I cannot find research on a $\beta$-Devil version of The Angel Game.
I am not sure if I cannot find it because I am not able to research it properly or if there is not much theory on this. In case there is not much theory I am interested in knowing why. Is there a process to transform a $\alpha$-Angle, $\beta$-Devil rule-set into a $\gamma$-Angel,1-Devil? Can we do the same for a $\alpha$-King (instead of $\alpha$-Angel) as well?
What I mean by $\beta$-Devil is that when it is Devil's turn to play, it can remove $\beta$ squares off the board. $\alpha$-Angel is an angel that can fly $\alpha$ squares in its turn.