Yesterday I offered a full breakdown of the French Open draw, with each player’s chances of advancing to each of several rounds. In any draw of this kind, there are winners and losers, thanks to the luck of the … well, draw.

In a 128-player field seeded in the manner that Grand Slams are seeded, there are nearly 100,000 permutations of the draw. The vast majority don’t matter — for instance, if you swapped Thiemo de Bakker and John Isner so that de Bakker played Nadal in the first round and Isner played Djokovic in the first round (instead of vice versa), no one’s chances of winning the title would change much.

But, of course, many of the possible permutations would matter a whole lot. Just ask de Bakker or Isner! Imagine how much better it would be for Isner to have a first round draw against, say, Yen-Hsun Lu, followed by a probable second-rounder against Sergiy Stakhovsky. In fact, that’s Kei Nishikori’s draw, and in that sense, Nishikori was very lucky that the chips fell where they did.

**Stepping back**

In my previous draw simulations–like the one I published yesterday–I took the actual bracket as a given. To generate the probabilities you see in yesterday’s chart, I had a computer program “play” the tournament 100,000 times, each time pitting Isner against Nadal in the first round, then the winner of that match against the winner of Giraldo/Andujar, and so on.

There’s a different way we could approach this. Instead of starting the simulation from the point at which the draw is set, we could start from the point at which the *field* was set and seeded. At that point, Isner would know that he is not seeded–and thus, that he would probably face a seed in the first or second round–but not which higher-ranked player he would face.

So, instead of 100,000 simulations of the actual French Open bracket, we can do 100,000 simulations of the draw itself, followed by simulating each ensuing bracket. Sometimes, Isner draws Nadal, sometimes he draws Hanescu, and so on.

**Measuring draw implications**

A good way to gauge a player’s overall chances at a tournament is his *predicted prize money*. Most players don’t have a significant chance of winning most tournaments (especially slams), so to compare Giraldo’s 0.01% chance of winning the title with Cuevas’s 0.02% chance doesn’t tell us much. But if we consider the possibility that each player reaches each round, we can estimate that Giraldo will take home E24,600, while Cuevas will collect E29,500. These numbers represent an average of the first-round prize money, second-round prize money, and so on, weighted by the probability that the player will reach each of those stages.

With this metric, we can compare the implications of the actual draw with the implications of the randomized draw, in which, for instance, Nadal could play any one of the 96 unseeded players in the first round.

Let’s compare the two outcomes in an extreme case. As we’ve seen, the draw was not kind to John Isner. My algorithm gives him a 12% chance of reaching the second round, and less than a 1% chance of reaching the semis. Crunch the numbers, and you have predicted prize money of E22,700. When you randomize the draw and he no longer has to beat Nadal in the first round, his chances of reaching the second round leap to 60%, and he has a 2% shot at a semifinal berth. Predicted prize money: E40,100.

As it turns out, Isner is our biggest loser. His predicted prize money fell more than 40% between the beginning and end of the draw ceremony. What’s remarkable is that the next four players on the list all come from the same 1/16th of the draw–you guessed it, Djokovic’s section.

The draw effect on Thiemo de Bakker is similar to that on Isner–it doesn’t get any worse than drawing Djokovic in the first round. Next on the list are Ernests Gulbis, Ivo Karlovic, and Juan Martin del Potro. Karlovic and Gulbis not only have the misfortune of drawing Delpo in the first two rounds, but if by some chance they get past the Argentine, *then* they face Djokovic! Each of those players lost more than 30% of their predicted prize money through the vagaries of the draw.

Del Potro is an interesting case. As is, his predicted prize money is E184,600. Before the draw was set, he could expect E266,000. The biggest difference, of course, is his chance of reaching the round of 16. In real life, he’ll need to beat Djokovic to get there, and he has a 30% chance of getting that far. Before the bracket was drawn, the expectation was that he’d need only to defeat someone in the top 16 (or possibly, a player who had upset someone in the top 16). He had a 63% chance of doing so.

**Winners**

Naturally, if there are so many players whose predicted prize money decreased, some players must have benefited from the way the draw played out.

One of the biggest winners was Andy Murray. As we’ve seen, plenty of dangerous players are concentrated in Djokovic’s quarter; in fact, Djokovic, Nadal, and Federer were all hurt by the draw. But Murray’s draw boosted his predicted prize money from E191,700 to E240,800. He’ll face a qualifier or lucky loser in each of the first two rounds, then no one more challenging than Milos Raonic in the third. Next would be Dolgopolov or Troicki–no walkovers, but compare that to Nadal’s possible fourth-rounder of Verdasco, and you see how the breaks went in Andy’s favor.

Murray’s quarter is the softest of the four, and other men benefit even more. In fact, the two players whose chances the draw boosted the most are Nicholas Almagro and Jurgen Melzer, who will likely play in the fourth round for a matchup with Murray in the quarters. Almagro, Melzer, and Juan Ignacio Chela (also in this section) all saw their predicted prize money jump by more than 40%. For example, Almagro went from E76,100 to E112,400–and more than doubled his chances of winning the title from 0.6% to 1.3%–by landing where he did in the bracket.

Regardless of any player’s specific placement, the best man will probably win. But the draw certainly has a say in how tricky the route to the title will be.

This is a fantastic way to settle a tough question about who has the harder draw. Did you think about using ranking points instead of cash for your expected value? I was thinking if you use that, you could then start to plug that into the current rankings to estimate weekly potential ranking moves.

Hell, if you used ranking points and knew the field at each tourney for the rest of the season, you could simulate the remainder of the season each week and make year-end #1 race projections every week.

Exciting, if not totally nerdy, stuff! Great work.

Thanks!

I have used ranking points in the past — this older article looks at what happens if you don’t use seeding:

http://summerofjeff.wordpress.com/2011/02/10/quantifying-the-bias-of-an-atp-draw/

In terms of results, there isn’t much difference — the ratio of ranking points to prize money is fairly consistent. Yep, you could use ranking points and predict the rankings.

You are right, I could do full-season predictions; I’d just need to predict who played which tournaments. I think the results would end up pretty boring — the #1 player would keep rising, since the draw biases end up canceling themselves out, at least for most players.

think you might like my entries on tennis http://schroeds.wordpress.com/