Saturday 26 May 2012

Luck or Conspiracy: Federer and Djokovic in Grand Slam Draws

A new Grand Slam is upon us, and the French Open draw was held yesterday. Despite the promised live showing of the draw, we were left to guess what was going on, while the tournament's official twitter account suggested helpful things like changing our Facebook language to French. Technical problems aside, however, we have the new men's and women's draws, and with them come the surprises.

Or do they? Once again this year, Victoria Azarenka and Agnieszka Radwanska are drawn into the same half. Once again in a Grand Slam tournament, Novak Djokovic and Roger Federer end up in the same half. The latter result had people crying 'Conspiracy' and 'Draw fixing!' long ago, claiming that the odds for that happening are near impossible.

I looked at the last 20 Slams, starting with Wimbledon 2007, which is when Djokovic started getting a top-4 seed. Since then, Federer and Djokovic were on the same side of the draw 15 times. If you assume that they have a 50% chance of falling in the same half, the possibility of that event occurring 15 times out of 20 is about 1.5%. A lot of the calculations I've seen before on the topic give even lower numbers than this one. However, in my opinion, this is the wrong way of looking at the numbers.

The main thing to remember is that the Slams don't draw specific players to face each other. Rather, the players are drawn out by their seeding in the tournament. And this is where the conspiracy theories start to fail.

The 1st and 2nd seeds are always placed on different sides of the draw. Following that, one of the next two seeds - #3 or #4 - is drawn into the #1's half. So, in every draw, we can get either #1-#3 (and #2-#4) or the other way around - #1-#4 and #2-#3. Those are the events that really have a 50% chance of happening, not any combination of specific top-4 players. And indeed, that's what you get when you analyze the last 6 years' draws. In the last 20 Grand Slam tournaments, there were 9 tournaments with the #1-#3 combination, and 11 tournaments with the #1-#4 combination. On the women's side, by the way, there were 11 tournaments with #1-#3 and 9 tournaments with #1-#4. Overall, those results make perfect sense for an event with a probability of 50%. In fact, this sample is too small to give it any statistical meaning, and it would be better to look father back for more exact numbers - but then, of course, you don't have Djokovic and Federer to conspire about.

The "problem" for Djokovic and Federer is that they never (not even once!) had a Grand Slam tournament where they were certain to avoid each other. Nadal has been inside the top2 for 19 out of the 20 Slams I looked at, thus ensuring that Federer and Djokovic couldn't be both #1-#2 or #3-#4 at the same time. The one Slam where Nadal was seeded 3rd (USO 2009) had Murray ranked 2nd, thus, again, ensuring that Federer and Djokovic could be drawn together. So, at the end of the day, if you're looking for someone to blame for this luck of the draw, you might as well blame Rafael Nadal.


Tournament Year1 2 3 4 Draw
W 2007 Federer Nadal Roddick Djokovic 1-3
USO 2007 Federer Nadal Djokovic Davydenko 1-4
AO2008 Federer Nadal Djokovic Davydenko 1-3
RG2008 Federer Nadal Djokovic Davydenko 1-4
W2008 Federer Nadal Djokovic Davydenko 1-3
USO 2008 Nadal Federer Djokovic Ferrer 1-4
AO2009 Nadal Federer Djokovic Murray 1-4
RG2009 Nadal Federer Murray Djokovic 1-3
W2009 Nadal Federer Murray Djokovic 1-3
USO 2009 Federer Murray Nadal Djokovic 1-4
AO2010 Federer Nadal Djokovic Del Potro 1-3
RG2010 Federer Nadal Djokovic Murray 1-4
W2010 Federer Nadal Djokovic Murray 1-3
USO 2010 Nadal Federer Djokovic Murray 1-4
AO2011 Nadal Federer Djokovic Soderling 1-4
RG2011 Nadal Djokovic Federer Murray 1-4
W2011 Nadal Djokovic Federer Murray 1-4
USO 2011 Djokovic Nadal Federer Murray 1-3
AO2012 Djokovic Nadal Federer Murray 1-4
RG 2012 Djokovic Nadal Federer Murray 1-3

2 comments:

  1. one more thing, consider that something like 50 players played the last 20 GS (didn't check and have no intension to) so you can have some 2500 combinations and it is expected (at 1.5%) to have 37 combinations of two players that were 15 times at the same half. it is only much more visible when its top players

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  2. We aren't trying to see whether 1-3, 1-4 etc combinations followed the rule of probability but whether Fed and Djoko being in the same half so often isn't a sign of fixing. To make it simple, let us look at expected values and assume there are 100 cases.
    Fed was seeded no. 1, p times and no. 2, q times i.e. p+q=100
    When Fed was seeded no. 1, Djoko was seeded no. 3, r times and no. 4, p-r times.
    When Fed was seeded no. 2, Djoko was seeded no. 3, s times and no. 4, q-s times.
    The expected no of times Fed and Djoko would be in the same half would be 1/2 *r +1/2*(p-r) +1/2*s + 1/2*(q-s)=1/2*(p+q)=1/2*100=50.
    In other words no matter what the distribution of no. 1 and 2 seeding for Fed and no. 3 and 4 for Djokovic, the probability of their being in the same half is 1/2.
    We could extend similar reasoning to cover cases when Fed is 3 or 4 and Djoko is 1 or 2.
    So their being in the same half 15 times out of 20 is a pretty rare event and could be a sign of fixing.

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