I've headcanoned Cadence as the nerdy type with a leaning towards RPGs (especially LARPing) since A Canterlot Wedding.
Celestia seems like the type to take to trying new things and Luna... Well, she'd probably play aggressively to win, whereas Celestia would play smart.
I haven't read the Solar Empire fic but it sounds good so I used the names. Sorry if it's irritatingly inaccurate! Also: The RPG isn't anything in particular. If you think it needs a Dungeon-Master-type (who isn't one of the players) then assume they're out of shot.
And the big brothers should hang out more!