I see now that my original comment on www.chessvariants.org/index/displaycomment.php?commentid=19541
is not available anymore, so I copy it here.
Chess for Three. Problems and proposals In the internet site Chess Variant Pages it is possible to find some chess variants for 3 players, and the more classical variant on a single board is the Jerzy Luberda's one : www.chess-for-three.com/
The idea of the chess board is brillant, but the rules of Jerzy Luberda have some problems.
(Let's say that A is much stronger than B, that is much stronger than C)
Problem 1: nobody has interest to attack.
A prepares a very good attack against B, for example A takes B's queen loosing a knight; it is not worth it, because comparing with C that did nothing, A has a knight less.
Problem 2: Kingmaker effect.
Let's say that A did a very good attack against C, using many pieces and a good strategy, and B is just trying to exploit the situation with one piece; if A is about to checkmate C, very often can happen that C must decide between be checkmated by A or by B. So, a player that has no more chances to win will decide the winner.
This problem is present in the standard game where the first that captures a king wins (http://www.chess-for-three.com/), or in the variant where who captures a king takes over the eliminated player's pieces, continuing with stronger forces (second option in the paper rules of the game).
Problem 3: cooperation against the strongest.
If A has an advantage comparing with the other two, B and C will probably cooperate, even if speaking is not allowed, until A will be not anymore the strongest. Now B has the best situation, so A and C will cooperate against B, and so on. So, the final victory is almost random, even if A plays very well, most probably he will not succeed to win.
After several tests with various players, I can say that a good set of rules to avoid or reduce the above problems is the following.
You need to have 2 (or better 3) sets of pieces for each color. When a
player captures a piece (unless a king), he changes it with another
identical of his color and puts it by his side of the board. At his round,
he will choose if he wants to do a normal move or if he wants to put
wherever on the board one of his captured pieces, like in Shogi (japanese chess). If a promoted pawn is captured, it becomes again a pawn, like in Shogi. In this way the problem 1 is solved, because in the above example A will loose a knight, but will gain a queen. The problem 3 is still present, but is reduced, because for the strongest player it is easier to gain quickly forces.
(To read an interesting way to try to solve the problem 3, see the rules
proposed by Jonathan Rutherford: www.chessvariants.org/multipla.../3-handed_chess.html)
The goal of the game is to be the last player in game and who captures the first king has no advantage. In this way the problem 2 is solved, because even if the first checkmater is random, then he will have to fight with the remaining player, and without having an advantage for having checkmated.
When a player looses his king, his pieces remain on the board without
moving anymore but they can still be captured by the other players.
(This rule 2 is the first option proposed in the paper rulement of Jerzy
Luberda. It should become mandatory, instead of optional).
The kings must be captured (not checkmated) and you can always leave or put your king in check. This is a less important rule just to make the things easier, otherwise it is not obvious when a move is allowed. (For example A moves the king, then B moves a piece leaving A's king menaced by a piece of C. Whose is the illegal move ?)