The only games saved in the archive are those played in Play > Live and the computer players there.
On Play vs Computer you can export the game PGN and save them locally or then import them into an Analysis board and save a link to that.
Saving Computer Games?
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I saw a thread where players are running up big winning streaks by playing the computer on level one. I have no interest in that. The thing I'm asking about is I actually went to their profile and the games were in their Archives. When I play against the computer the game is not saved in my archives and I do not see any way to save them. Is there a way to save these games?