You probably will not find it in English on the Internet: here is the “official” rule of the 3-player abalone game, translated from two documents in French found in boxed sets.
THE 3-PLAYER GAME
OBJECT OF THE GAME
To push six marbles (in all) out of play, into the board’s outer rim. The ejected marble can belong to one opponent or the other.
They are exactly the same as those of the 2-player rule. Each player can move in a same time 1, 2, or 3 of his marbles to a free point. All the directions are possible, forward, backward (“inline moves”), and even laterally (“side-step moves”).
You are not allowed to push more than 3 marbles of your colour in the same time. Marbles over 3 are not considered, so it is impossible to have Sumitos of 4 against 3, 5 against 2, etc. The maximum marbles of your colour which can be moved is 3.
THE PUSHING MOVES
You can push an opponent’s marbles if you are in a Sumito position:
2 marbles can push 1 opponent marble
3 marbles can push 1 opponent marble
3 marbles can push 2 opponent marbles
But you can push your opponent’s marbles regardless of their colour. For example, with 3 of your marbles, you can push 2 marbles of the same opponent or 1 marble of each opponent (diagram).
A marble is ejected when a push make it go out of the board by any of the 6 sides of the hexagon.
Please note that the first player who pushed off 6 marbles of his opponents wins the game which stops at this moment. The two remaining players do not continue the game, because the winner’s marbles are still on the board.
GAME VARIANT FOR THREE PLAYERS
Each of the three players have 11 marbles of one colour, placed in 2 rows, symmetrically disposed on 3 of the 6 bases of the hexagonal board.
Push the marbles of your opponents.
The game stops when one of the players succeeded in pushing off 6 opponent’s marbles of one or the other of his two opponents.
DEVELOPMENT AND PARTICULARITIES:
The players’ turns follow on clockwise.
A group of 3 marbles of the same colour can push 2 opponent marbles of different colours.
Every ejected marble is placed in front of the player who pushed it off in order to make the scoring easier.
Excepting these particularities, the other points of the rule stay the same as those for 2 players.
To increase and to balance the games, the players may agree to play it in 3 rounds, each begun by a different player.
The gaps between opponent marbles ejected and lost marbles of each player are noted in order to determine the winning player in case of a tie.
The rules are the same as those of 2-player games, with these few exceptions:
– each player has only 11 marbles (see diagram above).
– with 3 marbles of their colour, a player can push 2 marbles of different colours.
– a player has to eject 6 opponent marbles, without distinction of colour.
Naturally, as always with abalone, you can adapt the rules as your fancy!