Impartial games

Stream: combinatorial-games

[Violeta Hernández]

There's something quite annoying I've noticed recently: our definition of Impartial does not generalize to LGame.

[Violeta Hernández]

To be more specific. We define impartial games using the condition -x ≈ x rather than -x = x. I argue this is a good thing, since we're still able to prove Sprague-Grundy under this weakened condition. But I don't really think it's possible to do something analogous in the case of LGame, which is annoying, since IGame.Impartial x → LGame.Impartial x.toLGame should presumably hold.

[Violeta Hernández]

I was thinking. Perhaps we could split Impartial in two? Rename the current definition to WeaklyImpartial, and define Impartial so that it requires -x = x. Then LGame.Impartial would match IGame.Impartial.

[Violeta Hernández]

Though on the other hand maybe the split isn't really worth it? I don't know of any major theorems that Impartial would satisfy but WeaklyImpartial wouldn't.

[Violeta Hernández]

Wondering what you think

[Django Peeters]

I'm not strong on loopy games so idk. But I do favor the split, maybe we'll find a use for it.

[Violeta Hernández]

I guess we could do the split if for no other reason than to match mathematical convention