Three Dollar Quill

I want to make this quick, because it's late and I have a bunch of stuff I still want to do.

I was considering how to change the behavior of, let's call them the Eisenstein nimbers, and I managed to go from "no good options for * + * + * = 0" to "perhaps too many?" The basic thing I realized was that, instead of three players (complicates the strategy), I could have two players with asymmetric capabilities. Let's call the basic players Blue, Amber, and Red. The * + * = 0 model I had works by considering each pair of players separately. The new family of models instead picks one player to get two of the sets of moves, with the other one getting the remaining one; and crucially, the player with two sets of moves takes two moves on every turn, instead of one.

Now, for sure, this gives us 0 ≠ * ≠ * + * ≠ 0 = * + * + *. The problem I have is, there are a bunch of degrees of freedom now. I'm assuming that the two-move player may take their moves in either order, but could they be permitted to take any combination of moves, rather than my initial thought of two different ones? And, if the two-move player only has one move available, is that a simple loss, or is it a tie, a distinct outcome?

My intuitions are lightly steering me towards "any combination, ties possible" because I believe that simplifies the analysis, but I don't even know if I've found every degree of freedom I'll want to mess with, whether I'm using good criteria for choosing between the options, or even whether I'm judging accurately according to the criteria I'm using. (Maybe I need to be counting each move as a separate turn, opening up the possibility of the two-move player starting with a single move, or maybe this doesn't add anything.)

As I said, somehow I went from not many options, to a lot. My guess is that, once I've evaluated things, I should go for whichever one seems easiest to work with, with some consideration for questions like "do I like the aesthetics of the computed relation values for 1, 2, -1, -2, *, * + *, etc?" For now, I'm going to wrap up.

Good night.