As far as I have found in references to combinatorial game theory, impartial/partizan games are referring to games, where the goal/winning condition of the game is symmetric. In other words, if the same game-ending move was done by either player, the result would be the same (the player who made that move would win/lose).

However, can asymmetric games, in particular Maker-breaker games, be classified the same way? The only relevant thing I have found was on Wikipedia (impartial game), which states that impartial games are those where allowable moves are symmetric for both players (as in Maker-Breaker games, for example) but the payoffs are also symmetric. So by that definition Maker-Breaker games (and asymmetric games in general) are not impartial. Since partizan games are exactly the games that are not impartial (as far as I have gathered), all asymmetric games would be partizan.

However, is that definiton on Wikipedia cannon in combinatorial game theory or is it even reasonable to classify asymmetric games the same way?


There are a few things worth mentioning here.

Firstly, note that if you don't care about sums with other games, the distinction between impartial and partizan isn't very significant. For example, Chess is naturally thought of as partizan since one player moves the white pieces and one player moves the black pieces. But "Chess with a chess clock next to the board indicating which color of piece can be moved next" can be viewed as an impartial game with both players taking turns making moves of the form "move a piece of the color indicated by the clock, and then switch the clock".

Secondly, Maker-Breaker is not naturally viewed as "allowable moves are symmetric for both players" since the claimed positions/vertices are marked/occupied for the individual player. Player 1 has moves of the form "claim a position for player 1" and player 2 has moves of the form "claim a position for player 2".

Also, most games considered in combinatorial game theory, even partizan ones, are played with a win condition of "make the last move" or "don't make the last move". But Maker-Breaker games are not naturally considered with either of those conditions.

All that said, in the vein of my first point, you could turn Maker-Breaker into an impartial game with the normal play condition in a similar way to Chess: have a toggle indicating which sort of claim can be made next, and have the special rule "no moves can be made after a maker move (as indicated by the toggle) made a winning set".


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