Some things I see happening with this and some offshoot ideas:
- checkmating your opposite doesn't put you at a disadvantage as you win the game on the spot, so it wouldn't be discouraged.
- it encourages active play, because if you play passively and defensively you will miss out on first place; encourages play against the kings
Some possible ideas:
- To discourage players from playing solidly and building up a better position so that they can have an advantage in the fight for second after two others are gone, one possible scoring would be + points for 1st, 0 points for 2nd, - points for 3rd and 4th. Then you are really motivated to go for first.
- Another possibility is that when one person gets checkmated the game is simply over and only the person who won \ got checkmated have any changes to their ratings, while the other two stay put. Kind of weird and very different, more similar to Teams mode, but two players were essentially bystanders. I think this could lead to some really fun situations though, where two players are in a terrible position and trying to hold on as long as possible to save themselves.
- I think some variation of this could be a decent system in the case of all four players trying to win and being committed but I think it can become very problematic when players resign \ make huge blunders and hand somebody the win etc. How do you handle somebody leaving \ disconnecting? Perhaps one idea is you keep their pieces on the board and as long as there is a mate \ forced moves (so no ambiguity of what they would play) you keep them alive and have that move be played until they are checkmated. This can handle cases when people leave one or two moves before they are about to be mated, but what about the rest? I don't really have a good answer and maybe it's problematic enough that this wouldn't work at all. I suppose you could make it a race to capture the king like it is currently but that's pretty advantageous to the player following in order!
- Of course someone allowing a friend to checkmate them would be terrible 
- I haven't thought about all the possible ramifications and don't know if this is a good idea at all, but I think it's an obvious one and should be discussed!
Creative and brave.
This idea involves completely eliminating any system of points for one. The idea is simple: when a person gets checkmated, the attacker gets first place and the person who got checkmated gets last place. Both of their pieces become grayed out and the remaining two people remain to fight for second place.