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I hate the idea of having two tournament winners listed. If we are both winners, then we should be listed as co-winners. Which means Fencer would have to create a new category, Tournament Co-Wins, and check each player's tourney wins to see which are actually co-wins. Obviously not worth it.
In the tournament above, why not at least have a third round with just me and the opponent I tied with in the previous round? I can understand how some players would prefer not to play a fourth round, if the third should also be a tie, but as long as the next round is not a repeat of the previous, I see no reason why the round should not continue.
Of course, if you allow both players to advance from this round, I think you must do it in early rounds also for the sake of fairness.
As others have said better than I, we should only use tiebreakers when there is a specific reason, before a deadline or something like that. None of wich we really have here at BrainKing.