In their bestselling book on applications of game theory (Thinking Strategically), Avinash Dixit and Barry Nalebuff add a personal account of game theory gone wrong. On a visit to Jerusalem, one of the authors (along with another economist) employs his strategic game expertise on a local cab driver, only to find himself on the street. The lesson — think about the broader context of the setting before going too deep into game applications, or, don’t over-complicate a decision.
In watching the Patriots lately (along with others), the Dixit-Nalebuff lesson comes to mind, maybe not in its particulars, but in the overarching point. The 25 seconds before each snap has evolved into a sequence of moves and counter moves — setting protections, changing protections, route adjustments, play adjustments, … Over the past 10-12 years these kinds of pre-snap adjustments became routine with QB’s such as Brady, Manning, and Brees acting as master level strategists — offensive coordinators on the field.  Now, these QBs rarely call a play, get to the line, and run it. Of course, the pre-snap machinations counter defensive developments. The adjustments to protection counter the ever-more-sophisticated blitz schemes developed in the 1980s and 90s. The adjustments to routes/plays counters the ever-more sophisticated coverage schemes.
Defenses, or at least the really good ones, have countered the counter-measures, developing better means of disguising blitzes, coverages or changing them after the offensive changes. As a result, the offense can give away one of its key first-mover advantage by providing the defense with more time instead of less time to adjust. Sometimes, just as in the taxi example, the strategic moves become too clever and work perversely against their own interests. For example, in the Patriots loss to the Steelers, the Patriots trailed by more than a touchdown with 4 minutes to go, but held possession at the Steelers inside the Steeler 5 yard line. Over several plays, the pre-snap adjustments chewed up huge chunks of time — so much that by the time the Pats scored, they were forced into a low probability onside kick. In the Saints-Colts 2010 Super Bowl, the Colts kept wringing the strategic element every play in spite of what seems a dominant advantage of their wide receivers versus the Saints cornerbacks.
On obvious passing downs where the uncertainty of the offensive play, and, therefore, any first mover advantage, is limited, the pre-snap adjustments probably make a lot of sense. In less obvious situations, these adjustments (or, at least, their universal use) seem prone to the Dixit-Nalebuff cabbie problem — thinking so much that you out-think yourself.
Aren’t you assuming that the Patriots options were “Run clever/sophisticated plays to score a TD but take a lot of time doing it” and “Run simple but fast plays to score a TD and leave time on the clock”? What if the second option wasn’t an option at all? That is, what if reducing the time they consumed significantly decreases their chances of scoring at all?
This is a great point about wasting time making pre-snap adjustments when the play selection is limited (probably a pass) or when trailing with limited time left.
I wonder if, as the play clock winds down within 5 seconds, the defense gains an implicit advantage in knowing the ball must be snapped soon. Defensive linemen can get a better jump on the ball, linebackers/safeties can time their blitzes better, and things of that nature. If the ball is snapped at 15, not only do you avoid the issues raised above, but you also inherently benefit from the uncertainty of the time of the snap. With 2 or 1 seconds left on the clock, it’s like a shot clock in basketball; you know they have to shoot.
I have noticed in college football since everything is called from the sidelines these days and the offensive coordinator is in the pressbox that during last minute plays, it takes about 15 seconds to signal everything end and get the play off.
The college coaches are so used to the clock stopping on first downs that they waste a huge amount of time if the clock does not stop.