I just read about M,n,k-games and wondered if the following variation (with fixed $k$) has been studied as well and if there exists a name for it:
Two players consecutively mark elements of ${\bf Z}$ (but could be ${\bf Z}\times{\bf Z}$ as well) in their respective color and first player who owns elements $x,x+c,\dots ,x+(k-1)\cdot c$ with $x$ arbitrary and $c\neq 0$ wins the game.
I thought that this ruleset seems more elegant from a mathematical point of view since it generalizes straightforwardly to arbitrary groups.
Also, what if the game is played on an arbitrary graph and first player who owns $k$ connected vertices wins, does this make any sense or has any name?