Strategy in an infinite game

A strategy assigns one legal move to every finite position at which its player is to move. A play follows the strategy when every move of that player is the assigned move.

A strategy choosing an action from the current full game state rather than from the entire prior history. In a finite reachability game, the first-entry rank in the winning fixed-point iteration gives such a strategy by choosing a successor of smaller rank. The state must include all information affecting legal moves, such as ownership of reversible assignments and the player to move.

A strategy is winning when every play that follows it is won by its player.