Who is the best England manager?

Flickr user Eric Kilby, CC BY-SA 2.0

Paddy Moore

Myquest to answer this simple question began for the noblest of reasons—to win an argument with my wife. We are both football fans and have been following England all our lives (the phrase ‘long-suffering’ has never been more apt). Our house is usually an oasis of calm and tranquillity, but one thing is guaranteed to get things kicking off: was Sven-Göran Eriksson a good England manager?

This year, we have had more time than usual at home together and the discussion has become heated. I believe that Sven took a golden generation of England players and led them to disappointing performances in three major tournaments and she points to the team reaching the last 8 at consecutive World Cups under his stewardship. I’ve been a maths teacher for over 25 years, so surely, I can prove I am correct using maths, right?f How do I prove that I am right? Well, the first, and most obvious, thing to do is to look at the playing

Anders Henrikson, CC BY 2.0

Sven-Göran Eriksson wondering where his hat has gone.

f At this point, it isdefinitely not worth mentioning that in the 15 years we’ve been having this row, I never thought of applying any maths to the problem until my wife suggested it.

record of Sven-Göran Eriksson. He was manager of England from January 2001 until July 2006. In that time the team played 67 games and won 40 of them—a win percentage of 59.7%.

On its own that is a bit meaningless, so we need something to compare it to. It’s time for a spreadsheet.

I am going to tidy up the data a little though. Firstly, up to 1946, there was no England coach. Even under Walter Winterbottom, the players were selected by committee, so I am going to exclude them. Secondly, caretaker managers like Stuart Pearce or Joe Mercer (or Sam Allardyce who was sacked after one game for ‘reasons’) didn’t have enough games and so I’ll drop them from consideration. That gives us the trimmed list below right, ranked by winning percentage.

This shows that Sven had a pretty average record, only just reaching the top half of the table. I was feeling suitably pleased with myself for having come up with such a convincing statistic, only to be shot down with, “Yeah, but a lot of those games were meaningless friendlies.” I mean, you could argue that playing for England in any game is the pinnacle of a footballer’s career, and that international friendlies are always important games, but I decided to look at this as it seemed interesting (and I was confident that it would support my point even more).

P W %

Fabio Capello 2008–2011 42 28 66.7% Alf Ramsey 1963–1974 113 69 61.1% Glenn Hoddle 1996–1999 28 17 60.7% Ron Greenwood 1977–1982 55 33 60.0% Sven-Göran Eriksson 2001–2006 67 40 59.7% Gareth Southgate 2016–2020 49 29 59.2% Roy Hodgson 2012–2016 56 33 58.9% Steve McClaren 2006–2007 18 9 50.0% Bobby Robson 1982–1990 95 47 49.5% Don Revie 1974–1977 29 14 48.3% Terry Venables 1994–1996 23 11 47.8% Graham Taylor 1990–1993 38 18 47.4% Kevin Keegan 1999–2000 18 77 38.9%

Tonywalt, CC BY-SA 3.0

Walter Winterbottom

Since we were looking at competitive internationals, I decided to look at the overall results record rather than using just the win percentage. After all, there are three possible outcomes in football and a draw has value (although this value varies with the opponent—a draw against Brazil is generally seen as a fairly decent result, whereas as a draw against Greece is not).

To calculate this, I used 3 points for a win and 1 for a draw. This has been the standard across football since the 1980s as it rewards positive play. This may disadvantage the managers from before it was introduced because playing for a draw would have been more profitable in group games and qualifiers, but I feel it is the best of the options available (and I’m trying to prove that Sven was a negative manager and I think this will help me)...

Selected England managers’ full competitive international results (Correct to Nov 2020)

P W D L F A Win % Pts Pts available Pts %

Sven-Göran Eriksson 38 26 9 3 69 26 68.4% 87 114 76.3% Fabio Capello 22 15 5 2 54 16 68.2% 50 66 75.8% Ron Greenwood 26 17 5 4 48 17 65.4% 56 78 71.8% Roy Hodgson 31 19 9 3 73 18 61.3% 66 93 71.0% Glenn Hoddle 15 9 3 3 26 8 60.0% 30 45 66.7% Don Revie 10 6 2 2 22 7 60.0% 20 30 66.7% Alf Ramsey 33 20 6 7 56 29 60.6% 66 99 66.7% Gareth Southgate 36 22 6 8 80 29 61.1% 72 108 66.7% Bobby Robson 43 22 14 7 90 22 51.2% 80 129 62.0% Terry Venablesf 5 2 3 0 8 3 40.0% 9 15 60.0% Graham Taylor 19 8 8 3 34 14 42.1% 32 57 56.1% Kevin Keegan 11 4 3 4 17 10 36.4% 15 33 45.5%

Selected England managers’ full competitive international results (Correct to Nov 2020)

This did not go well and there was a significant amount of smugness, which I felt was inappropriate and irritating.

To be honest, this is a compelling result and I needed to come back strong if I was to maintain any credibility in this argument. I felt a little disappointed that I had done all that work to prove this important point and it wouldn’t be any use. I was starting to get concerned that manipulating statistics to get the result I wanted was not working, when a thought occurred to me—I might be able to use the Fifa ranking data to demonstrate that Sven-Göran Eriksson’s England team was only able to beat lesser teams and often struggled against higher ranking sides. In short, I chose to take a leaf from the Trump playbook—when you’re in trouble, smear the opposition.

Team Pld Pts Ranking England 10 25 9 Poland 10 24 23 Austria 10 15 72 (=) Northern Ireland 10 9 101 Wales 10 8 72 (=) Azerbaijan 10 3 113

OK, so it’s not classy but, in this case, I think it is a valid point to explore. Were most of Sven’s competitive games against weaker opposition? This is a possibility because qualifiers and group games are seeded, and so England would be facing socalled lesser teams. For example, let’s consider the 2006 World Cup qualifying group (right).

Only Poland finished ranked in the world’s top 50 international teams, which supports my contention that England were flat-track bullies under Sven. But this raised two interestingf questions:

1. How are the Fifa rankings calculated? 2. How can I use them to win this argument?

The final positions and 2005 Fifa world rankings of Uefa group 6 in the 2006 World Cup qualifying.

f Terry Venables’ stats are decimated here because in the run up to Euro 96, England were only playing friendlies because they had already qualified as hosts. f in my opinion

The Fifa ranking system

The Fifa ranking system was introduced in December 1992, and initially awarded teams points for every win or draw, like a traditional league table. However, Fifa quicklyf realised that there were many other factors affecting the outcome of a football match and, over timef moved to a system based on the work of Hungarian–American mathematician Árpád Élő (more on him in a moment).

The Fifa rankings are not helpful to me because they don’t cover all the managers I’m considering and because their accuracy, reliability and the many methods used to generate them were always questioned. Luckily, football fans have had these arguments before and there is an Elo ranking for all men’s international teams, which has been calculated back to the first international between England and Scotland in 1872 (a disappointing goalless draw).

Wikimedia commons user BaldL, CC BY-SA 4.0

Competitors in an esports event. The Elo rating system compares the relative performance of the competitors in two-player games. Although it was initially developed for rating chess players, variations of the system are used to rate players in sports such as table tennis, esports and even Scrabble. Strictly speaking, we should be saying an Elo system, rather thantheElo system as each sport has modified the formula to suit their own needs.

So how does an Elo system calculate a ranking? Well, at the most basic level, each team has a certain number of points and at the end of each game, one team gives some points to the other. The number of points depends on the result and the rankings of the two teams. When the favourite wins, only a few rating points will be traded, or even zero if there is a big enough difference in the rankings (eg in September 2015, England beat San Marino 6–0, but no Elo points were exchanged). However, if the underdog manages a surprise win, lots of rating points will be transferred (for example, when Iceland beat England at Euro 2016, they took 40 points from England). If the ratings difference is large enough, a team could even gain or lose points if they draw. So teams whose ratings are too low or too high should gain or lose rating points until the ratings reflect their true position relative to the rest of the group.

But how do you know how many points to add or take away after each game? Elo produced a formula for this, but there is a bit of maths—brace yourself.

Firstly, Elo assumed that a team would play at around the same standard, on average, from one game to next. However, sometimes they would play better or worse but with those performances grouped towards the average. This is known as a normal distributionf f or bell curve, where outstanding results are possible but rare. In the graph, the ��-axis would represent the level of performance, and the ��-axis shows the probability of that happening. So, we can see that the

f five years later f Over twenty years. I mean, why use an established and respected system when you can faff about making your own useless one? To be fair, the women’s rankings have used a version of the Elo system since their inception, which may make Fifa’s unwillingness to use it for the men even stranger. f f Elo uses a logistic distribution rather than the normal, but the differences are small (I mean, what’s a couple of percent between friends?).

chance of an exceptional performance is smaller than that of an unremarkable one and the bulk of games will have a middling level of skill shown.

This means that if both teams perform to their standard, we can predict an expected score, which Elo defined as their probability of winning plus half their probability of drawing. Because we do not know the relative strengths of both teams, this expected score is calculated using their current ratings and the formulas

and

Expected score ( �� ��.�� ��.�� ��.�� −��,������ −������ �� Difference in rating ( ������ ���� −����) ��,������ In these formulas, ���� and ���� are the expected results for the teams, and ���� and ���� are their ratings. If you plot a graph of the �� values for different values of ����− ���� you get the graph shown to the left. It’s interestingf to note the shape of this graph, which is a sigmoid, a shape that anyone who has drawn a cumulative frequency graph for their GCSE maths will recognise. It is an expression of the area under the distribution (ie the cumulative distribution function). The

The expected score for a range of differences in graph shows that if the difference between ratteam ratings. ings is zero, the expected result is 0.5. The system uses values of 1 for a win, 0.5 for a draw and 0 for a loss, so this suggests a draw is the most likely outcome. And if the difference is 380 in your favour, the expected score is 0.9, which suggests you are likely to winf . The system then compares the actual result to the expected outcome and uses a relatively simple calculationf f to calculate the number of points exchanged:

��′ ��= ����+ ��(����− ����). In this equation, ��′ �� is the new rating for team A, ���� is the actual result of the game, and �� is a scaling factor. We’ll come back to�� in a moment. Recently, England (rating 1969) played Belgium (rating 2087) at the King Power stadium in Leuven, Belgium. It is generally thought that the home team is at an advantage and to reflect this, the home team gets a bonus 100 points to their rating which means there is a 218-point difference between the teams. England are clear underdogs, and we can calculate the expected result as follows:

f f Again, interesting to me. An���� of 0.9 doesn’t necessarily mean you’ll win 90% of the games and lose the rest as other combinations also give an expected score of 0.9. For example, winning 80%, and drawing the rest or winning 85%, drawing 10% and losing 5% gives the same value. f f Honestly, it’s easier than it looks.

This shows that this will be a tricky game for England, and a draw would be a good result. Unfortunately, England lost the game 2–0, an ���� of 0 (still using 1 for a win, etc). Therefore we can calculate the rating change using the formula:

��′ ��= �������� + ��(�� − ��.����) Now we need to understand the�� value. In simple terms, the bigger the�� value we use, the more the rating will change with each result. We need to choose a suitable value so that it isn’t too sensitive, which would lead to wild swings, but also allows for teams to change position when they start to improve.

��′ ��= �������� + ����(�� − ��.����) ≈ �������� The world football Elo rankings adjust the �� value depending on the score and the competition. In our example, which was a Nations League game (a new competition between European teams with similar Fifa rankings), the base value for�� is 40. This is multiplied by 1.5 for a win by 2 clear goals giving a�� value of 60.

This is a change of−����points, and so Belgium would change by+����points to a new rating of 2100.

Although I have focused on the world football Elo rankings, the Fifa rankings now use a system which is basically similar, with slight variations in the weightings and allowances. This brings me to the second, and more important part, of the question: can I use this to prove that I’m right?

Unfortunately, this explanation shows that you can only use this type of ranking, whether it’s the Elo or the Fifa system, to compare with teams that were playing at that time. This means that trying to use it to look back over time is pointless. You can’t compare the performance of Alf Ramsey’s England with that of Steve McClaren using the Elo rankings, because it is not designed to do that.

What can I do?

I can, however, use a similar idea— looking at England’s performance against differently rated teams—to judge Sven.

To achieve this, I’ve collated all of England’s results in competitive games under Sven and used some spreadsheet magic to create the tables shown to the right.f This is conclusivef. Under Sven-Göran Eriksson, England were brilliant—if the team they were playing were outside the top twenty. Against good teams, England were awful. For comparison, in the 2020–21 season, Manchester United have a win percentage of 63.2% and a points percentage

P W D L F A Win % Points %

11 4 3 4 18 10 36.4% 45.5%

P W D L F A Win % Points %

27 22 4 1 51 15 81.5% 86.4%

Performance of England under Sven-Göran Eriksson against teams in the top 20 (top) and teams outside the top 20 (bottom). The full data is available at d chalkdustmagazine.com.

f Do not ask how long this took. f It is. Just trust me on this.

of 70.2%. On the other hand, Chelsea had a win percentage of 42.1% and a points percentage of 50.9% (based on results up to 27 January 2021), and they sacked the manager.

I can finally conclude that I was right. Sven was a rubbish manager who was worse than Frank Lampard.

Paddy Moore Paddy has been a maths teacher for over 25 years. He is a proud nerd and a perpetually disappointed football fan. a @PaddyMaths

My least favourite game Poker

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My favourite game Nim

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At first it’s a kinda fun, surprisingly complicated, strategy game. Then you Google the winning strategy and find it involves a pretty disappointing and unintuitive operation on binary numbers called XOR- or Nim-sum which comes out of nowhere.

Deflated but not defeated, after much head scratching it transpires that the losing states of the game form a beautiful��-dimensional Sierpiński’s triangle elegantly described by XOR-sum, and which is strategically intuitive. And then after more Googling you learn that XOR-sum actually works for a huge number of games in a really fascinating way… or is that just me?