Maker-Breaker game

A Maker-Breaker game is a kind of positional game.^{[1]: 13–24} Like most positional games, it is described by its set of positions/points/elements (${\displaystyle X}$) and its family of winning-sets (${\displaystyle {\mathcal {F}}}$- a family of subsets of ${\displaystyle X}$). It is played by two players, called Maker and Breaker, who alternately take previously untaken elements.

In a Maker-Breaker game, Maker wins if he manages to hold all the elements of a winning-set, while Breaker wins if he manages to prevent this, i.e. to hold at least one element in each winning-set. Draws are not possible. In each Maker-Breaker game, either Maker or Breaker has a winning strategy. The main research question about these games is which of these two options holds.