An $n$-in-a-row Type Game

We consider a Maker-Breaker type game on the square grid, in which each player takes t points on their tth turn. Maker wins if he obtains n points on a line (in any direction) without any of Breaker’s points between them. We show that, despite Maker’s apparent advantage, Breaker can prevent Maker from winning until about his nth turn. We actually prove a stronger result: Breaker only needs to claim ω(log t) points on his tth turn to prevent Maker from winning until this time. We also consider the situation when the number of points claimed by Maker grows at other speeds, in particular, when Maker claims tα points on his tth turn.