Who Benefits From Byes?

Roughly two-thirds of ATP tour-level tournaments have byes in the draw.  31 events–including the two this week, in Kitzbuhel and Los Angeles–have 28-man fields, with first-round byes for the top four seeds.

The obvious beneficiaries are the top four seeds.  They get free passes into the second round, eliminating the chance they’ll be handed a first-round exit.  It’s also a guarantee of greater prize money and more ranking points.  First-round byes are such a feature of the ATP tour, at least in part, because they help smaller tournaments convince big-name players to sign up.

Of course, you can’t simply hand an advantage to the top four seeds without affecting others.  In this most common format, a 28-man field with eight seeds and four byes, there are three important groups: The top four seeds, the bottom four seeds, and the rest of the field.

The top four seeds: The main effect of byes on the top four seeds is that, as noted, they don’t have play first-round matches.  The extent of that effect depends on how much of a threat the first-rounder would’ve been.

To quantify these effects, I ran simulations for the 2012 Estoril tournament.  First, I simulated the draw as the tournament was played, with 28 players and top seeds of Juan Martin Del Potro, Richard Gasquet, Stanislas Wawrinka, and Albert Ramos.  Second, I added the next four players on the alternate list to the draw in place of the byes.  To eliminate any bias stemming from the specific arrangement of the draw, I re-generated the brackets for each simulation.

In the 32-man field, Delpo won his first round match about 90% of the time, Gasquet and Wawrinka about 80%, and Ramos just under 60%.  Accordingly, Delpo didn’t benefit too much from the bye, but Ramos gained enormously.

However, when measured by expected ranking points, none of these four men gained as much as skipping the first round would suggest.  For instance, if Delpo would win only 90% of his first-round matches, removing that impediment would be expected to raise his other outcomes by (1/0.9 – 1), or 11%.  In fact, in the 28-man simulation, he gained only 9.5% over his 32-man expectation.

The slight difference is due to the other top seeds.  If Delpo is more likely to reach, say, the semifinals, then the same effect applies to Gasquet and Wawrinka, the two men who would be most likely to knock him out of the draw.  So while the bye itself increases Delpo’s expected ranking points by 11%, the increased probability of facing the other top seeds reduces it a bit.

Still, the net effect on the top four seeds is overwhelmingly positive.  For Gasquet and Wawrinka, the bye itself increases their expectations by 27% each, for a net effect of 24%, while for Ramos, the bye is a 74% increase, resulting in a net effect of 70%.

The next four seeds: The men seeded five through eight are the losers.  They must play a first-round match–which, in the Estoril example, they each have about a 60% chance of winning–but they are more likely to face one of the top four seeds later on.

The average effect of adding byes to the draw is a 5% decrease in expected ranking points for these lower four seeds.  They aren’t guaranteed to reach the quarterfinal, but in the 28-man version, if they do reach the quarters, they are at least 10% more likely to face a higher-ranked opponent.

The rest of the pack: Nearly everyone else benefits.  The effect of byes touches unseeded players in two ways, which work in opposite directions.  First, and most significantly, no one has to play a top-four seed in the first round.  In Estoril, the toughest first round opponent was 5th-seed Denis Istomin, not exactly a fearsome name in the locker room.  Because of the byes, nearly every player has a 40% chance of reaching the second round.

The countervailing force is a minor one–not enough to neutralize the advantage of missing top seeds in the first round.  When the field shrinks from 32 to 28, the average opponent is a bit better.  If four additional players were added to the Estoril field, they wouldn’t be automatically placed in the positions of the byes.  They would be randomly placed in the draw like everyone else.  Having those four lower-ranked players would give some players even easier first-round matches.

But on balance, for unseeded players, the goal is simply to win a match or two.  The best way to increase their chances of doing so is to keep the best players out of their path for as long as possible.  Byes take care of that.  The net benefit to unseeded players is an addition of 1% to 3% of their expected ranking points.  Generally speaking, the worse the player, the bigger the benefit.

The one exception to this rule is if an unseeded player is actually better than some of the seeds.  According to jrank, Igor Andreev was a better player than 8th-seed Flavio Cipolla going into Estoril.  Thus, the logic that applies to the bottom seeds applies to him.  He was likely to advance to the quarterfinals, so the effect of the byes was mainly to give him a tougher quarterfinal opponent.  In each tournament, this might affect one or two players–in Estoril, Andreev was the only one.

One more consideration: As we’ve seen, 23 of the 28 players benefited from the byes.  And the five players who were negatively affected didn’t lose too much.  How is that possible?

There’s one more group we haven’t talked about: The four players who aren’t included in a 32-man draw.  They don’t have much of a chance of reaching the final rounds, but they wouldn’t be much worse than the rest of the unseeded pack.

One of the players I used for this example, Igor Sijsling, just missed the cut, but in a 32-man draw, he would have been expected to take home 23 ranking points and about $9,000.  By adding four byes, the tournament is essentially taking what it would have given to Sijsling and three other players and divvying it up among the remaining 28.  The pie is the same size, but fewer players can claim a slice.

In the end, those four “missing” players are the only real losers, and they always have the option to head to a challenger for the chance of picking just as many points, even if they probably don’t come with as many dollars.

The winners, beyond the top seeds and the tournament organizers, are ultimately the fans.  When top players have more reason to play small tournaments, we get to watch more high-profile matchups, and ATP 250s look a bit less like Kitzbuhel and a bit more like Doha.