In Man City's hands?

Hi,
just so no-one gets misled by all the journalism suggesting the title was in Man City's hands now that they've beaten Aston Villa, putting them two points up with a much better goal difference, meaning that they only need a draw to win the title.
Here's the actual breakdown.
2 games that are interesting: Man City-West Ham and Liverpool-Newcastle.
That makes 3*3=9 possible result combinations:
1) City win, Liverpool win > City champions
2) City draw, Liverpool win > City champions
3) City lose, Liverpool win > Liverpool champions
4) City win, Liverpool draw > City champions
5) City draw, Liverpool draw > City champions
6) City lose, Liverpool draw > City champions
7) City win, Liverpool lose > City champions
8) City draw, Liverpool lose > City champions
9) City lose, Liverpool lose > City champions

1) is all down to City winning - a ninth of hands to City
2) is all down to City drawing - another ninth to City
3) is down to Liverpool winning and West Ham winning - so an eighteenth to each of them
4) you can either say well done City or well done Newcastle - so an eighteenth to each of them
5) ditto
6) well done Newcastle - a whole ninth to them
7) you can either say well done City or well done Newcastle - so an eighteenth to each of them
8) ditto
9) well done Newcastle - a whole ninth to them

Adding all that up, we get:

City: 1/9 + 1/9 + 1/18 + 1/18 + 1/18 + 1/18 = 4/9
Liverpool: 1/18
West Ham: 1/18
Newcastle: 1/18 + 1/18 + 1/9 + 1/18 + 1/18 + 1/9 = 4/9.

So as we see, West Ham and Liverpool have quite little bearing on the title, but it's quite extensively in Newcastle and City's hands!

Let's see what happens on Sunday!

Not as easy as it looks this hands business

Hello,
in the last post, I had a look at a hypothetical situation with the Premier League looking like this before the last day of the season.

Liverpool, played 37,  86 points, vastly superior goal difference.
Chelsea, played 37, 84 points, vastly inferior goal difference.
Man City, played 37, 82 points.

Liverpool play Stoke.
Chelsea play Hull.

This gave me nine possible result combinations:

1. Liverpool win, Chelsea win
2. Liverpool win, Chelsea draw
3. Liverpool win, Chelsea lose
4. Liverpool draw, Chelsea win
5. Liverpool draw Chelsea draw
6. Liverpool draw, Chelsea lose
7. Liverpool lose, Chelsea win
8. Liverpool lose, Chelsea draw
9. Liverpool lose, Chelsea lose

and I ended up saying the title was in:

Liverpool's hands: 12/18
Hull's hands: 4/18
Stoke's hands: 1/18
Chelsea's hands: 1/18

Possible error
I think I may have made an error here, because I put the first six possibilities in Liverpool's hands. However,  possibilities 2, 3, 5 and 6 are just as much under the influence of Hull's hands as they are under Liverpool's.

Revision
Adjusting the results for this error, we have to say that 2, 3 5 and 6 give an eighteenth to both Liverpool and Hull, making a net change of -4/18 for Liverpool and +4/18 for Hull. This gives us a final result of the title being in:

Liverpool's hands: 8/18
Hull's hands: 8/18
Stoke's hands: 1/18
Chelsea's hands: 1/18

What do you think about that?

They've got the whole title in their hands. Or have they?

Hello,
it's a common football punditry cliche to say that a team has got their fate in their hands if all they have to do to e.g. win the league is to win all their games. One recent example of this is Robbie Savage's piece from 13th April, written after Liverpool beat Man City.

"we saw why the title is now solely in their hands" he said.

Some critics, such as @MikeBenchcapon, dispute this approach as it ignores:
a) the fact that sometimes more than one team can win the league by winning all their games. Think about a cup final – there the above logic would suggest that the destination of the cup was solely in both teams' hands.
b) the teams playing the team in whose hands it supposedly is obviously also have some say in the result.

Useful, but inaccurate
Whilst I do recognise the usefulness of an expression that means "they win the league/stay up/etc. if they win all their games", the critique is valid that "in their hands" is somewhat inaccurate. To demonstrate in whose hands the title really is, I was going to do an analysis of all the teams still involved in potentially title-deciding matches to be played after Savage's article.

Noble, but unfortunate
Unfortunately, at the time of Savage writing, even Arsenal and Everton could still have won the league and only four teams, Spurs, Fulham, Stoke and Swansea, weren't involved in any matches which could possibly have had any bearing on the title. This large number of games remaining and indeed teams involved makes the calculations horrifically complicated – not difficult really but just very very many, which would make just setting up the spreadsheet take more time than I have.

So what I've done is I've worked out a couple of hypothetical but not unrealistic examples using a proposed model, just for interest really and I suppose to demonstrate just how spread the fate of a title can be across multiple hands.

Example №1: Two-horse race on final day of season
The first example situation sees the league table as follows:
Liverpool, played 37,  86 points, vastly superior goal difference.
Chelsea, played 37, 84 points, vastly inferior goal difference.
Man City, played 37, 82 points.

Liverpool play Stoke.
Chelsea play Hull.

It doesn't take long to work out that any draw will do for Liverpool whereas Chelsea need to win and hope Liverpool lose. Man City are out of it. Some would say the title is in Liverpool's hands, but to what extent is that true?

Breaking it down
Now, we have two games, each of which has three possible outcomes, a home win for the title-chaser, a draw and a defeat. This gives us a total of nine possibilities.

1. Liverpool win, Chelsea win
2. Liverpool win, Chelsea draw
3. Liverpool win, Chelsea lose
4. Liverpool draw, Chelsea win
5. Liverpool draw Chelsea draw
6. Liverpool draw, Chelsea lose
7. Liverpool lose, Chelsea win
8. Liverpool lose, Chelsea draw
9. Liverpool lose, Chelsea lose

So each of these results has a 1/9 contribution to the overall spread of the title across various hands.

The first six result possibilities (Liverpool win or draw) lead to  Liverpool winning the title and are down to Liverpool having made good use of their hands to such an extent that all other hands (particularly Chelsea's) are rendered irrelevant. So the amount of title in Liverpool's hands is:
6*(1/9)=6/9=2/3

Chelsea get the title in possibility 7, which is done by them winning and by Stoke beating Liverpool. This means that their hands share this 1/9, giving Chelsea's and Stoke's hands an 1/18 from possibility 7.

In possibilities 8 and 9, the Liverpool-Stoke result doesn't matter, and the title has been decided by Hull's salvaging a draw/landing a shock win. So their hands get both of these ninths.

To sum up, and converting to eighteenths to aid comparability, the title is in:

Liverpool's hands: 12/18
Hull's hands: 4/18
Stoke's hands: 1/18
Chelsea's hands: 1/18

Whilst Robbie Savage (I haven't asked him) might think it is a bit silly to say that Hull have more say in the title race than Chelsea, I think it makes complete sense given that they only need a draw or a win, whereas Chelsea need a win. I mean, each football match is contested by two teams with equal numbers of hands and an equal say in the outcome of the match. You could adjust for teams being better by saying that their hands are more secure, if you wanted to, but for now I'm weighting all teams' hands equally. 

The other example is a bit more complicated and I think I've made my main point here so I'll save the other example for a later post. 

Comments welcome as ever. Thanks.

FIFA rankings and football's interestingness

This week, the group stages of UEFA qualifying for the 2014 football World Cup came to an end. The draws for these groups were based on the FIFA world football rankings we know and love.

I read an article by Gabriele Marcotti, who sometimes writes reasonable things, written before the final round of qualifying matches. I've done a semi-close read of it here, followed by some of my own stuff. I'm focusing on the bit of the article that deals with the UEFA section of World Cup qualifying.

Here we go - Marcotti quotes in italics
"UEFA get 13 spots in the World Cup. Nine qualify directly and another four via two-legged playoff."

Fair enough.

"And of the 13 highest-ranked nations who went into qualifying, just two -- Denmark (eighth at the time) and the Czech Republic (13th) -- are out of the running."

Seems to make sense.

"Three of the top six -- Germany (second), Italy (fifth) and Holland (sixth) -- have already clinched a place. Another two -- Spain (first) and England (third) -- should also win their groups and guarantee passage."

This was written before the final round - Spain and England did indeed win their groups.

"The one team that have significantly underperformed are Portugal -- fourth at the time but who have had a tricky qualifying campaign -- though they are heavily favored [sic] to make the playoffs."

"Fourth at the time but have had a tricky qualifying campaign" doesn't seem to mean anything. Surely fourth because they've had a tricky qualifying campaign.

"Greece (10th at the time), Russia (ninth), Sweden (12th) and France (11th) have already made at least the playoffs, while, in addition to Portugal, Croatia (seventh) are also just about there."

This is where it really starts going downhill, as it exposes some clumsiness. Marcotti is clearly working with the rankings published in August 2011, rather than on 5th September 2011 (first qualifiers took place on 7th September).

"Some will point to Switzerland (14th) and Belgium (29th) and the fact that they've clinched qualifying as evidence of surprises. But really, Switzerland’s track record (and a fairly cream-puff group) and Belgium's deep and talented squad suggest that many saw this coming."

These might be points worth making but they have nothing to do with the rankings, but shoving them into a piece which you’re trying to write to say “the rankings work” (more on that later) doesn’t strike me as being overly scientific.

"Bosnia (19th) is a great tale given the nation's history." Actually 17th if you use the right rankings.

"They're guaranteed the playoffs and could yet win the group, but again, looking at the talent in the squad and the relative weakness of the group, it's not really a shock."

"It's not really a shock" is again something which makes any look at the rankings irrelevant if you can just claim that any time they don’t work then you might as well not look at them at all. It's a bit like selective reading of the bible (which I'm not suggesting Marcotti does).

"Iceland -- 46th in Europe, 130th in the world when qualifying kicked off –" actually 44th and 118th. "have a shot at the playoffs, which would be remarkable for a nation of just 300,000."

Seems to be a fair point – the least populous nations getting to playoffs in recent World Cup or Euro qualifiers would be Montenegro and Estonia, both of whom have populations well above 300,000.

"But given the seeded playoffs and the possibility of facing France or Portugal, the shock would be if their journey did not end there."

The seeded playoffs mean that Iceland can’t face France. And suggesting that the seeded playoffs mean it would be a shock if Iceland didn’t win gives an awful lot of credence to the ranking system which, elsewhere, can be dismissed simply by taking a look at talent in a squad. Or, on the other hand, you're assuming the effectiveness of the ranking system within your argement for its effectiveness.

"What all [Marcotti's all includes similar analyses of the other continental federations] this suggests is that maybe the FIFA rankings aren't as absurd as they sometimes seem. Or, at least, they weren't this time around. The vast majority of the teams that were supposed to make it did make it."

Clearly, Marcotti is confusing the rankings, which are supposed to recognise previous achievement, with some kind of tool for predicting future achievement.

I don’t think the word “supposed” is appropriate in this case, because he’s taken the rankings at a time which is irrelevant for the organisation of the competition, August 2012. (That he meant to take September 2012 doesn't matter in this instance.)

The only time when the rankings actually influence who teams face is at the draw, when the teams are sorted into pots – this is a deliberate attempt to improve the best-ranked teams’ chance of qualification by keeping them apart. If there is any “supposed to” anywhere, then it is here.

Of the nine teams that went into pot 1, five have qualified. This is about 55.56 % - a simple majority but not a vast one.

Of the nine teams that went into pot 2, one has qualified and two have made the playoffs, that’s 33% (a minority) doing anything remotely reasonable.

What if we try to be a bit more scientific?
As a little experiment, I did the following:

Gave all teams a score according to the pot they were in at the draw (pot 1 is 1 point, pot 2 is 2 points and so on).

Looked at the possible total points for all group winners and playoff reachers. Here, I doubled the points for a group win as a mechanism for weighting this more than reaching the playoffs. I think this works.

So the lowest possible total is ((9*1)*2)+(8*2) = 34

The highest possible total is ((8*6)+5)*2+(8*5) = 146

The lowest total would be if all pot 1 teams won their group and all playoff reachers came from pot 2. The highest would be if all pot 6 teams won their groups and all playoff reachers (and the group winner from the 5-team group) came from pot 5.

This gives us a possible range of 146 – 34 = 112 points.

The score of the actual group winners and playoff reachers comes out as 52.

This is 18 above the lowerst possible, in a range of 112, putting it at the:

18/112 = 16.07th percentile.

This gives us something I feel can be a useful figure for measuring how closely the rankings have “predicted” the results of qualification. I’m not suggesting any “supposed to” here, but a higher percentile would indicate that lower-ranked teams did better than the rankings would have “predicted”.

It’s up to the observer whether he wants to question the ranking system or applaud the interestingness of international football for not always having teams perform exactly as well as they have done over the past four year period used for ranking-point calculation.

If we get a bit more detailed and use the UEFA ranking of each team, rather than just a pot (so Spain are 1, Netherlands are 2 and so on), then the result comes out around the 13.49th percentile. And if you take the FIFA ranking, you get the 9.79th percentile.

This would seem to suggest that the more detailed you get (i.e. the more you reflect the relative strengths of the teams in the points you assign them), the better the rankings are at predicting the results and the less interesting the qualifiers become.

It’s only a one-off though, so all we can see at this stage is that in this case the better-ranked teams in each pot did better than the worse-ranked teams from the same pot. Generally speaking.

Anyway, having a nice figure like this also means that we can look, for example, at what the figure is if we take the rankings just before the qualifiers (not Marcotti’s August ones, the September ones).

If we do this and go through all the same calculations (being a little careful because, for example, there isn’t a team in the top 9 (pot 1) in every group), then we get the actual results as being at the 10.48th percentile. This is a marked improvement over the 16.07th percentile we had when we took the rankings at the draw.

Similarly, the percentiles when using the UEFA and FIFA rankings also both improve, from 13.49 to 8.84 and from 9.79 to a very respectable 6.57.

So the main issue this highlights is the fact that the draw is made based on rankings over a year before play begins, by which time the rankings have changed considerably. And, as you would probably expect, the rankings immediately before the games start are a better indicator than those at the time of the draw.

Sweeping statement check
It also lets us look sensibly at some things Marcotti said. He refers to Switzerland’s group as “fairly cream-puff”. Whilst the group as a whole is, by total ranking (at time of first game) of all teams, 6th out of 9 and slightly worse than average, Switzerland were the highest ranked side in it, despite being in pot 3.

This is, for my money, far more relevant than overall group strength because it explains why Switzerland were the ones to emerge from the (slightly) weak group.

Bosnia’s group is also said to display “relative weakness”. According to ranking, it is indeed 8th of 9, but, more importantly, Bosnia are the second best-ranked team in it, so have only actually overcome Greece (difference of 8 ranking points) to win the group.

And Belgium, far from simply having to rely on a “deep and talented squad”, were also 2nd best in their group and have only had to overcome Croatia (albeit making up 15 ranking places).

And if we go back to Iceland, regardless of their population, they have overhauled three better-ranked teams with a combined superiority of 64 ranking points. Next best, but a long way back are Romania, who also overhauled three teams, but only made up 27 ranking points in the process.

Hope you enjoyed all that.

[These italics aren't from Marcotti.] PS: If you're trying to recreate my calculations, it's possible you'll get minimally different results, as there are different ways of treating countries of level ranking. Don't worry about it.

Equilibrium in the Champions League III

This is just a very short post as a follow up to the two previous posts in this trilogy, which can be found here and here. Basically, those posts looked at how you could get complete equilibrium in the Champions League group stages, based on the teams' UEFA club rankings and then based on their coefficients.

This post is basically just a correction. I hadn't realised that Real Sociedad, although they had not been in European competition in the five years used as the basis for coefficient calculation, nevertheless have their coefficient calculated for the purposes of the draw. Whilst they don't get any points for their own achievements, they do get the 20% of Spain's national coefficient, which isn't too shabby at all and is actually enough to put them in 31st, above Vienna.

This therefore messes up the calculations I made in those posts for balanced groups and you can't even just make a simple swap of Real Sociedad and Vienna because I had Vienna in Barca's group. The word fiddlesticks comes to mind.

Another point is that, as Sociedad also don't have 0 coefficient points, but 17.605, the calculations for balanced groups would also be different. And it means that they wouldn't be 451st in Europe overall, but would be somewhere around 87th, meaning that balanced groups based on the overall UEFA club rankings are much more possible than I made out.

Whilst this is all quite disappointing, as it voids my results, it doesn't, I don't think, void the methodology, which I am convinced will stand the test of time.

Whilst I'm not going to redo all of the calculations, I can tell you that Sociedad's being ranked 31st of the Champions League qualifiers means that Man United aren't actually as lucky (l) as I'd made them out to be, as they get a luck score of just 1.33 and not 1.67. This doesn't however, affect the bigger picture of English luck, as they remain the only English side with positive (i.e. good) luck, based on the UEFA rankings.

Sorry about the error.

Equilibrium in the Champions League II

This post is the sequel to the previous post, hence the II in the title. So you should probably read that first, if you haven’t already done so

Anyway, the previous post ranked the Champions League qualifiers from 1 to 32 according to their UEFA club coefficients and used these rankings as a basis for analysing the balance of the groups and measuring each team’s luck in the draw.

There are two other obvious ways of measuring the strength of the teams:

- Using the UEFA club rankings;

- Using the club coefficient points.

Using the UEFA club rankings

The UEFA club rankings seem like a good idea, because they keep the numbers as nice manageable integers and they add some detail as the gaps between teams aren’t all the same, thereby taking into account that some teams are significantly lower-ranked than others. For example, the top 6 ranked teams in the Champions League are also ranked 1 to 6 overall, but Austria Wien, 31^st in the Champions League, are ranked 114 overall. 

The problem with using this ranking system is that Real Sociedad, the team ranked 32^nd and worst of all the Champions League qualifiers, don’t have a UEFA ranking at all, because they haven’t been in Europe in the last five years. As there are 450 teams with a UEFA ranking, this would make Sociedad joint 451^st. 

Unfortunately, this massive leap to the 32^nd team (all other 31 teams are in the top 114) skews the entire system. In the same way as, with a 1-32 ranking system, we had a total 528 points and 66 per balanced group, this system gives us 1,328 and 166 per balanced group. 

However, the group with Sociedad will automatically be far worse than average and most groups will be far better, with the only way to get a balanced group being to have Vienna (114) and three other teams with a total 52 between them, such as Bayern (2), PSG (19) and Dortmund (31). Even the third worst-ranked team, Plzen (74) cannot be put in a balanced group, with the most balanced scenario having a total 145, with Benfica (9), Juventus (20) and Leverkusen (42) completing Plzen’s group.

This is clearly unsatisfactory and, as such, I reject the use of the UEFA club rankings as a workable system for looking to achieve balance. They may be useful next year, if there is not such an anomaly, but, for 2013/14, they must be set aside.

What about using the club coefficient points?

Using the club coefficient points avoids the problem with Sociedad. Although they have zero points, this is only 16.575 fewer than Vienna, which does not represent such a dramatic and exaggerated drop as 114^th to 451^st in the ranking. In fact, this is not even the largest difference between two adjacently-ranked sides, as Arsenal have  17 fewer than Man U.

So this system seems to be workable. And it takes into account larger and smaller gaps between teams than the more simple ranking system. So it could be worth a look. 

Balance using the coefficients

Following an equivalent methodology to the one for the rankings, we have a total 2454.519 points, so balanced groups would each have 2454.519/8 = 306.815 points. 

It is unlikely that we will be able to get 8 groups with this exact score but we could consider them fairly balanced if we had no group with more than 306.815+(306.815/32) and none with less than 306.815-(306.815/32). 306.815/32 is 9.588 so this means groups between 297.227 and 316.403.

The choice of 32 is fairly arbitrary, basically being used because there are 32 teams. If anyone has a better suggestion for something more suitable, they are welcome to comment.

Anyway, let’s start out by looking at the 8 groups which were balanced with the ranking system. These have scores of:

Group	Total coefficient	Difference to balanced group
A	310.932	+4.117
B	316.865	+10.050
C	292.228	-14.587
D	295.280	-11.535
E	315.223	+8.408
F	301.727	-5.088
G	299.211	-7.604
H	323.053	+16.238

Based on what I’ve said above, this would make suggest that groups A, E, F and G are fairly balanced, whilst B, C, D and H are either too strong (B and H) or too weak (C and D).

By making two simple swaps:

• CSKA (D, 77.776) with Donetsk (H, 94.951
• PSG (C, 71.800) with Schalke (B, 84.922)

we redress this imbalance and come out with 8 balanced groups.

Of course, making these two changes brings about imbalance according to the 1-32 ranking system. However, if we fiddle about with the spreadsheet for long enough, then we can come up with the following groups:

Group A			Group B
Man U	130.592	5	Real	136.605	4
Schalke	84.922	12	Juventus	70.829	16
Basel	59.785	21	Zenit	70.766	17
Celtic	37.538	28	Bucharest	35.604	29
Totals	312.837	66	Totals	313.804	66
Group C			Group D
Benfica	102.833	8	Bayern	146.922	2
CSKA	77.766	14	Donetsk	94.951	10
Man City	70.592	18	Olympiakos	57.800	22
Napoli	46.829	26	Sociedad	0.000	32
Totals	298.020	66	Totals	299.673	66
Group E			Group F
Chelsea	137.592	3	Arsenal	113.592	6
Marseilles	78.800	13	Atletico	99.605	9
Dortmund	61.922	20	Leverkusen	53.922	24
Plzen	28.745	30	Anderlecht	44.880	27
Totals	307.059	66	Totals	311.999	66
Group G			Group  H
Porto	104.833	7	Barcelona	157.605	1
Milan	93.829	11	PSG	71.800	15
Galatasaray	54.400	23	Ajax	64.945	19
Copenhagen	47.140	25	Vienna	16.575	31
Totals	300.202	66	Totals	310.925	66

Now it took quite a long time for me to get such a distribution which suggests that there aren’t all that many possible draws which offer balance according to both ways of measuring. This is not, however, by any means to say that this is the only way.

Whilst it is nice to see that it is possible, given the fact that 1/32^nd deviation from the average was chosen rather unjustifiably, I could have chosen something less stringent and achieved balance with that more easily, if I’d wanted to.

And what about lady luck?

As we saw in the previous post, you can measure the luck of a team by comparing its actual draw, that is the strength of its actual opponents, with the strength its opponents would have in balanced groups. In the previous post, the luck (l) was calculated as:

l = (g-66)/3

where (g) was the total ranks of the group and 66 the rank of an average group.

The equivalent formula with the club coefficients, if we say k is now luck and the group total is h, would then be:

k = (h-306.815)/3

However, as we want good luck to be positive, we need to switch this around to take into account the fact that a high-numbered coefficient, unlike a high-numbered ranking, is given to a strong team. So we can make it:

k = (306.815-h)/3.

Luck (k) of the English

We saw in the previous post that, contrary to the claims of the BBC and, in particular, Phil McNulty, Arsenal and Chelsea were equally (and only slightly) unlucky, Man City had the worst luck and Man Utd were the only English side to get lucky.

If we measure luck (k) of the English teams, as formulated above, we get:

Man City with -5.74

Arsenal with +1.89

Chelsea with -3.70

Man Utd with +9.11

Clearly, whilst Man City and Man Utd retain their positions from the previous post as most unlucky and luckiest respectively, we see a significant change with Arsenal and Chelsea. 

Chelsea stay unlucky but Arsenal emerge as being a little bit on the lucky side of balanced. This is perhaps not a surprise as Arsenal’s, though ranked 6, are a long way behind the 5 top-ranked teams in terms of coefficient points. So measuring by coefficient points, they deserve, as it were, to get somewhat stronger opponents. 

Luck (k) of the Celtic

In the last post, Celtic were very unlucky. When measuring according to coefficient points, this doesn’t change much, as they come out with a luck (k) score of -15.70, making them 2.73 times more unlucky even than Man City, the unluckiest of the English sides. And, as with luck (l), Celtic (and the other teams in group H) have the most rotten luck (k) of all teams in the competition.

Arsenal, however, prove to be much more fortunate than a lot of pundits would have us believe.