Draw Fixing: An Official Study

First topic message reminder :

The link below shows a talk by an Estionian researcher Katarina Pijetlovic, giving an exposition on draw fixing at a Corruption in Sport Symposium in Koln.


Katarina's talk starts at around 13 minutes:

http://www.livestream.com/playthegame_dshs/video?clipId=pla_44809e94-aa04-46c7-9f1e-35b212ba9d46


She examines the pattern of draws at slam tournaments (French Open was not part of the study) between 2007-2011, drawing the conclusion that ITF organised draw fixing on behalf of Nike seeing that Djokovic fell in Federer's half of the draw statistically virtually impossible 12 out of 12 times.

Roland Garros was not taken into the study as it showed a healthy 50/50 pattern.



Interesting facts, e.g. I didn't know that seeds 3 and 4 are drawn by hand unlike all the other seeds/players that are computer drawn.
Draws are apparently public and televised, but not really accessible anywhere on Youtube.

To me, the most blatant example of draw fixing was the Isner Mahut match played in the first round last year ON COURT 18, just like at the record breaking match the year before!!!


Katarina did the research hoping it would interest sports journalists and encourage them to contact the players and ITF.


So far nothing came out of it.

Have a look with an open mind and share your thoughts.

Hm, in 2011 (after the study and the period which extended to 2010) #1 seed Djokovic drew Connor Niland a qualifier who never broke the top 100 in his life! And Nadal drew Golubev who was ranked barely inside the top 100.

#1 Wozniacki drew Nuria Llagostera Vives who was ranked 88 in July 2011, a month before the tournament, according to her wikipedia page. (Don't know how to look up rankings for a certain date on WTA page, may not be as easy as ATP.)

#2 Zvonereva drew Foretz Gacon, a qualifier, so would have been ranked fairly low.

The pattern continues! Even in 2011 after the draw was called into question! You would think if they were fixing it they might have fixed it this to have #1 and #2 draw #40 and #50, but apparently not!

laverfan wrote:

Henman Bill wrote:I think that whether there is draw fixing or whether this is an unlucky coincidence the US Open should now change to a new draw provider. The draw provider should be chosen by the ATP+WTA or another body without the US Open's input, and never communicate directly with the US Open, providing draw results directly to ATP authorities. If they just do that they can eliminate doubt for the future.

Did you mean ITF?

No, I did not mean the ITF but the ITF as well would be fine. In fact it might make more sense as this is an issue affecting both men's and women's draws and the ITF I believe could represent both.

Chydremion wrote:Really some people are so naive. Will use every argument they can find to say that nothing is wrong. Ok even if the study is not 100% correct, you never asked any questions when slam after slam Djokovic ended in Federer's half?
The study is incorrect because some material was left out (actually not some random and unfitting material, but the french open in its whole, so it gives a good view of the non-clay slams, and you should realise on clay it didnt matter if Djokovic ended in Nadal's half) but of course big business motives to fix the draw can't possibly be true because the world has never seen corruption before...
Long live fairy tales.

This is so true. They are looking for a leaf but cannot see the forest around them...and other prefer disappear than admitting they are wrong again.

I find it amusing that suddenly the study is split in 2. The 1 in 1/4096 argument becomes a joke and the 1 in 330k a more serious study ...not realising that together they become much stronger. The famous 1 in a billion!!!!! but no...just down to luck!

Henman Bill wrote:Hm, in 2011 (after the study and the period which extended to 2010) #1 seed Djokovic drew Connor Niland a qualifier who never broke the top 100 in his life! And Nadal drew Golubev who was ranked barely inside the top 100.

#1 Wozniacki drew Nuria Llagostera Vives who was ranked 88 in July 2011, a month before the tournament, according to her wikipedia page. (Don't know how to look up rankings for a certain date on WTA page, may not be as easy as ATP.)

#2 Zvonereva drew Foretz Gacon, a qualifier, so would have been ranked fairly low.

The pattern continues! Even in 2011 after the draw was called into question! You would think if they were fixing it they might have fixed it this to have #1 and #2 draw #40 and #50, but apparently not!

Why woudl they? Even with all the proof in the world, some don't want to see it! I woudl do the same if I were them. I woudl not want my top players getting exausted for the later stages.

Where did you get the 330k number? Is that part of the ESPN study? (Provide a link.)

Henman Bill wrote:The issues I do have with this are:

1 - Motive: While I do acccept that they wanted to keep the top 2 seeds in for as long as possible to boost attendance, TV revenue and media interest, and this IS motive for a fix, is it ENOUGH of a motive? My argument would be that seeds #1 and #2 ought to be able to despatch the say #40 and #50 ranked players with ease anyway most of the time, so yes an alleged fix would boost the chance of the top 2 players staying in the tournament, but only a little. So why risk your reputation for such minimal gain.

2 - It's obvious - why make it so obvious in ranking terms. If you are going to fix players #1 and #2 an easy draw why not be cuter about it..e.g. you could give them a #40 or #50 ranked player who is rubbish off clay, or is carrying an injury according to little known locker room talk, or is a player on the way down with a 0-10 record against his opponent. Then you could fix the draw and prevent it being statistically detetectable. (Mind you Costa, a clay courter, was the only player with a ranking under 60!)

These are not knockout blows though, maybe minor issues. In fact a simple explanation could be that the fixers are dumb.

I'll try and answer 2. I'd imagine IF there was any fixing going on it wouldn't be assigning #1 and #2 seed a specific player in R1, rather instead of choosing one of the remaining 96 players uniformly they would be weighting this draw towards those with higher ranking. Since it's computer generated it's relatively easy to construct a random variable on the set {33,...,128} of non-seeded players which is biased towards the higher numbers. For instance, the simple following algorithm.

Pick one uniformly at random. If the seed is higher than 80, keep the result. If not, reject and pick another uniformly at random. Keep the result of this second draw.

It's easy to see that in this case you'll pick a player ranked above 80 with probability 75%. The average seed of the player picked is 92 (=1/2(80+104)) give or take 1 or so. Judging by the fact that men and women's draws give an average rank of 97-98 or so, I expect that IF this is what the US Open are doing, it won't be this algorithm, but it could be something not much more complicated.

I'm off to bed in a short while. Would be interested to see Summer Blues and Barrystar and other cool and unbiased contributors on here giving their thoughts on this US Open thing.

Henman Bill wrote:Where did you get the 330k number? Is that part of the ESPN study? (Provide a link.)

yes it is. The link is up there in an earlier post. Not sure how they work out the 1/3000 and 3/3000 but that leads to 1/333k or so...and that is what the guy's saying in the link I posted earlier this afternoon.

HB, the 330k in fact is not a "chance" as such, rather the empirical evidence based on their simulations (1 in 1000 for women, 3 in 1000 for men). It's a bit wrong to "add" (should really be multiply ) probabilities with empirical evidence given by simulations.

Also, 1/3000 and 3/3000? It says 0/1000 for the women's and 3/1000 for the men's? I think you made a mistake there.

Of course, 0/1000 can't do the maths but we could use 1 in 1000 as an upper limit.

If you multiple one in 1,000 by 3 in 1000 you get odds of 1 in >333k which I guess is what you mean.

The greater than symbol being necessary because I've had to remove the zero with a 1.

Still, that's a sample size of only 3 "positive" results. Even 1 randomly different positive occurence would shift the odds significantly. So cannot be used to calculate the odds accurately if run from a simulation.

However it does say that stats prof ran "up to 1 million" repititions and got "nearly identical" results.

The only question is whether is really appropriate to multiply the 3/1000 and 1/1000. I think it is...can anyone else give an opinion?

By the way, I am not asking if the combined probability of two events occuring can be determined by multipling the individual probabilities of each since the answer to that is obviously yes. What I mean is whether it's appropriate to consider the men's and women's data as 2 separate "events" or whetther it should be somehow pooled into one data set and the results generated as if they were the men's results over a 20 year period. I imagine if they were the results might be the same as the multiplying anyway, just checking for a confirmation as I'm not 100% sure. Brain going a bit sleepy to be honest.

I'd have thought so HB, you can multiply independent events, and I think it's fair to assume men and women's draws are independent (certainly you should, as working out probabilities of non-independent events very quickly gets very sticky indeed).

My main gripe is that "chance" is the wrong word to use for empirical evidence as for me chance=probability. Like I said, I do reckon it should be reasonably easy (or at least doable) to find an upper bound for the odds of getting these results for random draws. Cramer's theorem should do the trick (this is typically a large deviation from the mean). I'll have a look into it, but not tonight (too tired).

EDIT: just seen what you've said, and TBH, still think multiplying has to be the way to go. Indeed if the draw is random then it's perfectly fine to consider men and women's draws seperately.

Mc4 - what puzzles me is that you seem to have the basic stat knowledge. I was pretty good at it in my youth but have forgotten a few formulae since. However, I can see straight away what's likely and what's not and work out the odds.

Very quickly we can see that 2 sets of really odd ones (or 3 if we consider men and women as different). Thta leaves close to zero chance of being luck.

I don't understand your suspicion being on the straight forward study and facts as opposed to the slams organisers rigging the draws.

TA, NITB, Chydremion and others saw it pretty quickly it coudl only be rigged.

Ah why is that you guys in UK seem to have the best tennis forums even though you have no tennis players to speak of ? By the time I get to read/post you are all fast asleep.

Tennisanorak wrote:Summerblues has got the gist of this thread, and I enjoyed his summary. Thanks for reading through this, man, and I'm happy it pleased the enrd in you.

Well thank you too. I will return the favor and say that I have found your explanations and reasoning as clear as anyone's here (will not say better than anyone because there are some other posters whose arguments I also really enjoyed reading and would not want to give the false impression to the contrary).

Tenez wrote:Sure Murray has won Nadal twice but quite unexpected. Nadal being very tired in that summer USO and then the knee.

Oh dear Tenez LOL,

When I was writing yesterday that it was not clear a few years back whether Djoker would be a tougher opponent for Nadal than Murray and when I supported it with Nadal's two losses to Murray in the slams, a thought occured to me "would it not be funny if Tenez now started to argue that Nadal was injured at the AO?". Did not think you would go that far but in fact you beat it - you now discounted both the US Open and AO losses. It is not that often that one hears you give excuses for Rafa's losses.

Don't get me wrong, I like many of your posts and think that you often make very good points, and would say it is often more your somewhat abrasive style than anything else that leads some members to discount your arguments. However, there are cases where I just cannot help but smile and this is one of those.

Now, with all that said, so perhaps a bit too late, I will admit that I do not know whether or not you have in the past espoused the view that Nadal's slam losses to Murray have been due to injury/fatigue. I was just assuming it would be unlikely. If you have done that consistently, then please accept my apologies.

Conspiracy alert... ITF has conspired since the dawn of Open Era to keep the two top seeded Mens and Womens players apart by rigging the draws so that #1 appears on Line #1 and #2 appears on line #128 of a 128 player draw.

I have definitive proof for this. I will send ESPN and ITF my proof with copies to ATP and WTA and ask for an explanation.



PS: Tenez, NiTB, Chydremion, TA and 'others' have all concurred. So it must be true. So say we 'all'.

summerblues wrote:

Tenez wrote:Sure Murray has won Nadal twice but quite unexpected. Nadal being very tired in that summer USO and then the knee.

Oh dear Tenez LOL,

When I was writing yesterday that it was not clear a few years back whether Djoker would be a tougher opponent for Nadal than Murray and when I supported it with Nadal's two losses to Murray in the slams, a thought occured to me "would it not be funny if Tenez now started to argue that Nadal was injured at the AO?". Did not think you would go that far but in fact you beat it - you now discounted both the US Open and AO losses. It is not that often that one hears you give excuses for Rafa's losses.

Don't get me wrong, I like many of your posts and think that you often make very good points, and would say it is often more your somewhat abrasive style than anything else that leads some members to discount your arguments. However, there are cases where I just cannot help but smile and this is one of those.

Now, with all that said, so perhaps a bit too late, I will admit that I do not know whether or not you have in the past espoused the view that Nadal's slam losses to Murray have been due to injury/fatigue. I was just assuming it would be unlikely. If you have done that consistently, then please accept my apologies.

Every player that loses a match must be tired, otherwise there is no reason to lose. Federer, Nadal, Melzer, Murray, Ferrer (must be tired), Seppi, Tomic, Cilic, Ranoic, Lopez, Berlocq, Ljubicic (he must be really tired).

There is no such concept as the better player winning the match. Nadal must have been really tired in the AO 2012 final after playing four sets of Tennis with Federer on Australia Day and oh the fireworks, too.

The company IDS that supplies software to USO, also supplies the same software to W, RG and AO (mentioned in the ESPN article).

IDS scoring and statistics systems and display boards and systems are on-site at such prestigious events as Wimbledon, the U.S. Open, Roland Garros, the Australian Open, the Olympic Games in London, The Open Championship, the Masters, the PGA Tour and the Dew Tour. IDS serves sports organizers, governing bodies and sponsors such as NBA, WTA Tour, WNBA, PGA Tour, UFC, USTA, USGA, CrossFIT, NHL, IBM and Sony Ericsson.

http://www.reuters.com/article/2012/02/15/idUS245787+15-Feb-2012+BW20120215

So why were FO draws excluded from the 12vs12 study? The ESPN study just focusses on USO.

I have posted the 2001-2012 R1 opponents of the Top 2.

Henman Bill wrote:By the way, I am not asking if the combined probability of two events occuring can be determined by multipling the individual probabilities of each since the answer to that is obviously yes. What I mean is whether it's appropriate to consider the men's and women's data as 2 separate "events" or whetther it should be somehow pooled into one data set and the results generated as if they were the men's results over a 20 year period. I imagine if they were the results might be the same as the multiplying anyway, just checking for a confirmation as I'm not 100% sure. Brain going a bit sleepy to be honest.

Good question. I was almost going to say that yes, you can simply multiply, but it actually is not the case. Let me illustrate on a slightly changed example:

Consider:
Case 1: Probability that the average score (using ESPN's "scoring") in both men's and ladies' draws is less than 0.3
Case 2: Probability that the average combined score is less than 0.3

The probability in Case 1 is just the product of the individual probabilities. However, one can see that Case 2 has to have a different probability (higher in my example):

Clearly if both averages are below 0.3, so is the combined average. However, it does not hold the other way around. There are ways in which the combined average can be below 0.3 even if one of the averages is above 0.3. Say, if the average on men's side is 0.32 and the average on the ladies' side is 0.26, the combined average will be 0.29.

As far as the appropriateness is concerned, looking at both Case 1 and Case 2 is equally appropriate, as long as one understands what probability one is calculating.

ESPN study is looking at something much like Case 1. More precisely, they are looking at the probability that men's average score would be below 0.326 and ladies' average would be below 0.313.

As M4C says, it is actually not that easy to calculate the exact probability of what ESPN was looking at. So I will try to estimate it. This will be quite technical, but I do want to give enough detail so that those of you who know and remember the formulas can have a chance to double check my numbers:

1. I will use "Central Limit Theorem" from probability. This theorem roughly says that if you take an average of a series of independently chosen numbers, this average behaves much like a "normally distributed" random variable. So, in my case I will be looking at the average of 20 numbers from the men's draw (and then the same for the ladies' draw). The 20 numbers are not quite independently chosen, but they are very close to it. Also, they are chosen from a distribution that looks much like a uniform distribution over the interval [ 0, 1 ], and for such distribution the central limit theorem works quite well.

2. I am assuming that the numbers that ESPN assigned to the players were 1 / [ 2 x 96 ], 3 / [ 2 x 96 ], 5 / [ 2 x 96 ], ... ,191 / [ 2 x 96 ]. This is consistent with their statement that the lowest ranked player would be assigned 0.005 and the highest one 0.095 (up to rounding) and it seems like the most likely way to do it. Even if they did something else, they would likely be very close to this pattern

3. The random numbers pulled from among the numbers above have mean 0.5 and standard deviation 0.28866.

4. The average of 20 such numbers has mean 0.5 and standard deviation 0.064546.

5. This average has approximately "normal distribution".

6. The men's average that the ESPN got was 0.326. This is 0.174 below the mean of 0.5

7. So the men's average is 0.174 / 0.064546 = 2.696 standard deviations below the mean.

8. Looking up in the values of the normal distribution, the probability of getting a number this small or smaller is about 1 in 285.

9. Similar calculation for the ladies' average of 0.313 gives probability of about 1 in 531.

10. The combined probability that both the men's average will be below 0.326 and ladies' average will be below 0.313 is the product of the two probabilities, i.e., about 1 in 150,000

11. This number is roughly consistent with what ESPN found. Note that both my and their numbers are approximations only so there is room for deviation from the true value but it seems that we do talk about 1 in 100,000 or more.

Ok, so I am in rough agreement with ESPN's numbers. What do I think of it? Quite frankly, I am not really sure. On one hand, the probability is indeed very low. On the other hand, we do need to remember that, even though they did their calculations properly, there is a bias in selection of the statisitcs.

If someone looked at the draw from a different angle and it did not show anything strange, that would obviously not get reported. And if different people look at the draw from say 1,000 different (and largely independent) angles than the probability that one of them will find something that has a 1 in 200,000 chance is about 1 in 200 - i.e., not nearly as bad. There is no way to mathematically quantify impact of this since we do not know how many people were digging into these draws from how many angles before one of them found this.

My personal hunch is that maybe 1 in 200 or so is not a bad estimate for how likely it is that someone would find something like this even if the draws are fair - but of course I have no way to really verify that number (though I would point out that the number is clearly going to be significantly less than 1 in 200,000).

With that, I am probably roughly where I was last night. That is, I do find this intriguing enough to keep it in mind but not quite sufficient to convince me that something nefarious is certainly afoot.

HB: if we combined the data, the combined average is 0.3195. If we then ask what is the probability that we get this average or a lower one, then, if my math is right, that probability would be about one in 26,000. So, as I mentioned above, the number depends on exactly how you ask the question.

Really interesting posts - as I am functionally innumerate, I don't have anything hugely intelligent to add My arguments are therefore based on gut feeling and not on mathematical reasoning at all

I would just like to play Devil's advocate and suggest that if we assume, for the sake of argument, that there is draw fixing, perhaps the purpose might have been to ensure two of the most competitive semi finals - just as lucrative for Wimbledon and more so for the other slams because of two sets of ticket sales. From 2007, the match up between Novak and Fed has generally been enthralling and very competitive and Murray had the distinction of knocking Nadal out at SF and QF stage of two slams. In other words, perhaps the idea of a semi final between TMF and Nole was as mouth watering a prospect as a final between Nadal and Fed??

I have to say I don't buy the theory because actually what tennis is in danger of suffering from is finals lacking the element of surprise and that does have a negative effect on the game in the long run, which I am sure the ITF is well aware of.

summerblues wrote:Now, with all that said, so perhaps a bit too late, I will admit that I do not know whether or not you have in the past espoused the view that Nadal's slam losses to Murray have been due to injury/fatigue. I was just assuming it would be unlikely. If you have done that consistently, then please accept my apologies.

I have put my argument about why Djoko was more of a threat than Murray in a much clearer way than the excuse for those 2 matches.

Yes I have given Nadal's excuses in the past and so did I in that USO 08. However I don't think the knee was an excuse as such in AO10 though none of us really know whether it hurt him from set 1 or on set 3. As I said, it's not what I think, it's about what Nadal thinks and there is little to suggest that Nadal feared Murray more than Djoko. I have made a great case why beginning of 2008 Djoko was by far the promising youngster....and certainly the threat for Nadal.

You apologies are accepted!

summerblues wrote:Ok, so I am in rough agreement with ESPN's numbers. What do I think of it? Quite frankly, I am not really sure. On one hand, the probability is indeed very low. On the other hand, we do need to remember that, even though they did their calculations properly, there is a bias in selection of the statisitcs.

If someone looked at the draw from a different angle and it did not show anything strange, that would obviously not get reported. And if different people look at the draw from say 1,000 different (and largely independent) angles than the probability that one of them will find something that has a 1 in 200,000 chance is about 1 in 200 - i.e., not nearly as bad. There is no way to mathematically quantify impact of this since we do not know how many people were digging into these draws from how many angles before one of them found this.

My personal hunch is that maybe 1 in 200 or so is not a bad estimate for how likely it is that someone would find something like this even if the draws are fair - but of course I have no way to really verify that number (though I would point out that the number is clearly going to be significantly less than 1 in 200,000).

With that, I am probably roughly where I was last night. That is, I do find this intriguing enough to keep it in mind but not quite sufficient to convince me that something nefarious is certainly afoot.

Do you really grasp the ESPN study? "Intriguing" shoudl not be the word. It's a clear proof of rigging...if not rigging a software coding malfunction (cough cough). There is no way around that. You elaborate as much as you can to sit in the middle of the fence. The truth on the other hand takes side.

We will see at the nest USO that suddenly this "bug" is fixed....even if that requires rigging the draw to "force" top seed players to have highly ranked opponents.

I think even without having a degree in statistics, everyone needs to admit that it's been pretty amazing that Nadal has avoided Djokovic (and later Fed when Roger dropped to #3) in pretty much all the Slams during the previous 4 years.

At the AO this year, Nadal finally got Federer in his half of the draw.
Next step to correct the statistics would be that Nadal ends up in top half of the draw at the French Open

time please wrote:I have to say I don't buy the theory because actually what tennis is in danger of suffering from is finals lacking the element of surprise and that does have a negative effect on the game in the long run, which I am sure the ITF is well aware of.

The ITF, or the tennis world in general was aware of one thing in 2008 is that Federer Nadal was what people wanted to see. They were the biggest tickets sellers by far and that for 4 years they tried to maximise that. It also coincides perfectly with the further slowing down of the courts, softening of time rule etc...to maximise the chance Nadal gets there. That AO09 with balls impossible to put away is comical from a purely tennis viewpoint.

That is many factors/chances to "multiply" which leaves no room for luck there....especially considering the lucrative side of it!

Tenez wrote:Mc4 - what puzzles me is that you seem to have the basic stat knowledge. I was pretty good at it in my youth but have forgotten a few formulae since. However, I can see straight away what's likely and what's not and work out the odds.

Very quickly we can see that 2 sets of really odd ones (or 3 if we consider men and women as different). Thta leaves close to zero chance of being luck.

I don't understand your suspicion being on the straight forward study and facts as opposed to the slams organisers rigging the draws.

TA, NITB, Chydremion and others saw it pretty quickly it coudl only be rigged.

well it's quite simple Tenez really. The only "oddity" the ESPN study found is in regards to the US Open draws. The others showed absolutely nothing odd about them. So IF you're working on that basis the oddity of the semi final draw at the US is only 1 in 16, which really isn't that hard to believe.

great work BTW summerblues. Your findings are pretty consistent with the ESPN simulations (1 in 285 isn't far off 3 in 1000)... Did/do you study this sort of thing? you got that very quickly...

Tenez wrote:

We will see at the nest USO that suddenly this "bug" is fixed....even if that requires rigging the draw to "force" top seed players to have highly ranked opponents.

Hm..not that long ago you were saying that it was no surprise to see low ranked players pulled out vs the #1s and #2s in 2011 (despite the organisers having being previously accused of the fixing by ESPN) since they wanted Djoko et al to have a rest. What are you saying? That ESPN won't cause enough waves to make them stop their fixing, but this thread will!

time please wrote:Really interesting posts - as I am functionally innumerate, I don't have anything hugely intelligent to add My arguments are therefore based on gut feeling and not on mathematical reasoning at all

I would just like to play Devil's advocate and suggest that if we assume, for the sake of argument, that there is draw fixing, perhaps the purpose might have been to ensure two of the most competitive semi finals - just as lucrative for Wimbledon and more so for the other slams because of two sets of ticket sales. From 2007, the match up between Novak and Fed has generally been enthralling and very competitive and Murray had the distinction of knocking Nadal out at SF and QF stage of two slams. In other words, perhaps the idea of a semi final between TMF and Nole was as mouth watering a prospect as a final between Nadal and Fed??

I have to say I don't buy the theory because actually what tennis is in danger of suffering from is finals lacking the element of surprise and that does have a negative effect on the game in the long run, which I am sure the ITF is well aware of.

Argument is not in terms of immediate ticket sales. Slam semi finals and finals (at least for the men's) are a guaranteed sell out and usually before we even know who's playing. If the organisers are fixing then they are doing to drive media and sponsor and TV interest, and perhaps ticket sales for the next year's edition.

By the way, on Wimbledon, I find it hard that they would fix the draw to drive revenues but not bother to drive ticket sales by legitimate methods. If they are so keen to boost ticket sales, why don't they:

1 Have night sessions every day - to sell almost twice as many tickets.
2 Have some £200 tickets for the final. The final costs about £100 and those tickets are like gold dust, they could easily boost them.
3 Increase all ticket prices by 30-40%. Wimbledon would still sell out easily.
4 Resell tickets in the grounds for £25 instead of £10, there is a huge queue for the £10 resell all of the time. People would easily pay £25 for centre court resells at 4pm/5pm.
5 Actually bother to promote the ticket sales at all in the media.
6 Allow ticket sales online, this would instantly cause a massive increase in sales even at higher pricing levels. The majority of would be applicants don't get past the initial hassle of posting an SAE far in advance.
7 Sell the 500 on the day centre court the tickets at a higher price (they are often at the front). You have to queue overnight to get these. They are like gold dust.

It seems to me that Wimbledon is either

a) not interesting in maximising profit in the slightest
OR
b) extremely poorly run

I suppose you could argue that they don't care about money and just want to fix the draws for prestige.

summerblues wrote:HB: if we combined the data, the combined average is 0.3195. If we then ask what is the probability that we get this average or a lower one, then, if my math is right, that probability would be about one in 26,000. So, as I mentioned above, the number depends on exactly how you ask the question.

Good work summer blues. I like one in 26,000 best for one thing as it has a larger sample size. I bet if the oddity had been only for one sex the ESPN study would have ignored the other sex. If the men's draw had looked more fixed and sceptics had retorted "what about the normal distribution of the womens'" conspiracy theorists would have proclaimed "who cares about the women's, the men's is where the money and 90% of the interest is".

Anyway, let's say it's one in 26,000.

Now let's say there are 26 ways of looking for oddities in data (rather than look at R1, look at R2/3/4/Q/S, rather than seedings, look at Federer/Nadal etc etc). Let's note that a drawing peculiarity has only been spotted in one of the 26 ways of looking at the data.

Now let's say that there are 10 important tennis tournaments (I won't count all ~100 ATP events because who cares about a draw fixing in Metz and it's obvious no-one has cherry picked a minor tournament here, I wont coun't only the 4 slams because I am convinced that conspiracy theorists would still highlight any discrepancies in the Olympics, Davis Cup, or World tour finals as a minimum). Masters series, debatable, so I've half included them. Let's note that drawing pecularities have only occurred in one of the top say 10 tournaments.

Now let's say that there are 5 main sports that have draws as a key component (e.g. football does, and no-one ever pointed to any oddities in the world cup or FA Cup - whereas golf doesn't have draws as a key component for example). Minor sports let's disregard as it's fair to say that we have not cherry picked a minor sport here. Now let's say that drawing peculaiarities have not been spotted in other sports.

So we've got 26 x 10 x 5 = 1,300 ways in which we could look for draw anomalies in sport. Therefore a 1 in 1,300 chance, occurring in only one event in one sport should not raise an eyebrow.

It's worth stopping at this point to admit that this is an extremely rough, mathematically impure and imprecise estimate. Are there really 26 different ways of looking at the data? Or is it 10? Or 100? Should we really be considering 10 tournaments, or the 4 slams, or all ~100? At the very best I have produced an order of magnitude estimate.

Continuing with the impure maths, we can expect to see something like 1 in 1,300 probabilities actually occuring approximately once, but we actually have a 1 in 26,000 (or 1 in 150,000, or 1 in >300,000) occurence. So, roughly, there is a one in 20 chance of such as 1 in 26,000 event occuring. (Don't think dividing one by the other is the correct methodology, can anyone correct, but suspect it's broadly similar to the true answer.)

So there is (only) an outside chance that this could have happened just due to random fluctuations. As conspiracy theories go, this is much better than the usual man didn't land on the moon/global warming is a hoax/ CIA caused 9-11 type stupid junk.

Mad for Chelsea wrote:
well it's quite simple Tenez really. The only "oddity" the ESPN study found is in regards to the US Open draws. The others showed absolutely nothing odd about them. So IF you're working on that basis the oddity of the semi final draw at the US is only 1 in 16, which really isn't that hard to believe.

But if you are logic, that's irrelevant cause no-one made a judgement on isolated slams. The ESPN case is extremely relevant cause it deals with 10 years and not 4. It's funny you are now trying to separate all slams cases to defend your point because at the beginning your main question point was that we isolated one, the FO!!! What really matters with the ESPN Study is that software can twick numbers to favour a draw. It's up to who is behind the application to decide what they want. Maybe the AO and W were not bothered about the first rounds but maybe they were about semis. It's irrelevant anyway. 1/4k and 1/300k is simply enough to proove a rigging case beyond reasonable doubt. Anything else is not "reasonable".

BTW, The chance for the other 2 to come up with a 8/8 semis is still pretty thin 1/256 too. Extremely unlikely. But I have always said that for 12/12 to happen one did not need to rig the 12 slams. Maybe there is a part of chance too...but chance clearly helped by some being rigged.

You are going great lengths trying to "banalise" events which are evidently extremely weird occurences.

Henman Bill wrote:[What are you saying? That ESPN won't cause enough waves to make them stop their fixing, but this thread will!

NO what I am saying is that they could make sure the rigging theory doesn't carry on. This Widmaier talking in the clip I posted already refers to the next USO as proof that it won;t be rigged. Almost an admission that the previous ones were rigged.

summerblues wrote:If someone looked at the draw from a different angle and it did not show anything strange, that would obviously not get reported. And if different people look at the draw from say 1,000 different (and largely independent) angles than the probability that one of them will find something that has a 1 in 200,000 chance is about 1 in 200 - i.e., not nearly as bad. There is no way to mathematically quantify impact of this since we do not know how many people were digging into these draws from how many angles before one of them found this.

That again is going to great lengths to reduce the chance by 1000 (1000 different angles). You will find a hard time looking at a draw from 1000 rigging valid point angles....unless you consider that one angle would be to rig the draw to favour lower ranked players.

It's almost a signature that you do not wish to see rigging there.

As we say in my native language "you are drowning in a glass of water". I woudl be curious to see who buys your argument there.

To be honest, I don't think anyone anywhere is looking at a draw from 1000 different angles. The ESPN study was an obvious one triggered when people thought they observed easy matches for top seed in grand slams. It's not something arbitrary like easy draws for players ranked 40-50 in grandslams, is it? It's a pretty straightforward thing to notice!

As for the 12 in 12 Fed- Djoker statistic, this one is one of the most obvious things to look for. Ever since the possibility arose that Fed and Nadal could meet in grand slam semis, people have been curious to see this since the only previous instance was in 2005 FO. Even before I heard about this 12/ 12 statistic, I remember being very puzzled that Federer and Djokovic kept meeting each other in semis when it could have been Federer- Murray or Federer- Nadal. As a fan, you want to see the top 4 seeds mixed around a bit.

So I really don't agree that these are the outcomes of looking for any minor irregularity in a draw. This is just an analysis of a glaring trend that any tennis fan would have noticed (Who among us hasn't been puzzled by the constant stream of Federer- Djokovic semis).

Let em reiterate once again that the rigging theory can be argued, but not really that the outcome was likely (since the probability was 1 in 4096 or 1 in 3000000) or that the trend was inconspicuous.

In fact, the trend was obvious enough to be spotted by pretty much any Federer/ Nadal/ Djoker/ Murray fan.

In fact, I've already challenged anyone in the forum to keep looking at draws from the past in a 1000 ways if they want and show me a comparable trend among top players. NITB has said the same too.

No one has been able to show any such trend. Laverfan picked up some scattered points across many years, but the best she managed was a 1/256 probability in spite of selective data picking and looking at many pairs.

This is way off a 1/4096 probability.

Her best defence was to then argue that picking something like Wimbledon (2003, 2007), USO (2005, 2009), Australian Open (2004, 2008) with so many gaps in between was somehow only arbitrary as looking at 4 continuous years data in 3 grand slams and only excluding the French Open each year.

She then argues that both are arbitrary. True- one is a tiny bit arbitrary and the other is hugely arbitrary. It is on such subtleties that arguments rest.

To say this yet again, if the AO had been removed and not the US Open or one year in between removed for all slams rather than removing all years for one slam, that yould have been a desparate attempt to find patterns in the data.

This one isn't- it's 4 straight years at all non-clay slams ie 4 continuous years of data for all slams where Nadal was less likely to reach the finals.

Imagine- if one had to leave out AO 2008, USO 2009, AO 2010 and Wimbledon 2010 to get this 12/12, that would have been totally arbitrary.

This most certainly isn't.

Let me know when this thread is over so we can get bback to tennis,.

To racap,

(a) Federer- Djoker met 12/12 times in non clay slams. The probability of this is 1/4096 (This is simple probability calculations and not is up for debate though several posters initially seemed to be arguing about this too!)

(b) The first objection was that FO data for the 4 years was not considered. The counter was that it was natural to not consider FO since the relative strengths of the players are different on clay

(c) The subsequent objection was that this was a result of a hunt for patterns. I am pretty sure that no one has hunted for patterns (like say Andreev and Gasquet being drawn on opposite halves in 10 consecutive slams) and that this Djoker- Fed statistic was in fact just a trend that almost everyone has discussed for months,

(d) This probability of 1/4096 for seeded draws becomes even more strange when you juxtapose it with the 1 in 3000000 probability of the USO top seed draws being random in the past decade.

(e) From here, the only two conclusions are that either something very unlikely has occurred or that it really wasn't unlikely in the first place i.e. that the draws were not random.

JM, I find it funny that you consider this to be not about tennis. We are all very passionate tennis fans. However, the next time you want to see who Djoker is drawn to play in the FO semis, you will be looking at something which is very pertinent to our current discussion. The excitement of waiting for a draw and analyzing it is very much part of the game, which is why this discussion is very much part of it too.

summerblues wrote:As M4C says, it is actually not that easy to calculate the exact probability of what ESPN was looking at. So I will try to estimate it. This will be quite technical, but I do want to give enough detail so that those of you who know and remember the formulas can have a chance to double check my numbers:

1. I will use "Central Limit Theorem" from probability. This theorem roughly says that if you take an average of a series of independently chosen numbers, this average behaves much like a "normally distributed" random variable. So, in my case I will be looking at the average of 20 numbers from the men's draw (and then the same for the ladies' draw). The 20 numbers are not quite independently chosen, but they are very close to it. Also, they are chosen from a distribution that looks much like a uniform distribution over the interval [ 0, 1 ], and for such distribution the central limit theorem works quite well.

2. I am assuming that the numbers that ESPN assigned to the players were 1 / [ 2 x 96 ], 3 / [ 2 x 96 ], 5 / [ 2 x 96 ], ... ,191 / [ 2 x 96 ]. This is consistent with their statement that the lowest ranked player would be assigned 0.005 and the highest one 0.095 (up to rounding) and it seems like the most likely way to do it. Even if they did something else, they would likely be very close to this pattern

3. The random numbers pulled from among the numbers above have mean 0.5 and standard deviation 0.28866.

4. The average of 20 such numbers has mean 0.5 and standard deviation 0.064546.

5. This average has approximately "normal distribution".

6. The men's average that the ESPN got was 0.326. This is 0.174 below the mean of 0.5

7. So the men's average is 0.174 / 0.064546 = 2.696 standard deviations below the mean.

8. Looking up in the values of the normal distribution, the probability of getting a number this small or smaller is about 1 in 285.

9. Similar calculation for the ladies' average of 0.313 gives probability of about 1 in 531.

10. The combined probability that both the men's average will be below 0.326 and ladies' average will be below 0.313 is the product of the two probabilities, i.e., about 1 in 150,000

11. This number is roughly consistent with what ESPN found. Note that both my and their numbers are approximations only so there is room for deviation from the true value but it seems that we do talk about 1 in 100,000 or more.

This entire calculation is flawed. This theorem roughly says that if you take an average of a series of independently chosen numbers, this average behaves much like a "normally distributed" random variable.

The 20 numbers are not quite independently chosen, but they are very close to it. - They are not. Also CLT requires significantly large numbers and there is the minor issue of number of samples [n] tending to infinity.

These numbers are a finite bound set. These numbers are not independent. If you choose player x, player x cannot be chosen again. The chosen number comes from a continuously depleting pool of 80 players (not 96 players).

Also, 32 seeds, 16 qualifiers, limited entries in the pool. The above at the best is an approximation.

PS: Printable US 2011 draw. http://2011.usopen.org/en_US/scores/draws/ms/msdraw.pdf. Notice the 'fixed' line numbers where 32 seeds are distributed (Line #, 1, 128, 8, 9, 16,17, 24,25, 32, 33, 40, 41, 48, 49, 56, 57, 64, 72, 73, etc.)

And no one is questioning draws before computers got involved, which were done by human hands. 12vs12 pales compared to 142 meetings between two players.

Sorry Laverfan, but I don't think you are being helpful here. They are close enough to being independent as makes no difference. The fact of one stat being removed from a pool of about 100 makes very little difference to anything and was rightly ignored to prevent excessive complexity. There is not much point, especially in such a casual amatuer analysis, striving to remove errors of 1% or less when you have much bigger margins on overall statistical relevance that you can do nothing about.

SB already stated that it was only an imperfect approximation anyway.

Henman Bill wrote:Sorry Laverfan, but I don't think you are being helpful here. They are close enough to being independent as makes no difference.

It does. An infinite series is very different from a finite sample set.

Henman Bill wrote:The fact of one stat being removed from a pool of about 100 makes very little difference to anything and was rightly ignored to prevent excessive complexity. There is not much point, especially in such a casual amatuer analysis, striving to remove errors of 1% or less when you have much bigger margins on overall statistical relevance that you can do nothing about.

The 12vs12, on which this is based, is also an amateurish attempt to propagate a 'conspiracy' theory, based on flawed analysis.

Henman Bill wrote:SB already stated that it was only an imperfect approximation anyway.

What is wrong in me reinforcing the same point?

So is ESPN's analysis. Tenez, et al are using a flawed and mathematically incorrect analysis to trumpet rigging accusations. It is laughable, IMVHO.

I agree with HB. When you pick two (necessarily different) numbers at random in a set of 100. these are very very close to being independent. As a first approximation there's nothing wrong with summerblues's approach.

TA:

(a) I don't think anyone disagrees with this per se.

(b) I (and others) find this counter argument invalid. The general thought is that Djokovic was a bigger threat to Nadal on non clay slams and therefore was better off in Federer's half. I believe Djokovic was more of a threat to Nadal on clay than on grass (and indeed for much of the period considered Murray has superior grass court results to Djokovic). Hence you either include all slams (getting roughly a 1 in 500 from the 14 in 16) or you exclude both grass and clay (and get 1 in 256).

(c) nothing to add to that.

(d) you've added a zero 300k not 3 million. First off though as explained previously this 1 in 300k isn't a "probability" as such rather an empirical results based on simulations. More importantly, I disagree with combining these events (IMO first round draws for top two seeds at the Us Open only is very different from semi-final draws at all non clay slams). I'd be happier if you just combined the Us Open oddities (but 1 in 16 doesn't look as impressive as 1 in 4096).

(e) possibly, but the evidence is much more damning for the Us Open in particular IMO.

I can understand that the probability of 1/2 happening 12 times in a row is 1/4096, but I'm afraid this other stuff is beyond my mathematical understanding, I did mechanics not statistics.

If there's a 1/300,000 chance of the draws having been as they were and they favoured the top players that needs explaining and it is an additional eye-cocking factor to the 1/4096.

Henman Bill wrote:

time please wrote:Really interesting posts - as I am functionally innumerate, I don't have anything hugely intelligent to add My arguments are therefore based on gut feeling and not on mathematical reasoning at all

I would just like to play Devil's advocate and suggest that if we assume, for the sake of argument, that there is draw fixing, perhaps the purpose might have been to ensure two of the most competitive semi finals - just as lucrative for Wimbledon and more so for the other slams because of two sets of ticket sales. From 2007, the match up between Novak and Fed has generally been enthralling and very competitive and Murray had the distinction of knocking Nadal out at SF and QF stage of two slams. In other words, perhaps the idea of a semi final between TMF and Nole was as mouth watering a prospect as a final between Nadal and Fed??

I have to say I don't buy the theory because actually what tennis is in danger of suffering from is finals lacking the element of surprise and that does have a negative effect on the game in the long run, which I am sure the ITF is well aware of.

Argument is not in terms of immediate ticket sales. Slam semi finals and finals (at least for the men's) are a guaranteed sell out and usually before we even know who's playing. If the organisers are fixing then they are doing to drive media and sponsor and TV interest, and perhaps ticket sales for the next year's edition.

You're quite right about ticket sales HB - I was very lazy about checking my post because I was really just playing Devil's advocate with my reply - I do mean that if the premise about tournaments trying to manipulate as far as possible a probable outcome then they are just as likely to do so to ensure competitive semis.

laverfan wrote:

Henman Bill wrote:SB already stated that it was only an imperfect approximation anyway.

What is wrong in me reinforcing the same point?

You were not reinforcing. You began by stating "This entire calculation is flawed" made some further criticism and then concluded "The above at the best is an approximation." with "at the best" implying it could be not even worthy as an approximation. The overall tone was to criticise not reinforce and if your comment on approximation was intended to reinforce it did not read like that following on from the criticism.

As for the FO, 12/12 etc, if anyone wants to see my opinion, please refer to the first few pages of the thread. Have a feeling we're going around in circles on that one though.

I nominate this thread for thread of the month.

Henman Bill wrote:... Have a feeling we're going around in circles on that one though.

I don;t think so, a few have moved on to the "right side" since.

I nominate this thread for thread of the month.

That I agree. Essentially cause it's a tester for logic, honesty and humility.

Tenez wrote:
That I agree. Essentially cause it's a tester for logic, honesty and humility.

So to disagree with your analysis is to be illogical, dishonest, and proud?