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Fencer: Yes, that is a good solution to be used by those who create tournaments. But it doesn't address the point I'm making, which perhaps I can clarify with a hypothetical situation.
Suppose there is a single-elimination chess tournament which uses normal games. Suppose it's a prize tournament, with a one-year Rook membership at stake. Player A is playing against player B in a semifinal round, and A's rating is slightly higher than B's. Their game reaches a position in which a draw is likely, and A becomes worried, because a draw would eliminate him from the tournament.
Now suppose that at the same time, A is also playing a game of chess against player C. This is just a regular game, not part of any tournament. Suppose A and C have both played well, and have reached a very interesting middlegame position in which any result is possible. Maybe A has a slight advantage.
But then A realizes that if C wins, it will cause A's rating to drop below B's, which would mean that in the event of a draw in the game between A and B, A would advance to the final round of the tournament. A considers the possibilities, and decides that he'll gladly give up a few ratings points to have a better chance at winning a valuable prize. He therefore begins to play poorly in his game against C. Maybe he even resigns prematurely.
Some might say that C should be pleased by this; after all, he won! But if C is like me, he will be upset, because a win isn't worth much if the opponent was trying to lose. C doesn't play games for such meaningless wins; he plays for the experience of trying always to find the best move, the best strategy, against a strong opponent who is doing the same. And C feels that he has been deprived of what could have been an excellent game, simply because the structure of a tournament (in which C was not participating!) gave A an incentive to throw away a game.
In my view, it's best if the rules of a tournament or other competition never give players an incentive to lose. Therefore I would prefer that the higher-rated player advance in the case of a draw.
Granted, situations like the one I've described probably will not occur very often, so I'm not inclined to worry about this much more. But I wanted to make this concern clear so that it could be considered for whatever it's worth.