similar to RPI.) These ratings were then used for all matches played in 1999. In the first week of 2000, the results from the first week of 1999 were discarded, replaced with the 2000 results, and new ratings were calculated using the formula described above.
One could set up the system such that the amount of points added to the average of your opponents' ratings was done on a linear scale. For example, we could say that a 50% record would equal to the average of your opponents' ratings, 60% were the average +100 points, 70% were the average +200 points, and so on. This would get rid of the problem of having infinity subtracted from the average for players who lost all their averages. However, I believe that a linear system is less fair. Let me give an example to show how I think a logarithmic scale treats players more accurately:
Suppose you have two players. Player A plays 14 tournaments, wins two,
and amasses a 48-12 record, or 80%. Player B plays 24 tournaments,
wins none, and amasses a 36-24 record or 60%. Then, each decides
to play an extra tournament. Let's consider two separate cases for
this extra tournament:
Result A's A's Log. A's Linear B's B's Log. B's Linear
Record Adjust Adjustment Record Adjust Adjustment
---------------------------------------------------------------------------
No extra 48-12 +230.5 +300.0 36-24 +67.9 +100.0
tournament
First-round 48-13 +217.5 +286.9 36-25 +61.0 + 90.2
loss
Wins tournament 53-12 +247.0 +315.4 41-24 +89.7 +130.8
Under a linear system, the better player is punished somehwat more for losing in the first round, but the punishment is much greater under a logarithmic scale. Likewise, under a logarithmic scale, the better player is helped slightly less by winning an extra tournament, but the lack of help compared to a lesser player is much less than under a linear system. In both cases, I believe that the logarithmic scale is fairer.
I can't stress this enough: in my system, unlike the WTA's current system, players are not defending points! They're defending one of two things: wins, or winning percentage.
For two examples: Martina Hingis started off the 2001 season with a 78-10 record. She played the Sydney warm-up tournament, where in 2000 she won two matches, losing to Amelie Mauresmo in the semifinals. In order to maintain her rating she would have had to reach the semifinals (and play the same caliber of opposition, of course). As it turned out, Hingis won the tournament outright, raising her record to 80-9 and increasing her rating by almost 30 points.
Monica Seles, on the other hand, was injured at the start of the 2000 season, and could not play until Oklahoma City at the end of February. She entered the 2001 Australian Open with a 52-13 record (and no penalty losses because she had three titles, making 16 events in all). In order to maintain her winning record, she would have had to win four matches, or in other words, reach the quarterfinal. She did indeed reach the quarterfinals, losing to Jennifer Capriati. However, the players she faced at the Australian Open were on average weaker than she had faced the rest of the year (not surprising, considering that most tournaments only have 32 players while there are 128 at the Slams). So her rating actually went down and she was passed by Serena Williams, who improved on her 2000 result (and unlike 2000, was actually beaten by a much higher-rated player -- Hingis in 2001 as opposed to Likhovtseva in 2000).
This is not the easiest question to answer. However, there are a few generalities: an extra win is worth the most to the players with the lowest winning percentages, and becomes less and less of an improvement as a player's winning percentage increases; an extra loss obviously hurts the player with the higher winning percentage much more.
Past experience can tell us something about the meaning of the size of a gap, however. At the start of 2000, there were four active players (Hingis, Davenport, and the Williams sisters) where were miles ahead of the rest of the players -- #4 Serena Williams was some 110 points ahead of Monica Seles. It took Serena playing poorly (losing her opening match at Amelia Island), along with Monica Seles winning two titles in the first four months of the year (Oklahoma City and Amelia Island), nearly four months to make up the gap. After the 2001 Australian Open, there was a gap of about 120 points between #5 Monica Seles and a group of players between #6 and #10. So, even if Seles were to hit a run of poor form and one of the lower players were to start playing extremely well, Seles probably wouldn't fall out of the Top Five until around the time of the French Open.
This brings up one of the big differences between my ranking system (and other alternative rankings) and the WTA system: Under the WTA's rankings, only a players' 17 best results count. A player could play 30 events, lose in the first round of 13 of those, and not have those 13 results count at all. Under my system, losses count. So, players who don't play as often, but are consistently very good, will in general be ranked rather higher than they are by the WTA.
The other reason for players being over-ranked under my ratings is by doing
very well in qualifying. The WTA rankings are used to determine which players
get automatic entry into WTA tournaments and which players have to go through
the qualifying rounds. With the WTA's rankings, a player could make it
through qualifying in several events and still not be ranked high enough to
receive direct entry into the main draw. Under my system, such a player would
have amassed a high enough winning percentage so as to have a rather high
ranking. If a system like mine were actually used to determine direct entry,
what would more likely happen is that a player would make it through
qualifying a few times, see her ranking rise enough to start getting direct
entry, and then start losing in the first round of the main draw as she faces
only higher-ranked players. After a few first-round losses, her ranking would
likely fall far enough to send her back to qualifying.
Note: This of course assumes that the player has reached
a stable level of performance. There are young players who are improving and
who would continue to win even when they have to stop going through
qualifying. Those players are over-ranked by my system, but usually
underranked by the WTA because of the WTA's age restrictions.
If you have any questions, please don't hesitate to e-mail me!