This is an attempt to describe, roughly, how the rating goes, so that I can copy-paste in other threads about it. I am not an expert, but I believe I have a good understanding of the basics of such systems. Feel free to correct me where required.
The Battleground’s may be a bit different though. Also, there could be bugs still, so please share your concerns if something looks odd.
The rating system is there to try to estimate your true strength, so that in a 1v1 situation you end up winning 50% of the time. If you win, your rating goes up, if you loose, your rating goes down. By how much? I depends on the strength of your opponent. If you win against someone with a higher rating than yours, then your rating goes up more than if you loose against someone lower than you. When you face someone rated higher than you, you are expected to loose more often than not. Extreme example: if a chess Grand Master plays against a 9 year old who just learned the rules, if they win, they will not go up in ranking, because that was a trivial win. If they loose though, their ranking would go down dramatically (assuming a game like this would be ranked at all!)
In an 8-way free for all, the simple example above gets more complicated. Honestly, I do not know how it is implemented, but here is a guess:
Your rating goes up or down based on each pairs between you and your opponents. 4 player example, with ratings 3000, 4000, 5000 and you at 4500. If you finish 2nd behind the player ranked 3000, then you rating will go up because you beat the one at 4000, it will go up even more because you beat the one at 5000 (who was ranked higher than you) and it will go down quite a bit because you lost to the player ranked 3000, which is much lower than you.
So, if the above is somewhat accurate, depending on the relative ratings of all players involved, you could end up getting a positive, neutral or negative adjustment to your rating, even though you finished 2nd out of 4 (or 4th out of 8).
The game matching algorithm is (or should be anyways) trying to match you with people with as close to the same rating as yours, so in most cases, you should be getting a small positive at 4th place, a slightly bigger positive at 3rd, etc., with negative starting at 5 and below. But since there is a limited number of players, if you end up playing against opponents a bit lower ranked, and they finish higher, you could end up with a small negative rank adjustment even with 4th place.
The “win” at 4th or higher is only really relevant for the quests (win X games) and the 3 wins = 10 gold.
Note that the above cannot explain why some people see a +0 adjustment to their ranking when they finish 1st. That should not happen, unless somehow they are very highly ranked, and playing against 7 players who are incredibly low rank, and such a win is trivial. That is likely a bug.
Gotta go, might edit this tomorrow.