I want to rank the 20 teams in the English Premier League, say that each team are assigned to the number 1 through 20, defining their ranking, no ties. There would be $20!$ number of permutations for the ranking assignment.
The final ranking would be the ranking assignment that minimizes
$Loss_1=$ number of matches where the lower ranked team beats the higher ranked team $-$ number of matches where the higher ranked team beats the lower ranked team
Among the 380 matches played in a season.
Does such a ranking algorithm exist? I tried to do it manually but iterating through $20!$ different permutations is very slow, I'm looking for a more efficient approach.
If possible, I'd also like ask the same question for these choices of Loss function:
$Loss_2=$ number of matches where the lower ranked team beats the higher ranked team
$Loss_3=$ $-$ number of matches where the higher ranked team beats the lower ranked team
And whether any of the 3 choices of Loss function are equivalent (always resulting on the same ranking assignment).
Yes, these are all equivalent. As the number of games the lower ranked team wins and the number of games the higher ranked team wins add up to all the games you can derive one from the other. The first will have twice the amplitude of the other two and the last two will be offset by a constant addition of the number of games.