Maybe they've changed it, but I believe 1st-3rd gain CSR and 4th-6th lose CSR. The amount of CSR gained or lost depends on your rank vs the ranks of those you beat or lost to (as well as your overall placement). If a Diamond player beats a Champ they gain more CSR than if they beat a Gold. If a Diamond player loses to a Gold they lose more CSR than if they lost to a Champ. I hope that helps.
work like this, but it doesn't. The vast majority of players that are mid diamond and up will lose CSR for placing anything but first. It starts pretty mild, but gets way worse the higher up the ranks you get. Mid diamond you may see a few CSR gain for placing second or third, but by the time you're past 1600 anything under first will result in a massive loss of CSR. My theory is as to why this happens is because FFA has such a low population count the matchmaking perimeters must be extremely wide just to find others to play with resulting in onyx players matching a wide variety of ranks. As FFA is right now only winning first counts truly as a "win". If the game sees that you a Diamond 5 player lost to a Platinum 6 player it will make the judgment that your rank is too high and it needs to be lowered even though you did well comparatively to the rest of the lobby.
Unfortunately the FFA playlist is in a downward cycle. It has a low population, so matchmaking perameters must be widened. High level players then start matching low level players and it becomes impossible to progress. High level players are frustrated because they can't progress so they stop playing, and low level players are frustrated because they get matched against players miles ahead of them and they quit playing too.
If 343 wants any shot at FFA becoming a enjoyable playlist again they are going to have to change something to attract more players. Personally I think if they added HCS settings it would give it enough traction to be able to set reasonable parameters and have reasonable wait times. I can't find it now, but Ske7ch made a twitter poll a while back asking what version of FFA fans would prefer to play, and the vast majority of people said HCS. I believe they haven't done it yet because they don't think the risk is worth it and that it would just end up splitting the population even more. Personally I think FFA is already at rock bottom, and almost anything is worth a shot. Very disappointing as FFA used to be one of my favorite modes and now I can't even stand to play more than a few matches.
Everything above is from my personal experience. I am certainly not an expert on the CSR system, so take what I said with a grain of salt.