Have you ever seen a football game (NFL or college) that was low scoring, with the teams punting the ball back and forth, but in the last few minutes each team managed to score quickly and repeatedly, making for a very exciting finish? The Patriots-Giants Super Bowl a few years ago was one such game. In my experience, there are many others. Is it just chance that the games end this way so often? Do the teams just choose to stop trying on defense, or is there a better explanation for why this occurs?
I think this is because of the teams' choice of play calls. Each team has a few offensive plays, or sequences of plays, which they developed and which they think are especially effective and unlikely to be anticipated by the defense. If they use these plays at some unimportant point of the game, their effect is wasted. So they are saved for the closing minutes of close games. At this point, both teams run all their special plays, which do tend to be more effective than normal plays. Thus, both teams often manage to score quickly. The defenses are trying just as hard as they have all game, but their task is harder and they tend not to do as well.
One consequence of this theory is to diminish the value of the “two-minute drive quarterback”. Not only is it easier to score at that point with all the special plays at hand, but the choice of plays is likely to have been substantially planned ahead of time. In the middle of the game, when there is no such planning, running an offense may be more difficult, not less.
Also, the clock management rules are such that more plays can be run in less game-clock-time toward the end of the game. (The clock stops on both out-of-bounds plays and incomplete passes in the last 5 minutes of the game, for example, but only on one or the other, or neither, earlier in the game.) Hurry-up offense, the two-minute warning, and timeout usage patterns are other reasons. I bet the effect you notice would be much less dramatic (though still present) if you look at the last X plays of a game rather than the last Y minutes.
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