These sequences are also known as "cubefree," so you might want to continue researching along those lines.
In particular, the game discussed is trying to find cubefree words over a two-letter alphabet. The sample infinite game seems to agree with the listed sequence on OEIS for the lexicographically earliest infinite cubefree word, though your method of generation appears to be different from the one in the comments. (I haven't analyzed it in detail.)
Comments
These sequences are also known as "cubefree," so you might want to continue researching along those lines.
In particular, the game discussed is trying to find cubefree words over a two-letter alphabet. The sample infinite game seems to agree with the listed sequence on OEIS for the lexicographically earliest infinite cubefree word, though your method of generation appears to be different from the one in the comments. (I haven't analyzed it in detail.)
https://oeis.org/A282317
Cubefree! Oh, that’s a good keyword to find more results about this.
From your link it seems it is a conjecture that this is the lexicographically earliest one. Very interesting!