It feels like this sequence should be related to the Thue–Morse sequence https://en.wikipedia.org/wiki/Thue–Morse_sequence - sometimes called the 'fair share' sequence

○ ● ● ○ ● ○ ○ ● ● ○ ○ ● ○ ● ● ○ ● ○ ○ ● ○ ● ● ○ ○ ● ● ○ ● ○ ○ ● ⋯

I don't think this contains any 'anti pattern game' losing sequences, but I may be missing something.