A version with Crazyhouse-like piece drops would do what you want, if you use the rule that the turn is transferred to a player when one of his pieces is captured. (So that he can always retaliate, and it gets more difficult to gang up on him.) on a non-capture the turn just goes to the next player in the clockwise order. Then it doesn't pay to remain passive while the 3 other teams 'fight it out'. Because the one who is winning that fight will have absorbed a large fraction of the armies of the opponents he beat, and will thus badly outgun you.
4-Player chess rules are dumb
Sort:
By that, I mean the points system. When chess is won on points it disincentivizes good play IMO. First of all, you gain points by taking but you don't lose points by being taken, so just trading all your pieces is generally a good idea if you're already ahead. Idk how many games I've seen with two players left, one with a full army and possibly multiple queens, but the guy who traded everything away wins on points. Playing a solid defensive line is just gonna lose you the game since you'll have zero points by the end.
Secondly, I feel like the most intriguing part of the concept gets taken away, which is that of natural cooperation and game theory. Without the points system it makes total sense to cooperate with another player to checkmate a third, like checkmating someone while being protected by another players piece. Or by simply not attacking a player because he is attacking someone who is a bigger threat to you or whatever. It makes no sense when you win on points though, since helping someone else get a checkmate just gives them 20 points but gets you nothing.
I would love to see how the complex strategies would evolve in a 4-player mode where It's all about being the last man standing. Has it been tried but was deemed unplayable or something? I know it would be slower but probably way more interesting.