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February 3, 2015

Super Bowl XLIX and the danger of oversimplifying game theory

The interception that effectively ended Super Bowl XLIX on Sunday has been derided as “one of the worst choices in pro football history” and “the worst Super Bowl play of all time.”

For those of you who haven’t heard about it by now: with 26 seconds to go, one timeout, and just one yard to go for a needed touchdown, Seattle Seahawks coach Pete Carroll called a slant route on second down, a passing play into an area stocked with defenders.

New England cornerback Malcolm Butler made an extremely athletic maneuver, picking off the pass to seal the Patriots’ win.

Much of the brouhaha immediately following the interception centered on Seahawks running back Marshawn Lynch, considered by some to be the best at that position in the league. Handing off the ball to Lynch would almost certainly score within the next two plays, they claim, often citing the offense’s power ranking.

A few economists have wielded game theory in an attempt to justify Carroll’s decision. Justin Wolfers at The New York Times, for one, says that game theory makes the play call “defensible.” He bases his analysis on the concept of a mixed-strategy equilibrium, a topic I’ve covered here.

At a high level, it’s a good analysis that explains why players should vary their strategies to keep their opponents guessing. He draws on rock-paper-scissors as an example: if the other side knows you’ll throw scissors, he’ll throw rock and beat you. The best decision is to choose your next move randomly.

In applying a similar mindset to football, however, Wolfers oversimplifies the decision to the point of confusion. The offensive coordinator’s call, as he suggests, cannot be boiled down to a binary choice of pass vs. run. There are several options in each set, and certainly some are better than others.

Wolfers writes that should New England coach Bill Belichick fear a rush play, he “would respond by piling players between Marshawn Lynch and the end zone.” That is, to say, the center of the field – exactly where Russell Wilson placed the fateful pass. If you’re passing to beat a rush defense, why put the ball where there are a lot of enemy hands?

Let’s say the Seahawks and Patriots decide to play rock-paper-scissors. Here’s what the normal payoff matrix would look like:

Rock Paper Scissors
Rock (0, 0) (-1, +1) (+1, -1)
Paper (+1, -1) (0, 0) (-1, +1)
Scissors (-1, +1) (+1, -1) (0, 0)

Now let’s say the Seahawks, represented in the rows, have a second option similar to Rock: Pebble.

Pebble offers the same opportunity to win against Scissors (equal probability of success if the Patriots counter with a run defense), but suffers lower damage against Paper (a reduced chance of an interception).

Rock Paper Scissors
Rock (0, 0) (-1, +1) (+1, -1)
Pebble (0, 0) (, ) (+1, -1)
Paper (+1, -1) (0, 0) (-1, +1)
Scissors (-1, +1) (+1, -1) (0, 0)

You would never use Rock, because any outcome you’d get with Pebble is equal to or better than (or, if you prefer, less worse than) the same outcome you’d get had you thrown Rock instead. We say that Rock is weakly dominated by Pebble.

Here’s my overview of the concept of dominance.

In this case, a passing play that avoids the middle of the field – say, a fade to the back corner, where at worst you’d have one-on-one coverage – is far less likely to result in a pick than what Pete Carroll actually chose.

Wolfers is right about the usefulness of a mixed strategy; where he errs, however, is in suggesting the play the Seahawks ran should have been one of the options in the first place. The risk was far too high.

In the end, that analysis will probably confuse more people than it should, and likely fail to properly convey the utility of this important aspect of game theory. That’s a shame, because The Upshot (the stats-heavy blog for which Wolfers writes) often does a great job of explaining high-level economic principles in easy-to-understand terms.

Note that I’m not arguing that a pass was even a good play, period; I’ll leave the parsing of those data for others better versed in football.

On another note, if you want a truly bad analysis, see Robert Lane Greene’s piece over at The Economist. Not only is the writing far too dense for the lay reader, but as Wolfers pointed out in response on Twitter, signaling has nothing to do with a mixed strategy.

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