I don't know if it was super obvious or anything, and Link spent more than 10 seconds on it, but basically it requires assuming that if everyone wanted everyone to live, and went for the easiest pattern to make that possible, then we'd all tie.

Basically it was every team pair would be numbered 1 or 2 in whatever order they were listed, and every team would vote for the three next teams with the matching number to them in the next three pairs.