• Anne


Do you know what three-player partizan game theory is?

That’s what I wrote my masters thesis on, based on combinatorial game theory principles I learned from Lessons in Play. Specifically, I focused my research on the three-player partizan game of Rhombination, a game my advisor created based on a Lozenge tiling version of Domineering. I’m going to try to describe this as briefly as I can (please note I wrote an approximately 50 page paper on it, so brevity is somewhat difficult).

Defining three-player partizan game theory

Three players obviously means the game is played between three people, who we conventionally call: Left, Center, and Right, for the respective players 1, 2, and 3. Partizan games are simply games where the different characters have different moves. For instance, Nim is not partizan, because given any specific game position, all players can make the same moves. Chess is a partizan game because you can’t move the same pieces as your opponent.

Basic Rhombination rules

You can read all about Rhombination here, but the basic premise is you start with a game board in any shape on an equilateral triangular grid.

These are the shapes I chose for my Rhombination prototype board.

All players are given tiles in the given colors: Blue for Left, Yellow for Center, and Red for Right. Player 1 (Left) can only place tiles that are tilted to the left, Center can only place tiles that are horizontally oriented, and Right can only place tiles that are tilted to the right.

They keep playing until one player is no longer able to move. At that point, the winner and loser are determined. The winner is always the person who played last and the loser is the one who was unable to move. If multiple rounds are to be played, the person who wins will receive two points, the person who loses will receive nothing, and the other player will receive one point.

Winner - Loser

Left - Center

Center - Right

Right - Left

In this sample game, Left just moved, but since Center doesn't have any moves, the game is over. As such, Left is the winner and Center is the loser. Please note that it is irrelevant if Right has a move or not since Center can't move.

That’s basically it! I recently sat down and built a prototype Rhombination game board, so I am finally able to play this game properly. If I ever figure out how to make it more accessible on an app or on a game board I can reproduce, you’ll be the first to know!


Obviously you can’t play Rhombination, but I encourage you to pull out an old board game you have and play a round. Even if you can’t play with anyone, play against yourself! Board games are great fun! Especially as you develop strategies to play!


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