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 | Year | 1 | 2 | 3 | 4 | Draw |
| W | 2007 | Federer | Nadal | Roddick | Djokovic | 1-3 |
| USO | 2007 | Federer | Nadal | Djokovic | Davydenko | 1-4 |
| AO | 2008 | Federer | Nadal | Djokovic | Davydenko | 1-3 |
| RG | 2008 | Federer | Nadal | Djokovic | Davydenko | 1-4 |
| W | 2008 | Federer | Nadal | Djokovic | Davydenko | 1-3 |
| USO | 2008 | Nadal | Federer | Djokovic | Ferrer | 1-4 |
| AO | 2009 | Nadal | Federer | Djokovic | Murray | 1-4 |
| RG | 2009 | Nadal | Federer | Murray | Djokovic | 1-3 |
| W | 2009 | Nadal | Federer | Murray | Djokovic | 1-3 |
| USO | 2009 | Federer | Murray | Nadal | Djokovic | 1-4 |
| AO | 2010 | Federer | Nadal | Djokovic | Del Potro | 1-3 |
| RG | 2010 | Federer | Nadal | Djokovic | Murray | 1-4 |
| W | 2010 | Federer | Nadal | Djokovic | Murray | 1-3 |
| USO | 2010 | Nadal | Federer | Djokovic | Murray | 1-4 |
| AO | 2011 | Nadal | Federer | Djokovic | Soderling | 1-4 |
| RG | 2011 | Nadal | Djokovic | Federer | Murray | 1-4 |
| W | 2011 | Nadal | Djokovic | Federer | Murray | 1-4 |
| USO | 2011 | Djokovic | Nadal | Federer | Murray | 1-3 |
| AO | 2012 | Djokovic | Nadal | Federer | Murray | 1-4 |
| RG | 2012 | Djokovic | Nadal | Federer | Murray | 1-3 |
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
ReplyDeleteWe 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.
ReplyDeleteFed 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.