I wonder if other sports could use a model similar to the Triwizard Tournament. The outcome of the first two events sets the handicap for the third event, which is the only one that really counts. Be ready for some bad ideas.
Adapting this to American Football, it could be that your score after the first three quarters is converted into an advantage in downs or yards in some fashion for the last quarter.
Baseball - each run in the first 8 innings counts as an extra out in the last inning.
In Everyone Else Football... umm... I'm having a hard time coming up with something not boring but less insane than "each goal you score in the first three quarters is an extra ball that goes on the field in the last quarter and can only be scored into your opponents' goal".
In a failed attempt to make the game "interesting", the NFL Pro Bowl would do the opposite, and help the team that was behind. They would get the ball again after they had scored if they were still behind. Naturally, that makes it easier to catch up.
In Magic: the Gathering and other popular card games, advanced players have developed the notion of a "win-more" card. A "win-more" card is one that works very well, but only if you're already winning. In other words, it never helps turn a loss into a win, but it is very good at turning a win into a blowout. This type of card seems strong at first, but since these games usually do not use margin of victory scoring in tournaments, they end up being a trap-- instead of using cards that convert wins into blowouts, you want to use cards that convert losses into wins.
This concept is useful and important and you should never tell a new player about it, because it tends to make them worse at the game. Without a more experienced player's understanding of core concepts, it's easy to make mistakes and label cards that are actually good as being win-more.
This is an especially dangerous mistake to make because it's relatively uncommon for an outright bad card to seem like a win-more card; win-more cards are almost always cards that look really good at first. That means that if you end up being too wary of win-more cards, you're going to end up misclassifying good cards as bad, and that's an extremely dangerous mistake to make. Misclassifying bad cards as good is relatively easy to deal with, because you'll use them and see that they aren't good; misclassifying good cards as bad is much more dangerous, because you won't play them and therefore won't get the evidence you need to update your position.
I call this the "win-more problem." Concepts that suffer from the win-more problem are those that-- while certainly useful to an advanced user-- are misleading or net harmful to a less skillful person. Further, they are wrong or harmful in ways that are difficult to detect, because they screen off feedback loops that would otherwise allow someone to realize the mistake.