It's not a complete coincidence. Very many sets of random numbers are uniformly distributed on a logarithmic scale. See for example Benford's Law.
Fibonacci numbers are just a rounded version of phi^x. So the only coincidences are 1. that the number of teams is such that phi is a good base, and 2. that the rounding all happened to go the right way.
There is no "the number of teams", because this league has a promotion/relegation system. There are 20 teams in the league at any point in time, but 150 or so professional clubs and tens of thousands of others who participate in the league pyramid and have at least a theoretical chance of promotion and ultimately winning the premier league.
There's a fair bit of churn in these numbers: 51 clubs in total have been in the premier league in the period of interest (the last 33 years).
After some small threshold, I think the number of clubs doesn't matter. You could get the same result if the top 100 or all 40,000 clubs played in the same league every year, ignoring the minor scheduling problem this would cause. Resources are distributed approximately in a power law, as you suggest. What matters is the level of inequality near the top, which is apparently such that each team has approximately phi times the resources (measured over a long period) of the team below.
In this case, it's a complete coincidence. I did this for UEFA Champions League winners over the last 33 years and found no Fibonacci sequence at all. I'm pretty sure it won't be found easily in other sports or competitions either.
A good observation, but it is also a coincidence of sorts that the sport lends itself to this distribution. Does it mean the league is especially fair? Or not fair? You wouldn’t expect to see this distribution in games of pure chance, or of pure monopolistic dominance.
It's absolutely not a game of pure chance, nor a level playing-field each year. Owners can decide how much to invest each year (unlike the US MLS) subject to some limits (UEFA Financial Fair Play Regulations) on how much. And many of these teams have been bought/sold/had stock market IPOs multiple times in that period. And also each team has widely differing revenue streams from TV rights (across all competitions, including Champions League and Europa League) and merchandise sales, with which to fund player salaries/transfers and stadium renovations.
If you redo this table to quote PL first/second/third/fourth position per £ pound invested, (or total points in a season (3 for a win, 1 for a draw, 0 for a loss)) you get a different picture, e.g. for 2023-4 season: https://www.dailymail.co.uk/sport/football/article-13446423/...
And even the Financial Fair Play regulations are nowhere close to the measures common in most American sports, like salary caps. They’re mostly intended as a speed bump for foreign oil magnates and Red Bull throwing all their money into their clubs and automatically winning everything.
Not even that, it's to prevent clubs going bankrupt all the time.
Without rules, every club is going to invest huge sums betting on achieving a top result that would give return on investment, and only a few can achieve that each year.
Investors throwing lots of money at the sport is a feature, not a bug, from the viewpoint of the people involved in the sport. So by itself that doesn't need to be prevented.
On the other hand you have the selection effect that someone only writes if there is a nice Fibonacci sequence somewhere in the top 30 or so leagues. So I agree, the distribution favors probably power laws with (1+ 2/3)^n but that it happened now is still kinda random.
When there are more teams but the min/max skill is unchanged, they must be closer together, so the base is lower. When there are fewer teams, farther apart, the base is higher.
The number of teams is constrained by "we need enough teams so there's actually variety" and "we can't have so many teams that we can't keep track of them all".
Benford's law is as it is because it reflects the structure of our decimal system having a logarithmic pattern where each order of magnitude gets its own column. It's purely a function of the way we represent the data itself, not the underlying numbers. I don't think these football scores have anything to do with it. Likewise yes in nature this pattern appears a lot due to its great space-filling without overlapping property which has been evolved towards. The football scores thing is pure fluke.
The common definition of Benford's Law is written in terms of base 10 (or 100, 1000, etc. for subsequent digits), but the underlying principle applies to any base (I'm pretty sure this includes non-integer bases, though I'm not sure about negative bases).
I feel I must point out we have twenty league championships and calling it two as if football started in 1992 while perfectly understandable for this amusing exercise still pisses me off.
You're correct to say Liverpool have won 20 titles. But it's also fair to differentiate between eras in which the game was fundamentally different. The Premier League's launch fundamentally altered the commercial side of the game, and the back-pass rule (same year) fundamentally altered the game itself. So it's understandable why the study confines itself to the Premier League era.
I don't really agree. You can also draw a semi-arbitrary line in the mid-2000s, when foreign oil money suddenly made Chelsea and City rise to the top of the league. The game evolves and that always has been the case. Other leagues underwent similar transformations, but they don't make this artificial bureaucratic distinction.
That's a stretch. Just because the rights were sold differently doesn't mean football was "fundamentally different" from one year to the next. Some tweaks to the game (and the ball) have been made over the years, but the game is fundamentally the same.
This really isn't true. The object of the game might be the same. But the backpass rule altered the game mechanics in a way that makes these separate eras impossible to compare in any kind of study.
> But the backpass rule altered the game mechanics
Nah, it didn’t. You’re overstating.
My memory of that time was general confusion of “why the name change?” And not much else. The game has certainly been tweaked, but your claim of it being fundamentally different doesn’t bear scrutiny.
Yes it did. The rule change ended the profoundly negative winning method of a team scoring a goal and then killing the game by passing the ball between defenders and the goalkeeper, who could scoop it up under any pressure. It isn't an overstatement to say this fundamentally altered the game.
> it's also fair to differentiate between eras in which the game was fundamentally different.
Nah, it's just a lazy justification for breathless pundits and journalists to hail "the best ever" this or that, dropping 20th-century football history into a memory hole. The likes of Chelsea or Manchester City can "break records" every other year, generating easy work for the commentariat classes, and reminding little people that the only thing that matters is money. "Oh, your club was a league founder? But you're in the bananarama league now, sucks to be you."
Football statistics are always hilarious with random cutoffs and clauses to make them work. The game commentators are like: this is the first player to have 5 yellow cards in a season with have a left foot and being born in the year of the snake
It’s not a sequence if you produce it by ordering aggregates. Next year no matter what happens the sequence will be broken until at best everyone advances one place (so best case scenario in 21 years), or a new team wins 21 straight titles.
Now if in twenty one years the distribution has been maintained, there’s one new team and all the other teams have won Fib(n+1) - Fib(n) titles, or one team has won 21 straight titles, reproducing the sequence, I’ll come back in 2046 and eat my hat.
My Premier League title predictions for the next 21 years: 2026 Man City, 2027 Man Utd, 2028 Newcastle (or Sunderland, to keep things interesting in the North East. But not Spurs), 2029 Man Utd, 2030 Arsenal, 2031 Chelsea, 2032 Man City, 2033 Man City, 2034 Man Utd, 2035 Arsenal, 2036 Man Utd, 2037 Chelsea, 2038 Man Utd, 2039 Man Utd, 2040 Leicester C (silver jubilee of Richard III reburial), 2041 Man City, 2042 Man Utd, 2043 Man City, 2044 Chelsea, 2045 Man Utd, 2046 Liverpool (to complete the mysterious cycle).
If anyone needs tonight's winning lottery numbers, let me know ...
I just extended the sequence - Utd needed 8 more wins to get to 21, City needed 5 to get to 13, etc. The only prediction I'll put money on is Leicester winning in 2040 because the only thing that explains their 2015 win was some divine intervention to reward the city for rescuing an anointed king from a car park and reburying him in their cathedral that year.
I'd go further and suggest that has tipped right over into insanity. I've been following them since 1985 and I'm pretty much resigned to never seeing another United EPL win in my lifetime (because it has been an absolute shit show since Ferguson left with no real signs of concrete progress.)
It's much less interesting than it seems because the data is aggregated and then sorted.
I don't know what the exact odds are, but if you waited until one team had won 13 times, you'd often end up with a distribution fairly similar to this.
> while fans of the club will no doubt be celebrating this moment of triumph, another astounding facet of their achievement has caught the attention of mathematicians
Its quite a good coincidence as the other 2 teams with 1 victory had no chance as 1 was already relegated previously and the the 2nd will be relegated . So any other team apart from Liverpool winning would have stopped the sequence from being created.
Reading the comment I can see a lot of people don't understand the meaning of the words coincidence or sequence.
...which may yet get undone if City are stripped of titles as a result of their 115 charges.
(I'm not holding my breath - my prediction would be the lightest slap on the wrist possible. Maybe a transfer ban for one window. Probably a 10 point deduction for Everton.)
I once had a room filled with an infinite number of monkeys typing on typewriters and one of them typed the source code of Windows ME. Now that's what I call mysterious!
It's not a complete coincidence. Very many sets of random numbers are uniformly distributed on a logarithmic scale. See for example Benford's Law.
Fibonacci numbers are just a rounded version of phi^x. So the only coincidences are 1. that the number of teams is such that phi is a good base, and 2. that the rounding all happened to go the right way.
There is no "the number of teams", because this league has a promotion/relegation system. There are 20 teams in the league at any point in time, but 150 or so professional clubs and tens of thousands of others who participate in the league pyramid and have at least a theoretical chance of promotion and ultimately winning the premier league.
There's a fair bit of churn in these numbers: 51 clubs in total have been in the premier league in the period of interest (the last 33 years).
After some small threshold, I think the number of clubs doesn't matter. You could get the same result if the top 100 or all 40,000 clubs played in the same league every year, ignoring the minor scheduling problem this would cause. Resources are distributed approximately in a power law, as you suggest. What matters is the level of inequality near the top, which is apparently such that each team has approximately phi times the resources (measured over a long period) of the team below.
In this case, it's a complete coincidence. I did this for UEFA Champions League winners over the last 33 years and found no Fibonacci sequence at all. I'm pretty sure it won't be found easily in other sports or competitions either.
OP says it is more likely than many other random sequences, not that it happens all the time. IMO he's right FWIW.
A good observation, but it is also a coincidence of sorts that the sport lends itself to this distribution. Does it mean the league is especially fair? Or not fair? You wouldn’t expect to see this distribution in games of pure chance, or of pure monopolistic dominance.
It's absolutely not a game of pure chance, nor a level playing-field each year. Owners can decide how much to invest each year (unlike the US MLS) subject to some limits (UEFA Financial Fair Play Regulations) on how much. And many of these teams have been bought/sold/had stock market IPOs multiple times in that period. And also each team has widely differing revenue streams from TV rights (across all competitions, including Champions League and Europa League) and merchandise sales, with which to fund player salaries/transfers and stadium renovations.
If you redo this table to quote PL first/second/third/fourth position per £ pound invested, (or total points in a season (3 for a win, 1 for a draw, 0 for a loss)) you get a different picture, e.g. for 2023-4 season: https://www.dailymail.co.uk/sport/football/article-13446423/...
And even the Financial Fair Play regulations are nowhere close to the measures common in most American sports, like salary caps. They’re mostly intended as a speed bump for foreign oil magnates and Red Bull throwing all their money into their clubs and automatically winning everything.
Not even that, it's to prevent clubs going bankrupt all the time.
Without rules, every club is going to invest huge sums betting on achieving a top result that would give return on investment, and only a few can achieve that each year.
Investors throwing lots of money at the sport is a feature, not a bug, from the viewpoint of the people involved in the sport. So by itself that doesn't need to be prevented.
> They’re mostly intended as a speed bump
Sadly aren't even succeeding at that given City's 115 charges (dating back to 2008!) and various clubs with their self-sponsorship shenanigans.
That’s how some people drive over speed bumps!
> subject to some limits (UEFA Financial Fair Play Regulations)
Unless you have enough oil money to just... not care about the rules.
Winners get money and money buys good players. So I think that's the source of the distribution.
On the other hand you have the selection effect that someone only writes if there is a nice Fibonacci sequence somewhere in the top 30 or so leagues. So I agree, the distribution favors probably power laws with (1+ 2/3)^n but that it happened now is still kinda random.
> that the number of teams is such that phi is a good base
Any theories why this might be the case?
> 2. that the rounding all happened to go the right way.
+1. See how close some title races are: 2013-2014 comes to mind.
When there are more teams but the min/max skill is unchanged, they must be closer together, so the base is lower. When there are fewer teams, farther apart, the base is higher.
The number of teams is constrained by "we need enough teams so there's actually variety" and "we can't have so many teams that we can't keep track of them all".
2018-2019 and 2021-2022 were decided by a point, 2011-2012 on goal difference.
But this isn't just evenly random, the teams most certainly do not have comparable chances of winning.
Benford's law is as it is because it reflects the structure of our decimal system having a logarithmic pattern where each order of magnitude gets its own column. It's purely a function of the way we represent the data itself, not the underlying numbers. I don't think these football scores have anything to do with it. Likewise yes in nature this pattern appears a lot due to its great space-filling without overlapping property which has been evolved towards. The football scores thing is pure fluke.
The common definition of Benford's Law is written in terms of base 10 (or 100, 1000, etc. for subsequent digits), but the underlying principle applies to any base (I'm pretty sure this includes non-integer bases, though I'm not sure about negative bases).
No matter what base you use, you're using a numeral system that has an inherent logarithmic distribution of one column per order of magnitude
The sequence starts with Spurs 0
Poor Spurs.
Now even Hacker News isn't a safe space for us Spurs fans :(
I feel I must point out we have twenty league championships and calling it two as if football started in 1992 while perfectly understandable for this amusing exercise still pisses me off.
It’s a funny thing, fandom.
You're correct to say Liverpool have won 20 titles. But it's also fair to differentiate between eras in which the game was fundamentally different. The Premier League's launch fundamentally altered the commercial side of the game, and the back-pass rule (same year) fundamentally altered the game itself. So it's understandable why the study confines itself to the Premier League era.
I don't really agree. You can also draw a semi-arbitrary line in the mid-2000s, when foreign oil money suddenly made Chelsea and City rise to the top of the league. The game evolves and that always has been the case. Other leagues underwent similar transformations, but they don't make this artificial bureaucratic distinction.
> fundamentally different
That's a stretch. Just because the rights were sold differently doesn't mean football was "fundamentally different" from one year to the next. Some tweaks to the game (and the ball) have been made over the years, but the game is fundamentally the same.
This really isn't true. The object of the game might be the same. But the backpass rule altered the game mechanics in a way that makes these separate eras impossible to compare in any kind of study.
> But the backpass rule altered the game mechanics
Nah, it didn’t. You’re overstating.
My memory of that time was general confusion of “why the name change?” And not much else. The game has certainly been tweaked, but your claim of it being fundamentally different doesn’t bear scrutiny.
Yes it did. The rule change ended the profoundly negative winning method of a team scoring a goal and then killing the game by passing the ball between defenders and the goalkeeper, who could scoop it up under any pressure. It isn't an overstatement to say this fundamentally altered the game.
> it's also fair to differentiate between eras in which the game was fundamentally different.
Nah, it's just a lazy justification for breathless pundits and journalists to hail "the best ever" this or that, dropping 20th-century football history into a memory hole. The likes of Chelsea or Manchester City can "break records" every other year, generating easy work for the commentariat classes, and reminding little people that the only thing that matters is money. "Oh, your club was a league founder? But you're in the bananarama league now, sucks to be you."
Football statistics are always hilarious with random cutoffs and clauses to make them work. The game commentators are like: this is the first player to have 5 yellow cards in a season with have a left foot and being born in the year of the snake
it does say premier league title wins though so it's technically correct.
Really cool!
But in your next interview, just know that this is not the most computationally efficient way to generate the sequence.
It’s not a sequence if you produce it by ordering aggregates. Next year no matter what happens the sequence will be broken until at best everyone advances one place (so best case scenario in 21 years), or a new team wins 21 straight titles.
Now if in twenty one years the distribution has been maintained, there’s one new team and all the other teams have won Fib(n+1) - Fib(n) titles, or one team has won 21 straight titles, reproducing the sequence, I’ll come back in 2046 and eat my hat.
My Premier League title predictions for the next 21 years: 2026 Man City, 2027 Man Utd, 2028 Newcastle (or Sunderland, to keep things interesting in the North East. But not Spurs), 2029 Man Utd, 2030 Arsenal, 2031 Chelsea, 2032 Man City, 2033 Man City, 2034 Man Utd, 2035 Arsenal, 2036 Man Utd, 2037 Chelsea, 2038 Man Utd, 2039 Man Utd, 2040 Leicester C (silver jubilee of Richard III reburial), 2041 Man City, 2042 Man Utd, 2043 Man City, 2044 Chelsea, 2045 Man Utd, 2046 Liverpool (to complete the mysterious cycle).
If anyone needs tonight's winning lottery numbers, let me know ...
That’s… a very optimistic prediction for United.
I just extended the sequence - Utd needed 8 more wins to get to 21, City needed 5 to get to 13, etc. The only prediction I'll put money on is Leicester winning in 2040 because the only thing that explains their 2015 win was some divine intervention to reward the city for rescuing an anointed king from a car park and reburying him in their cathedral that year.
I'd go further and suggest that has tipped right over into insanity. I've been following them since 1985 and I'm pretty much resigned to never seeing another United EPL win in my lifetime (because it has been an absolute shit show since Ferguson left with no real signs of concrete progress.)
It's much less interesting than it seems because the data is aggregated and then sorted.
I don't know what the exact odds are, but if you waited until one team had won 13 times, you'd often end up with a distribution fairly similar to this.
> while fans of the club will no doubt be celebrating this moment of triumph, another astounding facet of their achievement has caught the attention of mathematicians
mutual exclusivity most subtle
Its quite a good coincidence as the other 2 teams with 1 victory had no chance as 1 was already relegated previously and the the 2nd will be relegated . So any other team apart from Liverpool winning would have stopped the sequence from being created. Reading the comment I can see a lot of people don't understand the meaning of the words coincidence or sequence.
+---------------------+------------------------+
| Club | Premier League Titles |
+---------------------+------------------------+
| Blackburn Rovers | 1 |
| Leicester City | 1 |
| Liverpool | 2 |
| Arsenal | 3 |
| Chelsea | 5 |
| Manchester City | 8 |
| Manchester United | 13 |
+---------------------+------------------------+
I really like the observation that some very unlikely events happen all the time because there are just so many different ones that could happen.
When a pope dies, Avellino (italian football team) gets promoted. Fwiw.
It's funny, but not true. I apologze for the link in Italian.
https://www.ilpost.it/2025/04/22/avellino-promosso-morte-di-...
Get out of here with your "facts" - this is the internet and every meme is obviously true.
This could be a substory in Terry Pratchett's Discword. Some eccentric wizard combining mathematics with football.
Only the second time this has happened!!1!1!
...which may yet get undone if City are stripped of titles as a result of their 115 charges.
(I'm not holding my breath - my prediction would be the lightest slap on the wrist possible. Maybe a transfer ban for one window. Probably a 10 point deduction for Everton.)
Neat. We have some wiggle room to rig it by deciding what year to start the Premier League Table Wins (1992 or a more recent year).
I once had a room filled with an infinite number of monkeys typing on typewriters and one of them typed the source code of Windows ME. Now that's what I call mysterious!
Will there a new club pop-up getting 21 wins in a row declassing all other or are we at the end of times?
I absolutely cannot wait to hear what QAnon has to say about this mathematical pattern occurring here.
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