The only way I can picture it is with twelve. You, me, nine other writers, and one person who edits, streamlines, determines where the story goes, and probably decides who stays and who goes. That or the jobs of editing, writing the travel arc, and judging where everyone finishes may have to be split up. It is possible that people will volunteer to have their characters eliminated, meaning that we can tell how far a team goes by how into the story their author is. So if the quality of one character's story goes downhill, it's pretty obvious they're history.

Beyond that, it's just a matter of how well the storyline goes. And, of course, we would have to make a few rules of etiquette about keeping the storyline realistic - if ten people write about their characters acting in a realistic manner and some spoilsport writes, "Joebob and Rayray save the pilot's life in mid-air and fly the plane at 1,000 MPH into Rome and do a perfect 3-point landing! Then Joebob runs down to the Leaning tower of Pisa, tilts it back up, completes the Detour in ten seconds, pays the guy at the Roadblock, goes to the Pit Stop a whole day ahead of everyone else, and the prize is that they each get jet packs to use for the rest of the Race! Plus Rayray got elected POPE!" then the story is ruined because of one doofus, and all our hard work is down the drain.