The diameter game

In a generalized Maker-Breaker positional game, Maker and Breaker play in turns. Maker makes a moves in each turn and Breaker makes b moves in each turn. We choose Maker to be the first player, although this nearly always makes no difference in the outcome of the game. We call such games (a : b)-games. If a = b, the game is fair, otherwise it is biased. If a = b > 1, the game is accelerated. In [3], Beck asked about the behavior of accelerated versus unaccelerated games, having observed a game in which Maker can win the (1 : 1)-game, but the result of the (2 : 2)-game is inconclusive. It is well-studied in the literature that a game may have completely different outcomes if it is played as (1 : 1) or (a : b) game, where either a or b is greater than 1, see [1, 5, 6, 7]. Here, we investigate the so-called 2-diameter game and, in particular, we prove that in the (1 : 1)-game, Breaker wins; but, in the (2 : 2)-game, Maker wins.