Strong Games Played on Random Graphs

Did you ever play games such as Tic-Tac-Toe, its close relative n-in-a-row or Hex? Then you are already familiar with strong games. Strong games, as a specific type of Positional games, involve two players alternately claiming unoccupied elements of a set X, which is referred to as the board of the game. The two players are called Red (the first player) and Blue (the second player). The focus of both players is a given family H ⊆ 2X of subsets of X, called the winning sets of the game. While playing, Red and Blue take turns in claiming previously unclaimed elements of X, exactly one element in each round, with Red starting the game. The winner of such a game (X,H) is the first player to claim all elements of some winning set F ∈ H. If this has not happened until the end of the game, i.e. until all elements of X have been claimed by either Red or Blue, the game is declared as a draw.