League Tables and Head to Head

The following question appeared in The Knowledge in the Guardian (see bottom of this page):

“Has any team ever managed the statistical feat of holding a better head-to-head record against each of the other teams in their Champions League section and yet still finished bottom of the group?” muses Mark Wilson. “Presumably home 0-0s, away score draws and some terrible luck would be required but it is a possibility.”

By Team A ‘holding a better head-to-head record’ against Team B, Mark Wilson presumably means that if the scores were added together over the two games, Team A either scored more goals than Team B, or else they scored an equal number of goals but scored more of them as ‘away goals’ (i.e. goals scored in the away game). In particular, this means that Team A can beat Team B ‘head-to-head’ without actually beating them in a match. For instance, a 0-0 draw at home and a 1-1 draw away is enough.

Recall that groups in the Champions League are ordered first by points, and then by the restriction of the league to tied teams. If that is still equal, then goal difference and then goals scored is used – first in the mini-league and then in the whole league.

One way this can happen is for Team A to draw all their home games 0-0 and all their away games 1-1. Then if the rest of the games are home wins, Team A will finish bottom (with 6 points to the other teams 9 points) despite beating the other three teams head-to-head.

There are other possibilities too. For instance, the above still works with the modification that they lost to Team B 1-0 at home, but beat them 2-1 away. Then they have 8 points, still beat everybody else on head-to-head, and still finish last. Indeed, they can lose all their home games 1-0 and win all their away games 2-1 against all the other teams and still finish last. In this case, all teams will finish on 9 points and so the tie-break is goal difference and then goals scored. As Team A has a goal difference of zero, if we want them to finish last the other teams must also have a goal difference of zero (as goal difference sums to zero). By making the games between Teams B, C and D goal-rich but symmetric, we can make Team A finish last.

The final case is more complicated, but it is possible to win/draw against two of the teams and draw/draw the last. To be concrete, let us say Team A beat Team B head-to-head by a 0-0 draw at home and a 1-1 draw away. Against Team C and Team D, they lost 1-0 at home and won 2-1 away from home. This means Team A got 8 points. If , as in the previous examples, all other games were home wins, then Team B would have 8 points and Team C and Team D would have 9 points. This means that Team A and Team B finish with the same number of points, and so it goes to head-to-head record, which Team A won. Thus Team A did not finish last. The solution is for Team C and Team D to actually draw against each other in both their matches. Then each team finishes with 8 points, and again goals scored can be used to make Team A finish bottom.

Has it ever happened in the Champions League? The answer is no. Groups were introduced in 1991-92 and in that time no team has ever even finished last without losing more games than they did win, which makes this situation impossible.

In the 1998-99 Champions League, which was won by Manchester United against Bayern München in the last moments, there was however a very near miss. It was group B, which contained Juventus, Galatasaray, Rosenborg and Athletic Bilbao (of Italy, Turkey, Norwary and Spain respectively). The group finished in that order, letting Juventus qualify for the next round. The top three teams were all level on eight points, and Athletic Bilbao had six.

Nevertheless, Athletic Bilbao held Juventus 1-1 in Turin and 0-0 in Bilbao, beating them head-to-head. They also made up for their 2-1 loss in Turkey by beating Galatasaray 1-0  on Spanish soil – again, winning head-to-head. Unfortunately, their 1-1 home draw with Rosenborg was met with a 2-1 loss in Norway, and so they did not beat Rosenborg head-to-head. So they beat the top two teams (individually) on head-to-head, but not the third place team.

As the top three teams finished on equal points, their league position was decided upon the matches only including them. That is, games without Athletic Bilbao. In this sub-league Juventus came top and so qualified. However, suppose that Rosenborg vs Athletic Bilbao was 2-2 instead of 2-1. Then Athletic would have finished with 7 points, not 6. Rosenborg would have finished with 6, not 8. This wouldn’t be a solution to the posed problem, as then Athletic did not come bottom of the table. What it would do, however, is make Galatasaray go through instead of Juventus.

Galatasaray actually beat Juventus on head-to-head. It was 2-2 in Turin and 1-1 in Istanbul. Therefore, if only Galatasaray and Juventus were equal on points, then Galatasaray would have gone through instead of Juventus. Doesn’t this seem ridiculous? It seems like a fairly non-local effect of a goal.

Tie-break Criteria and the Manchester Clubs

The 2011/12 Premier League season finished with Manchester United and Manchester City at the top of the table with a joint number of point. Different leagues have different rules on what happens in this situation.

In the past the relevant tie-breaker was ‘goal average’. This is the ratio of goals scored to goals conceded. It has not been used in the English Football League since 1975-76. This seems like an odd measure, as intuitively it is the number of goals you win by which is important, irrespective of offsets. For instance, surely 2-1 is just as good as 1-0?

This is no longer used, presumably for the reason that it makes conceding goals too costly. Goal difference is used instead. This is the difference between the number of goals scored and the number conceded.

In tournaments like La Liga, Serie A, the Champions League group stage and the European Cup group stages, it isn’t immediately used. If two or more teams are equal in points then a new league is considered which contains only these teams and from the games between these teams. This is then ordered first by points, and then by goal difference (and then it will typically continue to goals scored, and so forth). The ordering of this league then orders the joint teams in the original league.

[As an aside, I find it slightly odd that the rules are not applied recursively.]

In other tournaments, like the Bundesliga, the Premier League and the World Cup group stage, then after points goal difference across all games is compared. This is what happened to Manchester United and Manchester City, where Manchester City become the first team to win the league on goal difference. (The only time there was been a draw on points at the top of the table since goal average was abolished was 1988-89, where Arsenal and Liverpool were drawn on both points and goal difference, and Arsenal won it on goals scored)

Both teams won, lost and drew the same number of games, giving them the same score. However, Manchester City had a goal difference of 64 and Manchester United had a goal difference of 54.

Goal difference is better than goal average in that it only cares what the difference in scores is. Nevertheless, it is perhaps too linear. Is winning four games 1-0 and then winning a single game 6-0 better than consistently winning 2-0 over five games? Goal difference treats them as the same. If they aren’t equal, it isn’t particular obvious which is better. The latter team were consistently better, but the former team achieved a large victory. There seems to be an argument either way.

The following is a graph of the distribution of wins of the two Manchester teams. As it is already decided that it is only goal difference in a match which matters, this is all that is graphed. Obviously the total left of ‘0’ denotes the number of losses, and right of it the number of wins.

It should be noted that when the Manchester clubs played each other, United won 1-0 away and lost 6-1 at home. Therefore if a head-to-head system were to be used which went to goal difference, City would still win. Goal average would not help United either – they conceded more goals and scored less, so any sensible combination of these two numbers will see City win. Had the United-City game been 1-0 instead of 6-1, the goal difference delta would have changed by 8, but City still would have won on goals scored.

The graph shows that United did achieve more very large wins than City though. United won of five occasions by five or more goals. City did this ‘only’ twice. Perhaps there is some home for United after all, if we change our values?

Is winning by two goals twice as good as winning by 1? Perhaps not. Winning by one can be luck, but winning by two seems like there might have been some skill involved. Perhaps instead of adding up the games goal differences, one should add up the squares of the games goal differences. Obviously, one would have to subtract instead of add if you lost by that amount, otherwise losing by 2 goals would be just as good as winning by 2 goals!

If this is done, Manchester United get a score of 188, but Manchester City get 212. They will still win.

Okay, let’s say winning by two goals is twice as good as winning by one. But then is winning by three goals only 1.5 times better than winning by 2? Perhaps it is twice as good again. Continuing  5 goals is 16 times better than scoring 1. If you lose, it should be the same as if you win except with the opposite sign, and if it was a draw the value is zero.

If we do this, then Manchester City get 112 points, but Manchester United get 121.

Nevertheless, this method is fairly silly, as it means the game Manchester United won 8-2 at the start of the season screws it all. Perhaps the truth is the opposite: the first goal is very important, but each goal after that becomes less and less important. After all, 5-0 and 6-0 are basically the same score. The last goal doesn’t matter very much at all. Perhaps it shows more like a logarithm than a power.

Using gives Manchester United 38.1 points, but Manchester City 32.3.

In summary, there is no easy answer to a tie-breaking criterium that rewards the things you want it to. At least goal difference is simple, and so it seems a good choice.

Fever Pitch Quotes

I’ve just finished reading Fever Pitch by Nick Hornby.

It is an autobiography told though Arsenal football matches, and how the authors life interacts with them and reflects them.

The following are my three favourite passages from the book:

A critical faculty is a terrible thing. When I was eleven there were no bad films, just films that I didn’t want to see, there was no bad food, just Brussels sprouts and cabbage, and there were no bad books – everything I read was great. Then suddenly, I woke up in the morning and all that had changed. How could my sister not hear that David Cassidy was not in the same class as Black Sabbath? Why on earth would my English teacher think that The History of Mr Polly was better than Ten Little Indians by Agatha Christie? And from that moment on, enjoyment has been a much more elusive quality

The following goes some way to explaining how playing and watching football are completely different things, and the latter is not merely a replacement for the former:

One thing I know for sure about being a fan is this: it is a vicarious pleasure, despite all appearances to the contrary, and those who say that they would rather do than watch are missing the point. Football is a context where watching becomes doing – not in the aerobic sense, because watching a game, smoking your head off while doing so, drinking after it has finished and eating chips on the way home is unlikely to do you a whole lot of Jane Fonda good, in the way that chuffing up and down a pitch is supposed to. But when there is some kind of triumph, the pleasure does not radiate from the players outwards until it reaches the likes of us at the back of the terraces in a pale and diminished form; our fun is not a watery version of the team’s fun, even though they are the ones that get to score the goals and climb the steps at Wembley to meet Princess Diana. The joy we feel on occasions like this is not a celebration of others’ good fortune, but a celebration of our own; and when there is a disastrous defeat the sorrow that engulfs us is, in effect, self-pity, and anyone who wishes to understand how football is consumed must realise this above all things.

And it is worth remembering how bad most of us are at football – even the ones who seem great:

This is how close I came to becoming a professional [footballer]: at college, one or two of the first team (I was in the first team in my final year) played for the Blues, a team consisting of the eleven best players in the whole of the University. To my knowledge, two of the Blues players in my time went on to play at a professional level. The best one, the university god, a blond striker who seemed to glow with talent in the way stars do, played as sub a few times for Torquay United in the Fourth Division – he may even have scored for them once. Another played for Cambridge City – City, Quentin Crisp’s team, the team with the wonky Match of the Day tape and a crowd of two hundred, not United – as a full-back; we went to see him, and he was way off the pace.

So… if I had ranked number one in my college, as opposed to number twenty-five or thirty, then I might have been able, if I had been lucky, to look bad in a very poor semi-professional team. Sport doesn’t allow you to dream in the way that writing or acting or painting or middle-management does: I knew when I was eleven that I would never play for Arsenal. Eleven is too young to know something as awful as that.

Evra and the Coin Toss (A little story hidden in the Suarez judgement)

The FA recently published the reasons for their judgement in the Suarez/Evra case. Suarez was found guilty of  using racist language to Evra during the Liverpool vs Manchester United match in October 2011, which finished 1-1.

I read the ‘Reasons’ report, which I did think was interesting (and surprisingly readable). Comments on the judgement are easy to come by, I’m sure, so I won’t give any more here.  However, I did like the following little story. It is paragraph 329, and it was from evidence put forward from the defence to show claim that Evra may have been angry enough to make the whole thing up. The evidence is mostly provided by Marriner, the referee for the game:

Mr Evra was seen to dispute the outcome of the coin toss with the referee. Mr Marriner explained that he used a FIFA coin which is blue on one side and yellow on the other. He asked Mr Evra, as the visiting captain, to call the colour. Mr Marriner tossed the coin, it came down yellow, and he awarded it to Steven Gerrard who elected to stay in their current ends. Manchester United had kick off. Mr Evra remonstrated that he had called correctly but, Mr Marriner said, he had not. Mr Evra then spoke to Ryan Giggs about it, and Mr Marriner walked over to Mr Evra to assure him that he (Mr Marriner) had got it right. Mr Evra’s evidence was that when such a coin was used, he always called yellow given that the alternative, blue, is a Manchester City colour, which he would never call. The toss came down yellow and so Mr Evra knew that he had won it. He particularly wanted to change ends at the start, he explained to the referee that he had called yellow, and why he had done so. Mr Evra was angry but the referee did not change his mind.

I liked the reason at the end. It seems funny to me that such things would influence him. It’s good to hear it, even though I’m an Arsenal fan.

“I think that usually with 10 points all the teams go through always, 99 percent” – The Champions League Group Stage

Manchester City failed to qualify from their group in the Champions League despite getting ten points, prompting their manager, Mancini, to say what is quoted in the title of this post.

A group in the Champions League has 4 teams in it. Each team player each other twice. There are three points for a win, one for a draw, none for a loss. Since 1999-2000, the top 2 teams (ordered by points, then in the case of ties head-to-head goal difference, and then general goal difference) in each group quality to the next round.

Is it true that ten points has guaranteed qualification 99% of the time? Well, it’s not too far out at 2.5%. On 158 occasions has a team achieved 10 or more points, since the 1999-2000 season (but excluding this season). Only four of those teams failed to qualify, and each of those got exactly 10 points.

These were:

• Werder Bremen in 2003-4 (Barcelona got second play with 11 points)
•  Olympiacos and Dynamo Kyiv in 2005-6 (Liverpool and Real Madrid went through in second place with 10 and 11 points respectively)
• PSV Eindhoven in 2006-7 (Deportivo had 10 points but went through instead of them).

Whilst in practice no team has ever failed to go through with 11 points, it is in fact possible to still miss out with 12 points. Imagine three teams A, B and C who win all their home games against each other (and so lose all their away games against each other), but all beat D both home and away. Then you get the following table, where C misses out on qualification:

13 points does ensure you qualification. As suppose you fail to qualify with 13 points. Then another two teams must get at least 13 points, and so there are at least 39 points on the board. But it is only possible to get 36 points on the board, as there are 12 games and each game can add up to three points on the board.

In reality, no team has qualified with less than 7 points. However, teams have qualified with exactly 7 points. Werder Bremen in 2005-6 and Lokomotiv Moscow in 2002-3 both came runners up to a high-scoring Barcelona. In theory, it is possible to get four points and still qualify. Imagine team A wins every game and teams B, C and D draw against each other. Then you get the following table:

Sir Alex Ferguson’s Long Stay

The following is a particularly nice diagram from The Observer showing how other teams have changed their managers whilst Manchester United (and to a lesser extent, Arsenal) have kept them the same. I just thought this was a good picture, and it must’ve taken ages for them to make.

Preston, however, have not kept with his son for quite so long (for good reason), and Ferguson is even attempting to take back a player from a season long loan to try and punish them for it. Nothing like abusing your power, ay?

A World Cup Of Upsets

My very first post on this blog showed the World Cup if there were no upsets at all (i.e. the highest ranked team always won). It would end up looking like this:

Now the group stage is over we can look at how it compares to reality, and it is fairly different.

Brazil vs Chile, Argentina vs Mexico and Spain vs Portugal are the only predicted matches that will happen, and they all sound great. 6 attacking teams, at least half of which with their own defensive issues.

England’s poor performances didn’t cost them as it did Italy or France, but it does mean the Round of 16 has England vs Germany, which will start in about half an hour.  The winner will have a road to the final that will probably include Argentina and Spain, so the winner has more issues.

The shock results in the group stage means that one of unfancied four of Uruguay, South Korea, the USA and Ghana will reach the semi-finals. (Well, since those games have been played, one of Uruguay or Ghana.) They will likely play either the Dutch or the Brazilians, but if Uruguay carry on as they are they could actually stand a chance in getting to the final. Ghana were bossed by Brazil in the last world cup, and there doesn’t seem to be any reason to suspect it would be any different this time.

The predicted quarter-finals of Netherlands vs Brazil and Argentina vs Germany can still happen, although the others cannot: Spain, if they reach the quarters, will get a theoretically easier test than Italy with either Paraguay or Japan, although both teams have been impressive and positive.

Premier League Upsets Graph

I mentioned in a previous post that a graph showing every ‘upset’ in the Premier League this year (August-December 2009) would be fairly interesting. And here it is:

A graph showing all the 'unexpected' Premier League results. Arrows on the graph point towards the team who were favourites. Click to see it larger.

How is an ‘upset’ defined? I split the teams in the Premier League into four groups. The ‘top teams’, ‘second tier teams’, ‘mid-table teams’ and ‘bottom teams’. I defined an ‘upset’ as either a ‘top team’ failing to win against a team which is not another ‘top team’, or a ‘second tier team’ failing to win against a ‘bottom team’.

I think this probably is a good definition, although it is open to arguments. Even if you accept it though there are bound to be arguments about which teams are in the various groups. I decided to use my expectations at the start of the season rather than how it turned out. Therefore I decided the following:

Top Teams: Man United, Chelsea, Arsenal, Liverpool

Second tier teams: Tottenham, Man City, Villa, Everton

Bottom teams: Burnley, Wolves, Hull

The rest were defined as ‘mid-table teams’. There are a few arguments to be had here. Many people though Man City were going to be a top team this year, but always thought that was unrealistic (although I would tip them as the most likely to win the FA Cup). Everton have done poorly in the end this season, but I think that is more due to injuries and so they keep there place in my mind as a second-tier team

The Premier League Wins Graph

I thought that with the new year in football, in would be interesting to draw a graph of all the football teams and all the results. Draws aren’t really directed in either way so I had to leave them off the graph, but this is what it looks like:

Premier league results for first half of 2009/10 season. Drawn games are not shown. Click on the picture to see a PDF which you can zoom into.

Now of course it is impossible to see anything on this graph! However, I programmed it in Mathematica, and so it was really easy to limit it into showing only a certain range of match days. For instance, the last four match-days of 2009 (17, 18, 19 and 20) are shown here:

Match-days 17 to 20 are shown. Again, draws are hidden.

Perhaps it would be interesting to come up of a graph of ‘upsets’, where perhaps that is a team from the bottom half of the table beating a team in the top half, or a draw in more extreme cases. Perhaps it could be decided by me ranking each team in the Premier League. Actually, that might be interesting. (Edit: I have now done this here)

Getting this together I used Excel’s ‘Import from web-page’ feature on the Footbo page. It was the first team I used it and it worked well, saving me from having to write an XML parser in C#. In Excel I formatted it to work with Mathematica and wrote some code to display it there. Most of the work is done by the GraphPlot function. You can download the workbook (with the data) here.

Arsenal fan and comedian Dara O Briain asked in the comments of his latest Guardian column how early in the season the first cycle of teams where each beats the last occurs. His column pointed out a couple of interesting cycles.

It obviously cannot happen on the first day. It is possible for it to happen on the second day, and in fact it nearly does: West Ham beat Wolves, who beat Wigan, who beat Villa. But then Villa ruined it all by losing against West Ham. The third day has a lot more possibilities.

The Premier League 2009/10 after the first three rounds. Draws are not shown.

By the third day it has indeed happened. Villa beat Liverpool 3-1, Liverpool beat Stoke 4-0, Stoke beat Burnley 2-0, Burnley beat United 1-0, United killed Wigan 5-0 and then Wigan beat Villa 2-0. In fact, this is the only such cycle by this time.

I wonder if that fact that a cycle happened so early was a sign that the Premier League is getting more unpredictable, or if such a cycle does tend to appear quite early in the usual case. It would be easy enough to check another couple of seasons but as the fixtures change every year it might not really tell us much.

Points needed to steer clear of relegation in the Premier League

The Premier League is England’s highest football league. Each of the 20 teams play each other twice. A win gains 3 points, a draw 1 point and a loss gains nothing. Teams that a drawn on points are sorted according to ‘goal difference’ (the number of goals scored subtracted by the number conceded) and after that the number of goals scored are used.

The team that finishes top wins the league title, whilst the bottom three teams are relegated to the Championship (the second highest football league). This begs the question: How many points do you need to win the league, and how many will the team with the most points achieve?

The following graph shows the values for every year that the Premier League has run in its current format:

Points achieved in the Premier League across different seasons

On average about 85 points have been required to win the Premier League. On average, 36 points will still get the team relegated.

Mathematically, anything less than winning every game (114 points) can see you losing out on the league table, for the team you lost or drew against can win all their other games and thus either win straight out or on superior goal difference. In reality nobody has ever got higher than Manchester United’s 91 points though, and it has been won with as few as 75 points (again by United).

The highest number of points that can be achieved whilst still being relegated is when everybody else in the league except the two teams below you are drawn on points (and so you go down on goal difference). Thus everybody wins both games against the two bottom teams and wins one of the games against everybody else. Thus gives 6+17×3 = 57 points. In this situation 58 points will win you the Premier League! In fact the most points a relegated team had was West Ham United’s 42.

Is there any reason to assume this year might be any different? I’ve taken the Christmas standings from the last few seasons and calculated the average number of points per game the top and 3rd bottom teams had:

Points per game at Christmas in the last few years

So far this year doesn’t look like it is significantly different.