A while back we had a note from a reader who suggested that it was not unreasonable to expect each year that Arsenal should win the title or at least be challenging for the title up to the last few weeks of the season.
In reply Untold showed that such a situation had never been achieved by any club year after year, and indeed in most seasons no club other than the eventual winner was looking like winning until the last few weeks.
To explore this further I started looking at another football situation which looks as if it ought to have a simple, predictable outcome, but in fact doesn’t.
I set up a simple model of a 20 team league. Team A averaged 2 goals per game, all 19 other teams average only 1 goal per game. And I had the computer set to let them play the same 38 game season the EPL plays. In fact I ran quite a few “seasons” to see if common sense prevailed.
Now you would obviously expect Team A to win the league every year in the same way that the German league is won most years by Bayern Munich. But what I wanted to know was what would happen if the computer followed the rule of Team A averaging two goals a game and the others one goal per game: would Team A always win?
Well, strange as it might seem, Team A doesn’t always win the league. The highest point total where Team A is not the outright winner is 77, in fact it is tied in that season with Team 2. Team 14 came in 3rd with 65 points.
In another random season, despite scoring on average two goals a game Team A ends with 73 points, but Team 16 (averaging only one goal a game) wins the league with 74 points. Team A has to settle for second, with Team 2 at 59 points and Team 4 at 58 points.
In another “season” at 72 points we have Team 5 winning one year. In another “season” Team 19 wins with 73 points, Team A having to settle for second with 72 points. And in yet another Team A drops to third with 71 points, with the league winner has 79 and the second place team 75.
Looking at the worst seasons for Team A, one season it only got 64 points and came 4th, six points behind the winner. And worst of all it actually went down 52 points one season, while the team in 6th place got 60 points.
So it goes on: even with the set formula that Team A always average two goals per game across the years it can get between 53 and 99 points.
Meanwhile at the bottom teams were able to be relegated with anything between 25 and 46 points.
So what does this tell us?
Basically that even with the league fixed in a way that common sense suggests should result in Team A always winning the league, it doesn’t always happen. The vagaries of chance get in the way and knock Team A down the league.
And these vagaries happen without ownership changes, with no management changes, while players never get tired or injured and play the same all season long. Oh, the officiating was perfect and impartial, not like what this crap PGMO provides. And each of these surveys was 50 times as long as Wenger has been at Arsenal. The aaa must be turning in their graves because it turns out that even with a league so fixed that one club averages two goals a game and the rest one goal a game, they are still not guaranteed to win the league each season.
Just to look at another variation instead of a German “one club” League approach I tried a “top 4” approach where four teams score an average of two goals a game and 16 get an average of one.
In this scenario the lowest observed point total difference between the teams in the “Top 4” is 9 points and the maximum is 37. And not surprisingly, there are circumstances where not all of the “Top 4” finish in the top 4. The lowest point total I have found 20th place, with 21 points. And that in a league where four clubs get an average of two goals a game and the rest an average of one goal a game.
What this shows is that the vagaries of even a totally fixed league can get in the way of endless winning.
Do like Tony suggests, and research things. Don’t be a spade and just shovel your advice at people.
Around 1 million Poisson random deviates were created to produce this document. Thanks to Math::Random::MT::Auto, the Mersenne Twister supplied all the randomness needed, and /dev/random never got depleted.
The purpose of computing is insight, not numbers. “Numerical Methods for Scientists and Engineers” by R.W.Hamming. Dover Press, 1962 and 1973
Or to put it another way, “To study, and when the occasion arises to put what one has learned into practice – is that not deeply satisfying?” Confuscious, Analects 1.1.1
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