The ultimate First World problem: did the Seattle Seahawks call the worst play ever?  Ask a game theorist.
The key insight of game theory for an N.F.L. coach is that when you think about what choice you should make, you need to also consider the response from the opposing coach, understanding that he is also thinking strategically. This line of thinking suggests that you should not necessarily call a run play, even if you’re blessed with a great running back. Likewise, it’s not clear that you should definitely pass. Rather, your choice should be somewhat random — a choice that game theorists call a “mixed strategy.”
Not somewhat random: your optimal mix is one that makes you indifferent among the pure strategies available to your opponent.  But your opponent knows that, and your opponent may also be mixing.
In the football context, your running back may be a better weapon than your quarterback, and so an optimal strategy does not dictate that you use them both with the same probability. Rather, you choose the probabilities in an optimal mixed strategy so that the payoff from a running play will be the same as that from a pass. This means that even with a great running back, an optimal strategy sometimes involves passing. Otherwise, your star running back, always facing a run defense, may end up less effective than a less great passer.
Yes, and within a game, offenses and defenses interact repeatedly, and teams can demonstrate tendencies.
Whenever I have taught economics students the idea of playing a mixed strategy, they respond incredulously, because it defies common sense to make the biggest decision of your football coaching life randomly. It may defy common sense, but it makes good strategic sense.

Perhaps you are like my students, and your advice is that maybe Carroll should follow a mixed strategy most of the time, but not in the dying seconds of the Super Bowl. But realize that if this were an optimal choice, Belichick would probably figure it out, and he would instruct his players to guard against the run. When most of the defenders focus only on stopping one running back, they usually succeed.

Or perhaps you believe that Lynch’s statistics show that he is so successful at bulldozing through opponents that he would succeed even against a defense set up only to stop the run. I disagree. A key reason that Lynch has been so successful is that his coach has been playing a mixed strategy all season. Lynch has accumulated impressive numbers in part because opposing defenses have had to be concerned about Russell Wilson’s passing. And so Lynch’s history of success when playing as part of a mixed strategy says nothing about how successful he would be if his opponents knew for sure his coach would call a running play.

Game theory points to the possibility that Carroll’s decisive call was actually the result of following the best possible strategy, and that this is a strategy that involves an element of randomness in play-calling. This leads to the intriguing possibility that if that fateful final play were to be run in a dozen parallel universes, with each coach continuing to play the same mixed strategy, the actual plays called would differ, as would their outcomes.
Yes, provided the universes are not parallel when it comes to making the actual choice.  To keep it simpler: mixing keeps the defense honest.  Line up on a short-yardage play as if you are going to smash straight up the middle and run the play-action Bart Starr special.  But sometimes you have to smash straight up the middle to make the bomb fly.  And Bart has to be careful not to tip off the play, something Russell Wilson might have done.  But even so, the Nash equilibrium involves optimal mixing by defense and by offense.

That's a simpler explanation for what happened than the tinfoil-hattery making the rounds in the fever swamps.
The conspiracy theory lies in the fact that Seahawks coach Pete Carroll believed that the last yard the Seahawks needed for that Super Bowl victory was a gimme and, all things being equal, much better to have the iconic Super Bowl moment go to Russell Wilson than to Marshawn Lynch. Coaches setting certain favored players up for glory is as old as football itself. In addition, the politics of race, respectability, public relations and what’s in the best interest of a $2 billion corporation all played into this.
Even a writer for The Nation isn't ready to buy it, noting that power running in short yardage situations doesn't always work.  Even so, "But in a locker room like Seattle’s where they truly do feel like it’s them against a world and an NFL power structure that wants to put them down, this is one theory that we can expect to find purchase in the months ahead." I have memories of the Randy Moss, Daunte Culpepper era Minnesota Vikings.  So it's time to step away from football during prime model railroading season, give the Seattle fans time to turn the page, and, once the draft arrives, start thinking about the next division title.

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