## Game theory says what? A game theorist's look at the Seahawks Super Bowl interception

*in*Et Cetera

This is a response and elaboration on Dr. Justin Wolfers' NYT post about "that play" (the Super Bowl interception at the 1-yard line) from the perspective of game theory. First of all, Dr. Wolfers is clearly a very well established economist and has been publishing and teaching in the realm of economics (including behavioral economics, which umbrellas many game theory fields of interest) for quite some time. This isn't a diss on him. Rather, good on him to say that there is a lot more to a decision than "give the ball to the dude who has been literally gaining 2 to 3 yards per carry". Because there is.

However, Dr. Wolfers tries to explain that this is a strategic situation to the like of a rock-paper-scissors game. I give him his due -- it's a very accessible analogy -- but it's a bit simple. So I want to take the next 1000ish words to describe a strategy or four and the process of decision making that could have happened.

Because that's what I study. Decision making and strategy formation. And I dabble in sports statistics.

### Some objective statistics

- Seattle's running back, Marshawn Lynch, was rushing 4.25 yards per carry.
- If you isolate only runs up the middle, he was rushing for 5.33 yards per carry.
- Out of Lynch's 24 rushing attempts, only two were stopped for no gain. Lynch literally was never stopped for a loss.
- Lynch's 4th quarter rush attempts gained 2 yards, 1 yard, 5 yards, and 4 yards in that order.
- One play before the interception at the 1 yard line was when he rushed for 4 yards. That was the last time he would touch the ball this season.

Dr. Wolfer's mentions "mixed strategy" or a strategy that is hard to predict due to a seemingly random nature of choices preceding a present choice. Let's isolate the last drive in terms of decisions to either pass or rush.

- Seattle passes 5 times, running off 52 seconds from the clock.
- Seattle then rushes once, running 40 seconds off of the clock. This puts them at the 1 yard line.
- Then with 26 seconds left, Seattle chooses to pass. Disaster ensues.

There are 7 plays Seattle ran, with 6 of them being within 2 mins left to play. This particular information is important because the rule of thumb for a professional football play is that each play takes 6 seconds and the clock only stops for timeouts (each team at this point had two left), if you run out-of-bounds, if there is a foul of some sort (some fouls stop the clock, others don't), or if you throw an incomplete pass.

Logically, you want to run enough time off of the clock and score a touchdown so your opponent has no time to score a touchdown for themselves. We'll talk about the finite nature of this drive later.

### Strategy 1: Couch Coach's strategy

The drive was built upon the pass, which was necessary due to two factors: time and the advantage to stop the clock. Seattle's quarterback, Russell Wilson, completed 12 passes for almost 250 yards in the game -- giving him a 20.58 yards per completed pass. He was completing 57% of his passes -- better than a coin flip. Seattle had to move the ball 80 yards.

Using some simple math here, If you gave Wilson this situation with these stats, he would need 8 passing plays to generate enough yards to equal a touchdown. They passed 6 times.

On the other hand, you would need 20 plays for Lynch to generate a similar amount of yards to equal the distance for a touchdown -- assuming time wasn't a factor (which it was).

So the strategy here seems somewhat simple from my couch: if you can fit in 8 passing plays in two minutes, you win the game. If you are within 5 yards of the goal line, rush with Lynch.

### Strategy 2: Something like Tit-for-Tat

In a game called The Prisoner's Dilemma, there is a strategy called tit-for-tat. Essentially, if your opponent does something to you, you do it back to them the next turn.

Let's consider the defense instead of the offensive statistics for a moment. The defense from the beginning of the final drive is considering that the plays called will be majority passes for one reason: you need a lot of yards in a small amount of time. And that's what happens with the last 6 plays within 2 mins left to play.

New England rushes 3 or 4 people and drops 7 or 8 people into a pass defense ("dropping into coverage") for every play except for the last two.

NE switches to a goal line defense, which is specific to stopping runs with pass coverage against outside receivers. They stop Lynch from a rushing touchdown because of this.

On the final play, NE expected a rush. All but 4 defenders try to collapse the offensive line. Those four are basically 1-on-1 with their receivers and are exposed to a pass because essentially no receiver is double teamed.

This is seemingly like a tit-for-tat strategy in Seattle's favor.

They present mostly pass plays up until the goal line. Then they rush, and the rush was positive (consider this the "tit"). New England responds with a more heavy rushing defense (consider this the "tat"). The best way to manipulate a tit-for-tat strategy? Coerce your opponent into a tit-for-tat strategy then exploit their predictability. If Seattle assumed New England was going to "tat", then exploiting their tat isn't a bad option -- especially when New England actually did "tat" and Seattle called against it correctly, based off of this strategy.

Dr. Wolfers mentions this in his NYT piece, where predictability would be a bane -- even if the team was amazingly great within their predictability. Strategically, Seattle actually had the advantage. Execution is a different story.

### Strategy 3: Why not win-stay lose-switch?

A third strategy would be the win-stay, lose-switch strategy. It is also fairly simple: if you are gaining positive results, stay with your choice. If you do not gain positive results, switch your choice.

Back to offense: Seattle couldn't perform an exact win-stay, lose-switch (WSLS) because of the finite time. The first play of the drive, Wilson connected with Lynch for a passing play. By WSLS standards, they would be wise to pass again. They do and it was incomplete. If you consider this a lose-switch situation, they should have rushed the ball.

Now, think about rushing the ball -- the clock would continue to run until you used all of your timeouts. They were around the 50-yard line after the passing play. Lynch would need 10 plays -- equaling (at best with no lag) 60 seconds. It takes 6 seconds per play, but an extra 5 to 7 seconds to get your offense lined up without committing a foul.

The WSLS strategy would be flawed here.

But consider the final three plays. First, a long pass. Second, a 4 yard rush. Third, a pass into an interception.

In a WSLS strategy, you would flip the last two plays. First, a long pass. Then, another pass. If we assume the pass is complete, it's a touchdown. If we assume it is incomplete, then switch to a rush. And if that doesn't do it? Then you can either switch back to a pass to stay true to the WSLS strategy, or hopefully exploit New England's defensive predictability and rush one more time.

### Strategy 4: Get statistical -- Monty Hall?

There were 4 scoring plays for Seattle, all within the "red zone" -- aka 20 yards away from the end zone. Three touchdowns and one field goal. Lynch rushed at most 3 times and at least once on all of these drives.

So, we can try to compute a probability of a touchdown given the amount of rushing attempts within the red zone. This would be a perfect fit for using the Bayes Theorem.

We want to know what the probability of a touchdown given rushing two or more times inside the red zone. We know that 66% of the time, when Lynch rushed two or more times inside the red zone, it led to a touchdown. We know that they scored a touchdown 75% of the time they were in the red zone. And we know Lynch rushed at least two times in the red zone 75% of the time, regardless of a touchdown or not.

Oddly enough, the traditional statistic and the Bayesian statistic are equal: Lynch would have a 66% chance of scoring a touchdown if he rushed twice inside the red zone. And you know what this is sort of like? The Monty Hall problem.

You have three doors. Behind one door is a car. The other two doors are goats. You make a choice. As a hint, the host reveals a door that you didn't choose and shows off a goat. Do you switch your door or stay? Statistics say you should switch, increasing your odds to... yep, 66%.

In Seattle's case, behind one door is a touchdown and the other two are not touchdowns. He was shown a goat his first attempt. Statistics say if he runs one more time, he has a 66% chance of a touchdown. Monty Hall says if they switch their choice after being shown a goat, you increase your odds to 66%.

One 66% versus another 66%.

### TL;DR strategy formation and decision making are extremely difficult, even in seemingly guaranteed situations.

Of these four strategies, most will see Couch Coach's strategy as the "most logical". And although statistically it sounds good, the fact is less about what *sounds* good but rather what is *actually* good. Gut instinct? Yeah, Couch Coach seems best. Statistically? I think either pass or rush had strong evidence. Strategically? I do like the idea of a win-stay, lose-switch method: two passes and a rush as opposed to pass-rush-pass.

But honestly, this is probably karma for The Fail Mary.