I suppose you are all familiar with the baseball elimination problem (it is about determining whether some team with particular number of points can finish in the first place, when there are still such and such games to be played by other teams).
The rules there are rather simple - the winner gets a point, while the loser gets nothing. But what if we changed those rules, so that the loser would get -1 point for each game lost. Could it still be brought down to the max-flow problem then? I've been thinking about this, but there's no way to do that in my opinion.