I've played several games with my tulpas, ranging from rhythm games to pokemon to (hotseat) chess, and I've found that in pretty much all of these games, at least one of them heavily outsmarted/outplayed me. For one, they seemed to learn faster than I did, and were more focused on the task than I was. For example, while playing Osu, Ea would often tell me that I pay too much attention to the health bar/combo/accuracy as opposed to the notes themselves, and said that that's why she's able to consistently break my highscores while she's playing, because she focuses solely on hitting notes and nothing more. A similar thing went for Valkyrie and chess, though she didn't provide me with much of an analysis as to why she was able to play better than me.
After lots of playing with tulpas I'd say the difference in their ability would be their mindset/ability to focus. Perhaps not having to deal with mundane stuff all the time makes you a better thinker?
I didn't play Tic Tac Toe with them yet, but here's Ea's score on Osu in contrast to my own
http://puu.sh/4mZ3Y (top two are Ea, bottom is me)