I am the GM for my groups. When I don't run the game, the game doesn't happen at all.
The poll choice, "We do something else," is the closest fit, but it might imply that the something else is still being done together, by the group (like, "we get together and play board games"). Generally, if I'm not running game, the group doesn't even gather.
Yes, that answer was meant to include both "nothing happens" and "we play cards, watch basketball or go to the bar."
When my current group first started meeting, on weeks that I couldn't DM due to work conflicts, they would still get together socially. I think mostly because they really wanted to keep the spot carved out in our schedule, so it didn't just all fall apart, but nobody else was comfortable DM'ing. Now that we've been meeting for a little while, there's a bit more confidence knowing that, if we have to cancel this week, the group will probably still get together next week.
I was the DM when this group started, but another player really wanted the opportunity to DM, so I stepped down to let him give it a shot. Unfortunately, while I enjoy playing, I really miss DM'ing, so I jumped at the opportunity to run some sessions while he was gone on vacation. The initial plan was to just pick up my original campaign where we left off, but, at the last minute (1 hour before everybody showed up), I decided to just run an adventure I'd read and liked. We used the same characters, but didn't make much effort to maintain any continuity beyond level and general gold levels from the last adventure. It worked really well, and I think I'd do it again. It feels like a stand-alone Conan or Fafhrd and the Gray Mouser story — we remember the gist of the characters, but we don't have to worry about exactly what happened whenever we last saw the characters, and if characters drop in or out, it has no impact. Then we can pick up where we left off when our regular DM takes over next week.
With a different group, when my work schedule got really bad (bit of a pattern here — my work is unpredictable but can easily blow up), a player offered to DM so everybody could keep playing, but he wanted to keep using the same characters. I gave him some parameters as for what he couldn't do, as this came in the middle of another arc, where going to certain places would trigger events that I needed to run myself, and then we discussed his hook and I worked with him to tie it into the setting. It went ok. I think everybody enjoyed his sessions, but when it was over, I had a hard time tying his loose ends together with my loose ends. I respect groups that can properly handle rotating DM's with the same group of characters, and Greenfield, it seems like your group has a great system in place for it. But it hasn't worked for me. (I also think your system wouldn't necessarily help my problem, which is an unreliable schedule which occasionally takes me off the board for a while, without much warning.)