Positional game

In the mathematical study of combinatorial games, positional games are games described by a finite set of positions in which a move consists of claiming a previously-unclaimed position. Well-known games that fall into this class include Tic-tac-toe, Hex, and the Shannon switching game.

A positional game may be described by a pair ${\displaystyle (X,{\mathcal {F}})}$ where ${\displaystyle X}$ is a finite set of positions and ${\displaystyle {\mathcal {F}}}$ is a family of subsets of ${\displaystyle X}$; ${\displaystyle X}$ is called the board and the sets in ${\displaystyle F}$ are called winning sets. The game is played by two players who alternately claim unclaimed elements of the board, until all the elements are claimed. The winner may be determined in several ways:

...
Wikipedia