There are 20 teams in the English Premier League (EPL) and each team plays 2 games against any other team (one at home and one as a guest). A win is rewarded with 3 points, a draw is 1 point and loss is 0 points. At the end of the season the bottom 3 teams are relegated (and replaced with the best performing 3 teams from the lower level).
There is a popular believe in the EPL, that you need to collect 40 points to avoid relegation. I want to find out:
- what is the real number of points which guarantee that you stay up;
or if this is too hard are 40 points really enough?
P.S. If two teams have equal number of points at the end of the season, other criteria take place, like better goal difference, away goal, etc. This means, that if the 17th and 18th team have the same number of points, it's not a guarantee by itself.
Here is how far I got so far:
- The number of games per season are 380.
- Each game can have one of 3 outcomes: win for the hosts, win for the guests, or a draw.
- Which means, that there are $3^{380}$ possible tables at the end of the season.
Of course, a lot of these are really duplicates: for example if t1 wins at home against t2 and looses away, is the same (for the final table) as if they lost at home and won away.
Also, "renaming" the teams (e.g. $t_1$ becomes $t_2$, $t_2$ becomes $t_3$, $t_3$ becomes $t_1$) would produce the same table (for our problem). That is, it doesn't matter (for us) if t1 finishes first or is relegated, only the points of the first team, the second team, etc.
We don't even care about the points of the first 16 teams, we only care about the points of the 17th and 18th places.
But I don't know how to use these features to make sure we don't get any duplicates.
So, how can I find that out?
P.S. If I can reduce the number of possible tables to something which a computer can process, (and a way to construct them of course), this would be fine.
[EDIT]
I just realized, that there's an easy way to prove that 40 points aren't enough: If each team win their home games (and thus looses it's away games), they all have 19*3 = 57 points at the end of the season. So even 57 points aren't enough to guarantee the 17th place.
Best Answer
The smallest number of points guaranteeing you to stay up is $64$.
The number is $>63$: In an extreme case, two teams lose all their games vs. the other $18$ teams, and in all the pairings of two of the other $18$ teams, one game is won and the other one lost. Then in the end, two teams have $\leq 6$ points, and all the other teams have $21\cdot 3 = 63$ points. So with $63$ points, you are still not safe, necessarily.
The number is $\leq64$: Assume you have $\geq 64$ points and still you end up in $\geq 18$th place. Then the first $17$ teams all have $\geq 64$ points. So in total, there are at least $18\cdot 64 = 1152$ points. However, the theoretical maximum is $3\cdot 380 = 1140$, contradiction.