Sorry, I was annoyed by the forums when I posted that, having lost a more through explanation. I would go about it as such:
During the week of the tournament, they each make contact and play a game like normal. They'll state before the match that it's for the tournament. They then report the outcome.
Whoever gets the most wins gets the grand prize. Ties are settled based on the sum total of the wins of the defeated. Further ties are settled based on the places of the defeated. If they're still unresolved, there's a rematch between the players who are tied.
Example: A, B, C, D, and E are playing.
A beats everyone, winning the grand prize.
B beats C and D.
C beats D and E.
D only beats E.
E only beat B.
B wins second place, because C+D = 3 wins, while D+E = 2 wins.
C wins third place.
E wins fourth, since B won second place.